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r>i'r,[,iN : 
 
 ^rintcti at ti)c Stntevsitp l@rcss, 
 
 BY M. H. Gll-L. 
 
 mi^ 
 
 1 7 2 1 5^ 
 
THE PROVOST AND SENIOR FELLOWS 
 <Bi ^tinitg ©oUcge, 15u6lin, 
 
 IN WHOSE HALLS THE SUBSTANCE OF THE FOLLOWING LECTURES 
 HAS BEEN DELIVERED,- 
 
 AND FROM WHOSE FUNDS AN 15IP0RTANT PART OF THE EXPENSE 
 OF THEIR PUBLICATION HAS BEEN DEFRAYED, 
 
 THIS VOLUME 
 
 IS RESPECTFULLY AND AFFECTIONATELY 
 
 DEDICATED, 
 
 BY THEIR OLD AND FAITHFUL SERVANT AND FRIEND, 
 
 THE AUTHOR. 
 
PREFACE. 
 
 [1.] The volume now oifered to the public is designed as an assis- 
 tance to those persons who may be disposed to study and to em 
 ploy a certain new mathematical method, which has, for some 
 years past, occupied much of my own attention, and for which I 
 have ventured to propose the name of the Method or Calculus of 
 Quaternions. Although a copious analytical index, under the 
 form of a Table of Contents, will be found to have been prefixed 
 to the work, yet it seems proper to offer here some general and 
 preliminary* remarks: especially as regards that conception from 
 which the whole has been gradually evolved, and the motives for 
 giving to the resulting method an appellation not previously in 
 use. 
 
 [2.] The difficulties which so many have felt in the doctrine 
 of Negative and Imaginary Quantities in Algebra forced them- 
 selves long ago on my attention ; and although I early formed 
 some acquaintance with various views or suggestions that had 
 been proposed by eminent writers, for the purpose of removing 
 
 * Some readers may find it convenient to pass over for the present these pre- 
 fatory remarks, and to proceed at once to the Volume, of which a large part has 
 been drawn up so as to suppose less of previous and technical preparation than 
 some of the paragraphs of this Preface. Indeed, great pains have been taken 
 to render the early Lectures as elementary as the subject would allow ; and it 
 is hoped that they will be found perfectly and even easily intelligible by persons 
 of moderate scientific attainments. It is true that some of the subsequent por- 
 tions of the Course (especially parts of the concluding Lecture) may possibly 
 appear difficult, from the novel nature of the calculations employed : but perhaps 
 on that very account those later portions may repay the attention of more ad- 
 vanced mathematical students. 
 
 .i 
 
(2), PREFACE. 
 
 or eluding those difficulties (such as the theorj^ of direct and in- 
 verse quantities, and of indirectly correlative figures, the method 
 of constructing imaginaries by lines drawn from one point with 
 various directions in one plane, and the view which refers all to 
 the mere play of algebraical operations, and to the properties of 
 symbolical language), yet the whole subject still appeared to me 
 to deserve additional inquiry, and to be susceptible of a more 
 complete elucidation. And while agreeing with those who had 
 contended that negatives and imaginaries were. not properly 
 quantities at all, I still felt dissatisfied with any view which 
 should not give to them, from the outset, a clear interpretation 
 andmeaning; and wished that this should be done, for the square 
 roots of negatives, without introducing considerations so expressly 
 geometrical) as those which involve the conception of an angle. 
 [3.] It early appeared to me that these ends might be at- 
 tained by our consenting to regard Algebra as being no mere 
 Art, nor Language, nor primarily a Science of Quantity; but 
 rather as the Science of Order in Progression. It was, how- 
 ever, a part of this conception, that the progression here spoken 
 of was understood to be continuous and uni dimensional : extend- 
 ing \wdQ^n\ie\y forward and backward, but not in any lateral 
 direction. And although the successive states of such a progres- 
 sion might (no doubt) be represented by points upon a line, yet 
 I thought that their simple successiveness was better conceived 
 by comparing them with moments of time, divested, however, of 
 all reference to cause and effect; so that the " time" here consi- 
 dered might be said to be abstract, ideal, or pure, like that "space" 
 which is the object of geometry. In this manner I was led, many 
 years ago,, to regard Algebra as the Science of Pure Time: 
 and an Essay,* containing my views respecting it as such, was 
 publishedf in 1835. If I now reproduce a few of the opinions put 
 
 * Theory of Conjugate Functions, or Algebraic Couples ; with a Preliminary 
 and Elementary Essay on Algebra as the Science of Pure Time. (Read Novem- 
 ber 4th, 1833, and June 1st, 1835) Transactions of the Royal Irish Academy, 
 
 Vol. XVII., Part n. (Dublin, 1835), pages 293 to 422. 
 
 1 1 was encouraged to entertain and publish this view, by remembering some 
 passages in Kant's Criticism of the Pure Reason, which appeared to justify the 
 expectation that it should be possible to construct, d priori, a Science of Time, 
 
PREFACE. (3) 
 
 forward in that early Essay, it will be simply because they may 
 assist the reader to place himself in that point of view, as regards 
 the first elements of algebra, from which a passage was gradually 
 made by me to that comparatively geometrical conception which 
 it is the aim of this volume to unfold. And with respect to any- 
 thing unusual in the interpretations thus proposed, for some sim- 
 ple and elementary notations, it is my wish to be understood as 
 not at all insisting on them as necessary,^ but merely proposino- 
 them as consistent among themselves, and preparatory to the 
 study of the quaternions, in at least one aspect of the latter. 
 
 [4.] In the view thus recently referred to, if the letters a and 
 B w^ere employed as dates, to denote any two moments of time, 
 which might or might not be distinct, the case of the coincidence or 
 identity of these two moments, or of equivalence of these two 
 dates, was denoted by the equation, 
 
 B = a; 
 which symbolic assertion was thus interpreted as not involving 
 any original reference to quantity, nor as expressing the result 
 
 as well as a Science of Space. For example, in his Transcendental ^Esthetic, 
 Kant observes : — " Zeit und Raum sind demnach zwey Erkenntnissquellen, aus 
 denen a priori verschiedene synthetische Erkenntnisse geschopft werden konnen, 
 wie vornehmlich die reine Mathematik in Ansehung der Erkenntnisse vom Raume 
 und dessen Verhaltnissen ein glanzendes Beyspiel gibt. Sie sind namlich beide 
 zusammengenommen reine Formen aller sinnlichen Anschauung, und machen 
 dadurch synthetische Satze a priori mbglich." Which may be rudely rendered 
 thus: — "Time and Space are therefore two knowledge-sources, from which 
 different synthetic knowledges can be a priori derived, as eminently in reference 
 to the knowledge of space and of its relations a brilliant example is given by the 
 pure mathematics. For they are, both together [space and time], pure forms of 
 all sensuous intuition, and make thereby synthetic positions a priori possible." 
 (Critik der reinen Vernunft, p. 41. Seventh Edition. Leipzig: 1828). 
 
 *For example, the usual identity (b - a) + a = b, which in the older Essay 
 was interpreted with reference to time, as in paragraph [8] of this Preface, the 
 letters a and b denoting moments, is in the present work (Lecture L, article 25) 
 interpreted, on an analogous plan indeed, but with a reference to space, the let- 
 ters denoting poMJis. Still it will be perceived that there exists a close connexion 
 between the two views ; a step, in each, being conceived to be applied to a state 
 of a progression, so as to generate (or conduct to) another state. And generally 
 I think that it may be found useful to compare the interpretations of which a 
 sketch is given in the present Preface, with those proposed in the body of the 
 work. 
 
(4) PREFACE. 
 
 of any comparison between two durations as measured. It cor- 
 responded to the conception of simultaneity or synchronism; or, 
 in simpler words, it represented the thought of the present in time. 
 Of all possible answers to the general question, " When" the 
 simplest is the answer, "Now :" and it was the attitude of mind, 
 assumed in the making of this answer, which (in the system here 
 described) might be said to be originally symbolized by the 
 equation above written. And, in like manner, the two formulae 
 of wow-equivalence, 
 
 B > A, B < A, 
 
 were interpreted, without any primary reference to quantity, as 
 denoting the two contrasted relations oi subsequence and o^ pre- 
 cedence, which answer to the thoughts oi the future and the past 
 in time ; or as expressing, simply, the one that the moment b is 
 conceived to be later than a, and the other that b is earlier than 
 A : without yet introducing even the conception of a measure, to 
 determine how much later, or how much earlier, one moment is 
 than the other. 
 
 [5.] Such having been proposed as \\iq frst meanings to be 
 assigned to the three elementary marks = > < , it was next sug- 
 gested that the^r*^ use of the mark -, in constructing a science 
 of pure time, might be conceived to be the forming of a complex 
 symbol b - a, to denote the difference between two moments, or 
 the ordinal relation of the moment b to the moment a, whether 
 that relation were one of identity or of diversity ; and if the lat- 
 ter, then whether it were one of subsequence or of precedence, 
 and in whatever degree. And here, no doubt, in attending to 
 the degree of such diversity between two moments, the concep- 
 tion of duration, as quantity in time, was introduced : the full 
 meaning of the symbol b - a, in any particular application, being 
 (on this plan) not known, until we know how long after, or how 
 long before, if at all, b is than a. But it is evident that the no- 
 tion of a certain quality (or kind) of this diversity, or interval, 
 enters into this conception of a difference between moments, at 
 least as fully and as soon as the notion of quantity, amount, or 
 duration. The contrast between the Future and the Past appears 
 to be even earlier and more fundamental, in human thought, than 
 that between the Great and the Little. 
 
PREFACE. . (5) 
 
 [6.] After comparing moments, it was easy to proceed to 
 compare relations; and in this view, by an extension of the recent 
 signification [4] of the sign = , it was used to denote analogy in 
 time ; or, more precisely, to express the equivalence of two marks 
 of one common ordinal relation, between two pairs of moments. 
 Thus the formula, 
 
 D- c =B- A, 
 
 came to be interpreted as denoting an equality between two 
 intervals in time ; or to express that the moment d is related 
 to the moment c, exactly as b is to a, with respect to identity 
 or diversity : the quantity and quality of such diversity (when 
 it exists) being here both taken into account. A formula of 
 this sort was shewn to admit of inversion and alternation 
 (c-D = A-B, d-b = c-a); and generally there could be per- 
 formed a number of transformations and combinations of equa- 
 tions such as these, which all admitted of being interpreted and 
 justified by this mode of viewing the subject, but which agreed 
 in all respects with the received rules of algebra. On the same 
 plan, the two contrasted formulae of inequalities of differences, 
 
 d-c>b-a, d-c<b-a, 
 
 were interpreted as signifying, the one that d was later, relatively 
 to c, than b to a ; and the other that d was relatively earlier. 
 
 [7.] Proceeding to the mark + , I used this sign primarily as 
 a mark of combination between a symbol, such as the smaller 
 Roman letter a, of a step in time, and the symbol, such as a, of 
 the moment /row2 which this step was conceived to be made, in 
 order to form a complex symbol, a + a, recording this conception 
 of transition, and denoting the moment (suppose b) to which the 
 step was supposed to conduct. The step or transition here 
 spoken of was regarded as a mental act, which might as easily be 
 supposed to conduct backwards as forwards in the progression of 
 time; or even to be a null step, denoted by 0, and producing no 
 effect (0 + a = a). Thus, with these meanings of the signs, the 
 notation 
 
 B = a + A, 
 
 denoted the conception that the moment b might be attained, or 
 
(6) PREFACE. 
 
 mentally generated, by making- (in thought) the step a from the 
 moment a. And it appeared to me that without ceasing to re- 
 gard the symbol b - a as denoting, in one view [5], an ordinal 
 relation between two moments, we might also use it in the con- 
 nected sense of denoting this step from one to another : which 
 would allow us (as in ordinary algebra) to write, with the recent 
 suppositions, 
 
 B-A = a; 
 
 the two members of this new equation being here symbols for 
 one common step. 
 
 [8.] The usual identity, 
 
 (b - a) + A = B, 
 
 came thus to be interpreted as signifying primarily (in the 
 Science of Pure Time) a certain conceived connexion between the 
 operations, of detennining the difference between two moments 
 as a relation, and of applying that difference as a step. And the 
 two other familiar and connected identities, 
 
 c - A = (c - b) + (b - a), c - b = (c - a) - (b - a), 
 
 were treated, on the same plan, as originally signifying certain 
 compositions and decompositions of ordinal relations or of steps 
 in time. A special symbol for opposition between any two such 
 relations or steps was proposed ; but it was remarked that the 
 more usual notations, + a and - a, for the step (a) itself, and for 
 the opposite of that step, might, in full consistency with the same 
 general view, be employed, if treated as abridgments for the more 
 complex symbols + a, - a : the latter notation presenting here 
 no difficulty of interpretation, nor requiring any attempt to con- 
 ceive the subtraction of a quantity from nothing, but merely the 
 decomposition of a null step into two opposite steps. But opera- 
 tions on steps, conducted on this plan, were shewn to agree in 
 all respects with the usual rules of algebra, as regarded Addition 
 and Subtraction. 
 
 [9.] One time-step (b) was next compared with another (a), 
 in the way of algebraic ratio, so as to conduct to the conception 
 of a certain complex relation (or quotient^, determined partly by 
 their relative largeness, but partly also by their relative direction. 
 
PRErACE. (7) 
 
 as similar or opposite; and to the closely connected conception of 
 an algebraic number (or multiplier)^ which operates at once on 
 the quantity and on the direction of the one step (a), so as to 
 produce (or mentally generate) the quantity and direction of the 
 other step (b). By a combination of these two conceptions, the 
 usual identity, 
 
 — X a = b, or b = a X a, it - = «, 
 a a 
 
 received an interpretation ; the factor a being a positive or a con- 
 tra-positive (more commonly called tiegative) number, according 
 as it prese?'ved or reversed the direction of the step on which it 
 operated. The four primary operations, for combining any two 
 such ratios or numbers or factors, a and b, among themselves, 
 were defined by four equations which may be written thus, and 
 which were indeed selected from the usual formulae of algebra, 
 but were employed with new interpretations : 
 
 (b + a) X a = {bx a) + (axa); (6 - «) x a = (5 x a) - (« x a) ; 
 (b X a) X a. = bx (axa); b -^ a= (bx a) -7- {a x a). 
 
 [10.] Operations on algebraic numbers (positive or contra- 
 positive) were thus made to depend (in thought) on operations 
 of the same names on steps; which were again conceived to in- 
 volve, in their ultimate analysis, a reference to comparison of 
 moments. These conceptions were found to conduct to results 
 agreeing with those usually received in algebra ; at least when 
 was treated as a symbol of a null number, as well as of a null step 
 [7], and when the symbols, + «, 0-a, were abridged to + a 
 and - a. In this view, there was no difficulty whatever, in in- 
 terpreting the product of two negatiife numbers, as being equal 
 to a positive number : the result expressing simply, in this view 
 of it, that two successive reversals restore the direction of a step. 
 And other difficulties respecting the rule of the signs appeared 
 in like manner to fall away, more perfectly than had seemed to 
 me to take place in'any view of algebra, which made the thought 
 of quantity (or of magnitude) the primary or fundamental con- 
 ception. 
 
 [11.] This theory of algebraic numbers, as ratios of steps in 
 time, was applied so as to include results respecting powers and 
 
(8) PREFACE. 
 
 roots and logarithms : but what it is at present chiefly important 
 to observe is, that because, for the reason just assigned, the 
 square of every number is positive^ therefore no number ^ whether 
 positive or negative, could be a square root of a negative num- 
 ber, in this any more than in other views of algebra. At least 
 it was certain that no single number, of the kinds above con- 
 sidered, could possibly be such a root: but I thought that with- 
 out going out of the same general class of interpretations, and 
 especially without ceasing to refer all to the notion of time, ex- 
 plained and guarded as above, we might conceive and compare 
 couples of moments; and so derive a conception oi couples of 
 steps (in time), on which might be founded a theory oi couples 
 ofnunibers, wherein no such difficulty should present itself. 
 
 [12.] In this extended view, the symbols Ai and Aj being 
 employed to denote the two moments of one such pair or couple, 
 and Bi, Bg the two moments of another pair, I was led to write 
 the formula, 
 
 (Bi, Bo) - (Ai, As) = (Bi - Ai, B^ - Ao) ; 
 
 and to explain it as expressing that the complex ordinal relation 
 of one tnoment-couple (bi, B2) to another moment-couple (Ai, Ag) 
 might be regarded as a relation-couple ; that is to say, as a sys- 
 tem of two ordinal relations, Bi- Aj and B2 — Aa, between the cor- 
 responding momeiits of those two moment-couples : the primary 
 moment Bi of the one pair being compared with the primary- mo- 
 ment Ai of the other; and, in like manner, the secondary moment 
 Bo being compared with the secondary moment Ag. But, instead 
 of this (analytical) comparison of moments with moments, and 
 thereby oi pair with pair, I thought that we might also conceive 
 a (synthetical) generation [7] of one pair of moments from ano- 
 ther, by the apiplication oi-d pair of steps [11], or by what might 
 be called the addition (see again [7]), of a step-couple to a mo- 
 ment-couple ; and that an interpretation might thus be given to 
 the following identity, in the theory of couples here referred to: 
 
 (Bi, B.)^ {(Bi, B2) - (Ai, A2)} + (Ai, A2). 
 
 And other results, respecting the compositions and decomposi- 
 tions oi single ordinal relations, or oi single steps in time, such 
 
PREFACE. (9) 
 
 as those referred to in paragraph [8] of this Preface, were easily 
 extended, in like manner, to the corresponding treatment of cow- 
 plex relations, and of complex steps, of the kinds above described. 
 [13.] There M'as no difficulty in interpreting, on this plan, 
 such formulae of multiplication and divisioii, as 
 
 a y. (ai, ag) = (aai, aaa) ; («ai, aaa) -j- (aj, a2) = « ; 
 
 where the symbols ai, ag denote any two steps in time, and a any 
 number, positive or negative. But the question became less 
 easy, when it was required to interpret a symbol of the form 
 
 (bi, bs) -f-(ai, 82), 
 
 where bi, b2 denoted two steps which could not be derived from 
 the two steps ai, 82, through multiplication by any single number, 
 such as a. To meet this case, which is indeed the general one 
 in this theory, I was led to introduce the conception [11] oi num- 
 ber-couples, or o^ pairs of numbers, such as (ai, a^ ; and to re- 
 gard every single number {a) as being 2i degenerate form of such 
 a number-couple, namely of {a, 0) ; so that the recent formula, 
 for the multiplication of a step-couple by a number, might be thus 
 written : 
 
 («i, 0) (ai, as) - («! ai, «i 82). 
 
 It appeared proper to establish also the following formula, for the 
 multiplication of a primary step, by an arbitrary number-couple: 
 
 («!, a^ (ai, 0) = («! ai, a^ dn) ; 
 
 and to regard every such number-couple as being the sum of two 
 others, namely, of a pure primary and a pure secondary, as fol- 
 lows: 
 
 («i, «2) = («i, 0) + (0, ao) : 
 
 the analogous decomposition of a step-couple having been already 
 established. 
 
 [14.] The difficulty of the general multiplication of a step- 
 couple by a number-couple came thus to be reduced to that of 
 assigning the product of one pure secondary by another : and the 
 spirit of this whole theory of couples led me to conceive that, for 
 such a product, we ought to have an expression of the form, 
 
 b 
 
(10) PREFACE. 
 
 (0, a^) (0, a2) = (7ia2a2, 72a2a2); 
 
 the coefficients yi and ya being some two constant numbers, in- 
 dependent of the step az, and of the number a^ : which two coef- 
 ficients I proposed to call the constants of multiplication. These 
 constants might be variously assumed : but reasons were given 
 for adopting the following selection* of values, as the basis of all 
 subsequent operations : 
 
 yi = - 1 ; 72 = 0. 
 
 In this way, the required law of operation^ of a general number- 
 couple on a general step-couple, as multiplier on multiplicand, 
 was found, with this choice of the constants, to be expressed by 
 the formula: 
 
 («i, ffa) (ai, as) = («i ai - Oj ag, «2 ai + «i as). 
 
 And in fact it was easy, with the assistance of this formula, to 
 interpret the quotient [13] of two step-pairs^ as being always 
 equal to a numher-pair, which could be definitely assigned, when 
 the ratios of the four single steps were given. 
 
 [15.] With these conceptions and notations, it was allowed to 
 write the two following equations : 
 
 (1, 0) (a, b) = (a, b) ; (0, 1) (a, b) = (- b, a) ; 
 
 and I thought that these two factors, (1, 0) and (0, 1), thus used, 
 might be called respectively the primary unit, and the secondary 
 unit, of number. It was proposed to establish, by definition, for 
 the chief operations on number-pairs, a few rules which seemed 
 to be natural extensions of those already established for the cor- 
 responding operations [9] on single numbers : and it was seen that 
 because 
 
 (0, 1) (- b, a) = (- a, - b) = (- 1, 0) (a, b), 
 
 we were allowed, as a consequence of those rules, or of the con- 
 ception which had suggested them, namely, (compare [33] ), by 
 a certain abstraction of operators from operand, to establish the 
 formula, 
 
 (0, 1)^ = (-1,0) = -1. 
 
 * In some of my unprinted investigations, other selections of these constants 
 were employed. 
 
PREFACE. (11) 
 
 A new and (as I thought) clear interpretation was thus assigned, 
 for that well-known expression in algebra, the square root of ne- 
 gative unity : for it was found that we might consistently write, 
 on the foregoing plan, 
 
 (0, i) = (- 1,0)^ = (-1)^ ='/::!; 
 
 without anything obscure, impossible, ox imaginary, being in any 
 way involved in the conception. 
 
 [16.] In words, if after reversing the direction of the second 
 of any two steps, we then transpose them, as to order; thus 
 making the old but reversed second step \)s\q^ first of the new ar- 
 rangement, or of the new step-couple ; and making, at the same 
 time, the old and unreversed first step the second of the same 
 new couple; and if we then repeat this complex process of rever- 
 sal and transposition, we shall, upon the whole, have restored the 
 order of the two steps, but shall have reversed the direction of 
 each. Now, it is the conceived operator, in this process of 
 ■passing from one pair of steps to another, which, in the system 
 here under consideration, was denoted by the celebrated sym- 
 bol V-lj so often called imaginary. And it is evident that the 
 process, thus described, has no special reference whatever to the 
 notion of space, although it has a reference to the conception of 
 PROGRESSION. The symbol - 1 denoted that negative unit of 
 number, of which the effect, as o. factor, was to change a single 
 step (+ a) to its own opposite step (- a) ; and because two such 
 reversals restore, therefore (see [10] ) the usual algebraic equa- 
 tion, 
 
 (-1)^ = + 1, 
 
 continued to subsist, in this as in other systems. But the symbol 
 V'-l was regarded as not at all less real than those other symbols 
 -1 or + 1, although operating on a different subject, namely, on a 
 pair of steps (a, b), and changing them to a new pair, namely, 
 the pair(-b, +a). And the/orm of this well-known symbol, 
 ■vZ-lj as an expression (in the system here described) for what I 
 had previously written as (0, 1), and had called (see [15] ) the 
 SECONDARY UNIT of number, was justified by shewing that the 
 effect of its operation, when twice performed, reversed each step 
 of the pair. 
 
(12) PREFACE. 
 
 [17.] The more general expression of algebra, ai + V^-l «2j 
 for any (so called) imaginary root of a quadratic or other equa- 
 tion, was, on this plan, interpreted as being a symbol of the num- 
 ber-couple which I had otherwise denoted by (ai, a-^ ; and of 
 which the law oi operation on a step-couple had already [14] 
 been assigned : as also the analogous law, thence derived,* of its 
 multiplication by another number-couple , namely, that which is 
 expressed by the formula, 
 
 (6l, 62) («!, tta) = (61 «1 - ^2«2J 62 «i + 61 «2). 
 
 In this view, instead of saying that the usual quadratic equation, 
 
 x^ -\- ax-\-b = 0, 
 
 where a and b are supposed to denote two positive or negative 
 numbers, has generally two roots, real or imaginary, it would 
 be said that this other form of the same equation, 
 
 {x, yy + {a, 0) (x, y) + (6, 0) = (0, 0), 
 
 is generally satisfied by two (real) number-couples ; in which, ac- 
 cording to the values of a and b, the secondary number (j/) might 
 or might not be zero. An equation of this sort was called a cou- 
 ple-equation, and was regarded as equivalent to a system of two 
 equations^ between numbers : for example, the recent quadratic 
 couple-equation breaks itself up into the two following separate 
 equations, 
 
 x^-y^ + ax + b = 0, 2xy + ay = 0, 
 
 which always admit ofreal and numerical solutions, whether ^a^-^ 
 be a positive or a negative number ; the difi'erence being only 
 that in the former case we are to take the factor 2/= 0, of the se- 
 
 * The principles of such derivation were only hinted at in the Essay of 1835 
 (see page 403 of the Volume above cited) : but it was perhaps sufficiently ob- 
 vious that they depended on the " separation of symbols," or on the abstraction 
 of a common operand. (Compare paragraphs [13], [33], of the present Preface.) 
 
 f M. Cauchy, in his Cours d' Analyse (Paris, 1821, page 176), has the re- 
 mark : — " Toute equation imaginaire n'est que la representation symbolique de 
 deux equations entre quantites r^elles." That valuable work of M. Cauchy was 
 early known to me : but it will have been perceived that I was induced to look 
 at the whole subject of algebra from a somewhat different point of view, at least 
 on the metaphysical side. As to the word " numbers," see a note to [33]. 
 
PREFACE. (13) 
 
 cond equation of the pair, whereas in the latter case we are to 
 take the other factor of that equation, and to suppose 2a; + a = 0. 
 And similar remarks might be made on equations of higher or- 
 ders : all notion of anything imaginary, unreal, or impossible, 
 being quite excluded from the view. 
 
 [18.] The same view was extended, so as to include a theory 
 of powers, roots, and logarithms of number-couples; and espe- 
 cially to confirm a remarkable conclusion which my friend John 
 T. Graves, Esq., had communicated to me (and I believe to 
 others) in 1826, and had published in the Philosophical Transac- 
 tions for the year 1829 : namely, that the general symbolical ex- 
 pression for a logarithm is to be considered as involving two ar- 
 bitrary and independent integers;'^ the general logarithm of 
 unity, to the Napierian base, being, for example, susceptible of 
 the form, 
 
 , ^ 2a> TT 
 
 l0gl=^ 7-^, 
 
 ^WTT — V ~ ^ 
 
 where to, lo denote any two whole numbers, positive or negative 
 or null. In fact, I arrived at an equivalent expression, in my 
 own theory of number-couples, under the form, 
 
 r' /, nx (0, 2a>'7r) 
 log . (1,0) = ^-— ^; 
 
 w (e, 0) \i-, ^(Oir) 
 
 and generally an expression for the logarithm-couple, with the 
 order w, and 7'ank w, of any proposed number-couple {y^, y^), to 
 any proposed base-couple {bi, bz), was investigated in such a way 
 as to confirmf the results of Mr. Graves. 
 
 • It is proper to mention, that results substantially the same, respecting the 
 entrance of two arbitrary whole numbers into the general form of a logarithm, 
 are given by Ohm, in the second volume of his valuable work, entitled : " Versuch 
 eines vollkommen consequenten Systems der Mathematik, vom Professor Dr. 
 Martin Ohm" (Berlin, 1829, Second Edition, page 440. I have not seen the first 
 Edition). For other particulars respecting the history of such investigations, 
 on the subject of general logarithms, I must here be content to refer to Mr. 
 Graves's subsequent Paper, printed in the Proceedings of the Sections of the 
 British Association for the year 1834 (Fourth Report, pp. 523 to 531. Lon- 
 don, 1835). 
 
 f Another confirmation of the same results, derived from a peculiar theory of 
 conjugate functions, had been communicated by me to the British Association 
 
(14) PREFACE. 
 
 [19.] After remarking that it was he who had proposed those 
 names, oi orders and ranks of logarithms, that early Essay of my 
 own, of which a very abridged (although perhaps tedious) account 
 has thus been given, continued and concluded as follows : — 
 "But because Mr. Graves employed, in his reasoning, the usual 
 ** principles respecting Imaginary Quantities, and was content 
 " to prove the symbolical necessity without shewing theinterpre- 
 " tation, or inner meaning, of his formulee, the present Theory of 
 " Couples is published to make manifest that hidden meaning : 
 "and to shew, by this remarkable instance, that expressions 
 " which seem, according to common views, to be merely symbo- 
 " lical, and quite incapable of being interpreted, may pass into 
 " the world of thoughts, and acquire reality and significance, if 
 "Algebra be viewed as not a mere Art or Language, but as the 
 " Science of Pure Time.* The author hopes to publish hereafter 
 
 at Edinburgh in 1834, and may be found reported among the Proceedings of the 
 Sections for that year, at pp. 519 to 523 of the Volume lately cited. The partial 
 differential " equations of conjugation," there given, had, as I afterwards 
 learned, presented themselves to other writers : and the Essay on "Conjugate 
 Functions, or Algebraic Couples," there mentioned, was considerably modified, 
 in many respects, before its publication in 1835, in the Transactions of the Royal 
 Irish Academy. 
 
 * Perhaps I ought to apologize for having thus ventured here to reproduce 
 (although only historically, and as marking the progress of my own thoughts) 
 a view so little supported by scientific authority. I am very willing to believe 
 that (though not unused to calculation) I may have habitually attended too little 
 to the symbolical character of Algebra, as a Language, or organized system of 
 signs : and too much (in proportion) to what I have been accustomed to consider 
 its scientific character, as a Doctrine analogous to Geometry, through the. Kan- 
 tian parallelism between the intuitions of Time and Space. This is not a proper 
 opportunity for seeking to do justice to the views of others, or to my own, on a 
 subject of so great subtlety : especially since, in the present work, I have thought 
 it convenient to adopt throughout a geometrical basis, for the exposition of the 
 theory and calculus of the Quaternions. Yet I wish to state, that I do not de- 
 spair of being able hereafter to shew that my own old views respecting Algebra, 
 perhaps modified in some respects by subsequent thought and reading, are not 
 fundamentally and irreconcileably opposed to the teaching of writers whom I 
 so much respect as Drs. Ohm and Peacock. The " Versuch," &c., of the former 
 I have cited (the date of the first Volume of the Second Edition is Berlin, 
 1828): and it need scarcely be said (at least to readers in these countries) 
 that my other reference is to the Algebra (Cambridge, 1830) ; the Report on 
 Certain Branches of Analysis, printed in the Third Report of the British Associa- 
 
PREFACE. (15) 
 
 " many other applications of this view; especially to Equations 
 "and Integrals, and to a Theory of Triplets and Sets of Mo- 
 tion for the Advancement of Science (London, 1834) ; the Arithmetical Algebra 
 (Cambridge, 1842); and the Symbolical Algebra (Cambridge, 1845): all by 
 the Rev. George Peacock. I by no means dispute the possibility of constructing 
 a consistent and useful system of algebraical calculations, by starting -with the 
 notion of integer number ; unfolding that notion into its necessary consequences ; 
 expressing those consequences with the help of symbols, which are already ge- 
 neral inform, although supposed at first to be limited in their signification, or 
 value : and then, by definition, for the sake of symbolic generality, removing the re- 
 strictions which the original notion had imposed ; and so resolving to adopt, as 
 perfectly general in calculation, what had been only proved to be true for a cer- 
 tain subordinate and limited extent of meaning. Such seems to be, at least in 
 part, the view taken by each of the two original and thoughtful writers who 
 have been referred to in the present Note : although Ohm appears to dwell more 
 on the study of the relations between the fundamental operations, and Peacock 
 more on the permanence of equivalent /orms. But I confess that I do not find my- 
 self able to frame a distinct conception of number, without some reference to the 
 thought of time, although this reference may be of a somewhat abstract and 
 transcendental kind. I cannot fancy myself as counting any set of things, with- 
 out first ordering them, and treating them as successive : however arbitrary and 
 mental (or subjective') this assumed succession may be. And by consenting to 
 begin with the abstract notion (or pure intuition) of time, as the basis of the ex- 
 position of those axioms and inferences which are to be expressed by the symbols 
 of algebra, (although I grant that the commencing with the more familiar con- 
 ception of whole number may be more convenient for purposes of elementary in- 
 struction,) it still appears to me that an advantage would be gained : because the 
 necessity for any merely symbolical extension of formulae would be at least consi- 
 derably postponed thereby. In fact (as has been partly shewn above), negatives 
 would then present themselves as easily and naturally as positives, through the 
 fundamental contrast between the thoughts of past a.nd future, used here as no 
 mere illustration of a result otherwise and symbolically deduced, without any 
 clear comprehension of its meaning, but as the very ground of the reasoning. 
 The ordinary imaginaries of algebra could be explained (as above) by couples ; 
 but might then, for convenience of calculation, be denoted by single letters, sub- 
 ject to all the ordinary rules, which rules -wouldi follow (on this plan) from the 
 combination of distinct conceptions with definitions, and would offer no result 
 which was not perfectly and easily intelligible, in strict consistency with that 
 original thought (or intuition) of time, from which the whole theory should (on 
 this supposition) be evolved. The doctrine of the n roots of an equation of the 
 nth degree (for example) would thus suffer no attaint as to form, but would ac- 
 quire (I think) new clearness as to meaning, without any assistance from geo- 
 metry. The quaternions, as I have elsewhere shewn (in Vol. XXL, Part ii., of 
 the Transactions of the Royal Irish Academy), and even the biquaternions (as I 
 hope to shew hereafter), might have their laws explained, and their symbolical 
 results interpreted, by comparisons of se^s of moments, and by operations on sets 
 
(16) PREFACE. 
 
 "ments, Steps, and Numbers, which includes this Theory of 
 "Couples."* 
 
 [20.] The theory of triplets and sets, thus spoken of at the 
 close of the Essay of 1835, had in fact formed the subject of va- 
 rious unpublished investigations, of which some have been pre- 
 served : and a brief notice of them here (especially as relates to 
 tripletsf) may perhaps be useful, by assisting to throw light on 
 the nature of the passage, which I gradually came to make, from 
 couples to quaternions. 
 
 Without departing from the same general view of algebra, as 
 the science of pure time, it was obvious that no necessity existed 
 for any limitation to pairs, of moments, steps, and numbers. 
 Thus, instead of comparing, as in [12], two moments, Bi and b,, 
 with two other moments, Ai and a^, it was possible to compare 
 three moments, Bi, Bj, B3, with three other moments, Aj, a,, A3 ; 
 that is, more fully, to compare (or to conceive as compared) the 
 
 of steps in time. Thus, in the phraseology of Dr. Peacock, we should have a very 
 wide "science of suggestion" (or rather, suggestive science) as ouv basis, on 
 which to build up afterwards a new structure of purely symbolical generalization,- 
 if the science of time were adopted, instead of merely Arithmetic, or (primarily) 
 the doctrine of integer number. Still I admit fully that the actual calculations 
 suggested by this, or by any other view, must be performed according to some 
 fixed laws of combination of symbols, such as Professor De Morgan has sought to 
 reduce, for ordinary algebra, to the smallest possible compass, in his Second 
 Paper on the Foundation of Algebra (Camb. Phil. Trans,, Vol. VII., Part ni.), 
 and in his work entitled " Trigonometry and Double Algebra" (London, 1849): 
 and that in following out such laws to their symbolical consequences, uninter- 
 pretable (or at least uninterpreted) results may be expected to arise. In the 
 present Volume (as has been already observed), I have thought it expedient to 
 present the quaternions under a geometrical aspect, as one which it may be per- 
 haps more easy and interesting to contemplate, and more immediately adapted 
 to the subsequent applications, of geometrical and physical kinds. And in the 
 passage which I have made (in the Seventh Lecture), from quaternions considered 
 as real (or as geometrically interpreted), to biquaternions considered as imaginary 
 (or as geometrically uninterpreted), but as symbolically suggested by the gene- 
 ralization of quaternion formulae, it will be perceived, by those who shall do me 
 the honour to read this work with attention, that I have employed a method of 
 transition, from theorems proved for the particular to expressions assuined for the 
 general, which bears a very close analogy to the methods of Ohm and Peacock: 
 although I have since thought of a way oi geometrically interpreting the biquater- 
 nions also. 
 
 * Trans. R. L A., Vol. XVIL, Part ii., page 422. 
 
 f These remarks on triplets are now for the first time published. 
 
PREFACE. (17) 
 
 homologous moments of these two triads^ primary with primary, 
 secondary with secondary, and tertiary with tertiary ; and so to 
 obtain a certain system or triad of ordinal relations, or a triad 
 of steps in time, which might be denoted (compare [5], [7], [12] ) 
 by either member of the following equation : 
 
 (Bi, Bz, Bs) - (Ai, Ag, As) = (Bi - Ai, Bj - Aa, B3 - A3). 
 
 And on the same plan (compare [7], [8], [12]), if we denote the 
 three constituent steps of such a triad as follows, 
 
 Bx — Ai = ai, Bj — A2=a25 B3 — A3 = as, 
 
 it was allowed to write, 
 
 (bi. Bo, B3) = (ai, a^, 83) + (Ai, A3, As) ; 
 
 a triad of steps being thus (symbolically) added (or applied) to a 
 triad of moments, so as to conduct (in thought) to another triad 
 of moments. It appeared also convenient to establish the follow- 
 ing formula, for the addition of step-triads, 
 
 (bi, bo, bs) + (ai, 82, as) = (bi + ai, h^ + 83, bs + 83), 
 
 as denoting a certain composition of two such triads of steps, an- 
 swering to that successive application of them to any given triad 
 of moments (Ai, Ao, A3), which conducts ultimately to a third 
 triad of moments, namely, to the triad (ci, Cg, C3), if 
 
 Ci-Bi=bi, C2-B3=b,, C3-B3 = b3. 
 
 Subtraction of one step-triad from another was explained (see 
 again [8] ) as answering to the analogous decomposition of a 
 given step-triad into others ; or to a system of three distinct de- 
 compositions of so many single steps, each into two others, of 
 which one was given ; and it was expressed by the formula, 
 
 (ci, C2, Cs) - (ai, ag, as) = (ci - ai, C3 - 83, C3 - 83) : 
 
 while the usual rules of algebra were found to hold good, respect- 
 ing such additions and subtractions of triads. 
 
 [21.] Multiplication of a step-triad by a positive or negative 
 number (a) was easy, consisting simply in the multiplication of 
 each constituent step by that number; so that I had the equation, 
 
 a (ai, 83, 83) = (aai, aa3, aas) : 
 c 
 
(18) PREFACE. 
 
 and conversely it was natural (compare [13] ) to establish the 
 following formula for a certain case of division of step-triads, 
 
 (aai, «ao, ao.^) -f- (ai, 82, as) = a. 
 
 But in the more general case (compare again [13]), where the 
 steps bj, b2, bs of one triad were not propoi'tional to the steps ai, 
 82, as, it seemed to me that the quotient of these two step-triads 
 was to be interpreted, on the same general plan, as being equal 
 to a certain triad or triplet of numbers, «i, a^, a^; so that there 
 should be conceived to exist generally two equations of the forms, 
 
 (bi, ba, bs) -^ (ai, 82, as) = («!, a^, a^) ; 
 (bi, b2, bs) = (ai, a2, «s) (ai, aj, as) : 
 
 the three (positive or negative) constituents of this numerical 
 triplet («!, 02, as) depending, according to some definite laws, on 
 the ratios of the six steps, aj a^ as bj b2 bj. 
 
 [22.] In this way there came to be conceived three distinct 
 and independent unit-steps, a primary, a secondary, and a ter- 
 tiary, which I denoted by the symbols, 
 
 ■*-i> J-2J ^3 ; 
 
 and also three unit-numbers, primary, secondary, and tertiary, 
 each of which might operate, as a species oi factor, or multiplier, 
 on each of these three steps, or on their system, and which I de- 
 noted by these other symbols, 
 
 xi, X2, X3 : 
 
 or sometimes more fully thus, 
 
 (1, 0, 0), (0, 1, 0), (0, 0, 1). 
 
 A triad of steps took thus the form, 
 
 rli+sla + ^ls, 
 
 where r, s, t were three numerical coefficients (positive or nega- 
 tive), although li I2 13 were still supposed to denote three steps 
 in time ; and any triplet factor, such as {m, n, p), by which this 
 step-triplet was to be multiplied, or operated upon, might be put 
 under the analogous form, 
 
 mxi + fix^ +JOX3. 
 
PREFACE. (19) 
 
 Continuing then to admit the distributive property of multipli- 
 cation, it was only necessary to fix the significations of the nine 
 products, or combinations, obtained by operating separately with 
 each of the three units of number on each of the three units of 
 step : every such product, or result, being conceived, in this 
 theory, to be itself, in general, a step-triad, of which, however, 
 some of the component steps might vanish. Hence, after writing 
 
 >^lJ-l = -l-lJl3 ^l-'^2 = -l2Jl5 ^^3 12= l2»35 ^31$= J-3?3> 
 
 I proceeded to develope these nine step-triplets into nine tri?io~ 
 mial expressions of the forms, 
 
 h, 9 = V, 9,-^ li + V, 5,2 ^2 + V, 5,3 I3) 
 where the twenty-seven symbols of the form 1^^^ ^i represented 
 certain Jixed numerical coefficients, or constants of multiplication, 
 analogous to those denoted by 71 and ys in [14], and like them 
 requiring to have their values previously assigned, before pro- 
 ceeding to multiplication, if it were demanded that the operation 
 of a given triplet of numbers on a given triplet of steps should 
 produce a perfectly definite step-triad as its result. 
 
 [23.] Conversely, when once these numerical constants had 
 been assigned, I saw that the equation of multiplication, 
 
 (»2Xi+ 91X2+ PX3) (^rli + sli + tli) = xli + yl^ + ^hi 
 
 was to be regarded as breaking itself up, on account of the sup- 
 posed mutual independence of the three unit-steps, into three or- 
 dinary algebraical equations, between the nine numbers, m, n, /?, 
 r, s, t, X, y, z; namely, between the coefficients of the multiplier, 
 multiplicand, and product. These three equations were linear, 
 relatively to m, n,p (as also with respect to r, s, t, and x, y, z) ; 
 and therefore while they gave, immediately , expressions for the 
 coefficients xyz of the product, and so resolved expressly the 
 problem of multiplication, they enabled me, through a simple 
 system of three linear and ordinary equations, to resolve also the 
 converse problem [21] of the division of one triad of steps by 
 another : or to determine the coefficients mnp of the following 
 quotient of two such triads, 
 
 wxi + WX2 +/?X3 = (a;li + 2/I2 + z\z) -^ (rli + 5I2 + tl^). 
 
(20) PREFACE. 
 
 [24.] Such were the most essential elements of that general 
 theory of triplets, which occurred to me in 1834 and 1835: but 
 it is clear that, in its applications , everything depended on the 
 choice of the twenty-seven constants of multiplication, which 
 might all be arhitrarily assumed, before proceeding to operate, 
 but were then to be regarded di^ fixed. It was natural, indeed, to 
 consider the primary number-unit Xj as producing no change in 
 the step or triad on which it operates; and it was desirable to de- 
 termine the constants so as to satisfy the condition, 
 
 X3 X2 == X2 X3, 
 
 for the sake of conforming to analogies of algebra. Accordingly, 
 in one of several triplet-systems which I tried, the constants were 
 so chosen as to satisfy these conditions, by the assumptions, 
 
 Xi ii = ii, Xj I2 = i2, Xl is = is, 
 
 Xalj^l., X2U=li + {b-b-^)l2, X2 13=613, 
 
 X3li = l3, X3l2=Z>l3, X3 l3 = li + &I2+ CI3; 
 
 which still involved two arbitrary numerical constants, b and c, 
 and gave, by a combination of successive operations, on any ar- 
 bitrary step-triad (such as rli + 5I2 + tl^, whatever the coefficients 
 r,s, t of this operand triad might be), the following symbolic 
 equations,* expressing the properties of the assumed operators, 
 Xj, X3, and the laws of their mutual combinations : 
 
 X22= (6-6-1) X2 + I; 
 
 X2 X3 = X3 X2 = O X3 ; 
 X3" = CX3+ 6x2 + 1; 
 
 while the factor x, was suppressed, as being simply equiva- 
 lent, in this system, to the factor 1, or to the ordinary unit of 
 number. But although the symbol Xj appeared thus to be given 
 by a quadratic equation, with the two real roots b and - 6'S I saw 
 that it would be improper to confound the operation of this pe- 
 culiar symbol X2 with that of either of these two numerical roots, 
 of that quadratic but symbolical equation, regarded as an ordi- 
 nary multiplier. It was not either, separately, of the two ope- 
 
 * These symbolic equations are copied from a manuscript of February, 
 1835. 
 
PREFACE. (21) 
 
 rations Xo - 6 and X2+ J-^, which, when performed on a general 
 step-triad, reduced that triad to another with every step a null 
 one : but the combmation of these two operations, successively 
 (and in either order) performed. 
 
 [25.] In the same particular triplet system, the three gene- 
 ral equations [23] between the nine numerical coefficients, of 
 multiplier, multiplicand, and product, became the following : 
 
 X = mr + ns + pt ; 
 
 y = tns + nr+ (b -b'^)ns+ bpt; 
 
 z = mt + pr + b (ni + ps) + cpt ; 
 
 whence it was possible, in general, to determine the coefficients 
 m,n,p, of the quotient of any two proposed step-triads. The 
 same three equations were found to hold good also, when the 
 number-triplet {x,y, z) was considered as the symbolical product 
 of the two number-triplets, {rn,n,p) and {r,s,t)', this product 
 being obtained by a certain detachment (or separation) of the 
 symbols of the operators from that of a common operand, namely 
 here an arbitrary step-triad. In other words, the same algebraical 
 equations between the nine numerical coefficients, xyz, mnp, rst, 
 expressed also the conditions involved in the formula of sym- 
 bolical multiplication, 
 
 {x, y, z) = {m, n, p) (r, s, t), 
 regarded as an abridgment of the ioWoWiBg fuller formula : 
 {x, y, z) (ai, as, 83) = {m, n, p) {r, s, t) (ai, a,,, a.^) ; 
 
 where ai, 82, O-z might denote any three steps in time. Or they 
 might be said to be the conditions for the correctness of this 
 other symbolical equation, 
 
 xy^i + 2/X2 + 2x3 = (wixi + nxi + p-Xz) (rxi + sxj + ^Xg), 
 
 interpreted on the same plan as the symbols x^^, X2X3, X3X2, x^^, 
 in [24]. 
 
 [26.] All the peculiar properties of the lately mentioned 
 triplet system might be considered to be contained in the^three 
 ordinary and algebraical equations, [25], which connected the 
 nine coefficients with each other (and in this case with two arbi- 
 trary constants). And I saw that these equations admitted of 
 
(22) PREFACE. 
 
 the three following combinations, by the ordinary processes of 
 algebra : 
 
 X -b'^l/ = (m-b~^n) (r-b'^s); 
 
 X + by + az= (m+ bn + ap) {r + bs + at) ; 
 
 x + by+az = (m + bn + a'p) (r + 6s + at) ; • 
 
 where a, a were the two real and unequal roots of the ordinary 
 quadratic equation, 
 
 a?= ca + 52 + l. 
 
 Here, then, was an instance of what occurred in every other tri- 
 plet system that I tried, and seemed indeed to be a general and 
 necessary consequence of the cubic form of a certain function, 
 obtained by elimination between the three equations mentioned 
 in [23], at least if we still (as is natural) suppose that Xi = l: 
 namely, that the product 0/ two triplets may vaiiish, without 
 either factor vanishing. For if (as one of the ways of exhibiting 
 this result), we assume 
 
 n = bm, r = -bSf t -Oy 
 
 the recent relations will then give 
 
 x = 0, y=Oy z = 0; 
 
 so that, whatever values may be assigned to m,p,s, we have, in 
 this systern, the formula : 
 
 (m, bm, p) (- bs, s, 0) = (0, 0, 0). 
 
 For the same reason, there were indeterminate cases, in the ope- 
 ration of division of triplets : for example, if it were required to 
 find the coefficients mnp of a quotient, from the equation 
 
 (m, n, p) (- bs, s, 0) = {x, y, z), 
 
 we should only be able to determine the function m-b'^n, but 
 not the numbers m and n themselves; while p would be entirely 
 undetermined: at least if x + by and z were each =0, for other- 
 wise there might come infinite values into play. 
 
 [27.] The foregoing reasonings respecting triplet systems 
 were quite independent of any sort of geometrical interpretation. 
 Yet it was natural to interpret the results, and I did so, by con- 
 ceiving the three sets of coefficients, (m, n^p), (r, s, t), (a;, y, z), 
 
PREFACE. , (23) 
 
 which belonged to the three triplets in the multiplication, to be 
 the co-ordinate projections, on three rectangular axes, of three 
 right lines drawn from a common origin ; which lines might (I 
 thought) be said to be, respectively, in this system of interpreta- 
 tion, the multiplier line, the multiplicand line, and the product 
 line. And then, in the particular triplet system recently de- 
 scribed, the formulae of [26] gave easily a simple rule, for con- 
 structing (on this plan) the product of two lines in space. For 
 I saw that if three fixed and rectangular lines, A, B, C, distinct 
 from the original axes, were determined by the three following 
 pairs of ordinary equations in co-ordinates : 
 
 x+ by = 0, z = 0, for line A ; 
 7/ -bx = 0, z - ax = 0, . . . B; 
 y-bx = 0, 2-a'x=0, . . . C; 
 
 we might then enunciate this theorem:* 
 
 " If a line L" be the product of two other lines, Z/, L', then 
 on whichever of the three rectangular lines A, B, C we project 
 the two factors L, L', the product (in the ordinary meaning) of 
 their two projections is equal to the product of the projections 
 (on the same) of L" and U, U being the primary unit-line 
 (I,. 0,0)." 
 
 [28.] I saw also that it followed from this theorem, or more 
 immediately from the equations lately cited [26], from which the 
 theorem itself had been obtained, that if we considered three 
 rectangular planes, A', B', C, perpendicular respectively to the 
 three lines A, B, C, or having for their equations, 
 
 y-bx=0, {A') ; x + bi/ + az=0, (B') ; x + by+ az = 0, (C) ; 
 
 then every line in any one of these three fixed planes gave a null 
 product line, when it was multiplied by a line perpendicular to 
 that fixed plane : the line A, for example, as a factor, giving a 
 null line as the product, when combined with any factor line in 
 the plane A'. For the same reason (compare [26] ), although 
 the division of one line by another gave generally a determinate 
 
 • This theorem is here copied, without any modification, from the manuscript 
 investigation of February, 1835, which was mentioned in a former note. 
 
(24) PREFACE. 
 
 quotient-line, yet if the divisor-line were situated in any one of 
 the three planes A, B\ C, this quotient-line became then in- 
 
 Jinite^ or indeterminate. And results of the same general cha- 
 racter, although not all so simple as the foregoing, presented 
 themselves in my examinations of various other triplet systems : 
 there being, in all those which 1 tried, at least one system of 
 line and plane, analogous to {A) and (^'), but not always three 
 such (real) systems, not always at right angles to each other. 
 
 [29.] These speculations interested me at the time, and some 
 of the results appeared to be not altogether inelegant. But I was 
 dissatisfied with the departure from ordinary analogies of algebra, 
 contained in the evanescence [26] [28] o{ z product of two trip- 
 lets (or of two lines), in certain cases when r\e\ihQv factor was 
 null ; and in the connected indeterminateness (in the same cases) 
 of a quotient, while the divisor was different from zero. There 
 seemed also to be too much room for arbitrary choice of con- 
 stants^ and not any sufficiently decided reasons for finally prefer- 
 ring one triplet system to another. Indeed the assumption of 
 the symbolic equation [24], Xj = 1, which it appeared to be conve- 
 nient and natural to make, although not essential to the theory, 
 determined immediately the values of nine out of the twenty-se- 
 ven constants of multiplication ; and six others were obtained 
 
 " from the assumptions, which also seerned to be convenient (al- 
 though in some of my investigations the latter was not made), 
 
 X2 il = 13, Xg ii = ig. 
 
 The supposed convertibility {^ee again [24] ), of the order of the 
 two operations Xj and X3, gave then the three following condi- 
 tions, 
 
 X3 ^2 ll ~ •^Z ^3 -l-lj ^3 ^2 -L2 = ^2 ^3 -'■2) ^3 ^2 -»-3 — ^2 ^3 -'■35 
 
 of which the first was seen at once to establish three relations be- 
 tween six of the twelve remaining coefficients of multiplication, 
 namely (if the subscript commas be here for conciseness omitted), 
 
 I23I = I32I5 -'-232 == -^3225 -'-233 = -'-323' 
 
 The two other equations between step-triads, given by the recent 
 conditions of convertibility, resolved themselves into six equa- 
 tions between coefficients, which were, however, perceived to be 
 
PREFACE. (25) 
 
 not all independent of each other, being in fact all satisfied by 
 satisfying the three following : 
 
 -•■321 — -1-233 ■'■333 ~ -1-233 ■'■322 } 
 
 ■^321 — -^333 (■'■233 ~ -'■222) + -'-223 (-'•322 ~ -'-333^ J 
 
 ■'-331 = ■'■332 (■'■233 "~ -'222) + -'■323 (■'•332 ~ ^ZZi) j 
 
 of which the two former presented themselves to me under forms 
 a little simpler, because, for the sake of preserving a gradual as- 
 cent from couples to triplets, or for preventing a tertiary term 
 from appearing in the product, when no such term occurred in 
 either factor, 1 assumed the value, 
 
 I323 = 0. 
 There still remained y?ue arbitrary coefficients, 
 
 •'■222) -'-3225 -1323; ■'■332J -'■333> 
 
 which it seemed to be permitted to choose at pleasure : but the 
 decomposition of a certain cubic function [26] of r, s, t mto fac- 
 tors, combined with geometrical considerations, led me, for the 
 sake of securing the reality and rectangular ity of a certain sys- 
 tem of lines and planes, to assume the three following relations 
 between those coefficients : 
 
 1 222 = -1-333 ~ 1 323) ■'•322 = 0) -1 332 = -^ 333 5 
 
 which gave also the values, 
 
 -'-231 = J- ) -^331 = ^) -IsSl^J-' 
 
 But the two constant coefficients I333 and I333 still seemed to re- 
 main wholly arbitrary,* and were those undetermined elements, 
 denoted by b and c, which entered into the formulae of triplet 
 multiplication [25], already cited in this Preface. 
 
 [30.] I saw, however, as has been already hinted [19] [20], 
 that the same general view of algebra, as the science of pure 
 time, admitted easily, at least in thought, of an extension of this 
 
 * The system of constants 6 = 1, c = 1, might have deserved attention, but I 
 do not find that it occurred to me to consider it. In some of those old investi- 
 gations respecting triplets, the symbol V-.l presented itself as a coefficient : but 
 this at the time appeared to me unsatisfactory, nor did I see how to interpret it 
 in such a connexion. 
 
 d 
 
(26) PREFACE. 
 
 ■whole theory, not only from couples to triplets, but also from 
 triplets to sets, of moments, steps, and numbers. Instead of two 
 or even three moments (as in [12] or [20]); there was no difficulty 
 in conceiving a system or set of n such moments, Ai, A2, . • a„, 
 and in supposing it to be compared with another eqiiinumerous 
 momental set, Bi, Bo, . . b„, in such a manner as to conduct to a 
 new complex ordinal relation, or step-set, denoted by the formula, 
 
 (Bj, B2, . . B„) - (Ai, A2, . . A;,) = (Bi - Ai, B2 - A2, . . Bn - A„). 
 
 Such step-sets could be added or subtracted (compare [20] ), by 
 adding or subtracting their component steps, each to or from its 
 own corresponding step, as indicated by the double formula, 
 
 (bi, ba, . . b„) + (ai, as, . . a„) = (bj + ai, bj ± aa, . . b„ ± a„) ; 
 
 and a step-set could be multiplied by a number (a), or divided by 
 another step-set, provided that the component steps of the one 
 were proportional to those of the other (compare [13] [21] ), by 
 the formulae : 
 
 a (ai, as, . . a„) = (aaj, aaj, . . aa„) ; 
 
 \adii, a&i, . . a&n) ~v~ (^15 ^2? • • ^n) ~ ^* 
 
 [31.] But when it was required to divide one step-set by ano- 
 ther, in the more general case (compare [13] [14] [21] ), where 
 the components or constituent steps aj, a^, • • ^n of the one set 
 were not proportional to the corresponding components bi, b2, . . 
 hn of the other set, a difficulty again arose, which 1 proposed still 
 to meet on the same general plan as before, by conceiving that a 
 numeral set, or set or system of numbers, {a.x, a^, . . ««), might 
 operate on the one set of steps, (ai, a^, . . a„), in a way analogous 
 to multiplication, so as to produce or generate the other given 
 step-set, as a result which should be analogous to a product. In- 
 stead of three distinct and independent unit-steps, as in [22], I 
 now conceived the existence oin such miit-steps, which might be 
 denoted by the symbols, 
 
 J-i) -1^29 • • -I-n j 
 
 and instead of three unit-numbers (see again [22] ), I conceived 
 n such unit-operators, which in those early investigations I de- 
 noted 
 
PREFACE. (27) 
 
 and of which I conceived that each might operate on each unit- 
 step, as a species of multiplier, or factor, so as to produce (gene- 
 rally) a new step-set as the result. There came thus to be con- 
 ceived a number, =w% of such resultant step-sets, denoted, on the 
 plan of [22], by symbols of the forms : 
 
 where the n^ symbols of the form l/,^,7i denoted so many numerical 
 coefficients, or constants of multiplication, of the kind previously 
 considered in the theories of couples [14], and of triplets [22], 
 which all required to have their values previously assumed, or 
 assigned, before proceeding to multiply a step-set by a number- 
 set, in order that this operation might give generally a definite 
 step-set as the result. 
 
 [32.] Conversely, on the plan of [23], when the n^ numerical 
 values of these coefficients or constants l/,^,ft had been once fixed, 
 I saw that we could then definitely interpret a product of the 
 form, 
 
 {mx,+ . .+m.gXg + . .Mn x„) (ri li + . . + r/l/+ . . + r„ 1„), 
 
 where mi, . . nig, . . ntn and ri, . . ?y, . . r„ were any 2n given 
 numbers, as being equivalent to a certain new or derived step- 
 set of the form, 
 
 where Xi, . . Xu, . . Xn were w new or derived numbers, determined 
 by n expressions such as the following : 
 
 Xh = ^mgrflf^gX, 
 
 the summation extending to all the n^ combinations of values of 
 the indices / and g. And because these expressions might in 
 general be treated as a system of n linear equations between the 
 n coefficients »2^ofthe multiplier set, I thought that the division 
 of one step-set by another (compare [14] [23]), might thus in 
 general be accomplished, or at least conceived and interpreted, 
 as being the process of returning to that multiplier, or of deter- 
 mining the numeral set which would produce the dividend step- 
 set, by operating on the divisor step-set, and which might there- 
 fore be denoted as follows : 
 
(28) PREFACE. 
 
 »2i Xj + . . + lUg X^ + . . mn ■>^n= {3C1I1+ . . + Xh\h+ • • -^ Xn In) 
 
 -f- (^1 li + . . r/ 1/+ . . + r„ 1„) ; 
 
 or more concisely thus, 
 
 2% X5 = ^xu Ih -^ 2?y 1/ : 
 while the numeral set thus found might be called the quotient of 
 the two step-sets. 
 
 [33.] It may be remembered that even at so early a stage as 
 the interpretation of the symbol bx a, for the algebraic product 
 of two positive or negative numbers,* it had been proposed to 
 conceive a reference to a step (a), which should be first operated 
 on by those two numbers successively, and then abstracted from, 
 as was expressed by the elementary formula [9], 
 
 (i X «) X a = 6 X (a X a). 
 
 Thus to interpret the product -2x-3as= + 6, I conceived that 
 some time-step (a) was first tripled in length and reversed in di- 
 rection ; then that the new step (-3a) was doubled and reversed; 
 and finally that the last resultant step (+ 6a) was compared with 
 the original step (a), in the way of algebraic ratio [9], thereby 
 conducting to a result which was independent of that original 
 step. All this, so far, was no doubt extremely easy ; nor was it 
 difficult to extend the same mode of interpretation to the case 
 [17] of the multiplication of two number couples, and to inter- 
 pret the product of two such couples as satisfying the condition, 
 
 (61, hz) («i, Ota) X (ai, a,) = (61, 60) X («i, oo) (aj, a.) ; 
 
 the arbitrary step-couple (ai, 83) being first operated on, and af- 
 terwards abstracted from. In like manner, in the theory of 
 ttnplets, it was found possible [24] [25] to abstract ft^om an ope- 
 rand step-triad, and thereby to obtain formulae for the symbolic 
 
 * This word " number," whether with perfect propriety or not, is used 
 throughout the present Preface and work, not as contrasted vfith. fractions (ex- 
 cept when accompanied by the word whole or integer'), nor with incommensura- 
 bles, but rather with those steps (in time, or on one axis), of some two of which it 
 represents or denotes the ratio. In short, the numbers here spoken of, and else- 
 where denominated '■' scalar s" in this work, are simply those positives or nega- 
 tives, on the scale of progression from - co to + cc , which are commonly called 
 reals (or real quantities) in algebra. 
 
PREFACE. (29) 
 
 multiplication of the secondary and tertiary number-units, Xg, X3, 
 and more generally of any two numerical triplets among them- 
 selves. But when it was sought to extend the same view to the 
 still more general multiplication of numeral sets, new difficulties 
 were introduced by the essential complexity of the subject, on 
 which I can only touch in the briefest manner here,* 
 
 [34.] After operating on an arbitrary step-set S r/ 1/ by a 
 number-set S?>?^ x^, and so obtaining [32] another step-set, 
 ^Xu Ih, we may conceive ourselves to operate on the same gene- 
 ral plan, and with the same particular constants of multiplication, 
 on this new step-set, by a neiv number-set, such as "Em'giXg^, and 
 so to obtain a third step-set, such as Saj'^, l/i, : which may then be 
 supposed to be divided (see again [32] ) by the original step-set 
 S/'/l/, so as to conduct to a quotient, which shall be another nu- 
 meral set, of the form '2,m"goXg„. Under these conditions, we may 
 certainly write, 
 
 S m'g, xg, (^ingXg.^ rf If) = S m"g. xg., . S rf V ; 
 
 but in order to justify the subsequent abstraction of the operand 
 step-set, ov the abridgment (compare [25] ) of this formula oi suc- 
 cessive operation to the following, 
 
 S m'g, Xg, . S mg Xg = S W"^-, X g« , 
 
 which may be called a formula for the (symbolic) multiplication 
 of two nmnber-sets, certain conditions of detachment are to be sa- 
 tisfied, which may be investigated as follows. 
 
 [35.] Conceive that the required separation of symbols has 
 been found possible, and that it has given, by a generalization of 
 
 * A fuller account of this theory of sets, with a somewhat different notation 
 (the symbols c,-, s, t and n,-, >■', ," being employed, for example, to denote the co- 
 efficients which would here be written as It, ,-, s and 1',., ,', /'), and with a special 
 application to the theory of quaternions, will be found in an Essay entitled: " Re- 
 searches respecting Quaternions. First Series." Trans. R. I. A. Vol. XXI., 
 Part II. Dublin: 1848. Pages 199 to 296. (Read November 13th, 1843.) This 
 Essay was not fully printed till 1847, but several copies of it were distributed in 
 that year, especially during the second Oxford Meeting of the British Associa- 
 tion. The discussion of that portion of the subject which is here considered is 
 contained chiefly in pages 225 to 231 of the volume above cited. 
 
(30) PREFACE. 
 
 the process for triplets in [24], a system of rC- symbolic equations 
 of the form, 
 
 X^, X^ = 2 1 g,g<,gi' X^„ ; 
 
 where I'^.p',^" is one of a new system ofn^ numerical coefficients, 
 and the sum involves n terms, answering to n different values of 
 the index g". Under the same conditions, the recent formula for 
 the multiplication of numeral sets breaks itself up into n equa- 
 tions, of the form, 
 
 m g„ = S lUg m g, 1 g^ g,^ g„ j' 
 
 the summation here extending to 71"" terms arising from the com- 
 binations of the values of the indices g and g'. For all such 
 combinations, and for each of the n values of y, we are to have 
 (if the required detachment be possible) the following equation 
 between step-sets : 
 
 and conversely, if we can satisfy these n^ equations between step- 
 sets, we shall thereby satisfy the conditions of detachment [34], 
 which we have at present in view. But each of these n^ equa- 
 tions between sets resolves itself generally into n equations be- 
 tween numbers : and thus there arise in general no fewer than w* 
 numerical equations, as expressive of the conditions in question, 
 which may all be represented by the formula,* 
 
 S 1/,^, h 1 h,g; Iv = S Vg^ g.^ k 1/, 7,^ ft, ; 
 
 all combinations of values of the indices/", g, g', h' (from 1 to « 
 for each) being permitted, and the summation in each member 
 being performed with respect to h. Now to satisfy these n^ 
 equations of condition, there were only 2w^ coefficients, or rather 
 their ratios, disposable : and although the theories of couples and 
 triplets already served to exemplify the possibility of effecting 
 the desired detachment, at least in certain cases, yet it was by 
 no means obvious that any such extetisive reductions] were likely 
 
 * A formula equivalent to this, but with a somewhat different notation, will 
 be found at page 231 of the Essay and Volume referred to in a recent Note. 
 
 f On the subject of such general reductions, some remarks will be found at 
 page 251 of the Essay and Volume lately cited. 
 
PREFACE. (31) 
 
 to present themselves, as were required for the accomplishment 
 of the same object, in the more general theory of sets. And I 
 believe that the compass and difficulty, which I thus perceived 
 to exist, in that very general theory, deterred me from pursuing 
 it farther at the time above referred to. 
 
 [36.] There was, however, a motive which induced me then 
 to attach a special importance to the consideration oi triplets, as 
 distinguished from those more general sets, of which some ac- 
 count has been given. This was the desire to connect, in some 
 new and useful (or at least interesting) way, calculation with geo- 
 metry, through some undiscovered extension, to space of three 
 dimensions, of a method of construction or representation [2], 
 which had been employed with success by Mr. Warren* (and 
 indeed also by other authors,! of whose writings I had not then 
 
 * " Treatise on the Geometrical Representation of the Square Roots of Ne- 
 gative Quantities, By the Rev. John Warren, A. M., Fellow and Tutor of 
 Jesus College, Cambridge." (Cambridge, 1828.) To suggestions from that 
 Treatise I gladly acknowledge myself to have been indebted, although the in- 
 terpretation of the symbol V— 1, employed in it, is entirely distinct from that 
 which I have since come to adopt in the geometrical applications of the quater- 
 nions. 
 
 f Several important particulars respecting such authors have been collected 
 in the already cited "Report on certain Branches of Analysis" (see especially 
 pp. 228 to 235), by Dr. Peacock, whose remarks upon their writings, and whose 
 own investigations on the subject, are well entitled to attention. As relates to 
 the method described above (in paragraph [36] of this Preface), if multiplication 
 (as well as addition) of directed lines in one plane be regarded (as I think it ought 
 to be) as an essential element thereof, I venture here to state the impression on 
 my own mind, that the true inventor, or at least the first definite promulgator of 
 that method, will be found to have been Argand, in 1806: although his " Essai 
 sur une Maniere de representer les Quantites Imaginaires," which was published 
 at Paris in that year, is known to me only by Dr. Peacock's mention of it in his 
 Report, and by the account of the same Essay given in the course of a subse- 
 quent correspondence, or series of communications (which also has been noticed 
 in that Report, and was in consequence consulted a few years ago by me), car- 
 ried on between Fran9ais, Servois, Gergonne, and Argand himself; which series 
 of papers was published in Gergonne's Annales des Mathematiques, in or about 
 the year 1813. My recollection of that correspondence is, that it was admitted 
 to establish fully the priority of Argand to Fran9ais, as regarded the method 
 [36] of (not merely adding, but) multiplying together directed lines in one plane, 
 which is briefly described above : and which was afterwards independently re- 
 produced, by Warren in 1828, and in the same year by Mourey, in a work enti- 
 tled : " La Vraie Theorie des Quantites Negatives, et des Quantites pretendues 
 
(32) PREFACE. 
 
 heard), for operations on 7nght lines in one plane : vihich. method 
 had given a species of geometrical interpretation to the usual 
 and well-known imaginary symbol of algebra. In the method 
 thus referred to, addition of lines was performed according to the 
 same rules as composition of motions^ or of forces, by drawing 
 
 Imaginaires" (Paris, 1828). If the list of such independent re-inventors of this 
 important and modern method of constructing by a line the product of two di- 
 reeted lines in one fixed plane (from which it is to be remarked, in passing, that 
 my own mode of representing by a quaternion the product of two directed lines 
 in space is altogether different) were to be continued, it would include, as I have 
 lately learned, the illustrious name of Gauss, in connexion with his Theory 
 of Biquadratic Residues (Gottingen, 1832). On the other hand, I cannot per- 
 ceive that any distinct anticipation of this vaeihoA. oi inidliplication of directed lines 
 is contained in Buee's vague but original and often cited Paper, entitled " Me- 
 moire sur les Quantites Imaginaires," which appeared in the Philosophical 
 Transactions (of London) for 1806, having been read in June, 1805. The inge- 
 nious author of that Paper had undoubtedly formed the notion of representing the 
 directions of lines by algebraical symbols ; he even uses (in No. 35 of his Memoir) 
 such expressions as V2 (cos 45°+ sin 45° V — 1) to denote two diiferent and di- 
 rected diagonals of a square : and there is the high authority of Peacock (Report, 
 p. 228), for considering that the geometrical interpretation of the symbol V — 1, 
 as denoting perpendicularity, was "first formally maintained by Buee, though 
 more than once suggested by other authors." In No. 43 of the Paper referred 
 to, Buee constructs with much elegance, by a bent line ake, or by an inclined 
 line AE (where ke is a perpendicular, = ^ a, erected at the middle point K of a 
 given line ab, or a), axi imaginary root (x) of the quadratic equation, x {a — x)= |a2, 
 which had been proposed by Carnot (in p. 54 of the Geometric de Position, Pa- 
 ris, 1804). But when he proceeds to explain (in No. 46 of his Paper) in what 
 sense he regards the two lines ae and eb (or the two constructed roots of the 
 quadratic) as having their product equal to the given value J a^ or i ab , Buee ex- 
 pressly limits the signification of such a product to the result obtained by multi- 
 plying the arithmetical values, and expressly excludes the consideration of the 
 positions of the factor-lines ivova his conception of their multiplication: whereas 
 it seems to me to belong to the very essence of the method [36] of Argand and 
 others, and generally to that system of geometrical interpretation whereon is 
 based what Professor De Morgan has happily named Double Algebra, to take 
 account of those positions (or directions), when lines are to be multiplied together. 
 My own conception (as has been already hinted, and as will appear fully in the 
 course of this work), of the ■product of two directed lines in space as a quater- 
 nion, is altogether distinct, both from the purely arithmetical product of nume- 
 rical values of Bu^e, and from the linear product (or third coplanar line), in the 
 method of Argand : yet I have thought it proper to submit the foregoing re- 
 marks, on the invention of this latter method, to the judgment of persons 
 better versed than myself in scientific history. A few additional remarks and 
 references on the subject will be found in a subsequent Note. 
 
PREFACE. (33) 
 
 the diagonal of a parallelogram ; and the multiplication of two 
 lines, in a given plane, corresponded to the construction of a 
 species oi fourth pi^oportional, to an assumed line in the same 
 plane, selected as the representative oi positive unity, and to the 
 two proposed /ac^or-/?>2e5 : such fourth proportional, or product- 
 line, being inclined to one factor-line at the sa)ne angle, measured 
 in the same sense, as that at which the other factor-line was in- 
 clined to the assumed unit-line ; while its length was, in the old 
 and usual signification of the words, a fourth proportional to the 
 lengths of the unit-line and the two factor-lines. Subtraction, 
 division, elevation to powers, and extraction of roots, were ex- 
 plained and constructed on the same general principles, and by 
 processes of the same general character, which may easily be con- 
 ceived from the slight sketch just given, and indeed are by this 
 time known to a pretty wide circle of readers : and thus, no doubt, 
 by operations on right lines in one plane, the symbol \/-l re- 
 ceived a perfectly clear interpretation, as denoting a second unit- 
 line, at right angles* to that line which had been selected to re- 
 
 * Besides what has been already referred to, as having been done on this 
 subject of the interpretation of the symbol V— 1 by the Abbe Buee, it has been 
 well remarked by Mr. Benjamin Gompertz, at page vi. of his very ingenious 
 Tract on " The Principles and Applications of Imaginary Quantities, Book II., 
 derived from a particular case of Functional Projections" (London, 1818), that 
 the celebrated Dr. Wallis of Oxford, in his "Treatise of Algebra" (London, 
 1685), proposed to interpret the imaginary roots of a quadratic equation, by 
 going out of the line, on which if real they should be measured. Thus Wallis (in 
 his chapter Ixvii.) observes : — "So that whereas in case of Negative Roots we 
 "are to say, the point e cannot be found, so as is supposed in ac Forward, but 
 " Backward it may in the same Line : M'e must here say, in case of a Negative 
 " Square, the point b cannot be found so as was supposed, in the Line ac; but 
 "Above that Line it may in the same Plain. This I have the more largely in- 
 "sisted on, because the Notion (I think) is new; and this, the plainest Declara- 
 "tion that at present I can think of, to explicate what we commonly call the 
 ^^ Imaginary Roots of Quadratick Equations. For such are these." And again 
 (in his following chapter Ixviii., at page 269), Wallis proposes to construct thus 
 the roots of the equation aa + 6a + «e = 0: — " On ACa = 6, bisected in c, erect a 
 " perpendicular cp = V<c. And taking pb = ^b, make (on whether side you please 
 "of cp), PBC, a rectangled triangle. Whose right angle will therefore be at c 
 " or B, according as pb or PC is bigger ; and accordingly, bc a sine or a tangent, 
 "(to the radius pb,) terminated in PC. The streight lines ab, Ba, are the two 
 " values of a. Both affirmative if (in the equation,) it be — ba. Both negative, 
 " if + ba. Which values be (what we call) Real, if the right angle be at c. But 
 
 e 
 
(34) PREFACE. 
 
 present positive unity. But when it was proposed to leave the 
 plane, and to construct a system which should have sotne general 
 analogy to the known system thus described, but should extend 
 to space,* then difficulties of a new character arose, in the endea- 
 
 *' Imaginary if at B." These passages must always (I think) possess an histo- 
 rical interest, as exemplifying the manner in which, in the seventeenth century, 
 one so eminent for his powers of interpretation of analytical expressions, as Dr. 
 Wallis was, sought to apply those powers to the geometrical construction of the 
 imaginary roots of an equation : and for the decision with which he held that such 
 roots were quite as clearly interpretable, as "what we call real" yaHues. His par- 
 ticular interpretation of those imaginary roots of a quadratic appears indeed to 
 me to be inferior in elegance to that which was long afterwards proposed by 
 Buee. But it may be noticed that, whether his point b were on or o^the line 
 ACa, Wallis seems (like Buee, and many other and more modern writers) to have 
 regarded that right line, as being in some sense a sum, or at least analogous to a 
 sum, of the two successive lines ab, bu; which latter lines conduct, upon the 
 whole, from the initial point A to the final point a ; and construct according to 
 him the two roots of the quadratic, whose algebraic sum is = b. Indeed, "Wallis 
 remarks (in the same page 269) that when those two roots are algebraically ima- 
 ginary, or are geometrically constructed (according to him) by the help of a point 
 B which is above the line ac«, then that straight line is not equal to the aggregate 
 of AB + Ba; but this seems to be no more than guarding himself against being 
 supposed to assert, that two sides of a triangle can be equal in length to the 
 third. In chap. Ixix., p. 272, he thus sums up : — " We find therefore, that in 
 " Equations, whether Lateral or Quadratick, which in the strict Sense, and first 
 '• Prospect, appear Impossible; some mitigation may be allowed to make them 
 " Possible; and in such a mitigated interpretation they may yet be useful." For 
 lateral equations (equations of the first degree), the mitigation here spoken of 
 consists simply in the usual representation of negative roots, by lines drawn back- 
 ward from a point, whereas they had been at first supposed to be Av&Y^n forward. 
 For quadratic equations with imaginary roots, Wallis mitigates the problem, by 
 substituting a hent line Asa for that straight line ACa, which constructs the given 
 algebraical sum (b) of the two roots of the equation, or parts of the bent line, 
 AB, Ba. It is also to be noticed that he appears to have regarded the algebraical 
 semi-difference of those two roots, ab, bct, as being in all cases constructed by the 
 line EC, drawn to the middle point c of the line a« : which would again agree with 
 many modern systems. Thus Wallis seems to have possessed, in 1685, at least in 
 germ (for I do not pretend that he fully and consciously possessed them), some 
 elements of the modern methods of Addition and Subtraction of directed lines. 
 But on the equally essential point oi Multiplication of directed lines in one plane, 
 it does not appear that Wallis, any more than Buee (see the foregoing Note), had 
 anticipated the method of Argand. 
 
 * At a much later period I learned that others had sought to accomplish 
 some such extension to space, but in ways different from mine. 
 
PREFACE. (35) 
 
 vour at surmounting which I was encouraged by the friend al- 
 ready mentioned (Mr. John T. Graves), who felt the wish, and 
 formed the project, to surmount them in some way or other, as 
 early, or perhaps earlier than myself. 
 
 [37.] A conjecture respecting such extension of the rule of 
 multiplication of lines, from the plane to space, which long ago 
 occurred to me (in 1831), may be stated briefly here, as an illus- 
 tration of the general character of those old speculations. Let 
 A denote a point assumed on the surface of a fixed sphere, de- 
 scribed about the origin o of co-ordinates, with a radius equal to 
 the unit of length; and let this point a be called the unit-point. 
 Let also b and c be supposed to be two factor -points, on the 
 same surface, representing the directions oa, ob, of the i^o fac- 
 tor-lines in space, of which lines it is required to perform, or to 
 interpret, the multiplication ; and so to determine, by some fixed 
 rule to be assigned, the product-point d, or the direction of the 
 product-line, od. Then it appeared that the analogy to opera- 
 tions in the plane might be not ill observed, by conceiving d to 
 be taken on the circle abc ; the arcs, ab, cd, of that (generally) 
 small circle of the sphere being equally long, and similarly mea- 
 sured ; so that the two chords ad, bc should he parallel: while 
 the old rule of multiplication of lengths should be retained : and 
 addition of lines be still interpreted as before. But in this sys- 
 tem there were found to enter radicals and fractions into the ex- 
 pressions for the co-ordinates* of a product ; and although the 
 case of squares of lilies, or products of equal factors, might be 
 rendered determinate by agreeing to take the great circle ab, 
 when the point c coincided with b, yet there seemed to be an es- 
 sential indetermination in the construction of the reciprocal of a 
 line : it being sufficient, according to the definition here consi- 
 
 * The rectangular co-ordinates (or projections") of the two factor-lines and 
 of the product-line being denoted by xyz, oc'y'z, x'y'z, if we also write, for con- 
 ciseness, 
 
 r = V (.1-2 + ^2 _j_ 22), ,.' = V(a:'2 + j/'2 + z'2), p^xx' + yy' + zz, 
 
 then the expressions which I found for x'y'z maybe included briefly in the 
 equations : 
 
 X —rr y z" rx' — r'x 
 
 rx' - r'x ~ ry - r'y rz -r'z p - rr' ' 
 
(36) PREFACE. 
 
 dered, to take the chord bc parallel to the tangent plane to the 
 sphere at the unit-point, in order to make the product point d 
 coincide with that point a. There was, also the great and (as I 
 thought) fatal objection to this method of construction, that it 
 did not preserve the distributive pi-inciple of multiplication ; a 
 product of sums not being equal, in it, to the sum of the products: 
 and on the whole, I abandoned the conjecture. 
 
 [38.] Another construction, of a somewhat similar character, 
 and liable to similar objections, for the product of two lines in 
 space, occurred to me in 1835, and also independently to Mr. J. 
 T. Graves in 1836, in which year he wrote to me on the subject. 
 It may be briefly stated, by saying that instead of considering, 
 as in the last-mentioned system, the small circle abc, and draw- 
 ing the chord ad, from unit-point to product-point, so as to be 
 parallel to the chord bc from one factor-point to the other, it was 
 now the arc ad of a great circle on the sphere, which was to be 
 drawn so as to bisect the arc bc, of another great circle, and be 
 bisected thereby. Or as Mr. Graves afterwards expressed to me 
 the rule in question : — " Bisect the inclination of the factor-lines, 
 and then double forward the angle between the linear unit and 
 the bisecting line:" the rule of multiplying lengths being under- 
 stood to be still observed. Mr. Graves made several acute re- 
 marks on the consequences of this construction, and proposed a 
 few supplementary rules to remove the porismatic character of 
 some of them : but observed that, with these interpretations, the 
 square-root of the negative unit-line^ or the triplet (-1, 0, 0)^, 
 would still be indeterminate, and of the form (0, cos 0, sin 0), 
 where remained arbitrary: while cases might arise, in which the 
 "minutest alteration" of a factor-line would make a "considerable 
 change" in the position of the product-line : and this result he 
 conceived to be, or to lead to, " a breach of the grand property 
 of multiplication," respecting its operation on a sum. He left to 
 me the investigation of the general expressions for the " consti- 
 tuent co-ordinates" of the resultant " triplet," or product-line, in 
 terms of the constituents of the factors : and in fact I had already 
 obtained such expressions, and had found them to involve radi- 
 cals and fractions, and to violate the distributive principle, as in 
 the system recently described [37] ; with which indeed the one 
 
PREFACE. (37) 
 
 here mentioned had been perceived By me to have a very close 
 analytical connexion.* 
 
 [39.] Mr. J. T. Graves, however, communicated to me at the 
 same time another method, which he said that he 'preferred^ 
 among all the modes that he had tried, " of representing lines in 
 space, and of multiplying such lines together." This method 
 consisted in considering such a line as a species of " compound 
 couple," or as determined by two couples, one in the plane of xy, 
 and the other perpendicular to that plane : it having been easily 
 perceived that the rules proposed by me for the addition and 
 multiplication [17] of couples, agreed in all respects with the pre- 
 viously known method [36], of representing the operations of the 
 same names on lines in one plane. From this conception of com- 
 pound couples Mr. Graves derived a " general rule for the multi- 
 plication of triplets," which I shall here transcribe,! only abridg- 
 ing the notation by writing p and pi to represent the radicals 
 -^/(a^^ + ^/O ^"^ y/ {p\' + yi')i or the projections of the factor-lines 
 on the plane of xy : "(a;, y, z) (.^i, yi, Zi) = {xz, y^, z^}, where 
 
 (xXi-yyA xy^ + yxi 
 
 «2 = {ppi - ZZi) , 2/2 = {ppi - ZZi) , Z2 = Zip + Zpi. 
 
 \ PPi J PP^ 
 
 This particular system of expressions he does not seem to have 
 developed farther, nor did it at the time attract much of my own 
 
 * With the notations recently employed, the expressions which I had found 
 for the co-ordinates of the product, in the case or system [38], are included in 
 the equations, 
 
 x" 4 rr' y z" rx + r'x 
 
 rx + rx ry + r'y rz + r'z p -\-rr' 
 
 ■which only differ from those for the former case [37], by a change of sign in the 
 radical r (or r), which represents the length of a factor-line. The conditions 
 for both systems are contained in these other equations, 
 
 xx" + yij" -f zz" = y-^ x, x'x" + y'y" + zz" = r"^ x, x"^ + ij"^ + z"2 = r^ r'2 ; 
 
 and the quadratic equation in a", obtained by elimination of ?/" and z", resolves 
 itself into two separate factors, each linear relatively to x, namely, 
 
 (p — rr') (.■»" — rr) — (rx — r'x')'^ = 0, 
 {p + rr') (/+ rr') - (rx + r'x)"^ = 0. 
 
 The first corresponds to the system [37] ; the second to the system [38]. 
 t From Mr. Graves's Letter of August 8th, 1836. 
 
(38) PREFACE. 
 
 attention : but I have thought it deserving of being put on re- 
 cord here, especially as, by a remarkable coincidence, it came to 
 be independently and otherwise arrived at by another member of 
 the same family, at a date later by ten years, and to be again 
 communicated to me.* And perhaps I may be excused if I here 
 leave the order of time, to give some short account of the train 
 of thought by which his brother, the Rev. Charles Graves, ap- 
 pears to have been conducted, in 1846, to precisely the same re- 
 lations between the constituents of three triplets. 
 
 [40,] Professor Graves employed a system of two new ima- 
 ginaries, i andj, of which he conceived that i had the effect of 
 causing a rotation (generally conical) through 90° round the axis 
 oi z, while J caused a line to revolve through an equal angle in 
 its own vertical plane (that is, in the plane of the line and of z) ; 
 and then he proceeded to multiply together the two triplets 
 X+ iy +JZ, x' -i iy'+jz\ by a peculiar process, and so to obtain a 
 third triplet x"+iy"+jz": the relations thus resulting, between 
 the co-ordinates or constituents, being (as it turned out) identi- 
 cal with those which his brother had formerly found. These 
 symbols ^ andj were each a sort of fourth root of unity : and the 
 first, but not the second, had the property of operating on a sum 
 by operating on each of its parts separately. Thus, as Profes- 
 sor Graves remarked, multiplication of triplets, on this plan, 
 would not be a distributive operation, although it would be a 
 commutative one. The method conducted him to an elegant ex- 
 ponential expression for a line in space, namely, re^e-^^, where r 
 was the radius vector^ and /, X might be called the longitude and 
 latitude of the line, so that the co-ordinate projections were 
 (some peculiar considerations being employed in order to justify 
 these expressions of them, as connected with that of the line) : 
 
 x = r cos / cos X, y = r sin I cos X, z = r sin A. 
 
 And then the rule for the multiplication oftivo lines came to be 
 expressed by the very simple formula : 
 
 * By the Rev. Charles Graves, Professor of Mathematics in the University 
 of Dublin, in a letter of November 14th, 1846. 
 
PREFACE. (39) 
 
 the lengths being thus multiplied (as in the other systems above 
 mentioned), but the longitudes and latitudes of the one line being 
 respectively added to those of the other : which was in fact the 
 rule expressed by Mr. J. T. Graves's co-ordinate formulae [39]. 
 [41.] It will not (I hope) be considered as claiming any me- 
 rit to myself in this matter, but merely as recording an unpursued 
 guess, which may assist to illustrate this whole inquiry, if I ven- 
 ture to mention here that thej^rs^ conjecture respecting geometri- 
 cal triplets, which I find noted among my papers (so long ago as 
 1830), was, that while lines in space might be ac/ofec? according 
 to the same rule as in the plane, they might be multiplied by 
 multiplying their lengths, and adding their polar angles. In the 
 method [36], known to me then as that of Mr. Warren, if we write 
 x = r cos 9, y = r sin 0, we have, for multiplication within the 
 plane, equations which may be written thus, r" = rr', 0"= + B\ It 
 hence occurred to me, that if we employed for space these other 
 known transformations of rectangular to polar co-ordinates, 
 
 x = r cos B, y- r sin 6 cos (f), z = r sin 6 sin 0, 
 
 it might be natural to define multiplication of lines in space by 
 the slightly extended but analogous formulae, 
 
 r" = rr', Q"=Q + &, 0"=0 + f : 
 
 which, however, conducted to radicals, as in the expression, 
 
 X" = XX - {if + 2:2)^ (?/'2 + 2:'^)*, 
 
 whereas within the -plane there were rational values for the rect- 
 angular co-ordinates of the product, namely (compare [17] ), 
 
 x" = XX - yy', y" = xy + yx. 
 
 But this old (and uncommunicated) conjecture of mine, which 
 was inconsistent with the distributive principle, though possess- 
 ing some general resemblance to the lately mentioned results [39] 
 [40] of Messrs. John and Charles Graves, cannot be considered 
 to have been an anticipation of them. For while we all agreed 
 in adding the longitudes of the two factors (in the sense lately 
 mentioned), they added latitudes also ; while 1, less happily, had 
 thought oi adding the colatitudes, or the angular distances from 
 a line {x), instead of those from a plane (xy). And this diffe- 
 
(40) PREFACE. 
 
 rence of plan produced a very important difference of results. 
 Indeed the two systems are totally distinct, although there exists 
 some sort of analogy between them. 
 
 [42.] I shall here mention one more system, which was com- 
 municated to me* in 1840, by the elder of those two brothers, 
 and which involved a method of representing the usual imagi- 
 nary quantities of algebra, each by a corresponding unique point 
 on the surface of a sphere, described (as in [37] ) about the ori- 
 gin with a radius = 1: whence it appeared that the ordinary ima- 
 ginary expression r (cos 9 + ^- 1 sin 0) might be denoted by a 
 triplet {x, y, z), under the condition, a"" + e/- + s"- = 1 : and that the 
 rules thus obtained, for the multiplication oi such triplets, might 
 perhaps afford some analogy, suggesting rulesj for the more ge- 
 neral case, where the constituents x,y, z are wholly independent 
 of each other. Mr. J. T. Graves's " mode of representing quan- 
 tity spherically" was stated by him to me as follows: — "All po- 
 " sitive quantities r may be represented by points on an assumed 
 " semicircle, by taking the extremity of the arc 2 tan-^ t (counted 
 "from one end (a) of the semicircle) to represent r. Next let us 
 " consider our sphere as generated by the revolution of the semi- 
 " circlet abc round the axis ac (forwards or backwards, according 
 " to arbitrary convention). When the semicircle has moved 
 " through an angle 0, let the position of a point on its circumfe- 
 <' rence denote r (cos + V ~ 1 sin 0), if the same point in its ori- 
 «« ginal position denoted r." I make a very easy transformation 
 of this statement, when I present it thus: — Construct all quan- 
 tities (so called), real and imaginary, according to the known me- 
 thod already described in [36], by drawing right lines from 
 the assumed point (a) of the unit-sphere, in the tangent plane at 
 that point; double all the lines so drawn, and treat the ends of 
 
 * In a letter of October 17th, 1840, from J. T. Graves, Esq, 
 + Mr. Graves appears not to have actually vs'orked out such rules, at least I 
 do not find that he communicated them to me. They would probably have been, 
 on the plan described in [42], to have multiplied (as before) the lengths, and (as 
 before) added the longitudes : but to have then multiplied the tangents of the halves 
 of the colatitudes of the factors, in order to obtain the tangent of the half of the 
 colatitude of the product. 
 
 t A figure, which it seems unnecessary here to reproduce, accompanied Mr. 
 Graves's Letter. 
 
PREFACE. (41) 
 
 the doubled lines as the stereographic projections of points upon 
 the sphere. Infinity was thus represented, in the particular system 
 of Mr. Graves here described, by the point diametrically oppo- 
 site to A. And in this endeavour of mine, to furnish faithfully a 
 record of every circumstance, which, even as remotely suggesting 
 to ^, friend a train of thought, may have indirectly stimulated 
 myself, I must not suppress the following acknowledgment of 
 Mr. J. T.Graves: — "What led me to this was a passage in 
 "a letter from De Morgan,* in which he expressed a wish to be 
 *' able to represent quantity circularly^ in order to explain the 
 *' passage from positive to negative through infinity." 
 
 [43.] The foregoing specimens may suffice to exemplify the 
 attempts which were made, a considerable number of years ago, 
 by Mr. Graves and by myself: on the one hand, to extend to 
 space that geometrical construction for the multiplication oi lines, 
 which was known to us from the work of Mr. Warren ; and on 
 the other hand, to render more entirely definite my conception of 
 algebraical triplets. 1 will not here trouble my readers with any 
 further account of the conjectures on those subjects which at va- 
 rious times occurred to him or me, before 1 was led to the qua- 
 ternions, in a way which I shall presently explain. But I wish 
 to mention first, that among the circumstances which assisted to 
 prevent me from losing sight of the general subject, and from 
 wholly abandoning the attempt to turn to some useful account 
 those early speculations of mine, on triplets and on setS;, was pro- 
 bably the publication of Professor De Morgan's first Paper on 
 the Foundation of Algebra,! of which he sent me a copy in 1841. 
 In that Paper, besides the discussion of other and more impor- 
 tant topics, my Essay on Pure Time was noticed, in a free but 
 friendly spirit ; and the subject of triplets was alluded to, in such 
 passages, for instance, as the following: — "But in this branch 
 of logical algebra" (that referred to in paragraph [36] of the pre- 
 sent Preface), " the lines must be all in one plane, or at least 
 afi'ected by only one modification of direction : the branch which 
 shall apply to a line drawn in any direction from a point, or mo- 
 
 * Augustus De Morgan, Esq., Professor of Mathematics in University College, 
 London. 
 
 f In Vol. VII., Part ii,, of the Cambridge Philosophical Transactions. 
 
 / 
 
(42) PREFACE. 
 
 dified by two distinct directions, is yet to be found." . . "An 
 extension to geometry of three* dimensions is not practicable 
 until we can assign two symbols, Q and w, such that a + bQ + C(o 
 = ai + 6ii2 + Ciw gives a = ai, b=bi, andc = Ci: and no definite 
 symbol of ordinary algebra will fulfil this condition." My sym- 
 bols X2, X3 (of 1834-5) had not then been published, nor other- 
 wise exhibited to him ; they were designed to fulfil precisely the 
 foregoing conditions : but I was not myself satisfied with them, 
 as not considering them " definite' enough (compare [29] ). 
 
 [44.] In the early numbers of the Cambridge Mathematical 
 Journal, there appeared some ingenious and original Papers, by 
 the late Mr. Gregory and by other able analysts, on the signs + 
 and-, on the powers of+, on branches of curves in diff'erent 
 planes, and on other connected subjects: but I hope that it will 
 not be thought disrespectful if I confess that I do not remember 
 their having had much influence on my own trains of thought. 
 Perhaps I was not sufficiently prepared, or disposed, to look at 
 algebra generally, and its applications to geometry, from the 
 same point of view, and was thereby prevented from studying 
 those Papers with the requisite attention. At least, if anything 
 in my own views shall be found to be inconsistent with those put 
 forward in the Papers thus alluded to, I wish it to be considered 
 as offered with every deference, and not in a controversial spirit. 
 And if for the present I omit all further mention of them, it is 
 partly because, without a closer study, I should fear to do them 
 injustice : and partly because I make no pretensions to be here 
 
 * Professor De Morgan proposed at the same time a remarkable conjecture, 
 which he maybe considered to have afterwards illustrated and systematised, by 
 his theory of cube-roots of negative unity, employed as geometrical operators, in 
 his Paper on Triple Algebra (Camb. Phil. Trans., Vol. VIII., Part, iii.); namely, 
 that "an extension to three dimensions" might "require a solution of the equa- 
 tion ^^ x = — X." I much regret that my plan wdll not allow me to attempt the 
 giving any further account, in this Preface, of that very original Paper of Pro- 
 fessor De Morgan, the first suggestion of which he was pleased to attribute to 
 the publication of my own remarks on Quaternions, in the Philosophical Maga- 
 zine for July, 1844: and a similar expression of regret applies to the independent 
 but somewhat later researches of Messrs. John and Charles Graves, in the same 
 year, respecting other Triplet Systems, which involved cube-roots of positive 
 unity, and of which some account has been preserved in the Proceedings of the 
 Royal Irish Academy. 
 
PREFACE. (43) 
 
 an historian of science, even in one department of mathematical 
 speculation, or to give anything more than an account of the pro- 
 gress of my own thoughts, upon one class of subjects. For the 
 same reasons, I pass over some other investigations having refe- 
 rence to the imaginary* symbol of algebra, which were not used 
 as suggestions by myself, and proceed at once to the quaternions. 
 [45.] With such preparations as I have described, I resumed 
 (in 1843) the endeavour to adapt the general conception of trip- 
 lets to the multiplication of lines in space, resolving to retain the 
 distributive principle, with which some formerly conjectured sys- 
 tems had been inconsistent, and at first supposing that I could 
 preserve the commutative principle also, or the convertibility 
 [24] [29] of the factors as to their o?'der. Instead of my old 
 symbols Xj, x^, xs (see [22] ), I wrote more shortly 1, i,J; so that 
 a numerical triplet took the form x + iy -vjz, where I proposed to 
 interpret x, y, z as three rectangular co-ordinates, and the trip- 
 let itself as denoting a line in space. From the analogy of cou- 
 
 * I am unwilling, howevei', to leave unmentioned here (although it did not 
 happen to supply me with any suggestion), a remarkable use of the symbol 
 V— I, which was made by the late Professor Mac CuUagh, of Dublin, whose great 
 and original powers in mathematical and physical science must ever be remem- 
 bered with admiration, and which he seems to have connected (in 1843) with in- 
 vestigations respecting the total reflexion of light. (See Proceedings of the 
 R. I. A. for the date of January 13, 1845.) This use of imaginaries was founded 
 on a theorem relative to the ellipse, which was expressed by him as follows, in 
 a question proposed at the Examination for the Election of Junior Fellows in 
 1842 (see Dublin University Examination Papers for that year, published in 
 1843, p. Ixxxiv.): — "Detur in spatio ellipsis, cujus centrum est origo co-ordina- 
 tarum. Puncta xy%, x y'z in ellipsi sint termini diametrorum conjugatarum. 
 Ostendendum est quantitates imaginarias 
 
 y+y' V^ z + 2 V~ 
 x-\-x\/-\ x + xV-l 
 
 constantes esse pi'o quolibet systemate diametrorum conjugatarum." This 
 elegant theorem of Professor Mac Cullagh may easily be proved, without em- 
 ploying any but the usual principles respecting the symbol V— 1, by observing 
 that the following expressions, for the six co-ordinates in question, 
 
 x = a cos v + d sin v, y = b cos v + b' sin v, z = c cos v + c' sin v, 
 X = a' cos v — a sinv, y' = b' cos v - b sin v, z' = c' cos « — c sin v, 
 give 
 
 a?+a!'V-l y+y'\/-l z + z'V-1 
 
 a + rt'V-1 6 + 6'V-l c + c'V-1 
 
 = cos »-sin vV-1. 
 
(44) PREFACE. 
 
 pies, I assumed i^=-\; and tried the effect of assuming also 
 j~--l, which I interpreted as answering to a rotation through 
 two right angles in the plane of ^2, as i^ = -l had corresponded 
 to such a rotation in the plane of oci/. And because I at first 
 supposed that if and ji were to be equal, as in the ordinary cal- 
 culations of algebra, the product of two triplets appeared to take 
 the form, 
 
 {a + lb +Jc) {x + iy +jz) = (ax -by- cz) + i {ay + hx^ 
 +j (az + ex) + ij (bz + cy) : 
 
 but I did not at once see what to do with ihe product ij. The 
 theory of triplets seemed to require that it should be itself a 
 triplet, of the form, 
 
 «>■ - a + ?/3 + jV» 
 
 the coefBcients a, /3, 7 being some three constant numbers : but 
 the question arose, how were those numbers to be determined, so 
 as to adapt in the best way the resulting formula of multiplica- 
 tion to some guiding geometrical analogies. 
 
 [46.] To assist myself in applying such analogies, I consi- 
 dered the case where the co-ordinates b, c were proportional to 
 ?/, z^ so that the two factor-lines were in one common plane, con- 
 taining the unit-line, or the axis of x. In that particular case^ 
 there was ready a known signification [36] for the product line, 
 considered as the fourth proportional to the unit-line (assumed 
 here on the last-mentioned axis), and to the two coplanar factor- 
 lines. And I found, without difficulty, that the co-ordinate pro- 
 jections of such a fourth proportional were here, 
 
 ax -by- cz, ay + bx, az + ex, 
 
 that is to say, the coefficients of 1, i,j, in the recently written 
 expression for the product of the two triplets, which had been 
 supposed to represent the factor-lines. In fact, if we assume 
 y = \b, z = \c, where X is any coefficient, we have the two iden- 
 tical equations, 
 
 {ax - Xb^ - Xc2)2 + (Xa + xf (b^ + c^) = (a^ + b'^ + c'-) (x^ + X^ b^ + X^ c^), 
 
 (Xa + X) (b- + c'-)^ (J^-t-c^)* ^ X(62 + c-)^ 
 
 tan-i ^^ :^yr; — ^ = tan'^ -^ —^ + tan-i— ^ -> 
 
 ax-X {b~ + c^) a x 
 
PREFACE. (45) 
 
 which express that the required geometrical conditions are satis- 
 fied. It was allowed then, in this case ofcoplanarity, or under 
 the particular condition, 
 
 hz - cy = 0, 
 to treat the triplet, 
 
 {ax -by - cz) + i {ay + hx) +j {az + ex), 
 
 as denoting a line which might, consistently with known analo- 
 gies, be regarded as the product of the two lines denoted by the 
 two proposed triplets, 
 
 a + ib +jCi and x + iy ^-jz. 
 And here the fourth term, 
 
 ij{bz-]rcy), 
 
 appeared to be simply superfluous : which induced me for a mo- 
 ment to fancy that perhaps the product ij was to be regarded as 
 = 0. But I saw that this fourth term (or part) of the product 
 was more immediately given, in the calculation, as the sum of 
 the two following, 
 
 ib .jz, jc.iy, 
 
 and that this su7n would vanish, under the present condition 
 bz = cy, if we made what appeared to me a less harsh supposition, 
 namely, the supposition (for which my old speculations on sets 
 had prepared me) that 
 
 ij = -ji: 
 or that 
 
 ij= + k,ji = -k, 
 
 the value of the product k being still left undetermined. 
 
 [47.] In this manner, without now assuming bz-cy^O, I 
 had generally for the product of two triplets, the expression of 
 quadrinomial form, 
 
 {a + ib +jc) {x + iy +Jz) = {ax -by- cz) + i {ay + bx) 
 -\-j {az + ex) + k {bz - cy) ; 
 
 and I saw that although the product of the sums of squares of 
 the constituents of the two factors could not in general be decom- 
 posed into three squares of rational functions of them, yet it 
 could be generally presented as the sum oi four such squares. 
 
(46) PREFACE. 
 
 namely, the squares of the four coefficients of 1, 2,7, k^ in the 
 expression just deduced : for, without any relation being assumed 
 between a, 5, c, a, ?/, z, there was the identity, 
 
 (a?' -\-b' + C-) (x- + 2/^ + 2'^) = {ax -hy - cz)- + {ay + bxf 
 + {az + cxf + {1)Z - cyf. 
 
 This led me to conceive that perhaps instead of seeking to con- 
 fine ourselves to triplets, such as a + ib +jc or (a, b, c), we ought 
 to regard these as only imperfect forms o/quaternions, such as 
 a + ib -{-Jc + kd, or (a, b, c, d), the symbol k denoting sotne new 
 sort of unit operator : and that thus my old conception of sets 
 [30] might receive a new and useful application. But it was ne- 
 cessary, for operating definitely with such quaternions, to fix the 
 value of the square k^, of this new symbol k, and also the values 
 oi the products, ik,jky ki, kj. It seemed natural, after assuming 
 as above that i^=f = - 1, and that ij = k,ji=-k, to assume also 
 that ki = -ik = - i^J = +j, and kj = -jk =j^ i = - i. The assump- 
 tion to be made respecting k- was less obvious ; and I was for a 
 while disposed to consider this square as equal to positive unity, 
 because i-f = + 1 : but it appeared more convenient to suppose, in 
 consistency with the foregoing expressions for the products of 
 i, j, kf that 
 
 ^2 = ijij = - iijj = -i'f = - {- I) (-1) = -I. 
 
 [48.] Thus all the fundamental assumptions for the multipli- 
 cation of two quaternions were completed, and were included in 
 the formulae, 
 
 j2=/=^2=-l; ij = -ji = k; jk=-kj=i; kt = -ik=j: 
 which gave me the equation, 
 
 («, b, c, d) (a, b\ c, d') = (a", b", c", d"), 
 or 
 
 (a + ib +Jc + kd) {a + ib' +jc' + kd') = a" + ib" +jc" + kd", 
 
 when and only when the following yo?^r separate equations were 
 satisfied by the constituents of these three quaternions : 
 
 «" = aa' — bb' — cc - dd', 
 b"={ab'+ba') + {cd'-dc), 
 c"= {ac + ca!) + {dh' - bd'), 
 d" = {ad'+ da) + {be - cb') . 
 
PREFACE. (47) 
 
 And I perceived on trial, for 1 was not acquainted with a theorem 
 of Euler respecting smns of four squares, which might have 
 enabled me to anticipate the result, that these expressions for 
 a, h'\ c", d" had the following modular "property : 
 
 a"" + V"" + c"2 + d'"" = (a^ + b^ + c^ + d^) (a'^ + h'^ + c^ + d'^). 
 
 I saw also that if, instead of representing a line by a triplet of 
 the form x + iy+jz^ we should agree to represent it by this other 
 trinomial form, 
 
 ix +jy + kz, 
 
 we should then be able to express the desired product of two lines 
 in space by a quaternion, of which the constituents have very 
 simple geometrical significations, namely, by the following, 
 
 (ix +jy + kz) {ix ■\jy' + hz) = w" + ix +jy" + kz", 
 where 
 
 w" --XX - yy - zz', 
 
 x" = yz' - zy, y" = zx! - xz! , s" = xy - yx ; 
 
 so that the part w", independent of ijk, in this expression for the 
 product, represents the product of the lengths of the two factor- 
 lines, multiplied by the cosine of the supplement of their inclina- 
 tion to each other; and the remaining part «V +72/" + ^«" of the 
 same product of the two trinomials represents a line, which is in 
 length the product of the same two lengths, multiplied by the sine 
 of the same inclination, while in direction it is perpendicular to 
 the plane of the factor-lines, and is such that the Quotation round 
 the multiplier-line, from the multiplicand-line towards the pro- 
 duct-line (or towards the line-part of the whole quaternion pro- 
 duct), has the same right-handed (or left-handed) character, as 
 the rotation round the positive semiaxis of ^ (or of ^;), from the 
 positive semiaxis of i (or of a:), towards that oij (or oi y). 
 
 [49.] When the conception, above described, had been so far 
 unfolded and fixed in my mind, I felt that the new instrument for 
 applying calculation to geometry, for which 1 had so long sought, 
 was now, at least in part, attained. And although 1 had left se- 
 veral former conjectures respecting triplets for many years uncom- 
 municated, except by name, even to friends, yet 1 at once pro- 
 ceeded to lay these results respecting quaternions before the 
 
(48) PREFACE. 
 
 Royal Irish Academy (at a Meeting of Council* in October, 
 1843, and at a General Meeting! shortly subsequent) : introducing 
 also a theory of their connexion with spherical trigonometry, some 
 sketch of which appeared a few months later in London (in the Phi- 
 losophical Magazine for July, 1844). On that connexion of quater- 
 nions with spherical trigonometry, and generally with spherical 
 geometry, I need not at present dwell, since it is sufficiently ex- 
 plained in the concluding Lectures of this Volume : but it may be 
 not improper that a brief account should here be given, of a not 
 much later but hitherto unpublished speculation, of a character 
 partly geometrical, but partly also metaphysical (or a priori), by 
 which I sought to explain and confirm some results that might 
 at first seem strange, among those to which my analysis had con- 
 ducted me, respecting the quadrinomial form , and non-commuta- 
 tive property, of the product of two directed lines in space. 
 
 [50.] Let, then, the product of two co-initial lines, or of two 
 vectors from a common origin, be conceived to be something which 
 has QUANTITY, in the sense that it is doubled, tripled, &c., by dou- 
 bling, tripling, &c., either factor; let it also be conceived to have 
 in some sense, quality, analogous to direction, which is in some 
 way definitely connected with the directions of the two factor lines. 
 In particular let us conceive, in order to preserve so far an ana- 
 logy to algebraic multiplication, that its direction is in all re- 
 spects reversed, when either of those directions is reversed ; and 
 therefore that it is restored, when both of them are reversed. On 
 
 • The Minutes of Council of the R. I. A., for October 16th, 1843, record 
 "Leave given to the President to read a paper on a new species of imaginary 
 quantities, connected with a theory of quaternions." It may be necessary to 
 state, in explanation, that the Chair of the Academy, which has since been so 
 well filled by my friends, Drs. Lloyd and Robinson, was at that time occupied 
 by me. 
 
 t At the Meeting of November 13th, 1843, as recorded in the '■^Proceedings" 
 of that date, in which the fundamental formulae and interpretations respecting 
 the symbols ijk are given. Two letters on the siibject, which have since been 
 printed, were also written in October, 1843, to the friend so often mentioned in 
 this Preface, Mr. J. T. Graves : and the chief results were also exhibited to his 
 brother, the Rev. C. Graves, before the public communication of November, 
 1843. These circumstances (or some of them) have been stated elsewhere : but 
 it seemed proper not to pass them over without some short notice here, as con- 
 nected with the date of the invention and publication of the quaternions. 
 
PREFACE. (49) 
 
 the other hand, for the sake of recognising what may be called 
 the symmetry of space, let this direction of the -product^ so far as 
 it can be constructed or represented by that of any line in space, 
 be conceived as not changing its relation to the system of those 
 two factor directions, when that system is in any manner turned in 
 space .* its own direction, as a line, being at the same time turned 
 with them, as if it formed a part of one common and rigid system ; 
 and the numerical element of the same product (if it have any 
 such) undergoing no change by such rotation. Let the product 
 in question be conceived to be G^ntueXy determined, when the fac- 
 tors are determined; let it be made, if other conditions will allow, 
 for the sake of general analogies, a distributive function of those 
 two factors, summation of lines being performed by the same rules 
 as composition of motions; and finally, if these various conditions 
 can all be satisfied, and still leave anything undetermined, in the 
 rules for multiplication of lines, let the indeterminateness be re- 
 moved in such a way as to make these rules approach as much 
 as possible to the other usual rules for the multiplication of num- 
 bers in algebra. 
 
 [51.] The square of a given line must not be any line in- 
 clined to that given line; for, even if we chose any particular 
 angle of inclination, there would be nothing to determine the 
 plane, and tlms the square would be indeterminate, unless we 
 selected some one direction in space as eminent, which selection 
 we are endeavouring to avoid. Nor can the square of a given 
 line be a line in the sa?ne direction, nor in the direction opposite; 
 for if either of these directions were selected, by a definition, then 
 this definition would oblige us to consider the square as reversed 
 in direction, when the line of which it is the square is reversed ; 
 whereas if the two factors of a product both change sign, the di- 
 rection of the product is always (by what has been above agreed 
 on) preserved, or rather restored. We must, therefore, consider 
 the SQUARE OF A LINE as having no direction in space, and there- 
 fore as being 7iot (properly) itself a line; but nothing hitherto 
 prevents us from regarding the square as a number, which has 
 always one determined sigti (as yet unknown), and varies in the 
 duplicate ratio of the length of the line to be squared. If, then, 
 the length of a line a contain a times the unit of length, we are 
 
 9 
 
(50) PREFACE. 
 
 led to consider aa or a' as a symbol equivalent to /a', in which I 
 is some numerical coefficient, positive or negative, as yet un- 
 known, but constant for all lines in space, or having one common 
 value for all. And, consequently, if a, j3 be any two lines in any 
 one common direction^ and having their lengths denoted by the 
 numbers a and b, we are led to regard the product aj3 as equal to 
 the number lab, I being the same coefficient as before. But if the 
 direction of j3 be exactly opposite to that of a, their lengths being 
 still a and &, their product is then equal to the opposite number, 
 — lab. The same general conclusions might perhaps have been 
 more easily arrived at, if we had begun by considering the pro- 
 duct of two equally long but opposite lines ; for it might perhaps 
 then have been even easier to see that, consistently with the sym- 
 metry of space ^ no one line rather than another could represent, 
 even in part, the direction of the product. 
 
 [52.] Next, let us consider the product aj3 of two mutually 
 •perpendicular lines, a and j3, of which each has its length equal 
 to 1. Let a', j3' be lines respectively equal in length to these, 
 but respectively opposite in direction. Then a j3 = - aj3 = a/3' ; 
 a'/3'= aj3. If the sought product aj3 were equal to any number, 
 or even if it contained a number as a part of its expression, then, 
 on our changing the multiplier a to its own opposite line a', this 
 product or part ought/or one reason{\he symmetry of space) to re- 
 main constant (because the system of the factors would have been 
 merely turned in space^ ; and for another reason (a'j3 =- aj3) the 
 same product or part ought to change sign (because one factor 
 would have been reversed) : but this co-existence of opposite re- 
 sults would be absurd. We are led therefore to try whether the 
 present condition {oi rectangularity of the two factors) allows us 
 to suppose the product a/3 to be a line. 
 
 [53.] Let 7 be a third line, of which the length is unity, and 
 which is at the positive side of /3, with reference to a as an axis 
 of rotation ; right-handed (or left-handed) rotation having been 
 previously selected as positive; let also -y' be the line opposite 
 to -y. Then any line in space may be denoted by ma-V n^-^ py, 
 we are therefore to try whether we can consistently suppose a/3 
 = ma + w/3 + P7, m, n, p being some three numerical constants. 
 If so, we should have (by the principle of the symmetry of space) 
 
PREFACE. (51) 
 
 a'/3 = ma + wj3 +^7'; and therefore (by a change of all the signs) 
 aj3 = ma + wj3'+ py ; therefore n[5' = nj3, and consequently - w = «, or 
 finally n = 0. In like manner, since aj3 = - a(5' = - (ma + n^'+py') 
 = /wa' + wj3 + joy , we should have md = ma, and therefore m = 0. 
 But there is no objection of this kind against supposing aj3 =pyi 
 p being some numerical coefficient, constant for all pairs of rectan- 
 gular lines in space : for the reversal of the direction of a factor 
 has the effect of turning the system through two right angles 
 round the other factor as an axis, and so reverses the direction of 
 the product. And then if the lengths of these two lines a, j3, in- 
 stead of being each = 1, are respectively a and b, their product aj3 
 will be =paby ; that is, it will be a line perpendicular to both fac- 
 tors, with a length denoted by pab, and situated always to the 
 positive or always to the negative side of the multiplicand line j3, 
 with respect to the multiplier line a as an axis of rotation, accord- 
 ing as the constant number p is positive or negative. 
 
 [54.] So far, then, without having yet used any property of 
 multiplication, algebraical or geometrical, beyond the three prin- 
 ciples : 1st, that no one direction in space is to be regarded as 
 eminent above another; 2nd, that to multiple/ either factor by any 
 number, positive or negative, multiplies the product by the same; 
 and 3rd, that the product of two determined factors is itself de- 
 termined ; we are led to assign interpretations: 1st. to the pro- 
 duct of two co-axal vectors, or of two lines parallel to each other, 
 or to one common axis ; and 2nd, to the product of two rectan- 
 gular veciovs ; which interpretations introduce only two constant, 
 but as yet unknown, numerical coefficients, I and p, depending, 
 however, partly on the assumed unit of length. And we see 
 that for any two co-axal vectors, a, j3, the equation a/3 - j3a = 
 holds good; but that for any two rectangular vectors, aj3 + j3a = 0. 
 K product of two rectangular lines is, therefore, so far as the 
 foregoing investigation leads us to conclude, not a commutative 
 function of them. 
 
 [55.] Since then we are compelled, by considerations which 
 appear more primary, to give up the commutative property of 
 multiplication, as not holding generally for lines, let us at least 
 try (as was proposed) whether we can retain the distributive pro- 
 perty. If so, and if the multiplicand line j3 be the sum of two 
 
(52) PREFACE. 
 
 Others, j3i and jSs, of which one (j3i) is co-axal with the multiplier 
 line a, while the other (fij) is perpendicular thereto, we must in- 
 terpret the product ai5 as equal to the sum of the two partial 
 products, aj3i and 0)82. But one of these is a number, and the 
 other is a line ; we are, therefore, led to consider a number as being 
 under these circumstances added to a line, and as forming with it 
 a certain swn, or system, denoted by aj3i + aj325 ov more shortly 
 by aj3. And such a sum of line and number may perhaps be 
 called a grammarithm,* from the two Greek words, ypafifin, a 
 line, and apidfxog, a number, A grammarithm is thus to be con- 
 cei ved as being entirely determined, when its two parts or elements 
 are so ; that is, when its grammic part is a known line, and its arith- 
 mic part is a known number. A change in either part is to be 
 conceived as changing the grammarithm : thus, an equation be- 
 tiveen two grammarithms includes generally two other equations, 
 one between two numbers, and another between two lines. 
 Adopting this view of a grammarithm, and defining that aj3 = a)3i 
 + ajSa, when j3 = /3i + jSg, j3i [[ a, jSs .L a, the product of any deter- 
 mined multiplier line and any determined multiplicand line will 
 be itself entirely determined, as soon as the unit of length and 
 the numbers / and p shall have been chosen ; and it remains to 
 consider whether these numbers can now be so selected, as to 
 make the rules of multiplication of lines approach more closely 
 still to the rules of multiplication of numbers. 
 
 [56.] The general distributive principle will be found to give 
 no new condition; and we have seen cause to reject the commu- 
 tative principle or property, as not generally holding good in the 
 present inquiry. It remains, then, to try whether we can deter- 
 mine or connect the two coefficients, I andp, so as to satisfy the 
 associative principle, or to verify the formula, 
 
 a . jSy = Mj3 . 7. 
 
 * The word "grammarithm" was subsequently proposed in a communication 
 to the Royal Irish Academy (see the Proceedings of July, 1846), as one which 
 might replace the word " quaternion," at least in the geometrical view of the 
 subject : but it did not appear that there would be anything gained by the sys- 
 tematic adoption of this change of expression, although the mere suggestion of 
 another name, as not inapplicable, seemed to throw a little additional light on 
 the whole theory. 
 
PREFACE. (53) 
 
 For this purpose we may first distribute tlie factors j3, 7 into 
 others, /3i/32 7i73'y3 which shall be parallel or perpendicular to 
 it and to each other; and then shall have to satisfy, if possible, 
 six conditions, Vt'hich may be reduced to the six following : 
 
 a .aa = aa . a; a . aa ~ aa . a ; a .aa = aa . a" ', 
 a . aa — aa.a] a . da = ad. d \ a . da ~ ad. a ; 
 
 a, a', a being three rectangular unit-lines, so placed that the ro- 
 tation round a from d to a is positive. Then, by what has been 
 already found, the following relations will hold good : 
 
 aa = dd = da = / ; ad = — da= pa" ; 
 aa = — da = - pd ; da" = — dd = +pa ; 
 
 and the six conditions to be satisfied become, 
 
 a. I -I. a; a.pd-l.d', a .-j^d = I . a" ; 
 
 a . -pd'=pd' . o ; a .1 = pd' . a' ; a.pa= pd' . a". 
 
 Of these the first suggests to us to treat an arithmic factor as 
 commutative (as regards order) with a grammic one, or to treat 
 the product " line into number" as equivalent to " number into 
 line;" the fourth and sixth conditions afford no new information; 
 and the second, third, and fifth become, 
 
 -p~d = Id ; - p'^ a" = la" ; ~p^ a = la. 
 
 The conditions of association are therefore all satisfied by our 
 assuming, with the present signification of the symbols, 
 
 al=la, and /= -p^; 
 
 and they cannot be satisfied otherwise. The constant I is, there- 
 fore, by those conditions, necessarily negative; and every line 
 in tridimensional space has its square (on this plan) equal to a 
 NEGATIVE number: which is one of the most novel but essential 
 elements of the whole quaternion theory. (Compare the recent 
 paragraph [48] ; also art. 85, pages 81, 82, of the Lectures.) And 
 that a grammarithm [55] may properly be called a quaternion, 
 appears from the consideration that the line^ which in it is added 
 to a number , depends itself upon a system of three numbers, or 
 may be represented by a trinomial expression, because it is al- 
 ways the sum of three lines (actual or null), which are parallel 
 
(54) PREFACE. 
 
 to three fixed directions (compare Lecture III.). Tiie coefficient 
 p remains still undetermined, and may be made equal to positive 
 one, by a suitable choice of the unit of length, and the direction 
 of positive rotation. In this way we shall have finally the very 
 simple values, 
 
 /? = + 1, l=-\; 
 
 and the rules for the multiplication of lines in space will then be- 
 come entirely definite, and will agree in all respects with the re- 
 lations [48], between the symbols ijk. 
 
 [57.] Another train of a priori reasoning, by which I early 
 sought to confirm, or (if it had been necessary) to correct, the 
 results expressed by those new symbols, was stated to the R. I. 
 Academy* in (substantially) the following way. Admitting, for di- 
 rected and copla7iar lines, the conception [36] o( proportion ; and 
 retaining the symbols ijk, or more fully, + ^, +j, + k, to denote 
 three rectangular unit-lines as above, while the three respectively 
 opposite lines may be denoted by - i, -j, -h; but not assuming 
 the knowledge of any laws respecting their multiplication, I 
 sought to determine what ought to he considered as the fourth 
 PROPORTIONAL, u, to the three rectangular directions^ j, i, ky 
 consisteiitly with that hiown conception [36] /or directions within 
 the plane, and with some general and guiding principles, respect- 
 ing ratios and proportions. These latter assumed principles 
 (of a regulative rather than a constitutive kind) were simply 
 the following: 1st, that ratios similar to the same ratio must 
 be regarded as similar to each other; 2nd, that the respec- 
 tively inverse ratios are also mutually similar; and 3rd, that 
 ratios are similar, if they be similarly compounded of similar 
 ratios : this similarity of composition being understood to include 
 generally a sameness of order. It seemed to me that any pro- 
 posed definitional! use of the word ratio, which should be in- 
 
 » See the Proceedings of November 11th, 1844. 
 
 •j- In the abstract published in the Proceedings, the words " South, West, 
 Up" were used at first instead of the symbols i, j, k ; and the sought fourth pro- 
 portional to jik, which is here denoted by m, was called, provisionally, " Forward." 
 
 % As an example of the use of the first of these very simple principles, in 
 serving to exclude a definition which might for a moment appear plausible, let us 
 take the construction [38], and inquii'e whether (as that construction would 
 
PREFACE. (55) 
 
 consistent with these principles, would depart thereby too widely 
 from known analogies, mathematical and metaphysical, and would 
 involve an impropriety of language: while, on the other hand, it ap- 
 peared that if these principles were attended to, and other analogies 
 observed, it was permitted to extend the use of that word ratio, and 
 
 suggest) we can properly say that four directions (or four diverging unit-lines), 
 a, /3, y, d, form generally a proportion in space, when the angles ad, |8y, between 
 the extremes and means have one common bisector (s). If so, when the three di- 
 rections a, (3, y became rectangular, we should have a: (3 :: y:~a, and y : — a 
 : : /3 : - y ; but we should have also, a: j3 :: (3:- a, and not a: (3 :: f3:- y -,80 that 
 the two ratios, a : j8 and /3 : — y, would be said to be similar to one common ratio 
 (y : — a), without being similar to each other, if the foregoing construction for a 
 fourth proportional were to be, by definition, adopted : and this objection alone 
 would be held by me to be decisive against the introduction of such a definition ; 
 and therefore also against the adoption of the connected rule mentioned in [38], 
 as having at one time occurred to a friend (J. T. G.) and to myself, for the mul- 
 tiplication of lines in space, even if there were no other reasons (as in fact there 
 are), for the rejection of thatrule. A similar objection applies, with equal decisive- 
 ness, against the rule mentioned in [37], as an earlier conjecture of my own. On 
 the other hand, an analogous and equally simple argument may serve to justify the 
 notation d — c = b — a, employed by me in the following Lectures, and elsewhere, to 
 express that the two right lines ab and cd are equally long and similarly directed, 
 against an objection made some years ago, in a perfectly candid spirit, by an 
 able writer in the Philosophical Magazine (for June, 1849, p. 410) ; who thought 
 that interpretation more arbitrary than it had appeared to me to be; and 
 suggested that the same notation might as well have been employed to signify 
 this other conception: — that the two equally long lines ab, cd met somewhere, at 
 a finite or infinite distance. I could not admit this extension ; for it would lead 
 to the conclusion that two lines ab, ef might be equal to the same third line cd, 
 without being equal to each other : which would (in my opinion) be so great a 
 violation of analogy, as to render the use of the word "equal," or of the sign =, 
 with the interpretation referred to, an embarrassment instead of an assistance. 
 But I do not feel that analogies are thus violated, by the simultaneous admis- 
 sion of the two contrasted proportions (see (3) (4) (3) of [57] ), 
 
 u:i ::j: k, u:j::i:-k; 
 
 for the elementary theorem called often " alternando," {kvaXkct^ \6yOQ, Euc. V. 
 Def. 1 3, and Prop. 1 6) is by its nature limited (in its original meaning) to the case 
 where the means which change places are homogeneous with each other : whereas 
 two rectangular directions, as here i and j, are in this whole theory regarded as 
 being in some sense heterogeneous. They have at least no relation to each other, 
 which can be represented by any ratio, such as Euclid considers, of magnitude 
 to magnitude; and therefore we have no right to expect, from analogy to old re- 
 sults, that alternation shdX\. generally be allowed in a, proportion involving such 
 directions : although, within the plane, alternation is found to be admissible. 
 
(56) PREFACE. 
 
 the connected phrase proportion, not only from quantity to direc- 
 tion, within one plane, as had been done [36] by other writers,* 
 
 * Since the note to paragraph [36], pp. (31) (32), was in type, I have had an 
 opportunity of re-consulting the fourth volume of the Annales de Mathematiques, 
 and have found my recollections (agreeing indeed in the main with the formerly 
 cited page 228 of Dr. Peacock's admirable Report), respecting the admitted 
 priority of Argand, confirmed. Fran9ais, indeed (in 1813), published in those 
 Annales (Tome IV., pp. 61, . . 71) a paper which contained a theory of "pro- 
 portion de grandeur et de position," with a connected theory of multiplication 
 (and also of addition) of lines in a given plane ; but he expressly and honourably 
 stated at the same time (p. 70), that he owed the substance of those new ideas 
 to another person (" le fond de ces idees nouvelles ne m' appartient pas") : 
 and on being soon afterwards shewn, through Gergonne, whose conduct in the 
 whole matter deserves praise, a copyof Argand's earlier and printed Essay (Paris, 
 1806), Fran9ais most fully and distinctly recognised (p. 225) that the true author of 
 the method was Argand ("il n'y a pas le moindre doute qu'on ne doive a M. Ar- 
 gand la premiere idee de representer geometriquement les quantites imagi- 
 naires"). Nothing more lucid than Argand's own statements (see the same 
 volume, pp. 136, 137, 138), as regards t)ie fundamental principles of the theory of 
 the addition and multiplication of coplanar lines, has since (so far as I know) 
 appeared ; not even in the writings of Professor De Morgan on Double Algebra, 
 referred to in former notes. But Argand had not anticipated De Morgan's 
 theory of Logometers ; and was on the contrary disposed (pp. 144, . . 146) to 
 
 _V~1 
 regard the symbol V— 1 , notwithstanding Euler's well-known result, as de- 
 noting a line (kp), perpendicular to the plane of the lines 1 and V— 1 : and to con- 
 sider it as offering an example of a quantity which was irreducible to the form 
 p + q^- 1, and was (according to him) as heterogeneous with respect to V— 1, as 
 the latter with respect to + 1 (" aussi heterogene" &c.). The word modulus 
 (" module"), so well known by the important writings of M. Cauchy, occurs in a 
 later paper by Argand, in the following volume of the Annales, as denoting 
 the real quantity Vp* + 52. If I have seemed to dwell too much on the specula- 
 tions of Argand (not all adopted by myself), it has been partly because (so far 
 as I have observed) his merits as an original inventor have not yet been suffi- 
 ciently recognised by mathematicians in these countries : and partly because one 
 of the two most essential links (the other being addition) between Double Algebra 
 and Quaternions, is Argand's main siadi fundamental principle respecting copla- 
 nar PROPORTION, expressed by him as follows (Annales, T. IV., pp. 136, 137) : — 
 " Si (fig. 2) Ang. AK'B = Ang. a'k'b', on a, abstraction faite des grandeurs abso- 
 lues, KA : KB : : k'a' : k'b'. C'est la le principe fondamental de la theorie dont 
 nous avons essaye de poser les premieres bases, dans 1' ecrit dont nous donnons 
 ici un extrait" (namely, Argand's printed Essay of 1806, exhibited by Gergonne 
 to Fran9ais, after the appearance of the first paper of the latter author on the 
 subject in 1813). Argand continued thus (in p. 137) : " Ce principe n'a rien 
 au fond de plus etrange que celui sur lequel est fondee la conception du rapport 
 geometrique entre deux lignes de signes difPerens, et il n' en est proprement 
 qu' une generalisation :" a remark in which I perfectly concur. 
 
PREFACE. (57) 
 
 but "also from the plane to space.* The supposed propor- 
 tion, 
 
 J:i::k:u, (1) 
 
 gave thus, by inversion, 
 
 u:k::i:j; (2) 
 
 but also, in the planes of y, ik, there were the two proportions, 
 
 * • J • • J • - h «ind k:i :'.-i:k; (3) 
 
 compounding therefore, on the one hand, the two ratios, u : k and 
 k:i, and, on the other hand, the two respectively similar ratios, 
 j:-i, and -i:k, there resulted the new proportion, 
 
 u:iz:j:k; (4) 
 
 which differed from the proportion (2) only by a cyclical trans- 
 
 * Although the observations in pan [57] relate rather to proportions than to 
 imaginaries, yet the present may be a convenient occasion for remarking that 
 Buee, and even Wallis, had speculated, before Argand and Fi-anfais, on inter- 
 pretations of the symbol V— 1, which should extend to space ,- but that the nearest 
 approach to an anticipation of the quaternions, or at least to an anticipation of 
 triplets^ seems to me to have been made by Servois, in a passage of the lately 
 cited volume of Gergonne^s Annales, which appears curious and appropriate 
 enough to be extracted here. Servois had been following up a hint of Gergonne, 
 respecting the representation of ordinary imaginaries of the form x + t/^—l 
 (x and y being whole numbers), by a table of double argument (p. 71) ; and 
 thought (p. 235) that such a table might be regarded as only a slice (une tranche) 
 of a table of triple argument, for representing points (or lines) in space. He 
 thus continued: — " Vous donneriez sans doute i chacune terme la forme trino- 
 ' ' miale ; mais quel coefficient aurait le troisieme terme ? Je ne le vols pas trop. 
 *' L' analogic semblerait exiger que le trinome fut de la forme, p cos a-{- q cos /3 
 ** +r cos y, a, §T y etant les angles d'une droite avec trois axes rectangulaires ; 
 ** et qu' on eut 
 
 " (p cos a+ g cos j3 + r cos y) (p cos a + 5' cos /3 + r' cos y) = cos2 a + cos2/3 + cos2 y = 1. 
 "Les valeurs de p, q^ r, p, q\ r qui satisferaient h cette condition seraient ab- 
 ^* surdes" (" quantites non-reelles," as he shortly afterwards calls them) : "mais 
 " seraient-elles imaginaires reductibles a la forme generale A + BV— 1 ? Voila 
 " une question d' analise fort singuliere, que je soumets a vos lumieres." The 
 six NON-BEALS which Servois thus with remarkable sagacity jToresaw, without 
 being able to determine them, may now be identified with the then unknown sym- 
 bols + i, +j, + k, — i, —J, — k, of the quaternion theory : at least, these latter 
 symbols fulfil precisely the condition proposed by him, and furnish an answer to 
 his " singular question." It maybe proper to state that my own theory had 
 been constructed and published for a long time, before the lately cited passage 
 happened to meet my eye. 
 
 h 
 
(58) PREFACE. 
 
 position of the three directions ijk. For the same reason, we 
 
 may make another cyclical change of the same sort, and may 
 
 write 
 
 u:j ::k:ii (5) 
 
 while, in this cycle of three rectangular directions, ijk, the right- 
 handed (or left-handed) character of the rotation, round the first 
 from the second to the third, is easily seen to be unaffected by 
 such a transposition. Again compounding the two similar ratios 
 (I) with these two others, which are evidently similar, whatever 
 the unknown direction u may be, 
 
 i : - i : : u : - u, (6) 
 
 we find this other proportion, 
 
 j :- i :: ki- u; (7) 
 
 and therefore, by (2) and (3), 
 
 uik : : li : - u. {^) 
 
 In like manner, 
 
 u '. i '. '. i '. - u, and u:j : :j t-ti; (9) 
 
 and in any one of these proportions, any two terms, whether be- 
 longing to the same or to different ratios, may have their signs 
 changed together. All these proportions, (2) . . (9), follow from 
 the original supposition (1), by the general principles above 
 stated, without the direction u being as yet any otherwise deter- 
 mined. 
 
 [58.] Suppose now that the two rectangular directions ^ and 
 k are made to turn together, in their own plane, round i as an 
 axis, till they take two new positions ji and ki, which will there- 
 fore satisfy the proportion, 
 
 j:k::j\:k,. (10) 
 
 We shall then have, by (4), 
 
 u:i ::j\i kr, (H) 
 
 and therefore, by a cyclical change of these three new rectan- 
 gular directions, 
 
 u:ji::hi:i'.:l: i^, (12) 
 
 if / and z'l be obtained from k^ and i by any common rotation 
 roundel. Another cyclical change, combined with a rotation 
 round the new line /, gives finally, 
 
PREFACli:. (59) 
 
 u:l :: ii :ji :: m:n; (13) 
 
 where /, m, n may represent any three rectangular directions 
 whatever, subject only to the condition that the rotation round 
 / from m to n shall be of the same character as that round i from 
 j to h. With this condition, therefore, the first assumed propor- 
 tion (1) may be replaced by this 7nore general one: 
 
 n:m::l:u; (14) 
 
 ■while for (8) and (9) may now be written, with the same signi- 
 fication of the symbols, 
 
 u'.l:'.l:-U', u: m :: m: -u; u :n :: n:-u; (15) 
 and because n: m :: m :-n,we have these other and not less ge- 
 neral proportions, 
 
 m : - n : : I : u ; m : n : : I : - u. (16) 
 
 If, then, there be atiy such fourth proportional, u, as has been 
 above supposed, to the three given rectangular directions j, i, k, 
 the same direction u, or the opposite direction - u, will also be, 
 in the same sense, the fourth proportional to any other three rect- 
 angular directions, ?i, m, I, or m, n, /, according as the character 
 of a certain rotation is preserved or reversed. 
 
 [59.] This remarkable result appeared to me to justify the 
 regarding the directions here called + u and - u rather as nume- 
 rical (or algebraical) than as linear (or geometrical) units; and 
 to make it proper to denote them simply by the symbols + 1 
 and - 1 ; because their directions were seen to admit only of a 
 certain contrast between themselves, but not of any other change: 
 all that geometrical variety, which results from the conception of 
 tridimensional space, having been found to disappear, as regarded 
 them, in an investigation conducted as above. And in fact it is 
 not permitted, on the foregoing principles, to identify the direc- 
 tion u with that oi any line (I) whatever: for in that case the 
 proportion (13) would give the result I : I :: m:n, which must be 
 regarded in this theory as an absurd one, the two terms of one 
 ratio being coincident directions, while those of the other ratio 
 are rectangular. But there is no objection of this sort against 
 our supposing, as above, that 
 
 + u = + l, -u = -l; (17) 
 
 and then the propo7'tions, derived from (13), (15), 
 
(60) PREFACE. 
 
 1:1 ::m:n :: n:-jn; 1:1 :: l:-l, (18) 
 
 maybe conveniently and concisely expressed by formulae oimul- 
 tiplication, as follows: 
 
 lm = n; ln=-m; P = -l. (19) 
 
 [60.] In this way, then, or in one not essentially different, 
 the fundamental formulse [48] of the calculus of quaternions, as 
 first exhibited to the R. I. A. in 1843, namely, the equations, 
 i' = -l,f = -l,k^ = -l, (a) 
 
 ij= + k,jk = +i, ki = +j, (b) 
 
 ji = -k, kj = -i, ik=-J, (c) 
 
 were shewn (in 1844) to be consistent with a priori principles, and 
 with considerations of a general nature ; a product being here 
 regarded as a fourth proportional, to a certain extra-spatial* 
 unit, and to two directed factor-lines in space : whereas, in the 
 investigation of paragraphs [50] to [56], it was viewed rather as 
 a certain function of those two factors, the form of which func- 
 tion was to be determined in the manner most consistent with 
 some general and guiding analogies, and with the conception of 
 the symmetry of space. But there was still another view of the 
 whole subject, sketched not long afterwards in another commu- 
 nication to the R. I. Academy,! on which it is unnecessary to say 
 more than a few words in this place, because it is, in substance, 
 the view adopted in the following Lectures, and developed with 
 some fulness in them : namely, that view according to which a 
 Quaternion is considered as the Quotient of two directed lines 
 in tridimensional space. 
 
 * It seemed (and still seems) to me natural to connect this extra-spatial unit 
 with the conception [3] of time, regarded here merely as an axis of continuous 
 and uni-dimensional progression. But whether we thus consider jointly time and 
 space, or conceive generallj^ any system of four independent axes, or scales of pro- 
 gression (?«, i,j, k'), I am disposed to infer from the above investigation the fol- 
 lowing LAW OF THE FOUR SCALES, as One which is at least consistent with 
 analogy, and admissible as a definitional extension of the fundamental equations 
 of quaternions : — " A formula o^ proportion hetween four independent and directed 
 units is to be considered as remaining true, when any two of them change places 
 with each other (in the formula), provided that the direction (or sign') of one be 
 reversed.'" Whatever may be thought of these abstract and semi-metaphysical 
 views, the formulas (a) (b) (c) of par. [60] are in any event a sufficient basis for 
 the erection of a calculus of quaternions. 
 
 t See the Proceedings of Feb. 10th, 1845. 
 
PREFACE. (61) 
 
 [61.] Of such a geometrical quotient,* b -i. a, the fundamen- 
 tal property is in this theory conceived to be, that by operating, 
 as a multiplier (or at least in a way analogous to multiplication), 
 on the divisor-line, a, \t produces (or generates) the dividend- 
 line, b ; and that thus it may be interpreted as satisfying the 
 general and identical formula (compare [9] ) : 
 
 (b -r-'a) X a = b. 
 
 The analogy to multiplication consists partly in the operation 
 being one which is performed at once on length and on direction, 
 as in the ordinary multiplication of a line by a positive or nega- 
 tive number ; or as is done in that known generalization [36] of 
 such multiplication, for lines within one plane, which (for reasons 
 assigned in notes to former paragraphs) ought (I think) to be 
 called the Method of Argand: and partly in the circumstance 
 that the new operation possesses, like that older one (from which, 
 however, it is entirely distinct,] in many other and important re- 
 spects), the distributive and associative,^ though not like it (ge- 
 nerally) the commutative properties, of what is called multipli- 
 
 * This view of a geometrical quotient was also developed to a certain extent, 
 in an unfinished series of papers, which appeared a few years ago in the Cam- 
 bridge and Dublin Mathematical Journal, under the head of Symbolical Geome- 
 try : a title adopted to mark that I had attempted, in the composition of that 
 particular series, to allow a more prominent influence to the general laws of 
 symbolical language than in some former papers of mine ; and that to this extent 
 I had on that occasion sought to imitate the Symbolical Algebra of Dr. Peacock, 
 and to profit also by some of the remarks of Gregory and Ohm. 
 
 f Among these distinctions of method, it is important to bear in mind that no 
 one line is taken, in my system, as representing the direction of positive unity: and 
 that, on the contrary, every vector-unit is regarded as one of the square roots of 
 negative unity. It is to be remarked, also, that the product of two inclined but 
 non-rectangular vectors is considered in this theory as not a line, but a quater- 
 nion : all which will be found fully illustrated in the Lectures. 
 
 J To this associative principle, or property of multiplication, I attach much 
 importance, and have taken pains to shew, in the Fifth and Sixth Lectures, that 
 it can be geometrically proved for quaternions, independently of the distributive 
 principle, which may, however, in a difi^erent arrangement of the subject, be 
 made to precede and assist the proof of the associative property, as shewn in the 
 Seventh Lecture, and elsewhere. The absence of the associative principle ap- 
 pears to me to be an inconvenience in the octaves or octonomials of Messrs. J. T. 
 Graves and Arthur Cayley (see Appendix B, p. 730) : thus in the notation of the 
 former we should indeed have, as in quaternions, ij = h, but not generally i.j(t> 
 = k(o, ifw represent an octave; for i.jl — iH——o = — kl = — ij.l. 
 
(62) PREFACE. 
 
 cation in algebra;* at least when a few definitional formulae 
 (resembling those in par. [9] ) are established. And the motive 
 (in this view) for calling such a quotient a Quaternion, or the 
 ground for connecting its conception with the number Four, is 
 derived from the consideration that while the relative length 
 of the two lines compared depends only on one number^ express- 
 ing their ratio (of the ordinary kind), their relative direc- 
 tion depends on a system of three numbers : one denoting the 
 angle (a ^ b) between the two lines, and the two others serving 
 to determine the aspect of the plane of that angle, or the direc- 
 tion of the AXIS of the positive rotation in that plane,y>"om the 
 divisor-line (a) to the dividend-line (b). 
 
 * The expression " algebra," or "ordinary algebra," occurs several times in 
 these Lectures, as denoting merely that usual species of algebra, in which the 
 equation ah = ha is treated as universally true, and not (of course) as implying 
 any degree of disrespect to those many and eminent writers, who have not hi- 
 therto chosen to admit into their calculations such equations as aj8 = — /3a, for 
 the multiplication of two rectangular lines, or for other and more abstract pur- 
 poses. It is proper to state here, that a species of non-commutative multiplication 
 for inclined lines (aussere Multiplikation) occurs in a very original and remark- 
 able work by Prof. H. Grassmann (Ausdehnungslehre, Leipzig, 1844), which I 
 did not meet with till after years had elapsed from the invention and communi- 
 cation of the quaternions : in which work I have also noticed (when too late to 
 acknowledge it elsewhere) an employment of the symbol /3 - a, to denote the 
 directed line (Strecke), drawn from the point a to the point /3. Nothwithstand- 
 ing these, and perhaps some other coincidences of view, Prof. Grassmann's system 
 and mine appear to be perfectly distinct and independent of each other, in their 
 conceptions, methods, and results. At least, that the profound and philosophi- 
 cal author of the Ausdehnungslehre was not, at the time of its publication, in 
 possession of the theory of the quaternions, which had in the preceding year 
 (1843) been applied by me as a sort of organ or calculus for spherical trigonome- 
 try, seems clear from a passage of his Preface (Vorrede, p. xiv.), in which he 
 states (under date of June 28th, 1844), that he had not then succeeded in ex- 
 tending the use of imaginaries from the plane to space; and generally that unsur- 
 mounted difficulties had opposed themselves to his attempts to construct, on his 
 principles, a theory of angles in space (hingegen ist es nicht mehr moglich, ver- 
 mittelst des Imaginaren auch die Gesetze f iir den Raum abzuleiten. Auch 
 stellen sich iiberhaupt der Betrachtung der Winkel im Raume Schwierigkeiten 
 entgegen, zu deren allseitiger Lbsung mir noch nicht hinreichende Musse gewor- 
 den ist). The earlier treatise by Prof. A. F. Mobius (der barycentrische Calcul, 
 Leipzig, 1827), referred to in the same Preface by Grassmann, appears to be 
 a work which likewise well deserves attention, for its conceptions, notations, 
 and results ; as does also another work of Mobius (Mechanik des Himmels, 
 Leipzig, 1843), elsewhere referred to in these Lectures (page 614). 
 
PREFACE. (63) 
 
 [62.] For the unfolding of this general view,* and the deduc- 
 tion from it of many geometriealf and of some physical^ conse- 
 quences, I must refer to the following Lectures ; of which a 
 considerable part has been drawn up in a more popular§ style than 
 this Preface : while the whole has been composed under the in- 
 fluence of a sincere desire to render the exposition of the subject 
 as clear and elementary as possible. The prefixed Table of 
 Contents (pp. ix. to Ixxii.), though somewhat fuller than usual, 
 will be found useful (it is hoped) not merely as an analytical 
 Index, assisting a reader to refer easily to any part of the volume 
 which he has once carefully read, but also as a general abridg- 
 ment of the work, and in some places as a commentary . I The 
 
 * I may just hint here that the biquaternions of Lect. VII. admit of being 
 geometrically interpreted (comp. note to [19] ), by considering each as a couple of 
 
 quotients (-, - j, constructed by a tribadial (a, ft, y), and multiplied by a com- 
 mutative factor of the form V— 1 (compare [16] ), when the line-couple (((3, y) is 
 changed to (— y, /3), or when the angle j3y is changed to an adjacent angle. 
 
 t Notwithstanding some references to works of M. Chasles, and other emi- 
 nent foreign geometers, my acquaintance with their writings is far too imperfect 
 to give me any confidence in the novelty of various theorems in the VII"' Lec- 
 ture and Appendix (such as those respecting generations of the ellipsoid, and 
 inscriptions of gauche polygons in surfaces of the second order), beyond what 
 is derived from the opinion of a few geometrical friends. 
 
 X Some such physical applications were early suggested by Sir J. Herschel. 
 
 § It had been designed that these Lectures should not go much more into 
 detail than those which have been actually delivered on the subject by me, in 
 successive years, in the Halls of this University ; and the First Lecture, printed 
 in 1848 (as the astronomical allusions at its commencement may indicate), was 
 in fact delivered in that year, in very nearly the form in which it now appears. 
 But it was soon found necessary to extend the plan of the composition : and it is 
 evident that the subsequent Lectures, as printed, are too long, and that the last 
 of them involves too much calculation, to have been delivered in their present 
 form : though something of the style of actual lecturing has been Iiere and there 
 retained. The real divisions of the work are not so much the Lectures them- 
 selves, as the shorter and more numerous Articles, to which accordingly the 
 references have been chiefly made. An intermediate form of subdivision into 
 Sections has however been used in drawing up the Contents, which the reader 
 may adopt or not at his discretion, marking or leaving unmarked the margin of 
 the Lectures accordingly. Some new terms and symbols have been unavoidably 
 introduced into the work, but it is hoped that they will not be found embarrass- 
 ing, or difficult to remember and apply. 
 
 11 For instance, as regards the formation of the Adeuteric Function (p. xliii.) 
 
(64) PREFACE. 
 
 Diagrams are numerous, and have been engraved* with care from 
 my drawings : some of them may perhaps be thought to have 
 been unnecessary, but it appeared better to err, if at all, on the 
 side of clearness and fulness of illustration, especially in the early 
 parts of a work based on a new mathematical conception, and 
 designed to furnish, to those who may be disposed to employ it, 
 a new mathematical organ. Whatever may be thought of the 
 degree of success with which my exertions in this matter have 
 been attended, it will be felt, at least, that they must have been 
 arduous and persevering. My thanks are due, at this last stage, 
 to the friends who have cheered me throughout by their conti- 
 nued sympathy; to the scientific contemporaries! who have at 
 moments turned aside from their own original researches, to no- 
 tice, and in some instances to extend, results or speculations of 
 mine; to my academical superiors who have sanctioned, as a 
 subject of public and repeated examination in this University, 
 the theory to which this Volume relates, and have contributed to 
 lighten, to an important extent, the pecuniary risk of its publi- 
 cation : but, above all, to that Great Being, who has graciously 
 spared to me such a measure of health and energy as was required 
 for bringing to a close this long and laborious undertaking. 
 
 William Rowan Hamilton. 
 Observatory of T. CD., June, 1853. 
 
 » By Mr. W. Oldham, whose fidelity and diligence are hereby acknowledged, 
 fin these countries, Messrs. Boole, Carmichael, Cayley, Cockle, De Morgan, 
 Donkin, Charles and John Graves, Kirkman, O'Brien, Spottiswoode, Young, and 
 perhaps others : some of whose researches or remarks on subjects connected with 
 quaternions (such as the triplets, tessarines, octaves, and pluquateriiions) have 
 been elsewhere alluded to, but of which I much regret the impossibility of giving 
 here a fuller account. As regards the theory of algebraic keys (clefs algebriques), 
 lately proposed by one of the most eminent of continental analysts, as one that 
 includes the quaternions (Comptes Rendus for Jan. 10, 1853, p. 75), it appears to 
 me to be virtually included in that theory of sets in algebra (explained in the 
 present Preface), which was announced by me in 1835, and published in 1848 
 (Trans. R.I. A., Vol. XXL, Partii., p. 229, &c., the symbols x,- being in fact 
 what M. Cauchy calls keys), as an extension of the theory of couples (and therefore 
 also of imaginaries) : of which sets I have always considered the quaternions 
 (in their symbolical aspect) to be merely a particular case. Before the publica- 
 tion of those sets, the closely connected conception of an " algebra of the n*^ cha- 
 racter" had occurred to Prof. De Morgan in 1844, avowedly as a suggestion from 
 the quaternions. (Trans. Camb. Phil. Soc, Vol. VIII., Part iii.) 
 
CONTENTS. 
 
 LECTURE I. 
 
 (Articles 1 to 36; Pages 1 to 32.) 
 
 ADDITIONS AND SUBTRACTIONS OF LINES AND POINTS. 
 
 Ihtroductort remarks (1848), Articles 1, 2, 3 ; Pages 1 to i. 
 
 § I. General views respecting the four signs, -\ — x -f- ; primary signification 
 proposed for the mark — in geometry, as a characteristic of ordinal ana- 
 lysis, or of analysis of position ; geometrical difference of two points, 
 point minus point ; analytic aspect of the symbol, b — a ; examples and 
 illustrations, Articles 4 to 14 ; Pages 4 to 14. 
 
 § II, Synthetic aspect of the same symbol, B — A, as denoting the step or vector, 
 a, from a to b ; distinction between vector and radius-vector ; the vec- 
 tor is simply a directed right line in space ; interpretations of the equa- 
 tions, B— A=a, B = a + A; proposed primary use of + in geometry, as 
 a characteristic of ordinal synthesis, or sign ofvection, or of the transport 
 of a point from one position to another ; geometrical sum of line and 
 point, line plus point ; synthesis of the conceptions of step and beginning 
 of step, producing the conception of end of step as their result ; this end 
 of step may in this view be equated to " step pZws beginning of step ;" 
 vector plus vehend equals vectum, vector minus vehend equals vector ; re- 
 vection, revector, revehend, revectum ; geometrical identities, b - A + 
 A = B, a + A-A = a, Articles 15 to 26 ; Pages 15 to 26. 
 
 § III. Provection (successive vection of a point, not generally along the same 
 straight line) ; provector = c — b = b ; provehend = vectum = b ; pro- 
 vectum = c identity, c = (c — b) + (b — a) + a, provectum equals pro- 
 vector plus vector plus vehend ; illustration, . Articles 27 to 29 ; Pages 25 to 27. 
 
 § IV. Transvection (transport of a point at once from A to c, substituted for two 
 successive transports, from a to b, and from b to c) ; transvector = c - 
 A = c ; transvehend = vehend = a ; transvectum = provectum = c ; 
 abridged identities, c — A = (c — b) + (b — a), c - e = (c — a) — (b — a) ; 
 
 TRANSVECTOR EQUALS PROVECTOR PLUS VECTOR ; prOVCCtor Cquals 
 
 transvector minus vector ; (c — a) + a = c, (b +- a) — a = b ; illustra- 
 tions, Articles 80 to 35; Pages 27 to 31. 
 
 b 
 
X CONTENTS. 
 
 § V. Addition and subtraction of lines corvesiioViAing to composition atidaecompo- 
 sition of vections, or of motions ; line plus line, and line minus line, each 
 equal to some third line ; these operations on lines are not peculiar to 
 QUATERNIONS, but are regarded here as secondary operations of ordinal 
 synthesis and anatysis, the primary combinations having been of the forms, 
 line plus point, and point minus point, .... Article 36 ; Pages 31, 82. 
 
 LECTURE 11. 
 
 (Articles 37 to 78 ; Pages 33 to 73.) 
 
 GENERAL VIEWS RESPECTING MULTIPLICATION AND DIVISION IN GEOME- 
 TRY; SQUARES AND PRODUCTS OF 2, j, Tc. 
 
 § VI. Recapitulation : quotient of two directed lines (which quotient is after- 
 wards shewn to be in this calculus a quaternion), /3-j-a = 5, 5Xa = /3; 
 the signs of division and multiplication, or -4- and x, are considered here 
 as marks of cardinal analysis and synthesis in geometry, expressing re- 
 spectively the investigation and the employment of a certain metrographic 
 relation, existing partly between the lengths, and partly between the di- 
 rections, of any two vectors, or steps, or rays in space ; faction, factor, 
 faciend, factum (the factor here introduced is afterwards shewn to be a 
 quaternion') ; identities, fS-i- ax a = l3, qx a-r- a. = q ; refaction, re- 
 factor, reciprocal cardinal relations, . . . Articles 87 to 44; Pages 83 to 39. 
 
 § VII. Profaction, profactor, y -i- f3=zr, rx/3=:y; transfaction, transfactor, 
 y-7-a = s, sxa = y — rxqxa, s = r x q ; transfactor equals 
 PROFACTOR multiplied INTO FACTOR, pvofactor equals transfactor di- 
 vided by factor ; (y -—fi) X (/3 -f- a) = y -7- a, (y -r- a) -r- (|S -r- a) 
 = (y -I- /3) ; (s -j- 9) X g' = s, {r X q) — q^r; triangle of vections, 
 pyramid of factions ; composition and decomposition of operations of the 
 factor kind, Articles 45 to 56 ; Pages 39 to 48. 
 
 § VIII. Examples ; case where the raj^s compared have all one common direction ; 
 operations on length, tension ; signless numbers, tensors ; null lines, 
 opposite lines, use oi plus and minus as factors, namely, as signs ot non- 
 version and inversion ; symbols 0, -I- 2a, — 2a ; rule of the signs ; positive 
 and negative numbers, scalaes ; these scalars are simply the reals of or- 
 dinary algebra, Articles 57 to 64 ; Pages 48 to 58. 
 
 § IX. Case where the rays compared have all one common length, operations on 
 direction ; version regarded as a species of graphic multiplication, 
 or as an operation of the factor kind, thus performed on the direction of a 
 line ; versor multiplied into vertend equals versum, versum divided by 
 vertend equals versor ; proversion, transversion, successive rotations of a 
 line, each rotation separately being performed in some one plane, but the 
 successive planes being different ; pro"\':ersor into versor equals 
 transversor ; composition and decomposition of versions, or of plane ro- 
 
CONTENTS. XI 
 
 tations of a line : to know fully what particular act of version has been 
 performed, we must know through what angle^ in what plane, and to- 
 wards which hand (or round what axis, and through what amount of 
 
 right-handed rotation), the line has been made to turn, 
 
 Articles 65, 66; Pages 58 to 61. 
 
 § X. Illustrations from meridional and extra-meridional transit telescopes, and 
 from the theodolite, or other instrument moveable in azimuth ; non-com- 
 mutative character of the composition of versions in rectangular planes ; 
 
 i X j = k, j X k = i, k X i =j ; 
 
 jxi = -k, kxj = -i, ixk=-j; 
 
 i X i =j xj — kxk = —l = (— ) ; 
 
 every quadeAntal veksoe is a semi-inveesoe, and as such is a geome- 
 trical square root of negative vnity, or of the sign minus ; every such 
 versor is represented, in the geometrical applications of this calculus, by a 
 VECTOE-UNiT, drawn in the direction of the axis of the version: thus the 
 symbols i, j, k come to denote here three rectangular vector-units (sup- 
 posed usually, in these Lectures, to be in the directions of south, west, and 
 M/>) ; and the formula ixj = k is foimd to receive two distinct but closely 
 connected interpretations, Articles 67 to 78 ; Pages 61 to 73. 
 
 LECTUEE III. 
 
 (Articles 79 to 120 ; Pages 74 to 129.) 
 
 OTHER CASES OP MULTIPLICATION AND DIVISION IN GEOMETRY; CONCEP- 
 TION OF THE QUATERNION; NOTATIONS, K, T, U. 
 
 § XI. Recapitulation ; additional illustrations of the effects of i, j, k, as operators ; 
 multiplication of any one line in space, by another perpendicular thereto ; 
 the product is (in this system) a third line, perpendicular to both the fac- 
 tors ; its length is numerically the product of their lengths ; and the di- 
 rection of the same product-line is obtained from that of the multiplicand 
 line, by a positive and quadrantal rotation, performed round the mul- 
 tiplier line as an axis ; non -commutative character of such multiplication, 
 equation of perpendicularity, ajS = — /3«, if /3 J. a ; these results are exten- 
 sions of those expressed by the formulae, ij — k,ji = — k, 
 
 Articles 79 to 82 ; Pages 74 to 79. 
 
 § XII. Th& product of a scalar and a vector, or of a number and a line, is a line, 
 of which the length and the direction are veiy easily assigned, and are 
 found to be independent of the order of the factors ; aa = aa; for example, 
 the symbols ix, jy, kz, denote the same three rectangular lines as xi, yj, 
 zk ;_. namely, when this system is brought into connexion with the Carte- 
 sian method of co-ordinates, the three rectangular projections of the line 
 
 drawn from the oingin (0, 0, 0), to the point {x, y, z), 
 
 Article 83 ; Pages 79, 80. 
 
XU CONTENTS. 
 
 § XIII. The product of two parallel lines is a number^ namely, the numerical pro- 
 duct of the lengths of the factors ; but this number is taken negatively or 
 positively (in this calculus), according as they agree or differ in their di- 
 rections ; thus, the square of evekt vector is a negative scalar, 
 a?<Q (as we had i^ -p = A^ = - 1) ; this remarkable result is a simple 
 geometrical consequence of the composition of two successive and quadran- 
 tal rotations about any common axis in space ; commutative character of 
 the multiplication of parallel vectors, equation of parallelism., a/3 = ^a, 
 if^lla Articles 84, 85 ; Pages 80 to 82. 
 
 § XIV. Powers of unit-vectors ; symbols t*, t'/c, where t is such an Unit-line in 
 space, and k a vector j_ i ; the first of these two symbols (i*) denotes a 
 versor, not generally quadrantal ; the second (I'/c) denotes a line, which is 
 formed from (c by a positive and plane rotation of t quadrants, round i re- 
 garded as an axis ; examples, Article 86 ; Pages 82, 83. 
 
 § XV. Multiplication of two inclined lines ; their product kX (which is afterwards 
 shewn to be a quaternion') may also be considered as th^e product of a ten- 
 sor and a versor; whereof the tensor is the numencsl product of the 
 lengths of the two factor lines ; while the versor has its axis in the direc- 
 tion of the axis of positive (namely, in these Lectures, right-handed) rota- 
 tion, /rojn the multiplier line/c to the multiplicand line X, and has its angle 
 equal to the supplement of the angle of this last rotation ; examples ; ver- 
 sor and reversor; conjugate VERSOKS, conjugate products, characte- 
 ristic OF conjugation K ; K.t*=t-*, K.kX = Xk, 
 
 Articles 87 to 89 ; Pages 83 to 87. 
 
 ''I XVI. Resolution of every act oi faction into a metric and a graphic element, or 
 into an act of tension, and an act of version ; the letters T and U are em- 
 ployed in this calculus as characteristics of the tivo separate operations, of 
 taking THE TENSOR, a7id TAKING THE VEEsoR, Or of taking Separately 
 the two factor-elements, Tq and Ug", of any proposed factor q, or of any 
 product or quotient of two lines, when regarded as such a factor ; identi- 
 ties, 5=T5xU5 = UgxTg; T.Ug=l, \].'Iq = + ; T.T3 = Tg, 
 U . Ug = U?, Article 90 ; Pages 87 to 89. 
 
 § XVII. The tensor Tg (by §§ viii., xvi.) is always to be conceived as a single 
 number, expressing the ratio in which the factor q changes the length of 
 the line a on which it operates ; but (by §§ ix., xvi ) the versor JJq, which 
 may generally be put (see § xiv.) under the form of a power i* of an unit- 
 vector I, with a scalar exponent, t, requires for its complete nmnerical de- 
 termination a system of three numbers ; namely, the number (f) of qua- 
 drants contained in the angle of the version ; and some two angular 
 co-ordinates or other equivalent system of two numbers, to fix the direc- 
 tion in space of the axis (i), or to identify on a globe or chart the star, or 
 to fix the region of infinite space, towards which that axis is pointed ; it 
 follows therefore that the lately considered product of tensor and versor, 
 Tq . J]q, or (see §xvi.) the equivalent /ac^or q, depends upon, and con- 
 versely includes within itself, a system of four nujibeks, as necessarv 
 
CONTENTS. XIU 
 
 for its complete identification, or full immerical determination ; and there- 
 fore that a GEOMETRICAL FACTOK of this sort may properly be called a 
 Quaternion, Article 91 ; Pages 89, 90. 
 
 § XVIII. When the factor, q, is regarded (see § vi.) as a geometricai. quotient 
 = /3 -f- a = DB -~ DA, it may conveniently be pictured or constructed by a 
 BiRADiAL, ADB, with a cwved arrow inserted, and directed y^owi theiwi- 
 tial ray DA (the faciend, or divisor-line, a), towards the fnal ray db (the 
 factum, or dividend-line, (3) ; the point d, from -which the two rays di- 
 verge, is the vertex of the biradial ; a biradial has a shape, or species, de- 
 pending on the ratio of the lengths of its two rays, and also on the anffle 
 which they include ; two biradials may be similar, namely, by their agree- 
 ing with each other in these two respects ; but a biradial has also a. plane, 
 and an aspect, determined by and directed towards that star, or region 
 of infinite space, which the plane may be said to face, and as seen from 
 which the rotation from the initial to the final ray would appear to be po- 
 sitive (right-handed) ; condirectional and contradirectional (or opposite) 
 biradials, included in the class of parallel biradials ; two biradials, which 
 are at once similar and condirectional, are said to be equivalent bira- 
 dials ; examples ; it is proposed to employ (see § xx.) the conception 
 and construction of such biradial figures to assist in determining the con- 
 ditions of equality between twogeometiical quotients, (3 -r- a, and S ~ y ; 
 and also in enumerating the modes of possible inequality, of any two such 
 quotients, Articles 92 to 95 ; Pages 90 to 95. 
 
 § XIX. Analogous determinations for differences of points (see § i.), constructed or 
 pictured by straight lines, with straight arrows attached ; interpretations 
 of the two equations d — c = b — a, d = b — a+c; d is here the/owrfA 
 corner of a parallelogram, of which c, a, b are three successive corners, 
 and of which the altitude may vanish ; inversion and alternation of an 
 equation between diiFerences of points, c — a+b = b — a + c; vectors are 
 equal, when they diflTer only in their situations in space ; addition of vec- 
 tors still corresponds to composition of vections, although they are not 
 now given as successive (compare § v.) ; such addition is commutative and 
 associative, a + /3=/3+a, (y-t-/3) + a=y+(/3+a); the sum of any 
 set of vectors is simply that one resultant vector which produces the same 
 total OT final effect, in changing the position of a point, as all the pro- 
 posed summand vectors would do, if the motions, of which they are sup- 
 posed to be the instruments, were simultaneously or successively per- 
 formed ; the sum of two directed and co-initial sides of a. parallelogram is 
 the intermediate and co-initial diagonal ; most of the foregoing results 
 of this section (xix.) are common to several other modern theories ; a vec- 
 tor (in space) is a species of natural triplet, suggested by geometry, 
 and found to be capable of a triple variety, or to depend upon a system of 
 three distinct elements, which admit of being expressed numerically, and 
 correspond to the tridimensional character of spa.ce; in the present 
 calculus (compare § xii.), a vector may be represented generally by the 
 TRINOMIAL FORM, o = ix ^ jy + hz, whci'e 3c, y, z are three scalar (or Car- 
 
XIV CONTENTS. 
 
 tesian) co-ordinates, while i,j, k are those three rectangular vector- units, 
 
 ■which were introduced (see § x.) in the foregoing Lecture, 
 
 Articles 96 to 101 ; Pages 95 to 106. 
 
 § XX. Equivalent biradials (see § xviii.) con-espond to equal quotients ; 
 examples ; in fact a biradial may be turned round in its own plane, or 
 transported parallel to itself, or its legs may be altered proportionally, 
 without changing the relative direction, or the relative length, of those 
 two legs, or rays, or vectors, and therefore without aifecting that complex 
 (metrographic) relation between the two rays which has been considered 
 (in § VI.) as determining their geometrical quotient ; hence in this calcu- 
 lus, as in many other modern systems, the equation S —- y = l3-r- a, be- 
 tween two quotients, is interpreted as signifying a proportionality of 
 lengths, combined with an equality of angles in one plane, between the two 
 pairs of lines, a, /3, and y, S; but, when we come to take account of the 
 PLANE OP THE AITGLE, between any two such lines a, (3, and to regard 
 that plane as variable in space, there arises a new double variety, 
 in the geometrical quotient j3-r- a, or in the numerical elements on which 
 it depends ; because we introduce hereby the consideration of the aspect 
 (see § xviii.) of the plane, or of the biradial, and thus bring into play (or 
 at least may be conceived to do so) a new pair of nujibers, such as 
 those which determine in astronomy the inclination of the plane of 
 the orbit of a planet or comet to the ecliptic, and the longitude of its node, 
 in addition to that former pair of numbers, which determine the ratio 
 of the lengths of the two lines compared, and the magnitude of the angle 
 between them : the Geometrical Quotient of two vectors is found 
 therefore again (compare § xvii.), in this new way, by consideration of its 
 representative biradial, to involve or depend upon a systejm of four 
 numbers (two for shape, and^a'o for plane^, and to be (see again § xvii.), 
 in that sense, a Quaternion, . . . Articles 102 to 107 ; Pages 106 to 112. 
 
 § XXI. Multiplication of two arbitrary quaternions, eifected by means of their re- 
 presentative biradials, prepared so that the_^wa/ ray of the multiplicand 
 may coincide with the initial ray of the multiplier, as factum andprofaciend ; 
 and therefore so that the identity (y -j- /3) x (/3 -f- a) = y -7- a, of § vii., 
 may be employed to form the product ; this process is absolutely /ree^ow 
 vagueness in its conception, and altogether definite in its results, wliich 
 therefore are adapted to become the subject matter of theorems ; exam- 
 ple, here stated by way of anticipation, q q • q= q ■ q q; this is the as- 
 sociative principle of multiplication of quaternions, and will be afterwards 
 fully discussed (in Lectures V., VL, VII.) ; Division of Quaternions may 
 
 obviously be eifected by an entirely analogous process, 
 
 Article 108; Pages 112, 113. 
 
 § XXII. Before entering on the general theory of operations on quaternions, we 
 may perform operations on numbers, and on lines, regarded as particular 
 cases of quaternions ; for example, we can shew that the tensor of a sca- 
 lar is the absolute (or arithmetical) value of that scalar, T (+ 3) = 3 ; 
 
CONTENTS. XV 
 
 and that the tensor of a vector is the number expressing the length of that 
 vector, Ti = Tj=Tk = l; T.k\=Tk.TX, T (\ -^ «:) = TX -f- Tk : 
 T(0= \/ — p~; Tw— yf+w"'; it will be proved (in § lxiii.) that gene- 
 rally the tensor of a quaternion q is 
 
 T5 = T (w + p) = V(w2 - p2) ; 
 
 examination and explanation of a formula -which may seem at first a pa- 
 radox, Articles 109 to 112 ; Pages 113 to 117. 
 
 § XXIII. The versor of a positive scalar is the sign +, or the factor + 1 ; the 
 versor of a negative scalar is the sign — , or the factor — 1 ; the versor Up, 
 of a vector p, is the vector-unit in the direction of that vector, JJp = p 
 -r- Tp = p -i- V (- p2)j (Up)2= - 1 ; the versor of zero, UO, is generally 
 an indeterminate symbol, but it may become determinate, if we know, in 
 any particular investigation, the law according to which the scalar or vec- 
 tor tends to vanish ; a tensor mai/ be treated as a positive scalar (instead 
 of a signless number) ; the conjugate of a scalar is the scalar itself, but 
 the conjugate of a vector is equal to that vector reversed, Kw = + w, 
 Kp = - p ; it may be remarked by anticipation, that the conjugate of a 
 quaternion is, generally, see § lxiii., 
 
 Kg = K (w + p) = w — p, 
 
 Articles 113, 114; Pages 118, 119. 
 
 § XXIV. Powers of vectors, the exponents being still scalar s, but the vector bases 
 being not now unit-lines (compare § Xiv.) ; such powers are quaternions ; 
 examples : the tensor of the power is the power of the tensor, and the ver- 
 sor of the power is the poiver of the versor ; T . p* = (Tp)* = Tp, U . p' 
 — (Up)* = Up* ; the power p*, when operating as a factor on a line ff j. p, 
 produces another line r = p*cr, which also is perpendicular to p ; the direc- 
 tion of this new line r is formed from that of <t by a rotation through t 
 quadrants round p, and its length bears to the length of (t a ratio expressed 
 by the ?*'' power of the number Tp which expresses the length of p ; the 
 power, or quaternion, or quotient, p* = r -j- cr, degenerates into a scalar 
 when t is any even integer ; p°, for example, is positive unity, and p2 is a 
 negative number, = — Tp2 (compare §§ xiii., xxii.) ; on the other hand 
 the power p' degenerates from a quaternion into a vector, when the ex- 
 ponent t is any odd whole number, for example, p^ = p ; another and 
 more important example is the reciprocaZ of p, or the power p-i ; this 
 power is a line, which, when operating as a factor on a line u perpendicu- 
 lar to p, has the effect of dividing the length of ff by the number Tp, and 
 of causing its direction to turn negatively (or left-handedly) through a 
 quadrant, round p as an axis ; the tensor and versor of the reciprocal are 
 respectively the reciprocals of the tensor and versor, T (p-i) = (Tp)-i, 
 U(p'0 = (Up) ^ =-Up, p-i = -Tp-i.Up ; any two recipeocal 
 VECTORS, p and p-i, have their directions opposite, and their 
 LENGTHS reciprocal; the product /3xa'i is equal to the quotient 
 
 /3 -*. a, and may be denoted more concisely by /3a-i or by — , while the re- 
 
XVni CONTENTS. 
 
 nerally by the symbols, /3» a*, /3* a^ ; the tensor and versor of the cube 
 root of a quaternion may be regarded as being respectively the cube-roots 
 of the tensor and the versor ; in general we may interpret the powek q* of 
 any quaternion q, with any scalar exponent t, as being a new quaternion, 
 of which the tensor and the versor are respectively the same (t*^') powers 
 of the tensor and the versor of the old or given quaternion, which is pro- 
 posed as the BASE of the power ; thus (compare § xxiv.), 
 
 T.qt= (Tqy = Tqf, V . qt = (Vq)* = JJqt ; 
 
 and we may conceive that this latter power of a versor is itself another 
 versor, which has the effect of turning any line a, in a plane perpendicular 
 to the axis of JJq, or of q, through an angle, or amount of rotation, posi- 
 tive or negative, represented by the product txLq; but in order to deve- 
 lope and apply this general conception, we must first fix definitely what is 
 to be understood in general by the angle, or amplitude, /. q, of a quater- 
 nion, or of a versor, Articles 135, 136; Pages 151 to 153. 
 
 § XXXI. If we allow this amplitude Z. 5 to take any one of the values included in 
 the formula Lq = q + 2l7r, where g denotes an Euclidean angle, g>0, 
 < TT, we shall then have two values for a square root, three for a cube root, 
 &c., as in the usual theory of roots of unity, and as in those modern geo- 
 metrical systems which represent all such powers or roots by lines, whereas 
 with us they are quaternions ; examples : this view oi Lq would give 
 L (gO = tq + 2(lt+ l) TT, L (?«) = uq\2 (mu + m) tt, /_.q«+t= (u-\-t) q 
 + 2p (u^ t) n + 2p'Tr, L (q" . q*) = (u + t) q + 2 (It + mu + w) tt ; and in 
 order that we should have generally q(" q* = q" + ', it would be necessarj' 
 and sufiicient to assume p = m = l, or, in other words, we should assimie 
 one common value q-\-2lTr ior L q, in forming the three powers here com- 
 pared ; and after making this assumption, it would still be necessary, in 
 general, to retain that value t (q + 2Z7r) of the power q*, which was im- 
 mediately given by the multiplication t x Lq, and not to add to this pro- 
 duct any multiple 2l'ir of the circumference, before proceeding to form, by 
 a second multiplication, the angle of the power of a power of a quater- 
 nion, if we wish that this new power shall satisfy generally the equation 
 {qt)u = qut, Articles 137 to 147 ; Pages 153 to 163. 
 
 § XXXII. But for the sake of avoiding as much as possible all multiplicity of 
 value of elementary symbols, it appears convenient to define that the nota- 
 tation Z. q shall represent the simplest value of the angle, or that one 
 which most conforms to ordinary geometrical usage, namely, the angle in 
 the first positive semicircle, which was lately denoted by q, admitting 
 however and it as limits, and therefore writing* ■^ 9 ^ 0, < tt ; so that 
 the prefixed mark L comes to be the characteristic of a definite operation, 
 which may be said to be the operation of taking the angle of any pro- 
 posed quaternion q ; this view agrees with our earlier definitions (§§ xiv., 
 
 TT 
 
 XXIV.) respecting powers of vectors, and gives /_p—~, so that the angle 
 
CONTENTS. XIX 
 
 of a vector is a right angle ; the angle of a positive scalar is zero, and the 
 angle of a negative scalar is two right angles ; with the single exception of 
 powers of negatives (for which /)owers, as well as for their bases, the axes 
 are indeterminate'), the same definition assigns a determinate quaternion 
 as the value of the t^f^ power of any proposed quaternion q ; and the equa- 
 tion q^'qt=:q'^ + i is satisfied, each member representing a quaternion, of 
 which the versor has the effect of tm-ning a line perpendicular to the axis 
 of q through an amount of rotation represented hy {u+f) iq, . . . . 
 
 Articles 148 to 150 ; Pages 163 to 166. 
 
 § XXXIII. On the other hand, although the rotation produced by the operation 
 of the power q* is now correctly and definitely expressed by the product 
 ty. /_q, yet because this product is not generally confined between the 
 limits and tt, we cannot now consider it as being generally equal to the 
 angle of the power, because we have agreed (in § xxxii.) to confine the 
 ANGLE of every quaternion, and therefore of the power q* among the rest, 
 within those limits ; thus with the present definite signification of 
 the mark Z., we must not write generally Z (^O =tx Lq, but rather 
 l^ (c[f^ = 2w7r +t Lq, the axis of the power being in the same direction as 
 the axis Ax . q of the base, or else in the opposite direction, according as 
 it becomes necessary to take the upper or the lower sign ; the square root, 
 qh, of a (non-scalar) quaternion is acute-angled, and so are the cube-root, 
 qi, &c., while the axes of these roots coincide with the axis of their com- 
 mon power ; but the square q^ of an obtuse-angled quaternion q has its 
 angle Z. (5^) equal to the double of the supplement of the obtuse angle Z q, 
 and has its axis in the direction opposite to that of the axis Ax . q ; with 
 this definite view of powers and roots, although three distinct quaternions 
 may have one common cube, yet only one of them is (by eminence) the 
 cube-root of that cube ; examples : in like manner the symbol {q^)i de- 
 notes now definitely +q, or — q, according as the angle of q is acute or 
 obtuse ; (p^)! denotes a vector, with a length = Tp, but with an indeter- 
 minate direction, because p^ is a negative scalar ; we must not now write 
 generally (2*)" = 5"', but may establish this modified formula, (<7*)" = 
 (Ax . g)*"" . 5''<, Articles 161 to 161; Pages 166 to 174. 
 
 § XXXIV. Reciprocals and conjugates of quaternions (compare §§ xxiv., xxx.) ; 
 
 T(5-i) = (T?)-i = T5-i, V(q-i) = iUqyi=Vq-i; 
 
 L(q-^) = Lq, Ax . {q-^) = - A:s.. q ; Ug - 1 = KU7 = reversor : 
 
 ZKU5=ZUg, Ax.KUg = -Ax.Ug; 
 
 LKq=Lq, Ax . Kg = -Ax . g, TKq = Tq; 
 
 the reciprocal and conjugate of q may be thus expressed, 
 
 g-i=Tg-i.KUg', Kq^Tq.lJq-i; 
 
 in general qKq = Tq'^, so that the product of any two conjugate 
 quaternions is a positive scalar, namely, the square of their common 
 
 tensor; Tq—{^qYiq)i, U5' = ± (s* -7- K^) 5? according as Z g" ^ - ; exam- 
 
XX CONTENTS. 
 
 pies ; when q is a vector = p, so that Lq=--, then Kg = - 5 (compare 
 
 § xxiii.) ; and although (g-4- Kg')i is in this case an indeterminate vec- 
 tor-unit, yet we have still Ugs = g -j- 'Kq, each member being = - 1, . . 
 
 Articles 162 to 165 ; Pages 175 to 178. 
 
 § XXXV. More close examination of the case of indeteemination, mentioned 
 in several recent sections, when the base of a power becomes a negative 
 scalar ; Z. (- 1) = tt ; Ax . (— 1) is indeterminate ; the symbol (— 1)* or 
 (— )' denotes a versor, which has the effect of producing a given and defi- 
 jinite amount of rotation =i7r, but in a wholly arfitYrary /)?ane ; in parti- 
 cular, z (— 1)1 = -, so that (- 1)3 or \/— 1 represents in this theory 
 
 (compare §§ x., xxix., xxxii, xxxiii.) a quadrantal versor with an arbi- 
 trary axis, and therefore also a vector-unit with an indeterminate 
 DIRECTION ; this perfectly real hut partially indeterminate interpre- 
 tation, of the symbol V—1, is one of the c^ze/ peculiarities of the 
 present calculus, so far as its connexion with geometry is concerned ; ex- 
 amples of its use, in forming certain equations of loci ; if o be origin 
 of vectors, and p a point upon the unit-sphere, then the vector of that 
 point may be expressed as follows : 
 
 p-o = p = V-l, 
 
 so that p2 -I- 1 = is a form for the equation of a spheric surface ; this 
 form is extensively useful in researches of spherical geometry ; the ex- 
 pression |0 = /3+6V— 1 represents the vector of a point upon another 
 sphere, whose radius is h, and the vector of whose centre is j8 ; the equa- 
 tion of this new sphere may also be thus written, 
 
 (jO - j8)2 + 62 :3: 0, or thus, T (p - /3) = 6 ; 
 
 the equation pa-'^ — '^— 1, or (pa-i)2 = — 1, may be interpreted as repre- 
 senting a circular circumference, namely, the great circle in which the 
 plane through o, perpendicular to a, cuts the sphere which has the origin 
 o for its centre, and has its radius = Ta ; the indefinite plane of the same 
 circle may be represented by the equation U. pa-i = V— 1, and a parol- 
 lelplane by U. (p — /3) a-i = V— 1 ; the equation pa-i = (— 1)1 repre- 
 sents another circle, namely, the locus of the summits of all the equilate- 
 ral triangles which can be described upon the given base a ; and the 
 equation U . pa"i = (— 1)1 represents a sheet of a right cone, with its ver- 
 tex at the origin, and with the last-mentioned circle as its base, . . . 
 
 Articles 166 to 174 ; Pages 178 to 186. 
 
CONTENTS. XXI 
 
 LECTURE V. 
 
 (Articles 175 to 250 ; Pages 186 to 240.) 
 
 ASSOCIATIVE PRINCIPLE FOR THE MULTIPLICATION OF THREE LINES IN 
 SPACE ; QUATERNION VALUES OF THEIR TERNARY PRODUCTS, /3a7, AND 
 FOURTH PROPORTIONALS, /3a' '7; VALUES OF ijk, hji\ GENERAL CON- 
 STRUCTION FOR THE PRODUCT OP TWO VERSORS, BY A TRANSVECTOR 
 ARC UPON A SPHERE. 
 
 § XXXVI. Proof that for any three coplanar vectors, a, /3, y, the product /3 . a-i y 
 represents the same fourth line 5 in their plane as the product /3a-i . y ; 
 thus /3 . a-i y = /3a-i . y, at least when a 1 1 1 j8, y (this last restriction is 
 afterwards shewn to be unnecessary) ; the proof is given for the three 
 cases, 1st, when the product a"iy is a vector; 2nd, when it is a scalar; 
 and 3rd, when it is a quaternion ; in treating these cases, we avail our- 
 selves of the formulae, a-i . a£-i = £-1, y£.£-i = y, Zl ■ t]-^9 = ^9, 
 which are indeed included in the general associative principle of multipli- 
 cation (stated by anticipation in § xxi.), but can be separately and more 
 easily proved ; in general, by the conceptions of reciprocal and product, 
 it can easil}' be shewn that for any two quaternions q and r, we have, as 
 in algebra, the identities, r-^ . rq = q, rq . q-'^ = r; another general for- 
 mula for the multiplication of any two quaternions is juX-i . X/c-i = )uk-i, 
 
 Articles 176 to 182 ; Pages 186 to 192. 
 
 § xxxvii. Negatives of quaternions, 
 
 T(-5)=T9, L{-q) = ^- Lq = 7r-L'Kq, Ax . (-g) =- Ax. gr = Ax .Kg ; 
 
 the axes of the negative and conjugate coincide, but their angles are sup- 
 plementary ; 
 
 T(-Kg)=Tg', Z (- Kg) = tt - Z ?, Ax . (- Kg) = Ax, g ; 
 
 the negative of the conjugate has the effect of turning the line on which it 
 operates, round the same axis as the original quaternion, but through a 
 supplementary angle ; (these results are seen at a later stage, to admit of 
 being connected with the form Tg (cos + y — 1 sin) L q, to which every 
 
 quaternion g may be reduced, but in which the v — 1 is regarded as re- 
 presenting a vector- unit, in the direction of Ax . g) ; KKg = g, K^ = 1 ; 
 K (— g) = — Kg ; if this = + g, then g must be a vector, and vice versa ; 
 the tensor and versor of a product or quotient of any two quaternions 
 are respectively ih& product or quotieiit of the teiisors and versor s, 
 
 T . rg = Tr . Tg, U . rg = Ur . Ug, 
 T (r -f- g) = Tr ^ Tg, U (r -^ g) = Vr ^ Ug ; 
 
 this result is connected with the mutual independence of the two acts or 
 
XXll CONTENTS. 
 
 operations of tension and of version ; the conjugate and the reciprocal of 
 the product of any two quaternions are respectively equal to the product 
 of the conjugates, and to the product of the reciprocals, but taken in an 
 inverted order, K. rq = Kq .Kr, (rq')-^ = q'^r-^ ; ii d = (3a-^ . y = 
 ya-i. /3 (see § xxvii.), then j8 . a-iy = K (-i6).K (ya-i) = -K(ya-i./3) 
 = — Kd = S ; the result of the foregoing section, that /3 . a"i y = (3a"i . y, 
 when a, /3, y are three coplanar vectors, is therefore confirmed in this new 
 way, Articles 183 to 193 ; Pages 192 to 198. 
 
 § XXXVIII. The associative principle therefore holds for the multiplication of any 
 three coplanar vectors, such as the recent lines y, a-i, and j8, with a 
 partial validity of the commutative principle also ; so that we may dis- 
 miss the point in the notation, and may write either J = j8a-iy, or 
 ^= ya-i/3 ; the line £ may still be called (see § xxvii.) the Fourth Pro- 
 portional to a, 13, y, or to a, y, /3 ; but it may also be said to be the 
 co7itinued product of y, a~i, /3, or of |3, a"i, y ; without introducing — 1 
 as an exponent of the middle factor, if /i 1 1 1 X, ;c, we have the following 
 equation of coplanar ity, ij,\ic = K\fi ; each of the symbols here equated 
 denotes a line, coplanar with the lines k, X, [i, which fourth line in their 
 plane may at pleasure be called the fourth proportional to \-i, fi, k, or to 
 X'l, K, fx, or the continued product of k, X, /t, or of )u, X, k; (X-i)-i = X, 
 (gf-i)-i = gf; /3ay = a^ . /Ja-i y ; and because a^<0(by § xiii.), the 
 continued product fSay of three coplanar vectors, y, a, (3, has the direc- 
 tion opposite to that of the fourth proportional to the lines a, fS, y ; the 
 continued product (a — c) ( c — b^ (b — a) of the three successive sides, 
 AB, Bc, CA, of any plane triangle abc, represents by its length the product 
 of the lengths of those three sides, and by its direction the tangent at A to 
 the segment ABC of the circumscribed circle (contrast with this the cor- 
 responding result in § xxviii.); this construction of a continued product 
 appears to be peculiar to quaternions ; case where the three points a, b, C 
 are situated on one straight line ; if A, b, c, d be the four successive cor- 
 ners of an uncrossed and inscribed quadrilateral, the continued product 
 (d — c) (c — e) (b — a), of the three successive sides ab, bc, cd, is con- 
 structed in this calculus by a line which has the direction of the fourth 
 side, DA or A — D ; but the same product represents a line in the direction 
 opposite to that of the fourth side, if the quadrilateral be a crossed one ; 
 these results also (which may again be contrasted with those of § xxviii.) 
 appear to be peculiar to quaternions ; the formula, 
 
 U. (d-cJ (c- b) (b-a) = + U(a-d), 
 
 expresses, in the present calculus, a property which belongs only to plane 
 
 and inscriptible quadrilaterals, . . . Articles 194 to 200 ; Pages 198 to 203. 
 
 § xxxix. Interpretation of the fourth proportional |8a-i . y, or j8 -^ a x y, for 
 the cases where the three lines a/3y are not coplanar, y not \\\ a, (3, but 
 ■where a is perpendicular either to y or to j8 ; for each of these two cases, 
 the associative property of multiplication holds, /3a-i . y = j3 . a-i y, and 
 
CONTENTS. XXm 
 
 the point may therefore be omitted; but the symbol /3a-i y does not now 
 represent any line but a quaternion; the symbol j8ay denotes another 
 quaternion, which is still (as in the last section) = a^ . /3a "^ y ; the ver- 
 sors of these two quaternions are negatives of each other, U . j3ay = — 
 U./3a~iy; for any multiplication of any number of quaternions, the 
 tensor of the product is equal to the product of the tensors (compare 
 § xxxvii), TIT = nX; in the case where the three lines a/3y compose a 
 rectangular system, the fourth proportional /3a "ly degenerates from a 
 quaternion to a scalar, which is a negative or a positive number, according 
 as the rotation round a from /3 to y is of a positive or a negative charac- 
 ter; on the contrary, the continued product fiay is positive in the first of 
 these two cases, and negative in the second ; thus /3ay = — ya/3 = + Tj3 . 
 Ta . Ty, if ^ _u a, y _i_ a, y -U ^, the upper sign holding when the ro- 
 tation round y from a to (8 is positive ; if da, db, dc be thi-ee co-initial 
 edges of a right solid, then 
 
 (c - b) (b — d) (a - d) = + volume of solid, 
 according as the rotation round the edge da from db towards dc is di- 
 rected to the right hand or to the left ; examples from the unit-cube, k-^j 
 yi = -l, kji = +l,ijk = -l, . . . Articles 201 to 210; Pages 203 to 208. 
 
 § XL. More general cases, where a, (3, y are neither coplanar, nor rectangular ; 
 each of the two symbols, /3a ~i . y, /3 . a~i y, represents a determined 
 quaternion, but it remains to prove (§§ XLii., xLiii.) that these two qua- 
 ternions are equal ; it is sufiicient for this purpose to establish the equality 
 of their versors, and therefore the lines a, /3, y may be supposed to be 
 three unit-vectors, OA, OB, oc, terminating at three given points a, b, c 
 on the surface of the unit-sphere (§ xxxv.) ; the quaternion quotient /3a - 1 
 becomes then a versor, with aob for its representative hiradial (§ xviii.) ; 
 and the great-circle arc, ab, which subtends the angle aob, may be said 
 to be the representative arc of the same quaternion or versor, /3a "i; 
 it is proposed to construct the representative arc of the quaternion j3a-'^.y, 
 
 Articles 211 to 216; Pages 208 to 212. 
 
 § XLI. Equality of any two versors corresponds to equality of their represen- 
 tative arcs, such arcual equality being defined to include sameness of 
 direction on the spheric surface, of the vector arcs compared, so that 
 EQUAL ARCS are always supposed to be portions of one common great cir- 
 cle ; but an arc may be conceived to slide or ttern, in its own plane (com- 
 pare § XX.), or on the great circle to Avhich it belongs, without any change 
 of value ; constructions for multiplication and division of versors, bj' pro- 
 cesses which may be called addition and subtraction of their representa- 
 tive arcs ; if any multiplicand versor q, and any multiplier versor r, be 
 represented by two successive sides kl, lm, of a spherical triangle klm, 
 the product versor rq will be represented by the base km of the same tri- 
 angle; i\ms, versor, proversor, and transversor (see § ix.), are represented 
 by what may be called an arcual vector, an arcual provector, and an ar- 
 cual transvector respectively (compare First Lecture) ; we may write the 
 formula '-^ lm + '-kl = — km, and the arcual sum of two successive 
 
XXIV CONTENTS. 
 
 sides of any spherical triangle, regarded as two successive vector arcs, may 
 in this sense be said to be equal to the base (compare §§ iv., v.); such 
 ADDITION (of vector arcs) corresponds to, and repi'esents, a composition of 
 two successive versions (§ ix.), or plane rotations of a line (the radius) ; 
 the sum of the three successive sides of a spherical triangle, or generally 
 the stem of all the successive sides of any spherical polygon, may be said 
 to be a null are, or to be equal to zero, ^ mk + -s lm + '^ kl = ; to go 
 on the surface of the sphere successively from k to l, from l to m, and from 
 M toK again, produces no final change of position; subtraction of vector 
 arcs, corresponding to division ofversors, is very easily effected, on the 
 same general plan of construction, and represents (compare again § ix.) 
 a decomposition of a given version into two others, of which the first in 
 order is given, namely, the one represented by the subtrahend arc; in 
 short, for arcs as for lines, the relations of § iv., between vector, provector, 
 and transvector, hold good in this manner of speaking ; the provector arc 
 is regarded as the remainder, in the arcual subtraction of vector from 
 transvector ; addition of arcs is not a comsiutative operation ; for if 
 two arcs kk', m'm bisect each other in l, we shall have 
 
 ^ KL + '- LM = — Lk' -f ^ m'l = ^ m'k', 
 
 and this arcual szim ^ m'k' is indeed equally long with the arc ^ kji, 
 which was found to be = '-- lm + ^ kl, but it is part of a different great 
 circle, and therefore these two sums are not arcually equal to each other, 
 in the sense of the present section ; this result answers to and illustrates the 
 general non-commutativeness of the operation oi multiplication ofversors, 
 whereby qr is not generally =r^ (§§ x., xi., xxix. &c.) ; it is necessary to 
 distinguish in writing between two such symbols as '--'+ '-> and '-^ + — ■' ; 
 the rule adopted in this calculus is to write the symbol of the addend arc, 
 like that of the multiplier quaternion, and generally the symbol of the 
 
 operator, to the LEFT of the SYMBOL OF THE OPERAND, that is, in this 
 
 case, to the left of the symbol of the arc to which another is to be added; 
 thus we s^j'ZZ write "provector plus vector," and not, generally, vector plus 
 provector ; several other general properties of multiplication and division 
 of quaternions may be illustrated by the same method of arcual construc- 
 tion, Articles 217 to 222; Pages 212 to 217. 
 
 § xLii. Application of the method of the last section to the problem proposed 
 at the end of § xl., namely, to the construction of the representa- 
 tive arc of the fourth proportional j3a' i. y to three unit- vectors, a, f3, y, 
 or OA, OB, oc, which are not rectangular, nor in one common plane 
 (§ XL.), but which shall at first be supposed to make acute angles with 
 each other, so that the sides of the triangle abc shall each be less than a 
 quadrant ; the vector arc representing y is here a quadrant kl with c for 
 its positive pole ; the provector arc representing the other factor /3a -i, is 
 the arc ab, or an equal arc lm ; the transvector arc loi, which represents 
 the required fourth proportional, under the form of the product jSa-^.y, 
 is found to have its pole at a new point d, which is a corner of a new cir- 
 cumscribed spherical triangle DEF, whose sides EF, FD, DE are respec- 
 
CONTENTS. XXV 
 
 tively bisected by the three corners A, B, c of the old or given triangle ; 
 and the kepeesentative angle, kdm, at this pole d, which corresponds 
 to the representative arc, km, and mai/ replace it, as representing the 
 fourth pi-oportional to the three vectors a, (3, y, is equal to the semisum of 
 the angles of the auxiliary triangle, def, or to the supplement of that 
 semisum, according as the rotation round a from (3 to y is positive or ne- 
 gative ; hence the two quaternions /3a"i . y and ya" i . /3 have one common 
 axis, namely, the radius od, but have their angles supplementary ; but 
 these were the conditions assigned in § xxxvii., as necessary and suffi- 
 cient, in order that one quaternion should be the negative of the conjugate 
 of the other ; we have therefore, as in the last cited section, 
 
 i8a-i.y = -K(ya-i.|S) = /3.a-iy, 
 
 and the associative principle is again found to hold good for the three 
 vectors y, a-i, /3, although these three lines are not now coplanar (as 
 they were in §§ xxxvi., xxxvii.), and do not form a wholly or even par- 
 tially rectangular system (as they did in § xxxix.), 
 
 Articles 223 to 235 ; Pages 217 to 228. 
 
 § XLiii. Other proof of the same theorem, by means of an analogous construc- 
 tion for the product /? . w'^y ; the case where jSj^a may be treated as a 
 limit of a case lately discussed, the arc ab becoming a quadrant, and the 
 triangle def becoming a lune ; case where the arc ab is greater than a 
 quadrant; value of /3a-i. y', when y'= — y, and when the sides of the 
 new triangle abc' are each greater than a quadrant ; we have 
 
 i8a-i . y' = _ K (y'a-i . /3) = /3 . a-i y' ; 
 
 in EVERY case^ the associative principle of multiplication holds good 
 for any system o/ three vectors, and we may alwats write in this 
 calculus (as in algebra) the formulae, 
 
 /3. a-iy = ^a-i.y = j8a-iy ; /3. ay = /3a .y = /3ay ; 
 
 to establish this result has been the main object of the present Lecture, . 
 
 Articles 236 to 240 ; Pages 228 to 233. 
 
 § XLiv. Partial indetermination of the constructed triangle def, when the given 
 triangle abc is triquadrantal ; the point d may take infinitely many po- 
 sitions on the sphere, but the semisum of the angles at d, e, f is always 
 equal to two right angles ; the scalar character of the fourth proportional 
 to three rectangular vectors, which had been established in § xxxix., may 
 in this way be proved anew, as a particular or litniting case of a much 
 more general result ; when a scalar is treated as a quaternion, its axis is 
 indeterminate ; the rule of § xxxix. for determming the sign of the scalar 
 is also reproduced, Articles 241 to 244 ; Pages 233 to 237. 
 
 § XLV. Illustrations of the equations (of § xxxix.), kji — -]- 1, ijk = -l; the 
 former may be interpreted as expressing that if a line \ be suitably chosen, 
 namely, so as to be perpendicular to the (meridional) plane of k and i, 
 and be then operated on successively by i, by 7, and by k, considered as 
 
 d 
 
XXVI CONTENTS. 
 
 three quadrantal and mutually rectangular versors (§ x.), the final direc- 
 tion of this revolving line \ will be the same as the initial direction ; the 
 latter equation (ijk = — 1) may in like manner be interpreted as expres- 
 sing that if the same (westward or eastward) line X be operated on suc- 
 cessively by k, by j, and by i, it will take at last that (eastward or west- 
 ward) direction which is opposite to the initial direction ; and because 
 each of the vector-units i, j, k, when thus regarded as a quadrantal versor, 
 is evidently (see again § x.) a semi-inversor, we have in this way ex- 
 tremely SIMPLE INTEKPRETATIONS ybr ALL THE PARTS OF THE FORMULA, 
 
 ii =ji = k2 = ijk=-l; 
 
 which continued equation may be considered as including within itself all 
 the laws of the combination of the symbols, i, j, k ; and therefore 
 ultimately, on the symbolic side, the whole theory of quaternions, 
 because these are all reducible to expressions of the quadrinomial form, 
 
 g = w + ix +jy + kz, 
 
 Articles 245 to 250 ; Pages 237 to 240. 
 
 LECTUEE VI. 
 
 (Articles 251 to 393 ; Pages 241 to 380.) 
 
 GENERAL ASSOCIATIVE PROPERTY OF THE MULTIPLICATION OF QUATER- 
 NIONS; REPRESENTATION OF THE PRODUCT OF TWO VERSORS BY THE 
 EXTERNAL VERTICAL ANGLE OP A SPHERICAL TRIANGLE; CONNEXION 
 OF TERNARY PRODUCTS OF QUATERNIONS WITH SPHERICAL CONICS; 
 CONTINUED PRODUCTS OF THE SIDES OF PLANE OR GAUCHE POLYGONS 
 INSCRIBED IN A CIRCLE OR IN A SPHERE ; COMPOSITION OF CONICAL 
 ROTATIONS ; THEORY OF SPHERICAL POLYGONS OF MULTIPLICATION, 
 WITH THEIR SYSTEMS OP INSCRIBED CONICS, AND RELATIONS OF FOCAL 
 ENCHAINMENT. 
 
 § xLVi. Postponement of the proof of the distributive principle of the multiplica- 
 tion of quaternions ; additional illustrations of the general theoiy of the 
 fourth proportional to three vectors, which was assigned in the foregoing 
 
 Lecture ; case of coplanarity, regarded as a liynit, 
 
 Articles 251 to 257; Pages 241 to 247. 
 
 § XLVii. The product of the square roots of the successive quotients of the vectors 
 d, S, Jj, of the corners of a spherical triangle def, is a quaternion, 
 
 of which the angle is the semi-excess of the triangle, 
 
 and the axis of the same quaternion product has the direction of + S, that 
 
CONTENTS. XXVU 
 
 is of OD or of DO, according as the rotation round S from Z towards «, or 
 
 that round d from f towards b, is positive or negative, 
 
 Articles 258 to 263 ; Pages 247 to 252. 
 
 § xLviii. General construction for the multiplication of any two quaternions, by a 
 process analogous to addition of their KEPEESENTATi^rE angles (compare 
 §§ XLi., xLii.) ; if these be made the base angles of a spherical triangle, 
 and if the rotation round the vertex of this triangle, from the base 
 angle which represents the multiplier, towai'ds the base angle which 
 represents the multiplicand, be positive, then the product is repre- 
 sented by the external, vertical angle ; if we agree to call the ex- 
 ternal vertical angle of a spherical triangle generally the spherical sum 
 OF THE TWO BASE ANGLES, when ihe. positions of the vertices of these seve- 
 ral angles on the sphere are taken into account, and when the addend 
 angle answers to the multiplier quaternion, according to the ride of rota- 
 tion above given, we may enunciate a general rule /or the multiplica- 
 tion of any two quaternions, as follows : " the tensor of the product is the 
 arithmetical product of the tensors (§ xxxvii.), and the angle of the pro- 
 duct is the spherical sum of the angles of the factors ;'' this new sort of 
 SPHERICAL ADDITION OF ANGLES IS connected With a certain composition 
 of rotations of arcs ; such addition of angles (like that of arcs in § XLI.) 
 is a non-commutative operation ; this result furnishes a new illustration of 
 the non-commutative character of the general multiplication of quater- 
 nions ; the rotation round the axis or round the pole of the multiplier, 
 from that of the multiplicand, towards that of the product (compare 
 §§ XL, XV., XXVI ), is always jooszYitie, . Articles 264 to 272 ; Pages 252 to 261. 
 
 § XLix. Corollaries from the general construction for multiplication assigned in 
 the foregoing section (xi,viii.) ; interpretations by it of the symbols a/3, 
 /3a~i, j8a-i/3, agreeing with the results previously obtained respecting 
 the product, quotient, and third proportional of any two vectors ; inter- 
 pretations of (8i«J, /3Jal, /35a5, as denoting quaternions (compare §§ xxix., 
 XXX.) ; analogous interpretation of the more general sj'mbol 5 = /3* a^-*, 
 when a and /3 are sxipposed to be unit- vectors ; the unit axis Ax . q = op, 
 of this quaternion q, describes by its extremity p a curve apb upon the 
 unit- sphere, which curve is the locus of the vertex p of a spherical triangle 
 APB, whose base-angles are complementary; this curve is a spherical 
 conic ; for any spherical triangle, with a, /3, y for the unit vectors of its 
 corners a, b, c, and with x, y, z for the (generally fractional) numbers 
 of right angles at those corners, the rotation round c from b to a being 
 supposed to be also positive, we have the three equations 
 
 y-^va" = - 1 ; a'^y-^y = - 1 ; (Sva'^y- = - 1 ; 
 
 any one of which will be found to include, when interpreted and developed, 
 by the principles of the present calculus, the whole doctrine of spherical 
 trigonometry ; with the phraseology recently proposed, the spherical sum 
 of the THREE ANGLES of a7iy Spherical triangle, if taken in a suitable order 
 of succession, is always equal to two right angles, ...... 
 
 Articles 273 to 280 ; Pages 261 to 268. 
 
XXVIU CONTENTS, 
 
 § L. Interpretation of the symbol rqr-^, where q and r are any two quaternions; 
 this symbol denotes a new quaternion, with the same tensor, and same 
 magnitude of angle, as the original or operand quaternion, q, 
 
 1 .rqr-'^ — Tq, /_ . rqr-'^ — L q] 
 
 but the axis of the new quaternion ^(^r-i is generally different from K^.q, 
 and is formed or derived from this latter axis, by a conical and positive 
 KOTATION round the axis Ax . r, of the other given quaternion, r, through 
 DOUBLE the ANGLE ofthat quaternion ; analogous interpretations of 5 ~ i rq, 
 qtrq-i ; the latter symbol denotes a quaternion formed from r, by making 
 its axis revolve conically roimd the axis of q, through a rotation expressed 
 by the product 2f x Z. g' ; by employing arcs instead of angles, we may in- 
 terpret the symbol q (^ ) 5-1, in which q may be said to be the ope- 
 rating quaternion, as denoting the operation of causing the Aec which 
 represents the operand quaternion, and whose symbol is supposed to be 
 inserted within the parentheses, to move along the doubled arc of the 
 operator, without any change of either length or inclination (like the equa- 
 tor on the ecliptic in precession) ; if i be still a scalar exponent, {qrq-^y = 
 qr'q-^ ; the symbol qpq'^ denotes a vector formed from the vector p, and 
 the analogous symbol q^q~^ may be used to denote a bodg derived from 
 the body B, by a conical and finite rotation, through 2 Iq roimd Ax . q ; 
 to express that this body has afterwards been made to revolve through 
 2 Lr round Ax . r, we may employ the following symbol for the new po- 
 sition of the body, or system of vectors, r . gB^-i . r-i ; and so on for atii/ 
 number of successive and finite rotations, round ani/ axes drawn from or 
 through one common origin o ; interpretations of the symbols 5- (a + p) 5-1, 
 q (a + B) ^ ~ 1 ; expression for rotation of a body round an axis which does 
 not pass through the origin of vectors ; symbols qi (^ ')9'i> 7 ( )y"'^; 
 the former represents a rotation through the angle itself of q ; the latter 
 represents a reflexion with respect to the line y, or a conical rotation 
 of the operand (whether vector or body), round y as an axis, through two 
 right angles ; the formula /3 . a"i fa . /3-i = /3a-i . £ . aj8-i, expresses that 
 two successive reflexions, with respect to any two diverging lines a and /3, 
 are equivalent upon the whole to a single conical rotation, roimd an axis 
 perpendicular to both those lines, through twice the angle between them, 
 
 Ai-ticles 281 to 292 ; Pages 268 to 277. 
 § LI. The general demonstration of the associative property of the multiplication 
 of any three quaternions (mentioned by anticipation in § xxl), may be 
 made to depend on the corresponding principle for the multiplication of 
 any three versors, q, r, s ; when these versors are represented by arcs 
 (§ XL.), we may propose to prove that a certain arcual equation (§ XLI.) 
 is a consequence of five other equations of the same sort ; first proof by 
 spherical conies ; the two partial or binary products rq and sr are re- 
 presented by portions of the two cyclic arcs of a conic circumscribed about 
 a quadi-ilateral, whose successive sides, or portions of them, represent the 
 three proposed /actors, q, r, s, and their ternary product, srq ; other and 
 more elementary geometrical proof of the associative principle, not intro- 
 
CONTENTS. XXIX 
 
 ducing the conception of a cone; second proof by spherical conies ; certain 
 angles at the corners of a new spherical quadrilateral abcd represent the 
 three factors and their total product, while certain other angles at the /oci 
 EF of an inscribed conic represent the two binary products ; three equa- 
 tions between spherical angles are thus shewn to be consequences of three 
 other equations of the same sort, in such a way as to establish the pro- 
 perty above proposed for investigation; it is therefore proved geo- 
 metrically, in several different Avays, that the associative pkinciplb 
 OF MULTIPLICATION holds good for any three versors, and thence for any 
 THREE QUATEKNiONS, sr . q = s . rq = srq ; (in the Fifth Lecture this 
 theorem was established only for the multiplication of ani/ three vectors') ; 
 extension to the case of awy number of factors; arcual addition (§ xll), 
 and angular summation (§ xlviii.)) are also associative operations, 
 although they have been seen to be not gejierally commutative, .... 
 
 Articles 293 to 304 ; Pages 277 to 290. 
 
 § Lii. Other forms of the associative prmciple ; if the first, third, and fifth sides of a 
 spherical hexagon be respectively and arcually equal to the three successive 
 sides of a spherical iriaw^'/e, then the second, fourth, and sixth sides of the 
 same hexagon will be respectively and arcually equal to the three succes- 
 sive sides of another triangle ; or if the arcual sum of three alternate sides 
 of a hexagon (fifth plus third plus first) be equal to zero (see § xll), 
 then the corresponding sum of the three other alternate sides (sixth plus 
 fourth plus second) will likewise vanish ; symbolical transformations of the 
 same principle ; if a^-i = y£-i, then ?^-i. aj3-i = ^£-1 . y/S"! ; if ^£-1 = 
 (cX-i.e?;-!, then ^k-i= sjy-i. 0X-i ; if (e^ . y/3) a = ?, then (a/3 . y^) £ 
 = Z ; remarks on the necessity that existed for demonstratiiig the general 
 associative principle of multiplication, notwithstanding that to a certain 
 extent the principle had been previously defined to hold good ; we may be 
 said to have virtually used the definitional associative formula, 
 rq. a = r . qa, for the case where a, qa, and r . qa were lines, in order 
 to interpret the product, rq, of any two geometrical ^ac^ors, or qua- 
 ternions ; but the very fact of the perfect definiteness (§ xxi.) of this in- 
 terpretation of a binary product made it necessary that we should not as- 
 sume but prove the corresponding formula respecting a general ternary 
 product, Articles 305 to 316 ; Pages 290 to 303. 
 
 § Liii. If the continued product oi any odd number of vectors be a line, it is 
 equal to the product of the same vectors, taken in an inverted order ; and 
 reciprocally, if the continued product of an odd number of vectors be not 
 a line, it will tiot remain unaltered by such inversion of the order of the 
 factors ; on the other hand, if the number of vectors thus multiplied be 
 even, the product will be changed to its own negative, if it be a line, and 
 not otherwise, by such inversion ; if the continued product of an even 
 number of vectors be a scalar, the inversion produces no change ; and re- 
 ciprocally if the continued product of an even number of vectors receive 
 no change by inversion of order, that product must be a scalar ; conjugates 
 find reciprocals of products oi any number of vectors or quaternions, are 
 
XXX CONTENTS. 
 
 the products of the conj ugates or reciprocals of the factors, taken in an in- 
 verted order ; in § xxxvii. this was only established for the case of two 
 factors ; the formulse Ka = — a, K . j8a = + a/3 (see §§ xxm., xv.), may 
 now be extended as follows, K . y^a = — aj8y, K . 5y/3a = + «^y<^) &c., 
 the signs of the results being alternately — and + ; the construction of 
 § XXXVIII., for the continued product of the three sides of an inscribed 
 triangle, may now be extended so as to shew that the product of the suc- 
 cessive sides of a polygon inscribed in a circle is equal either to a scalar, 
 or to a tangeiitial vector, at the first corner of the polygon, according as 
 the number of the sides is even or odd ; thus the continued product of the 
 four successive sides of an inscribed quadnlateral abcd is a scalar, 
 
 U . (a - d) (d - c) (c - b) (b - a) = + 1, 
 and the upper or lower sign is to be taken, according as the quadrilateral 
 is an uncrossed or a crossed one (compare §§ xxviii., xxxviii.) ; this 
 symbolical result appears to be peculiar to the present calculus, and con- 
 tains a characteristic property of the circle, corresponding to the known 
 and elementary relations between angles in alternate segments, or in the 
 same segment ; the versor of any product of quaternions is equal to the 
 product of the versors, Un = nU, . Articles 317 to 322 ; Pages 303 to 309. 
 
 § Liv. To interpret the continued product of the four sides of a gauche quadri- 
 lateral, ABCD, we may conceive it to be inscribed in a sphere ; the 
 product is a quaternion, of which the axis has the direction of the out- 
 Avard or inward normal to the sphere at the first corner A, according to 
 the character of a certain rotation ; the angle of the same quaternion pro- 
 duct is the angle of the lunule, abcda, or the angle between the two 
 small-circle arcs, ABC, ADC ; this includes as a limit the case of a qua- 
 drilateral in a circle ; an analogous construction holds for the continued 
 product of the sides of a gatjche hexagon, octagon, or other polygon 
 with an even number of sides, inscribed in a sphere ; the product is still a 
 quaternion, of which the axis is normal, or the plane tangential, to the 
 sphere, at the first coi-ner of the polygon ; construction for the contmued 
 product of the sides of a gauche pentagon, heptagon, &c., inscribed in a 
 sphere; this product is a tangential vector, drawn at the first corner; 
 conversely, if the continued product of the sides of a gauche pentagon 
 ABODE be a line, when this product is constructed according to the rules of 
 the present calculus, the pentagon is inscriptible in a sphere ; hence is de- 
 rived the following equation of homosph^ricism, or condition for 
 five points A, B, C, d, e, being situated upon one common spheric surface, 
 
 AB . BC . CD . DE . EA = EA . DB . CD . BC . AB ; 
 
 this vector character of the product of the sides of a pentagon in a sphere 
 includes, as a limit, the scalar character of the product of the sides of a 
 quadrilateral in a circle (§ Liii.), which latter relation may be expressed 
 by the following equation of concikcularity, 
 
 AB . BC. CD . DA = DA . CD . BC . AB, 
 
 Articles 328 to 328 ; Pages 309 to 315. 
 
CONTENTS. XXXI 
 
 LV. One form of the equation of the tangent plane at A to the sphere abcd is 
 the following: 
 
 AB . BC . CD . DA . AP = AP . DA . CD . BC . AB ; 
 
 the two equations, 
 
 AB . BC . CD . DB . EA = EA . DE . CD . BC . AB, 
 
 and 
 
 AB . BC . CD . DA . AE = AE . DA , CD . BC . AB, 
 
 must therefore be incompatible, except under the supposition that either 
 the point e coincides with A, or that the four points A, b, c, d are copla- 
 nar ; in fact when the distributive principle shall have been established 
 (in § Lxxv.), it wiU become clear that the addition of these two equations 
 
 AB . BC . CD X AE . EA = AE . EA X CD . BC . AB, 
 
 and therefore that either 
 
 ae2 = 0, AE=0, E = A, 
 or else 
 
 AB . BC . CD = CD . BC . AB, 
 
 which are respectively (compare § xxxviii.) conditions of coincidence and 
 coplanarity ; problem of inscription in a given sphere, of a gauche quadrilate- 
 ral ABCD, whose four successive sides ab, ... da shall be respectively parallel 
 to four given radii oi, ok, ol, om ; problem of expressing an re*'* radius, 
 OP,j, or pn, of a given sphere, considered as a function of an initial radius 
 OP or p, and of n other radii, oii, . . . oi„, or ti, . . . („, to which the n 
 successive and rectilinear chords ppj, . . . p,}-i p,j are required to be pa- 
 rallel ; if a and j3 be any two equally long and diverging lines, OA, ob, 
 and if y have either of the two opposite directions of the lines ab, ba con- 
 necting their extremities, then j6 = — 7«7"^ ; hence in the recent question, 
 jOi = — iijOtri, |02 = — t3|Oii2"^) &c., and if we introduce the quaternion, 
 qn = hi • • ' t2ti) the solution of the problem will be expressed by the for- 
 mula p„ = (.—)"qiip9n~^ ; the same expression will hold good, if we regard 
 the quaternion q,i as the continued product 
 
 5')i=(a«-p(i-i) («n-i-|0,t-3) • . • (ai-p), 
 
 of the n first segments PAi, P1A2, . . . &c., of the n successive chords, on 
 which Aj, A2, &c., are n points arbitrarily taken, but not supposed to be 
 situated upon the surface of the sphere ; relation to a conical rotation (see 
 § L.) ; EQUATION OF CLOSURE, |0,j = jo ; for an inscribed and even-sided 
 polygon, pqn =- qnP, Ax . q,i || p, with inclusion of the limiting case for 
 which the product qn is a scalar ; for an odd-sided polygon, pq,i = — q„p, 
 and the same product g„ must reduce itself to a vector j_ p ; these last 
 results agree -with those of § Liv. ; if, in a sphere, the five successive sides 
 of an inscribed gauche pentagon, Abcde, be respectively parallel to the 
 five radii drawn to the five corners of a svperscribed spherical pentagon, 
 IKLMN, then the fifth corner n of the second pentagon is situated some- 
 where upon that great circle fh, of which a portion coincides with the 
 
XXXll CONTENTS. 
 
 areual sum, -- lm + --^ IK (see § xt,i.) of the, first and third sides of that 
 second pentagon ; this theorem involves and expresses a geaphic pko- 
 PEETT OF THE SPHEEE, "which is sufficient to characterize that surface, 
 and is analogous to the well-known and elementary relation letween the 
 DIRECTIONS of the sides of a quadrilateral inscribed in a circle ; indeed 
 this graphic property of the circle can be derived as a limit fi-om the lately 
 Stated and graphic property of the sphere ; theorem respecting a general 
 relation of an inscribed gauche polygon of 2n sides, to a certain other in- 
 scribed polygon of 4m + 1 sides ; examples, 
 
 Articles 329 to 340 ; Pages 315 to 325. 
 
 § LVI. Composition of conical rotations; the symbol srqB {srq)-'^ denotes the 
 position into -which the body B is brought, by three successive and finite 
 rotations, round the three successive axes, Ax . g, As . r, Ax . s, all 
 drawn from the origin o, through the three successive angles denoted by 
 2Z.5, 2 Lr, 2Zs; but the same final position of the body, or of the sys- 
 tem of vectors operated on (compare § l.), can also be attained by a sin- 
 gle resultant rotation, round Ax . srq, through 2 L . srq ; in like manner 
 any number of successive and conical rotations of a line p, or body B, 
 round axes passing through one common point o, can be compounded into 
 one, by multiplying together, in the given order, the quaternions which 
 represent, by their axes and angles, the halves of the given rotations, and 
 then taking the axis and the doubled angle of the quaternion product ; 
 examples: the identity /3-r-a = /3xa-i of § xsxv., since it gives 
 Q3^a) p(a-i-j8) = |3.a-ipa./3-i, may be interpreted (see again § l. ) 
 as expressing that two successive reflexions of an arbitrary hue p, with 
 respect to two given lines a, /3, are jointly equivalent to the double of the 
 conical rotation represented by the arc ab ; the identity, y -f- a = 
 (y -^ j6) X (/3 -f- a), of § VII., conducts in hke manner to the conclusion 
 that a conical rotation thus represented by the double of an arc ab, if fol- 
 lowed by another conical rotation represented by the double of a successive 
 arc BC, produces on the whole the same eiFect as that third and resultant 
 conical rotation, which is on the same plan represented by the double of 
 the arc ac ; that is, by the double of the aecual sum (see § xli.) of 
 the HALVES of the arcs which represent the two component rotations ; 
 three successive and conical rotations, represented by the doubles of the 
 three successive sides of any spherical triangle, produce on the whole no 
 effect ; geometrical illustrations and confirmations of these results ; exten- 
 sion to spherical polygons, and to any mnnber of successive rotations, re- 
 presented by the doubles of the sides ; rotations may be represented also 
 by spherical angles (instead of arcs); the equation y»'/3ya= = — 1, of 
 § XLix., shews that if the double of the rotation represented by the angle 
 CAB be followed by the double of the rotation represented by the angle 
 ABC, the result will be the double of the rotation represented by the angle 
 ACB, or the opposite of the double of the rotation represented by bca ; two 
 successive reflexions, with respect to two rectarigular lines, are equivalent 
 to a single reflexion M'ith respect to a line perpendicular to both ; if a body 
 
CONTENTS. XXXlll 
 
 he made to revolve through at}y number of successive rotations, represented 
 as to their axes and amplitudes by the doubles of the angles of any sphe- 
 rical polygon, the body will be thereby brought back to its original posi- 
 tion, Articles 341 to 349 ; Pages 325 to 334. 
 
 § L.VII. The system of the two successive rotations represented by the two succes- 
 sive sides DF, FE, of any spherical triangle, is equivalent to a single rota- 
 tion, represented by the double of the arc which is the common bisector of 
 those two sides ; the arcual sum ^'^-ED+i^FE-f^^ DF, of the halves 
 of the three successive sides of any such triangle def, is an arc which has 
 the fii"st corner d of that triangle for its positive or negative pole, accord- 
 ing as the rotation round d from f towards e is positive or negative ; the 
 length of the same sum-arc represents the spherical semi-excess, or semi- 
 area, of the triangle ; extension to any spherical polygon, and even to 
 ANY cl6Se» figuee ON A SPHERE ; case of negative areas ; successive 
 rotations, represented by the successive sides of any spherical triangle or 
 polygon (and not now by the doubled sides), or even by the successive 
 elements oi any closed perimeter on a sphere, compound themselves into a 
 single resultant rotation round the first corner or point of the tigure, or 
 round the radius drawn to it, through an angle which is numerically equal 
 to the TOTAL AREA of the figure (the case of negative elements of area 
 being attended to when necessary) ; if a body, or system of vectors, be 
 made to revolve in succession round any number of different axes, all pass- 
 ing through one fixed point, so as first to bring a moveable line a into 
 coincidence with a fixed line j5, by a rotation round an axis perpendicular 
 to both ; secondly, to bring the same moveable line a from the position j8 
 to another given position y, by revolving in a new plane ; and so on, till 
 after bringing it to coincide successively with any number of lines given 
 and fixed, and finally after turning from k to X, the line a is brought 6acA 
 from X to its own original position ; then the body will be brought, by 
 this succession of rotations, into the same final position as if it had re- 
 volved B-OVNT) THE ORIGIN AJj POSITION ofthe moveable line (a), as an 
 axis, through an angle of finite rotation which has the same numerical 
 measure as the spherical opening ofthe pyrajmid (a, /3, y, . . . k, X), 
 whose edges are the successive positions of the line ; in symbols, for the 
 case of five given lines, including the original position of a, if we form the 
 quaternion product, 
 
 and if the rotations round a, from j3 to y, from y to S, and from ^ to £ be 
 positive, then 
 
 Tq=l, Ax. 5 = a, Lq=l {A + B + C ^ D + E -^ir), 
 
 the addition ofthe five angles of the pentagon being performed in the 
 usual way (and not here by such spherical summation as was mentioned 
 in § XLviii.) ; extension to the product ofthe square roots of any number 
 
 e 
 
XXXIV CONTENTS. 
 
 of successive quotients of vectors ; even if that number be infinite, this 
 product of square roots is still a definite quaternion, of which the angle 
 represents the semi-area of a closed figure on a sphere, while the axis of 
 this latter product is still the radius drawn to the first point of the figure ; 
 interpretation of the symbols, 
 
 fi y a 
 a (3 p 
 
 if Casin § xlii.) the corners a, b, c of owe spherical triangle bisect respec- 
 tively the sides opposite to the corners d, e, f of another, and if a body be 
 made to revolve in succession through three rotations represented respec- 
 tively by 2 ^ CA, 2 "- Bc, 2 ^ ab, or by the doubles of the three 
 SIDES of the first triangle abc, taken in an inverted order, this body 
 will on the whole have revolved round the corner d of the second triangle, 
 as round a negative pole, tlurough an angle which is numerically equi- 
 valent to the DOUBLED area of the same second triangle, def, . . . 
 
 Articles 350 to 357 ; Pages 334 to 343, 
 
 § Lviii- New elementaiy proof of the associative property of multiplication of 
 three quaternions ; six double co-arcualities may be assumed to exist by 
 construction, and then the theorem is, that three arcual equations are con- 
 sequences of three others ; this corresponds to the second proof by spheri- 
 cal conies in § li., which shewed that three equations between angles 
 were consequences of three others : if q, r, s, t, be any four given quater- 
 nions, and u their total or quaternary product, u = tsrq, while v, ic, x 
 denote respectively their three binary products, rq, sr, is, and y, z denote 
 their two ternary products, srq, tsr ; if also these ten factors and products 
 q, r, s, t, u, V, w, x, y, z, be represented by ten angles at ten points 
 A, B, c, D, E, F, G, H, I, K upon the unit-sphcre, then since y = sv, z = tw, 
 u = ty, we can, by six triangles, answering to six binary multiplications, 
 construct successively the six points f, g, h, i, k, and E, the four points 
 A, B, c, D being here regarded as given, and also certain angles at them ; 
 in this process of construction, /_r is represented by two different angles 
 at B, giving one equation of condition; Z s is represented by iAree dif- 
 ferent angles at c, giving two other such equations ; Z t gives two equa- 
 tions ; Lv, Lw, and L y give each one other equation : but the angles of 
 q, X, z, M, are each only once employed in the construction ; on the whole 
 then there are eight equations of construction, required for the cor- 
 rectness of the figure ; but the associative principle gives ^/b?*?" other binary 
 products, y = wq, z = xr, u = xv, u = zq, axiAfour other triangles; there 
 are thus ten triangles in the completed figure, representing ten binary 
 multiplications (on the plan of § xlviii.), and it is found that each of the 
 ten points A ... K is a common corner oi three of those ten triangles; at 
 each point three angles are equal, and there are thus as many as twenty 
 EQUATIONS between angles, including the eight equations of construction ; 
 the remaining twelve equations are therefore consequences of those eight, in 
 virtue of the associative principle, . Articles 358 to 364; Pages 343 to 350- 
 
CONTENTS. XXXV 
 
 § Lix. In general, if there be any number, n, of quaternions (or versors), qu • • • qn, 
 represented by angles at n points, Qi, . . . Q« on a sphere, and if the total 
 product q = q,, qn-i • • • q^li t*® represented at another point Q, we may 
 conceive these points to be the successive corners of a certain spherical po- 
 lygon of jp = w + 1 sides, which may be called a polygon of multiplica- 
 tion ; this conception includes the cases of the triangle of hinary multipli- 
 cation in § xLViii., the second quadrilateral of ternary multiplica- 
 tion, ABCD, in § Li., and the pentagon of quaternary multiplication, 
 ABODE, in § LViii. ; in general we may form fi — 1 binary products, 
 fi = q2qi, &c., n — 2 ternary products, Si = qiq^qi, &c., and so on ; the 
 number of these intermediate or partial products, or of their represen- 
 tative points on the sphere, is i (m + 1) (w - 2) ; along with the p former 
 points, they make up altogether J (» + 1) » points in the completed 
 figure ; each point may be supposed to have tux) spherical co-ordinates, 
 but between these (ra + 1) n co-ordinates there exist generall}' n {n — 2') 
 relations, or equations of condition, because they are all determined by the 
 H versors 51 . . . qn-, and therefore by on numbers (compare § xvii.) ; 
 other proof of the general existence oi n(n — 2) equations of condition, or 
 equations between certain angles in the figure; each of the h(n+V)n 
 points of the figure is a common corner of w — 1 different triangles, re- 
 specting so many hinary multiplicatiotis ; at each point, n— 1 angles are 
 equal, and thus there are in all in (w+ 1) (n — 2) equations between an- 
 gles ; of these, re (re - 2) are true by construction (as above), and the re- 
 maining angular equations are true by the associative principle ; there 
 are therefore Jw (n — 1) (re — 2) equations of association, which are 
 consequences ofn(n — 2) equations of constkuction ; and the de- 
 pendent equations are more numerous than those on which they depend, 
 whenever the number n of the proposed factors exceeds three ; in the com- 
 plete construction oi a polygon of multiplication, with /) = re + 1 corners, 
 and ^p (/> — 3) inserted points (representing /jarie'a? products), is involved 
 (bj' the associative principle) the construction of a number of auxiliary 
 spherical polygons of inferior degree, expressed by the formula 
 
 M/>- ) {P- ) ■ • KP-P+ ) jj , j^g jjjg number of sides of the 
 1.2.3.. p ^ 
 
 auxiliary and inferior polygon ; this result is not to be confounded with the 
 
 elementary theorem of combinations, expressed by the same formula, . . 
 
 Articles 365 to 378 ; Pages 351 to 366. 
 
 § Lx. The /oca/ character, mentioned in § li., of the points e, f which represent 
 the two binary products rq, sr, in any case of ternary multiplication, srq, 
 namely, tha.t they are foci of a spherical conic inscribed in the quadrila- 
 teral ABCD, if A, B, c, D be the four points which represent the three fac- 
 tors, q, r, s, and their total or ternary product, may be denoted bj' the for- 
 mula, 
 
 EF (. .) ABCD, 
 
 which admits of various ti'ansformations ; in the complete construction of 
 the/)-sided polygon of multiplication, there arises a systein of such conies, 
 
XXXVl CONTENTS. 
 
 in number amounting to -^^p (jo — 1) (jo - 2) {p — 3), and inscribed in 
 so manj' quadrilaterals ; their /oci are the ^p ( p — 3) inserted points (of 
 § Lix.), which represent the partial products ; these points may therefore 
 be called the focal points of the polygon of multiplication ; and if they 
 be conceived to be the comers of a certain other polygon or polygons, 
 there -will exist, between these different polygons, a species of focal en- 
 chainment ; examples ; table oi fifteen focal relations, for the case of the 
 general hexagon of multiplication ; this hexagon is in this way connected 
 or enchained with a certain other hexagon, and also with a triangle on the 
 sphere, the nine corners of which auxiliary hexagon and triangle ai&foci 
 of a system of fifteen spherical conies, inscribed in fifteen spherical qua- 
 drilaterals of the completed figure ; geometrical and numerical illustra- 
 tions ; the general pentagon of multiplication abcde (of § lviii.) is in 
 an analogous way focally enchained with another pentagon figkh (or 
 with fghik), by a system of five conies, giving the five following focal 
 relations : 
 
 EG (. .) ABCI ; GH (. .) BCDK ; 
 hi (. .) CDEF ; IK (. .) DEAG ; KF (. .) EABH ; 
 
 each conic has its foci at two corners of the second spherical pentagon, 
 and touches two sides of the first ; elementary illustration, taken from the 
 limiting case where the pentagons become regular and plane, .... 
 
 Articles 379 to 393 ; Pages 366 to 380. 
 
 LECTURE VII. 
 
 ADDITION AND SUBTRACTION OF QUATERNIONS ; SEPARATION OF THE SCA- 
 LAR AND VECTOR PARTS; NOTATIONS S AND Y; DISTRIBUTIVE PRIN- 
 CIPLE OF MULTIPLICATION OF QUATERNIONS; NEW PROOF OP THE AS- 
 SOCIATIVE PRINCIPLE; GEOMETRICAL APPLICATIONS OF THESE PRIN- 
 CIPLES, INCLUDING SOME NEW GENERATIONS AND PROPERTIES OF THE 
 ELLIPSOID; NEW REPRESENTATIONS OF LOCI; CONNEXIONS OF QUA- 
 TERNIONS WITH CO-ORDINATES, DETERMINANTS, TRIGONOMETRY, LO- 
 GARITHMS, SERIES, LINEAR AND QUADRATIC EQUATIONS, DIFFEREN- 
 TIALS, AND CONTINUED FRACTIONS; INTRODUCTION OF THE BlQUATER- 
 NFON. 
 
 § Lxi. Recapitulation, Articles 394 to 400 ; Pages 381 to 386. 
 
 § LXii. Addition of a number to a line; interpretation of the symbol 1 + A; we 
 look out for some common operand, that is, for some one line such as i, on 
 which the two proposed summands, k and 1, can both operate separately 
 as factors, in ways already considered, so as to produce two separate re- 
 sults or partial products, which shall themselves be or denote lines, 
 namely, in this casej and i; we then add these two lines (§§ v., xix.), 
 so as to form a new line (i -\-j') ; finally we divide the sum by the common 
 operatid, and we take the quotient (i +j) -^ i, obtained by this division, 
 
CONTENTS. XXXVll 
 
 which qvotient is in general (see §§ vi., xx.) a quaternion, as the alne 
 of the proposed sum, 
 
 1 + A = (li + Ai) H- i = (i + j) -T- i ; 
 
 the effect of 1 + A, as a. factor, is to change the side of a horizontal square 
 to that diagonal of the same square which is more advanced than it in 
 azimuth by 45° ; 
 
 T(l+^) = 2* U(l+A)-A^ 1 + A = 2^A«; 
 
 this plan of interpretation of the symbol 1 + A is analogous to that em- 
 ployed in the calculus of finite differences for the interpretation of the sym- 
 bol 1 + A, in which also the tivo summands appear at first as heteroge- 
 neous, but are incorporated by being made to operate on one common 
 function fx; more elementary illustration of the process ; in general the 
 symbol w± p, where w denotes a scalar, and p a vector, can on the same 
 plan be interpreted as a quotient of two lines, and therefore as a quater- 
 nion, by taking some line a J_ p, and defining that w+ p = (wa + pa} -i- a, 
 when wa and pa are lines ; addition of this sort is a perfectly definite 
 operation, and has the commutative character, w \- p = p + w, . . . . 
 
 Articles 401 to 405 ; Pages 387 to 391. 
 
 § Lxm. Conversely, an arbitrary/ quaternion q can always be definitely decomposed 
 into two parts, such as w and p, of which one shall be a number and the 
 other a line, although it is possible that one of these parts may vanish ; if 
 q=: f3 -^ a, and if we decompose the dividend line /? by projection into , 
 two partial vectors, or summand lines, (S, jS', respectively parallel and 
 perpendicular to the divisor line a, and divide each part separately by 
 that line a, the partial quotients thus obtained will be respectively the 
 scalar part and the vector part of the total quotient or quaternion q ; in- 
 troducing then the letters S and V, as characteristic of the two operations 
 of TABJNG THE SCALAR and TAKING THE VECTOR of a quaternion, we 
 shall have S (w-^ p) = w, Y {w+ p) = p, and S (/3-f- a) =/3'-^ a, 
 V(/3-^a)=/3"-^a, if j8 = ^'+ /3", /3'|| p, /3" J.p ; q = ^q+Yq=Yq 
 + S^, 1 = S + V=V+S; also (compare § XVI.), S2 = S, SV = VS=0, 
 V2 = V; thus, Sw = w, Sp = 0, Vw = 0, Yp = p; conjugate quaternions 
 have equal scalars but opposite vectors, SKq = +Sq, YKq = —Yq, 
 SK=S, VK = -V; K{w + p)=w-p (§ xxiii.); Kq = Sq-Yq, 
 K=S-V; TK = T (§ xxxiv.), T (w + p)=T (m;-p) = (M;2_p2)| 
 (§ xxu.) ; if a? be a scalar, Yx = 0, then S . xq = xSq, Y. xq = xYq ; 
 for example, 
 
 S(-9)=-S?, Y{-q) = -Yq; 
 
 S(-K5) = -Sg, V(-K<7) = +V?, -K=V-S; 
 
 x{w + p) = xw + xp; STg^ + Tg-, YTq=0; 
 
 S? = Tg . SU?, Yq^Tq. YVq ; YVq = VYq . TYVq ; 
 
 VYq = Ax.q, (UV<7)2 = -1, UV? = V-1: 
 
 quaternions are connected with trigonometry, by the relations, 
 
 SU? = cos L q, TVU? = sin z <? ; 
 
XXXVlll CONTENTS. 
 
 these reproduce the following general expression of well-known _/%r/n, as 
 representing in this system the versor of a quaternion, 
 
 \Jq = SUg' +Y\Jq = cos Lq + \/ -^ sin Z 5 5 
 
 but the symbol V - 1 here denotes (compare § xxiii.) the particular vec- 
 tor-unit which is drawn in the direction of UVg' or of Ax . q, that is, in 
 the direction of the axis of the versor ; the indetermination mentioned in 
 the Fourth Lecture (§ xxxv.) thus disappearing, when \]q is a determined 
 versor, Articles 406 to 411 ; Pages 391 to 397. 
 
 § LXiv. Expressions for geometrical loci, supplied by the symbols S and V ; 
 the scalar of a quaternion is positive, null or negative, according as the 
 angle of the quaternion is acute, right, or obtuse ; S (p -j- a) = S . pa i = 0, 
 
 <r'7V '■ 
 
 according as ap = - , if the symbol ap here denote the aiigle hetween the 
 
 directions of the two lines a, p, and therefore the angle of their quotient, 
 regarded as a quaternio7i (but not the angle of that other quaternion which 
 is their product') ; to write the equatipn S (p -7- a) = 0, or S . pa " 1 = 0, is 
 therefore to express, by the notations of this calculus, that the line p is per- 
 pendicular to the line a, and consequently that the locus of the point p is 
 a PLANE through the origin O, perpendicular to the given line OA, if 
 a = OA, p = OP ; if also /3 = ob, the equation S . (p — /3) a-i = expresses 
 the perpendicularity p — (3 A-a, and gives, as the locus of p, a plane 
 through B, perpendicular to OA, or parallel to the former plane ; such a 
 parallel plane may also be denoted by the equation S. pa"i = a, where 
 the scalar a is such that aa denotes the constant projection p = op' of the 
 variable vector p on the fixed vector a ; the equation S . ap - 1 = 1 ex- 
 presses that the projection of a on p is the line p itself, or that the angle 
 OPA is right ; it gives, therefore, as the locus of p, a sphere tvith oa for 
 diameter ; the same spheric surface may also be denoted by either of the 
 equations, 
 
 S.(a-p)p-i = 0, T|^p--J = iTa; 
 
 methods of transforming, by calculation, any one of these eqiii-significant 
 forms into any other, will be explained at a later stage (in § Lxxvi.) ; 
 more generally the two equations, 
 
 T{p-i(a + /3)} = T{K«-/3)}, S^=0> 
 
 each represent a sphere described on ab as diameter, - 
 
 Articles 412 to 415 ; Pages 397 to 402. 
 
 § LXV. The system of the two equations S . pa-i = 1, S . (3p-'^= 1, represents a 
 CIRCLE, namely, the mutual intersection of the plane through A, perpen- 
 dicular to OA, and the sphere on OB, as diameter ; the product of the 
 same two equations, namely, the equation S . pa-i . S . /3p~i = 1, re- 
 presents a CONE, with the last described circle for its base; if this last 
 
CONTENTS. XXXIX 
 
 equation be combined with the equation of a new plane, S . py-^ = 1, the 
 resulting system represents a plane conic, considered as a curve in 
 space ; the equation of the cone may also be thus written, 
 
 under this form it gives the subconteaey circular section of the cone, 
 namely, as the intersection of the sphere described on a"i as diameter, 
 with the plane S . jO/3 = 1 ; the, parallel plane through the vertex, S . pjS = 0, 
 touches the former sphere S . jSp - 1 = 1, which contained \he, former circular 
 base; this latter plane, and the plane S.pa = 0, are the two cyclic 
 planes of the cone ; the equations of these two planes may also be thus 
 written, S . /8p = 0, S . ap = ; for in general (by §§ xv., lxiii.), S . pa = 
 SK . pa = S . ap ; thus, in taking the scalar of the product of any two vec- 
 tors, we are allowed to alter their order ; more generally it will be found 
 (see § Lxxxix.), that tinder the sign S we may alter cyclically the 
 ORDER of any number of factors, even if those factors be quaternions ; a 
 SPHERICAL CONIC may be expressed by combining either of the two forms 
 above assigned for the equation of the cone with any one of the three fol- 
 lowing forms for the equation of the concenteic sphere, 
 
 Tp = c, p2+c2 = 0, S^-^^ = 0; 
 
 y is here the vector of some one point upon the sphere, and c is the length 
 of the radius ; we might also represent the same concentric sphere by the 
 equation Tp = Ty, or p2 = y2 ; one cy'Clic arc may be represented by the 
 two equations S . ap = 0, Tp = c, and the other cyclic arc by the equa- 
 tions, S . /3p = 0, Tp = c, . . . . Articles 416 to 421 ; Pages 402 to 407. 
 
 § Lxvi. If a given sphere with a for radius have its centre at the origin o, and if 
 we conceive t to be a sought point of contact of the sphere with a rectili- 
 near tangent from a given external point s, and make cr = os, r = ot, 
 we shall have the two equations T^ = -a'^, S . irr- 1 = 1, the first denoting 
 the given sphere round o, and the second an auxiliary sphere on os ; the 
 POLAR PLANE of the point s, or the plane of which s is the pole, with re- 
 spect to the given sphere, is the plane of the circle of intersection of the two 
 spheres, and its equation (obtained by suitably multiplying their equa- 
 tions) is S . ffr =- a2, or S. 7-jU-i= 1, ifwe make /x = OM = — a^ff-i ; ris 
 here treated as a variable vector, but rr and fj, as tixed vectors ; JJfi = Uc, 
 T;u = a2T(7-i ; M is the centre of the circle of contact of the given sphere 
 with the ENVELOPING CONE of tangents drawn from S ; if p = op be the 
 variable vector of a point p upon this cone, then 
 
 {(S.<T(p-<7)}2 = (ff2+a2)(p_^)2; 
 
 but a simpler form of the equation of the enveloping cone will be assigned 
 afterwards (in § lxxvii.) ; the cone which cuts this enveloping cone per- 
 pendicularly along the above-mentioned circle of contact, and has its ver- 
 
Xl CONTENTS, 
 
 tex at the centre of the given sphere, is (S . ffpY + a^p^ = ; the equation 
 S . (TjO = — a2 expresses that the points p and s are conjugate points, 
 with respect to tlie given sphere ; the equations S . per = — a^, S . pu' = — a^, 
 represent jointly a EIGHT LINE, which is the polar of the line ss' ; the 
 continued equation, 
 
 S.pcr= S.p(T' = S.p'(T = S. pV =-a2, 
 
 expresses that the two lines pp', ss', are reciprocal polars of each other, 
 with reference to the same given sphere as before ; in general, for any two 
 vectors p and cr, 
 
 S . piT = Tp T(T cos (7r - pa) ; 
 
 the scalar of the product of any two lines is equal to the rectangle under 
 the lines, multiplied by the cosine of the supplement of the angle between 
 
 A 
 
 their directions ; Z..p(T = 7r — per =7r — Z..p<T"i; 
 
 SU. pcr-i = + cos p(7, SU. per =— cos per; 
 
 this supplementary relation between the angles of the product and quo- 
 tient of two lines (compare § Lxiv.), is one which it is important to re- 
 memher in this calculus, from the principles of which it was deduced so 
 early as in § xv. ; it may also be considered as connected with the negative 
 character of the square of a vector (§ xiii.), since /3a = a^.jSa ~i = — T 
 a-.j8a"^, U.j8a = — U. jSa'i, and the angle of the negative of a quater- 
 nion is the supplement (by § xxxvii.) of the angle of the quaternion itself; 
 if (S be (as in § lxiii.) the projection of ^ on a, then S.(3a = (3'a = a/3', 
 and this scalar product (see again § xiii.) is positive or null or negative, 
 according as the angle between a and jS is obtuse, or right, or acute (con- 
 trast again § lxiv.) ; the projection (5' may be expressed in terms of /3 and 
 
 a, by writing /3'=a-i S. j8a, or /3' = a S./3a-i, 
 
 Articles 422 to 426; Pages 407 to 416. 
 
 § Lxvii. Vector of the product of two lines «, /? ; if /3" denote (as in § lxiii.) the 
 component of /3 which is perpendicular to a, then V,/3a =j8"a = a line 
 perpendicular to the plane of the two given factors a, (3 ; V. /3« j_ a, V. 
 /3a _l_ (8 ; the rotation round this vector of the product, from the multiplier 
 line /3, towards the multiplicand line a, is positive ; whereas the positive 
 rotation round the vector of the quotient /3 -f- a, or (ia-'^, is directed from 
 a towards /3 ; UV./3a =— UV./3a"i ; the length of the vector of the pro- 
 duct of two adjacent sides of a parallelogram represents the area of that 
 parallelogram, 
 
 TV.j8a= /I7AOB = T/3Tasm/3a; 
 
 A 
 
 TVU . /3a = sin jSa (compare § lxiii.) ; Y.a[3 = -Y.[3a,the vector of the 
 product of two lines changes sign (or direction) tvhen the two factors are 
 interchanged (whereas, by § lxv., S.a/3 = + S./3a) ; the perpendicular 
 component j8" may be expressed in any one of the following ways, 
 
 (8" = V.jSa-f-a=-a-iV./3a = a-iV.a/3 
 = V./3a-ixa = -aV./3a-i = a V.a-i /3 ; 
 
CONTENTS. xli 
 
 new proof (compare § l.) that when ya = a/3, then y is the reflexion 
 of the line j3 with respect to a ; the equation V. pa = V . jSa, or V . (p — /3) 
 rt = 0, expresses that the termination p of p is situated on the right line 
 through n, which is parallel to a, or to OA ; the same rectilinear locus 
 of p may be expressed by writing p = /3 + xa, where x denotes a variable 
 scalar ; the equation V . pa = denotes the indefinite right line through the 
 origin o, of which the given line OA is a part; Y .pa = Y.al3 denotes 
 another indefinite right line, parallel to the line oa, and passing through 
 a point c, which is the reflexion of the point b with respect to the line OA; 
 the equation V (p V . /3a) =0, or V . p V . /3a = 0, expresses that p is per- 
 pendicular to the plane AOB of a and j8 ; whereas the equation S . p V . /3a 
 = (afterwards abridged, see § lxxxvi., to the form S . p/3a = 0), expresses 
 that the three lines a, /3, p, are coplanar, and gives therefore a plane as 
 the locus of p ; the equation, 
 
 (V. pa) 2= (V. /3a) 2, or TV. pa = TV. /3a, 
 
 denotes a cylinder of revolution, with a for axis^ and T ^' iov radius ; 
 in like manner the equation (V.p/3 -1)2 +62- o, or TV.p/3-i = 6, repre- 
 sents another cylinder of revolution, with /3 for axis, and 6T|8 for radius, 
 
 Articles 427 to 431 ; Pages 416 to 423. 
 
 § Lxviii. If we cut the last cylinder by the perpendicular plane S.p/3-i =a, the 
 section is a circle, contained on the sphere Tp = (a? + 62) j T/3 ; the sphere 
 round origin with radius T/3, namely, the sphere for which Tp = T/3, or 
 T.p/3"i = l, may have its equation thus transformed, (S.p/3- 1)2 — (V. 
 p/3-i)2=l, and maybe regarded as the loeus of a varying circle, for 
 which S .p/3-1 = a:, TV . pj8"i = (1 — a?2)i; the first of these two equations 
 of the circle represents here a varying plane, and the second represents a 
 varying cylinder of revolution ; if a be inclined to |8, the cylinder TV . 
 pj3-'^ = b is cut obliquely by the plane S . pa - 1 = o in an ellipse ; in like 
 manner the equations, S.pa-^=x, TV.p/3-i = (1 — a'2)J, represent a va- 
 rying ellipse, of which the locus (obtained by elimination of .r) is an 
 ellipsoid, represented by the equation, .... 
 
 (S.pa-i)2-(V.p/3-i)2-l; 
 
 geometrical illustration of this mode of generating an ellipsoid by a cer- 
 tain deformation of a sphere (^ellipses being substituted for circles, by sub- 
 stituting oblique for perpendicidar sections of a certain varying cylinder^ ; 
 the ellipsoid is enveloped by the cylinder of revolution, whose equation 
 is (V.p/3~i)2 = — 1 ; the plane of the ellipse of contact is S.pa-i = 0; 
 the equation of the ellipsoid may also be thus written, (S. pa -1)2+ (TV . 
 p/3-i)2 = 1 ; or thus, T (S . pa-i + V . p/3-i) = 1 ; this last form will be 
 found to furnish (in §§ lxxviii., &c.) a 7iew mode of generating the ellip- 
 soid (or rather a number of such new modes), 
 
 Articles 432 to 436 ; Pages 423 to 430. 
 
 § Lxix. Analogous deformations of other surfaces of revolution ; the locus of the 
 varying circle, S . p/3-i = x, TV . p/3-i = (a-2 - 1)*, is an equilateral 
 
 f 
 
xlii CONTENTS. 
 
 AND DOUBLE-SHEETED HYPERBOLOID OF REVOLUTION, whoSe equation is 
 
 (S . pj8~^)^ + (V . pp'^y= 1 ; the locus of the connected and varying el- 
 lipse, S . pa-i = ar, TV . jO|8-i = (ar^ — l)i, where a is still supposed to be 
 inclined to (3, is another double- sheeted hyperholoid, which is not one of 
 revolution, and which has for its equation the following, 
 
 CS.pa-ip+(V.p/3-i)2=l; 
 
 geometrical illustrations : the right and oblique cones, which are respec- 
 tively ASYMPTOTIC to thesc two hyperboloids, have their equations formed 
 by changing 1 to in the second members of the equations of those two 
 surfaces ; by changing 1 to — 1 in the same second members, we get the 
 equations of two single-sheeted hyperboloids, T\dth the same asymp- 
 totic cones, of which two hyperboloids the first is equilateral and of revo- 
 lution, while the second touches the ellipsoid of § Lxviii. along the ellipse 
 of contact mentioned in that section, namely, the ellipse whose equations 
 are, 
 
 S.pa-i = 0, TV.|0/3-i = l; 
 
 the second of the two double-sheeted hyperboloids touches the same ellip- 
 soid at the extremities of the two opposite vectors which have the directions 
 of + jS, the common tajigent planes at those two points being given by the 
 formula S . pa-i =+ 1 ; the equations, 
 
 S . |0/3-i + (V . ,0/3-1)2= 0, S . pa-i + (V . p;8-i)2=: 0, 
 
 represent two elliptic paraboloids, Avhereof the first is a surface of re- 
 volution ; the equation S.pa-iS.pj8-i = S.py-i represents an hyperbo- 
 lic paraboloid ; an arbitrary surface of revolution may be 
 represented by the formula, TV. p/3-i=/(S . p/^-i), and then the con- 
 nected equation, TV • p/3-i =/(S . pa-^) will represent the result of a cer- 
 tain deformation of that surface, whereby ellipses are still substituted 
 for circles ; but if a be supposed to be not inclined to /3, but only to be 
 longer or shorter, the results of all the foregoing deformations will them- 
 selves be surfaces of revolution, . . . Articles 437 to 440 ; Pages 430 to 435. 
 
 § lxx. Mac CuUagh's modular generation of surfaces of the second order, ex- 
 pressed in the language of quaternions ; origin being on a directrix, a being 
 vector o? a focus, f3 vector of another point of directrix, and y perpendicular 
 • to a directive plane, the following equation may be established, T (p — a) = 
 T (pS . y/3 - /3S . yp) ; it will be found (see § xci.) that this equation ad- 
 mits of being put under the form 
 
 T(p-a)=TV.yV./3p, . . . 
 
 Article 441 ; Pages 485 to 437. 
 
 § LXXI. The symbol V (V . a/3 . V . yd) denotes a line situated in the intersection 
 of the tivo planes of a, (3, and of y, ^ ; if there be si.v diverging vectors a, ■ 
 «',... a", and if we form from them three others, /3, /3', /3", by the 
 formulije, 
 
CONTENTS. xliii 
 
 /3 = V (V . aa' . V . a'a^"), 
 /3'=V(V.a'a".V.a'''a^), 
 |8"=V(V.aV". V.a'a), 
 
 then the equation, = S . j8 /8' /3", expresses the condition for the six diverg- 
 ing lines, a, a\ . . . a^, being six sides of one common cone of the second 
 degree, and may therefore be called the equation op homoconicism ; the 
 scalar function S . jS /3' j6" may be called the Aconic Fukction of the six 
 vectors a . . a", or of the hexagon (plane or gauche) at whose corners 
 they terminate, because it vanishes when they are homocotiic, by a form 
 of the theorem of Pascal ; hence may be derived an expression by quater- 
 nions, for what may be called the Adeuteeic Function of ten vectors, 
 a, a, . . . a^^, or of the (generally gauche) decagon at whose corners 
 they terminate, because this function vanishes, when those ten points are 
 on one COMMON deuteric surface, or common surface of the second 
 order ; the Adeuteric may be thus expressed, 
 
 S (+ ABCDEF. GHIK), 
 
 if A ... K be the ten points, while the symbol abcdef here denotes the 
 aconic function of six of them, with respect to any eleventh point o arbi- 
 trarily taken as an origin, and ghik denotes the pyramidal function of the 
 other four, that is, the sextupled volume of the pyramid of which they are 
 the comers, taken with a proper algebraic sign ; in symbols, this pyramidal 
 function of four points, g, h, i, k, or of four vectors, a", a*", a'''", a'" may 
 be expressed by quaternions as follows : 
 
 S.(ai''-a") (a""- a'') {a"'^^ - a"') (compare § lxxxix.) ; 
 
 the ten points are supposed to be combined in all possible ways, as groups 
 of four and six (namely in 210 ways), by successive mutual interchanges 
 of points or of letters between the two groups ; for every such binary inter- 
 change the sign + prefixed to the product varies ; this formation of the 
 adeuteric function is only alluded to in the text of the Lectvu-e, . . . 
 
 Article 442 ; Pages 437 to 439. 
 
 § LXXII. The general addition of any two quaternions can always be easily and 
 definitely effected by the rule of the common operand, or by the formula 
 (y -7- a) + ((8 -7- a) = (y + /3) -h a ; subtraction of quaternions may in like 
 manner be effected by the formula (y -j- a) — (^ -f- a) = (y — /3) -f- a ; 
 
 Articles 443 to 447 ; Pages 439 to 444. 
 
 § Lxxiii. Properties of such addition ; it is a commutative and associative opera- 
 tion ; the scalar, vector, and conjugate of a sum of quaternions are respec- 
 tively the sums of the scalars, vectors, and conjugates, SS=SS, VS = 2V, 
 K2 = SK ; similarly for differences, S A = AS, VA = AV, KA = AK ; it is 
 useful to be familiar with the two following general expressions, for the 
 scalar and vector parts of the product of any two vectors, S . <x/3 = ^ (aj8 + 
 iSa), V . a/3= 1 (a^ - /3a), .... Articles 448, 449 ; Pages 444 to 447. 
 
 § Lxxiv. The general quadrinojiial form, q — u}-\- ix +jy + kz, for a- quater- 
 
xliv CONTENTS. 
 
 nion, may now be more fully understood ; q — w + ix! +jy' + kz being 
 another quadrinomial of the same sort, the sum and difference of these two 
 quaternions are formed by taking the sums and differences of their consti- 
 T CENTS, w, a-, y, z and w, x, y\ z ; in symbols, (^ ±q = w' ±w-\-i (js ±x) 
 +j (y ± y) + '' i^' ± *) ; a^ qnaternion cannot vanish, except by its four con- 
 stituents separately vanishing ; nor can two quaternions become equal, 
 without their constituents becoming separately equal ; a?i equation q' = q 
 between tivo quaternions includes thus a system of four equations be- 
 tween scalars ; namely, w=w^x=x,y' = y,z—z, 
 
 Article 450 ; Pages 447 to 449. 
 
 § Lxxv. General proof of the disteibutive pkinciple of multiplication of 
 
 quaternions ; "Er .'2q — "2 .rq; , . . Articles 451 to 455 ; Pages 449 to 455. 
 
 § Lxxvi. Elementai-y applications of the distributive principle ; transformations 
 by means of it, referred to in § lxiv. ; the equation or identity, 
 
 (a-jS)2=a2-2S.a/3i-,82, 
 
 is equivalent to \hQ fundamental formula of plane trigonometry, or to the 
 equation, 
 
 BA2 = CA^ - 2CA . CB . cos ACB + CB^ ; 
 
 centre of mean distances, or of gravity, n—'2.aa-~-^a\ investigation of 
 the (spherical) locus of the vertex of a triangle, of which the base and the 
 ratio of the sides are given ; T (ct — My) = T (wff — -y ), if To- = Ty, . . . 
 
 Articles 456 to 459 ; Pages 455 to 460. 
 
 § Lxxvii. Intersections of right line and sphere ; the locus of all the tangents to 
 the sphere p'i-]-c'^ = 0, which can be drawn from the extremity of /3, has 
 for equation, c2 (p — j6)2 = (V . /3|o)2 ; this form of the equation of the en- 
 veloping cone is simpler than that which was obtained in § lxyi., but the 
 one can be transformed into the other ; new investigation of the equation 
 of the polar plane, S. I3p= — c^ (compare again § lxa'i.); proof by qua- 
 ternions, of the known harmonic property of this plane ; harmonic mean 
 betv^'een any two vectors ; fourth harmonical to any three points (not 
 necessarily on one straight line) ; extension hereby given to the usual no- 
 tion of harmonic conjugates ; circular harmonic group (four points on a 
 circle, for which what is called the anharmonic quotient becomes unity') ; 
 interpretations of the sum and difference of the reciprocals of any tivo 
 rectors, Articles 460 to 464; Pages 460 to 466. 
 
 § lxxv III. Equation of ellipsoid resumed (from § lxviii.), and transformed to 
 
 T (tp + pk) = /c2 - 1« ; 
 
 geometrical equality hence deduced, 
 
 generation of the ellipsoid, hence derived ; if Abe a. superficial point 
 of a. fixed sphere with centre c, and b an external point, and if a secant 
 bdd' be»drawn, and on the guide-chord ad, or on that chord cither way 
 
CONTENTS. xlv 
 
 prolonged, a portion ae be taken, which in length is equal to bd', the lo- 
 cus of the point E ivill he an ellipsoid, with A for its centre, and b for a 
 point of its surface ; abc in this construction may be called the generat- 
 ing TRIANGLE, and the sphere round c the diacentkio sphere ; the 
 points D and d' on that sphere may be said to be conjugate guide-points ; 
 geometrical deductions from the formula, ae = bd' ; constractions for the 
 lengths and directions of the three principal semi-axes of the ellipsoid, a, 
 b, c ; expressions for the lengths of the sides of the generating triangle, 
 
 BC = ^ (a + c), CA = 1 (a — c), Ab = acb-^ ; 
 
 enveloping cylinder of revolution, with the side ab for axis, andBG=& 
 for radiixs, if g be the second point of intersection of ab with the diacentric 
 sphere ; the two other sides, bc, ca, of the triangle are perpendicular to 
 the two cyclic planes of the ellipsoid; the one that is _Lk, or4_ ca, 
 touches the diacentric sphere at A ; these planes are also shewn by this 
 construction to be (as is known) the cyclic planes of all the concentric 
 cones, that rest on those spherical conics in which the ellipsoid is cut 
 by a system of concentric spheres ; mean sphere, containing the two dia- 
 metral and circular sections ; the construction exhibits also geometrically 
 the known mutual rectangular ity of the semi-axes aei, aEo of any other 
 diametral section of the ellipsoid, and conducts easily to the known ex- 
 pression for the difference of the squares of their reciprocals, namely, 
 
 aeo-s — AEi-2= (c-2_a-2) gin J, sin v, 
 
 where v and v' are the inclinations of the cutting plane to the two cyclic 
 planes ; the equations of these latter planes are, respectively, S . ip = 0, 
 S . K(0 = ; the equation of the mean sphere is 
 
 Tp =: 6 = (k2 - i2)T (t - k)-1; 
 a=Tt4-TK, c = Tt-TK:, ac=K3-i2, ac6-i = T(t-K) ; 
 
 equations of a spherical conic on the ellipsoid ; expressions for the two new 
 
 vectors, t, k, as functions of the vectors, a, (3, of § lxviii., 
 
 Articles 465 to 470 ; Pages 466 to 475. 
 
 § Lxxix. Introduction of two new vectors, X, ju, with two new scalars, h, K, and 
 two new points, l, m, which all depend upon and vary with the vector p, 
 or the point e, and satisfy the equations, 
 
 X = (/cp + pK) (k — t)-i = A (i — k) = al = A . ab, 
 H = (tp + pi) (i — k)-i = h' (^K — i) = AM = h' .BA; 
 
 to each given value of h (between certain limits) answers a circle on the 
 ellipsoid, for which 
 
 S.Kp = iAT(t-K)2,LE = T(p-X) = 6; 
 
 in like manner, to each given value of h' (suitably limited) there answers 
 another circle on the ellipsoid, determined by the equations, 
 
 S.(p = ^AT(i-jc)2, ME=T(p-jU.) = 6; 
 
Xlvi CONTENTS. 
 
 these two suhcontrary and circular sections of the ellipsoid have their 
 planes perpendicular to the sides, CA, CB of the generating triangle 
 (§ Lxxviii.), and therefore /jaraZ/e/ (as is known) to the two cyclic planes; 
 everj' such pair of subcontrary circles (h, h) is contained (as by known 
 results it ought to be) on one common sphere ; this sphere, in these calcu- 
 lations, is given by the formula, 
 
 T (p - 5) = NE = K, 
 
 where the vector ^, the positive scalar n, and the point n, may be deter- 
 mined by the equations, 
 
 AN = f = At+A'K, b^-n'2={h+K) (/u2+A'k2); 
 
 and if we make en = ^— p = b'^v, then n is the foot of the normal to the 
 elhpsoid drawn at the point e, and terminated by the plane of the gene- 
 rating triangle, or by the plane of the greatest and least axes, while n de- 
 notes the length of that normal ; the new vector v is parallel to the normal, 
 and satisfies the equation S . vp=l ; its expression as a function of p is, 
 
 i, = (k2-i2)-2 {(t-K)2p + 2t. S./cp + 2KS.tp}; 
 
 the equation of the ellipsoid may be put under the form, p^-\- b' = \fi, 
 while that of the mean sphere may be thus written, p^+b^ = 0, . . . 
 
 Articles 471 to 474 ; Pages 476 to 479. 
 
 § Lxxx. If we make for abridgment v = (p (p), or simply v = (pp, the vector Junc- 
 tion (j) will be linear or distributive, 
 
 and if we agree to write/(p, -sj-) = S . p(p'ST, the scalar function f viiW be 
 at once commutative or symmetric with respect to the two vectors on which 
 it depends, and linear or distributive relatively to each of them, so that 
 /(w> p) =/(p) '^)./ iP + p', -ra- + ^') =/(p. '^) +f(p, ^') +f(p', '^) +f 
 (P) ^')) f{.^Pi y^^ = ^yfipi '^) i if then we farther abridge /(p, p) to/ 
 (p) or to fp, this neio scalar function of one vector will, relatively to it, be 
 of the second dimension, and we shall have 
 
 /(p+p')=/p + 2/(p, p')+/p',/(^-p) = ^Yp ; 
 
 the equation of the ellipsoid reduces itself in this notation to the formula, 
 fp = l; and if a cylinder (not generally of revolution) be circumscribed 
 about the ellipsoid, with its generating lines parallel to a given vector ot, 
 the equation y(p, 'ct)= represents ihe diametral plane of contact, and 
 the normal to that plane has the direction of the vector (p-sr ; va. general 
 the last equation denotes that the directions of p and •tjj are conjugate, re- 
 latively to the ellipsoid; reciprocal relations of bisection, conjugation of 
 line and plane, system of three conjugate semi-diameters, equation ar2 + y2 
 + 22=1, Articles 475 to 480 ; Pages 480 to 485. 
 
 § Lxxxi. Theequation/(p,OT) = l,or S. VOT =1, expresses that the vector ot ter- 
 minates on the tangent plane to the ellipsoid, drawn at the extremity of the 
 
CONTENTS. xlvii 
 
 semi-diameter p ; the vector v, or 0p, may be called the vectok of proxi- 
 mity, namely, of the tangent plane to the centre, because its reciprocal 
 V - 1 represents in length and in direction the perpendicular let fall from 
 that centre on that plane ; in general the formula / (p, ot) = 1 may be said 
 to be the equation of conjugation between the two vectors p and v:, be- 
 cause it expresses that they terminate m two conjugate points ; the same 
 equation represents the polar plane of either of those two points, when the 
 other is treated as variable ; if ot be treated as the vector of the vertex of 
 an enveloping cone^ the equation of that cone is 
 
 {/(p, w)-lp = (/p-l)aw-l): 
 
 when the vertex goes off to infinity, there results an enveloping cylinder, 
 Aviththe equation/ (p, ztj)^= (/p — 1)/^ ; verifications for th e case oft 
 sphere, for which k = 0, (pp — i-~p; general harmonic property of the polar 
 plane, Articles 481 to 486; Pages 485 to 491. 
 
 § Lxxxii. The triangles lmn, abc, are similar and similarly situated in one com- 
 mon plane ; the points b, d, e, l are concircular ; the triangle lem is isos- 
 celes ; the lines ln, mn are portions of the axes of the two circles on the 
 ellipsoid which pass through the point e, . Articles 487, 488 ; Pages 491, 492. 
 
 § LXXXIII. New proof of the associative principle of multiplication of quater- 
 nions, derived from the distributive principle ; importance of combining 
 these two principles, Articles 489, 490 ; Pages 493 to 495. 
 
 § Lxxxiv. Transformed equation of the ellipsoid, 
 
 T (t'p + pw') = k'2 — 1'2 ; ik'—lk = T.ik; 
 new generating triangle ab'c', and new diacentric sphere round c', touch- 
 ing at A the cyclic plane a_ i (compare § lxxviii.) ; ab' is the axis of 
 asecond enveloping cylinder of revolution ; if we make (compare § lxxix.), 
 
 al' = X'=2 (k'-i')-iS . k'p, Am= fi' = 2(i - /c')-iS. t'p, 
 
 the two new triangles, l'm'n and ab'c' are similar and similarly situated in 
 one common plane, namely, in the principal plane oi the ellipsoid; the 
 symbols V-i 0, S-iQ, denote respectively a scalar and a vector; when 
 three points are collinear, the vector part of the quotient of the differences 
 vanishes and conversely ; lmbi'l' is a quadrilateral in a circle, whereof the 
 diagonals lm', ml' intersect in n, that is (§ LXXix.), in ihQ foot of the 
 normal to the ellipsoid ; genekation of a system of two recipeocal 
 ellipsoids, by means of a moving sphere ; generation of the same sys- 
 tem of two ellipsoids by means of a fixed sphere ; if the sides of a plane 
 quadrilateral inscribed in the fixed sphere move parallel to four fixed 
 lines, one pair of opposite sides will intersect in a point on one ellipsoid, 
 and the other pair of opposite sides will intersect in the corresponding 
 point on the other or reciprocal ellipsoid ; these two ellipsoids have one 
 common mean sphere, namely, the fixed sphere employed in the construc- 
 tion ; other geometrical relations of the fixed sphere and lines to the two 
 ellipsoids thus generated, .... Articles 491 to 495 ; Pages 495 to 502. 
 
xlviii CONTENTS. 
 
 § Lxxxv. Generation of an ellipsoid by means of a pair of sliding spheres ; 
 if two equal spheres slide within two cylinders of revolution, whose axes 
 intersect each other, in such a manner that the right line joining their cen- 
 tres moves parallel to a fixed line, the locus of their circle of intersection 
 is an ellipsoid, inscribed at once in both the cylinders ; the same ellipsoid 
 may also be generated as the locus of the circular intersection of another 
 pair of sliding spheres, inscribed within the same two cylinders, but with 
 their line of centres parallel to a different straight line ; the diameter of 
 each sliding sphere is equal to the mean axis 26 of the ellipsoid ; an arbi- 
 trary curve on the surface of the ellipsoid may he described by the vertex 
 iL of an isosceles triangle lem' (or l'em), the common length of whose 
 two sides el,, em' (or el', em) is constant, and — b, while its base lm' (or 
 l'm) moves parallel to a given line ac (or Ac'), and is inscribed in a given 
 angle bab'; or a rhombus of constant perimeter, =46, maybe employed to 
 generate, in an analogous way, by the motions of two opposite corners, two 
 curves on the ellipsoid, Article 496 ; Pages 502, 503. 
 
 § LXXXVi. Introduction of two new fixed vectors, j; =:Tt U (t— /c), 0=Tk U 
 (i'— k') ; making </ = — /t' T (1 — ki"^), we have fi=gri, X=g9, and the 
 equations of one pair of sliding spheres become 
 
 T{p-gr,) = T{p-ge) = b; 
 
 for any one value of the variable scalar g, the plane of the circle of inter- 
 section is represented by the equation, 
 
 ^(02_,,2) = 2S.(0-,,)p, 
 
 and we have the value, >/ - = 6 Ui ; elimination of g gives for the ellip- 
 soid, regarded as the locus of these circles, the transformed equation, 
 
 TV-?p4-=02_,2,or,TV.'"'-P^- ^'-"' - 
 
 V(r,-e) " ' r]-e T(,,-0)' 
 
 other mode of obtaining this last equation from the form in § lxxviii., 
 namely, T (ip + ps) = k^ - 1^ ; in general, for any three vectors a, j3, y, we 
 have the identities, 
 
 S.a/3y = -S.y/3«, V . a/3y = + V . y/3rt, 
 
 with analogous results (compare §§ liii., lxiii.) for the scalar and vector 
 of the product of any odd mimber of vectors; we have also, generally, 
 
 S.yV.j8a=S.y/3a, S.yYq = S.yq; 
 
 a fraction in this calculus may generally be transformed (as in Algebra)^ 
 by dividing both numerator and denominator by any common vector or 
 quaternion distinct from zero ; or, in other words, by midtiplying each into 
 (but not generally 6y) the reciprocal of any such vector or quaternion, , 
 
 Articles 497 to 500 ; Pages 503 to 509. 
 
 § Lxxxvii. Geometrical significations of the two new fixed vectors, rj, 0; i] + 
 = w is the vector of an umbilic of the ellipsoid, and the equation of the 
 
CONTENTS. xHx 
 
 tangent plane at that umbilic (found by making p = 2) is S . (0 — jj) p = 
 02— }j2 ; the umhilicar normal there has the direction of r] — 9, or of the 
 cyclic normal i; 0-i — >j-i has the direction of the other cyclic normal k; 
 
 «=T>}U(»;-6)), K = T9U(0-i-jj-i); 
 a = Tjj + T0, 6 = T (»/ - 0), c= Tjj -T0 ; 
 
 the sum and difference U») + U0 are respectively equal to U (i — k) + U 
 (t'— k'), and have the directions of the greatest and least axes of the ellip- 
 soid ; the length of an umbilicar vector, or umbilicar semi-diameter of the 
 ellipsoid, is 
 
 M=Tw=T(>; + 0) = V(o2-62^.c2); 
 
 the length of the perpendicular from the centre on the umbilicar tangent 
 plane is 
 
 ;>=(02_^2)T(j;-0)-i = ac6-i; 
 
 these values of u and p agree with known results ; another umbilicar vec- 
 tor is 
 
 w' = T9j U0 + T0 U?; = - T . JJ0 .(»;-! + 0-1) ; 
 
 — w, — w' are also umbilicar vectors; thus ?j-i + 0-' has the direction of 
 
 such a vector ; 
 
 w + w' = (Tq + T6)) (Uq + U0), 
 
 w - w' = (T»7 - T0) (Uj; - U0), 
 
 the angles between the umbilicar diameters are seen to be bisected by the 
 greatest and least axes, Articles 601 to 503 ; Pages 509 to 511. 
 
 § Lxxxviii. For the square of any quaternion we have the following scalar, vec- 
 tor, and tensor, 
 
 S.q^ = Sq^ + Yq^,Y.q'^ = 2\qSq,T.q^ = Sq^-Yq^; 
 
 hence for the scalar of the square root of any other quaternion q we have 
 the expression, 
 
 SV5' = VaS?' + iTg'); 
 
 this is only one out of a vast number of general transformations, in which 
 the present calculus abounds, and which may be deduced from the 
 laws of the symbols S, T, U, V, K ; applied to the ellipsoid, in combination 
 with the recent values for a, h, c, it enables us to infer that the linear ec- 
 centricities of the two sections, perpendicular respectively to the mean and 
 greatest axes, are, 
 
 (a? - c2)^ = 2T V (jj0), (62 _ c2)3 = 2S V (jt^) ; 
 
 if we change at once Q to tQ and jj to «-i »j, where t is any positive scalar, 
 we pass to a confocal ellipsoid, the focal ellipse and focal hy- 
 PEEBOLA remaining stUl unchanged ; the focal ellipse may conveniently 
 be represented by the system of the two equations 
 
 S . pUjy =. S . pW, TV . pU»j = 2S V {rjG), 
 
 which represent separately the plane of the ellipse, and a cylinder of revo- 
 
1 CONTENTS. 
 
 lution on which the ellipse is contained ; or we may combine the same plane 
 with this other cylinder of revolution, 
 
 TV.pU0 = 2SV(j70); 
 
 the focal hyperbola is adequately represented, as a curve in space, by the 
 single equation, 
 
 V.j;p.V.p9=(V.jj0)2; 
 
 because this equation will be found to include within itself the equation 
 of the plane of the hyperbola, namely, S . p>j0 = 0, as well as the constancy 
 of the product of the projections on the asymptotes, which asymptotes are 
 here the lines r],9, ox (as is known) the axes of all the cylinders of revo- 
 lution circumscribed about the ellipsoid and its confocals ; 
 
 Articles 504, 505 ; Pages 511 to 513. 
 
 § LXXXIX. In general, in this Calculus, a scalar equation, fp — c, involving one 
 variable vector p, represents a surface ; in fact it is equivalent to an ordi- 
 nary algebraic equation between the three Cartesian co-ordinates x, y, z, 
 and may be changed to such an equation by substituting for p its trino- 
 mial value ix+jy+kz (see §xix.); examples; the actual process of 
 squaring the last-mentioned trinomial gives p^— — x^ — y^ — z^; if we make 
 a = ia +jb + kc, a! = iu! +jb' + kc, then actual multiplication gives ex- 
 pressions for the products ap, dap, of which the scalar parts are, respec- 
 tively, S . ap = — {ax + by-\- cz), and S . ciap = the detershnant 
 
 a, b, c, 
 a, b', c, 
 
 or = a (b'z — c'y^ + b (c'x — dz) 4 c {ay — h'x) ; 
 we have the two identities, 
 
 pS . y/3a = yS . p^a + j3S . ypa + aS . y^p, 
 pS . y/3a =V. jSaS . yp + V. «yS . /3p + V. y/3S . ap, 
 
 of which the second shews that the elimination of p between the three 
 equations S . ap = 0, S./3p = 0, S.yp = 0, conducts to the equation 
 S. y/3a =.0 ; co-ordinates and quaternions may thus be employed to as- 
 sist and illustrate each other ; additional examples ; the symbol S . y/3a 
 denotes the volume of the parallelepipedon of which a/3y are edges, this 
 volume being taken positively or negatively, according as the rotation 
 round y from j8 to a is negative or positive (compare § xxxix.) ; we 
 niight in this way see (compare § lxxxvi.) that this function S . y/3a 
 changes sign, when any two of its factors are interchanged ; the scalar of 
 a product does not alter, when its factors are cyclically permuted, 
 
 S . y(Sa=S. ^ay, S . srq=S .rqs, SiC, 
 
 Articles 506 to 512 ; Pages 513 to 621. 
 
 § xc. An equation o{ vector fortn, (pp = \, where denotes a vector function, 
 and X a given vector, may in general be resolved into three scalar equa- 
 tions, which suffice (theoretically speaking) to deteimine generally x, y, z, 
 
CONTENTS. ll 
 
 and therefore also p, or at least to restrict those co-ordinates, and this 
 vector, to a finite variety of values ; examples ; if 5 be a given quaternion, 
 
 V 
 
 the equation V . 5p = X gives pSg'= X + 5~iV.XVg'; notations -, &c. ; 
 
 other form for the solution of the last equation in p; the equation 
 
 V. pp7 = X gives p = —^ — ; interpretation of this expression, 
 
 in connexion witli the results of § xlii. ; the sine of the semisum of the 
 angles of the spherical triangle def is equal to the cosine of the com- 
 mon bisector ab of two sides, divided by the cosine of cd, namely, of the 
 half of the third side ; for any three vectors, we have the following trans- 
 formation, which is very often useful in this calculus, 
 
 V.|8p7 = /3S.7p-pS./37;+7S.|Sp, 
 
 Articles 513 to 518 ; Pages 521 to 526. 
 
 § xci. Other mode of deducing this general and useful equation of transforma- 
 tion ; if n' be used as the characteristic of the operation of taking a pro- 
 duct, with an inverted order of the factors, then (by §§ Liii., LXin.), 
 
 Kn = n'K, s=ki + k:), Y=h(i-K); 
 
 hence 
 
 sn = in+in'K, vn^jn-in'K; 
 
 thus, whatever vectors a, (3, 7, 5, may be, we have 
 
 S . 7/3a = i {yl3a - a/37), V. 7/3a = i iy(3a + a^y) ; 
 S . ^y^a=l {dy[3a+ a/iyS), V. dyl3a= I (dyf^a- a^yS), &c. ; 
 
 and the identity, i (yf3a + ajSy) = iy ((3a + aj8) - J (7a + ay)l3 + 
 ia (7/3 + I3y), gives V. yl3a= 7S . f3a- /3S . ya+ aS . [3y, a result agree- 
 ing with the last section ; we have also (compai'e § lxx.), these two other 
 formulae of transformation, 
 
 V. 7V. l3a = aS. I3y-(3S. ay ; V(V. yjS . a) = yS . /3a -/3S . a7 ; 
 
 the student ought to make himself very /amiViar with the three last for- 
 mula, which are valid for any three vectors ; we have also, for any four 
 
 vectors, 
 
 S . a"a"a'a = S . a"'aS . a'a" — S . a"'a'S . a!' a + S . a"'a"S . aa ; 
 S (V. a"'a".Y. a'a") = S . a"'a . S . a'a"— S . a'a . S . a" a ; 
 
 the comparison of the two expressions for V (V. a'a'.Y. da) conducts to 
 the first identity of § lxxxix. ; as included in which, it is shewn that if 
 a, d be two non-parallel vectors, and a"= V. da, then an arbitrary vec- 
 tor jO may be expressed as follows, 
 
 „ dp ,^ pa S . dp 
 
 p = aS -^ + a S S, + ^, 
 
 a a a 
 
 Articles 519 to 623 ; Pages 526 to 529. 
 
Hi CONTENTS. 
 
 § xcii. Connexion of quaternions with spherical trigonometric ; the expression 
 recently given for the scalar part of the product of the vector parts of two 
 binary products of vectors may be interpreted as equivalent to the follow- 
 ing theorem of Gauss, 
 
 cos ll". cos l'l'" — cos ll"'. cos j^il'— sin ll'. sin l"l"' cos A, 
 
 where A is the spherical angle between the arcs i.l', l"l"'; there are various 
 ways of deducing from quaternions the fundamental formula, cos b = 
 cos c cos a + sin c sin acosB; if the rotation round (3 from a towards y 
 be positive, 
 
 V. yjS .V. /3a = sin a sin c (cos + (3 sin) B ; 
 
 tan a/sV = tan 5 = j8-i-(V.y/3.V. /3a), 
 
 Articles 524 to 526 ; Pages 529 to 532. 
 
 § XCIII. Connexion of quaternions with goniometry, or with the doctrine of func- 
 tions of angles ; a and i being any two unit-vectors, and t any scalar, we 
 have S . a* = S . t* =/(0 =fi = a scalar and even function oft; a* =ft 
 + af(t-l), it =ft + if{t - 1) ; /(- =ft, f{2 + t)= -ft ; /(« + t) 
 
 ^Mt -/ (« - 1) /(^ - 1) ; C/O^ + {/(< - 1) } ^ = 1 ; /(iO = (i + i/0» ; 
 
 the values of ft may be numerically calculated and tabulated ; the func- 
 tion / of a multiple of t may be transformed by the help of the equation, 
 
 2/(«0 = {fl'r f(t--^)}n+{ft-lf{t-l)]n ; 
 
 the consideration of a small rotation gives the differential expression, 
 
 d.t« = |i*+idi; hence /'« = |/(^ + l),/"< + f^y/*f=0;yi3 = l,/0 = 0; 
 
 developements for ft and /(i — 1) ; 1* = eh'^ti^ this exponential symbol being 
 here employed merely as a concise expression for a series of well-known 
 
 'TTt "JTt 
 
 form ; with the usual notations for cosine and sme, ft — cos —, i* = cos ^ 
 
 "Kt 
 
 + ^ sin — ; the equation y~^ya'^ = — 1, of § xlix., under the form y2-« = ■ 
 
 pva", maybe expanded into the following, cos (7r — C) + y sin (tt — C) 
 = (cos £ + (3 sin B') (cos A + a sin A') ; the comparison of scalars gives a 
 known and fundamental formula of spherical trigonometry, from which all 
 others might be deduced, namety, — cos C = cos BcosA — cos c sin 5 sin ^ ; 
 the comparison of vectors gives 
 
 y sinC=a sin A cosi?+/3sin5cos^ + V. jSa. sin ^4 sin 5, 
 
 which may be interpreted as a theorem respecting the construction of a pa- 
 rallelepipedon, connected with a spherical triangle; addition of quater- 
 nions, and the distributive character of their multiplication, might be illus- . 
 trated by spherical trigonometry, . Articles 527 to 629 ; Pages 532 to 537. 
 
 § xciv. Brief account of some early investigations by the present writer, whereby 
 he was led (in 1848) to results agreeing in substance with those lately 
 mentioned, respecting the connexions of quaternions with spherical trigo- 
 
CONTENTS. liii 
 
 nometry ; symlolic multiplication table, for the squares and products of 
 t, _/, k ; developement of a product of two quaternions, under their quadri- 
 nomial forms ; reproduction of a theorem of Euler, respecting the products 
 of sums of four squares ; subsequent extension (in the same year) by 
 J. T. Graves, Esq., to a theorem respecting sums of eight squares, and to a 
 theory of certain octaves, involving seven distinct imaginaries ; allusion to 
 subsequent publications of Professor De Morgan, and other mathematicians 
 of these countries, in the same general field of research, or at least on ana- 
 logous subjects, such as the triplets, tessarines, and pluquaternions ; the 
 writer regrets that it is not possible for him here to analyze, or even to 
 enumerate, those important and interesting publications ; the quaternions 
 early conducted him to a general theorem respecting spherical polygons, 
 ■which includes as a particular case the following theorem respecting a 
 spherical triangle, and may in turn be derived from it, 
 
 (cos C + 7 sin C) (cos 5 + jS sin B) (cos ^ + a sin ^) = — 1 ; 
 
 this particular theorem may be expressed by the lately cited formula of 
 § XLix., y^jS'-Za^ = — 1 ; the more general theorem for a polygon may be 
 
 expressed by an analogous equation, namely, «„"i^. • • ai"i «" = (— 1)" ; 
 
 another early and general theorem of this calculus, respecting spherical 
 polygons, which is a sort o? polar transformation of the foregoing, may 
 be expressed by a connected formula, . Articles 530 to 536 ; Pages 537 to 545. 
 
 i xcv. Exponential Functions, direct and inverse ; the tensor of the sum of any 
 number of quaternions cannot exceed the sum of the tensors ; if we write 
 
 11.2 1 . 2 . . . »i 
 
 the number m may be assumed so large, however large the given tensor of 
 the quaternion q may be, that the last term (reading here from left to 
 right) may, have its tensor less than any given and positive quantity, 6 ; 
 and not only so, but that the quaternion sum of the n following terms of 
 the same series, or the quaternion difference Ym+n {jf) — F,n (g), shall also 
 have its tensor <6, however large the number n of these new terms may 
 be ; the finite series Ymq converges to a definite quaternion limit, F^ q 
 or Fq, when the number m of terms increases indefinitely ; the resulting 
 function, Fq, has the well-known exponential character, whenever 
 the condition of commutativeness is satisfied ; Fr . F5 = F (r + 5) if rg' = 
 qr ; for example, we have, generally, Fg' = FSg . FYq, where it is found 
 that rSg' is a positive scalar, and FYq is a versor, so that TFg = FS?, 
 
 TFVg = 1 ; UFg = FV^ = (cos + U Vg sin) T Vg ; F ( Vg + ^ UVg) = U Vg 
 
 . FYq, F (Yq + TrUVg) = - FYq = (cos - VYq sin) (tt - TYq) ; the 
 function FYq is a periodic one, in the sense that it only changes sign, 
 when we add + tt to TYq ; ant versor, Ur, may be considered as an ex- 
 ponential function of a vector, and put as such under the form FVg, where 
 the (positive) tensor TYq shall not exceed tt, and may therefore be treated 
 
Ivi CONTENTS. 
 
 or we may deduce and employ the equation, (hq - qb) Sb = Y.YbYc ; or 
 may regard the proposed equation as a case of the following, 
 
 aq + qb = c, 
 which gives, q (62 + 2bSa + Ta^) = a'c + cb, if a'=Ka; if we make r=g 
 + y, and S . /3S . ap + V. ■yp = <pp, ^ = ^ + ff, the general linear and vec- 
 tor equation of the present section becomes ipp = ff, and the problem of its 
 solution comes to inverting the function ■>p i the functional characteristic 
 <p is found to satisfy a symbolic and cubic equation, O = w + w'0 
 + m"03 + 03, where n, n, n" are three scalar coefficients, of which the va- 
 lues are assigned, in terms of the given vectors, a, (3, a', /3', . . and y ; the 
 characteristic ip must therefore satisfy this other symbolic and cubic equa- 
 tion, 
 
 Q=-^3 — }7i"ip2 _j- m'lp — m, where in=g^ — ng'^ 
 
 + n'g — n,m— 3g^ — 2ri'g + w', m" =Zg — n ; 
 the solution of the linear equation, ipp = ff, comes thus to he found anew 
 under the form, 
 
 mp = mip-^a = (jii — m"\jj + ^z^) a^a" — gd + g'^'O, 
 
 where <r' and tr" are vectors derived from the given vector a, by assigned 
 operations, involving the given vectors a, ^, a, ^', . . and y, but not the 
 scalar g ; theorem of the parallelepipedon of derivation, obtamed 
 by interpreting the lately written symbolic and cubic equation ; for any 
 proposed mode of linear deforjiation, represented by the operation t//, 
 if we form the three successive derivative lines, -^p, \p^p, rp^p, and then 
 decompose, by projections, the original line p into three others, in these 
 three directions, or in their opposites, the 7-atio of each component -to the 
 corresponding derivative line will depend only on the mode of deri- 
 vation, and not generally on the length, nor on the direction, of the line 
 p thus operated on; we have m4'"iO = 0, and therefore generally i//-i 
 = ; but if it happen that g is a, root, gi or go or gs, of the ordinary cubic 
 equation, = m = ^'3 _ ^"^2 ^_ n'g — n, then the function \pp may vanish, 
 without p itself vanishing; if, after assuming any arbitrary vector <r, we 
 derive from it three others by the formula, 
 
 Pi = <y" -g\.(! + 5'i^o", pi = <^"- ffz<^' + 5'3^0-j p3 = (y" - 9z<y + 9^<y-, 
 we shall have 
 
 Spipi = x//3P2 = 'I'SPS = »l(7 = ; 
 
 that is, /or these three directions, pi, p2, pa, we shall have 
 ^Pi=-9iPh <pPi=~9-2P2^ <ppz--9zp3; 
 
 this analysis might be developed so as to include the theories of the axes 
 of a surface of the second order, and the axes of inertia of a body, . . 
 
 Articles 554 to 567; Pages 559 to 569. 
 
 § xcvili. Definition of the differential of a function of a quaternion, 
 
 Afq = lim . « {/(9 + w - 1 &q) ~fq } ; 
 
CONTENTS. Ivii 
 
 q and dg are here any two quaternions, Tdj being not necessarilj-^ small, 
 but the positive Avhole number n being conceived to increase without li- 
 mit ; the third quaternion difq, which results as the limit of this process, is 
 a, Junction of the two assumed quaternions, q and d^, of which the parti- 
 cvlaxform depends on the form of the proposed function, f, but which is 
 always linear, or distributive, with respect to the quaternion dg ; but this 
 differential Afq is not in general reducible in this calculus, to a product of 
 the form/'g' . dg, itf'q denote a function of the quaternion q alone ; when the 
 function / (g' + dg') can be developed in a series, involving terms or parts 
 of successively higher and higher dimensions, with respect to the quater- 
 nion dg, the part of this developement which is of the first dimension, re- 
 latively to &q, is (as in the ordinary differential calculus) the required 
 differential Afq ; but it is proposed to avoid, in this calculus, adopting this 
 as the fundamental property of a differential, because the recent definition 
 can often be applied more easily than the developement can be found; 
 examples ; 6L.q^ = q .Aq-\-&q.q,ox more concisely, d . g2 = q^q ^ ^qq^ ^^ 
 being treated as a simple symbol, or as if it were a single letter ; d . g-i 
 = — g-idgg-i ; in differentiating any product of quaternions, we simply 
 differentiate each factor in its own place ; we may extend Taylor's series 
 to quaternions, under the form /(g + dg) = ey^, where dg is treated as 
 constant ; examples ; Articles 568 to 573; Pages 569 to 572. 
 
 § xcix. Geometrical applications ; if a vector p be a given function fj>t of a varia- 
 ble scalar t, we may express its differential under the usual form, dp = A(j>t 
 = <l>'t . dt=:p'dt, where p'=<p't = a certain derived vector, which is parallel 
 to the tangent to the curve in space, which is the locus of the extremity 
 of p ; the length of this new vector is unity, T(p't= 1, if the arc be the in- 
 dependent variable ; in mechanics, if t denote the time, and if a second 
 differentiation have given dp' = d0'f = (p"t . d^ = p"d^, then p' may be called 
 the vector of velocity, and p" the vector of acceleration, while p may be 
 named the vector of position ; in geometry, if t be again the arc of the 
 curve, p — p"-i is the vector of the centre of the osculating circle, and p" 
 may therefore be called the vector of curvature ; when a surface is ex- 
 pressed, as in § Lxxxix., by an equation of the form /p = const., where/ 
 denotes a scalar function, we may then, by cyclical permutation under the 
 sign S (see the same section lxxxix.), express the differentiated 
 equation of that surface under the form d/p = 2S . vdp = ; the logic 
 of this process will be more closely considered in § cr. ; y is a nor- 
 mal VECTOR, and if we oblige it to satisfy the condition S ,vp=l, 
 then (compare § lxxxi.) its reciprocal v-i will represent, in length 
 and in direction, the perpendicular let fall from the origin of vec- 
 tors on the tangent plane to the surface, so that v itself may be called, 
 under the same conditions, the vector of proximity ; without oblig- 
 ing V to satisfy the equation S . vp = 1, if we only choose it so as to 
 give generally S . vdp — 0, it will still be, as before, a normal vector, and 
 this symbol v may be used to form equations of classes of sur- 
 faces ; thus an arbitrary cone (with vertex at origin) may be d*ioted 
 
 h 
 
Iviii CONTENTS. 
 
 by the equation S . vp = 0, an arbitrary cylinder by S . va= 0, and an 
 arbitrary surface of revolution by S . fSvp = ; this last equation is ana- 
 logous to an EQUATION IN PARTIAL DIFFERENTIALS, and may be treated 
 as such by a species of integration, eliminatmg v, and introducing an 
 arbitrary function, under the form |02 = F(S.^p), or TV. p/3"i = 
 /(S . p/3-1), which last form was assigned in § lxix. ; conversely, by a 
 process of differentiation, we can eliminate the arbitrary function, f, from 
 this last equation, and so recover the formula of the present section, 
 S./3vp=0, Articles 574 to 578 ; Pages 572 to 575. 
 
 § c. Geodetic lines ; the normal property of the osculating plane gives the following 
 general equation of a geodetic, S . v&pd^p = 0, or S . vp'p" = 0, p being re- 
 garded as a fmiction of some scalar variable ; we have also this other ge- 
 neral formula, V. vdUdp = 0, where dUdp denotes the differential of the 
 versor of the differential of p, and is treated as a simple symbol; if we 
 take the arc of the geodetic as the independent variable, or suppose that 
 Tdp is constant, the last general form may be reduced to V. v6?p = 0, or 
 V. vp" = ; examples ; geodetics on a sphere, and on an arbitrary cylin- 
 der, cone, and surface of revolution ; variations in quaternions ; for- 
 mula for the differential of the tensor of an arbitrary vector c, dTc = 
 — S . Uffdff = S . U(7-i dtr ; this result will be extended in § ci. ; M = d^, 
 d^=lb; the variation of the length of the arc of a curve, on any given 
 surface, is expressed by the formula, 
 
 ^jTdp = J^Tdp = - AS . Udp^p + JS (dUdp .Sp); 
 
 hence the varied equation of the surface being S . vSp = 0, the general diffe- 
 rential equation of a shortest line is V. vdUdp = 0, as above ; equations 
 of limits ; for a geodetic on an ellipsoid, with the same significations of/ 
 and V as in § lxxx., if Tdp be assumed as constant, the differential equa- 
 tion of the geodetic becomes, 
 
 = J— + S — , and gives 1v V(/Udp) = const. : 
 2/dp V 
 
 this reproduces the well-known theorem of Joachimstal, P. Z) = const, 
 because Tj/ = P-i, and V(/Udp) = i3-i, if P be the /)er/jewdicM?ar let fall 
 from centre on tangent plane, and D the semidiameter parallel to the ele- 
 ment dp ; geodetic on a developable suTt&ce ; proof of the rectilinear form 
 which the curve assumes, when the surface is flattened into a plane ; the 
 general theorems of Gauss, respecting the spheroidical excess (or defect) of 
 a geodetic triangle on an arbitrary surface, admit also of being proved bv 
 quaternions (see the investigation in § cvi.) ; reproduction of some geome- 
 trical properties, discovered by M. Delaunay, of the curve which on a given 
 surface, and with a given perimeter, includes the greatest area ; it is pro- 
 posed to name a curve of this kind a Didonia; Xhe isoperimetrical for- 
 mula for its detennination is 
 
 #• JS.UvdpJp + c^jTdp^O, 
 
CONTENTS. Jix 
 
 which gives the following differential equation of a Didonia, 
 
 c-idp = V.Ui/dUdp; 
 
 geodetics are that limiting case of Didonias, for which the constant c is 
 infinite ; in general, that constant may have its expression in various ways 
 transformed, and may receive various geometrical interpretations j among 
 which the most remarkable is connected with the known property of the 
 curve, that if a developable surface be circumscribed about a given sui-face, 
 so as to touch it along a Didonia, and if this developable be then unfolded 
 into a plane, the cm-ve will at the same time be flattened generally into a 
 circular arc, of which the radius =c, . Articles 579 to 590 ; Pages 575 to 584 
 
 CI. More close examination of the logic (compare § xcix.) of the process of dif. 
 ferentiating the equation of a surface, and so obtaining the equation of its 
 tangent plane, and the normal vector v, without necessarily supposing for 
 that purpose the differential dp to be small ; differential of a, function of a 
 function of a quaternion ; d/(05') = d (/0) 5 ; examples of the process ; 
 case of the ellipsoid •, differentials of the tensor and versor of a qua- 
 ternion, and of their logarithms: dT^^ S . dgUg'-i, dlTg' = S. dg'5-1, 
 dlUg' = dU5Ug'-i = V. dg'g'-i ; incidental notice of the general transfor- 
 mations, r-i (r^(f)^ g'-i = U (Sr Sg- + \r Yq) = \J (rq + KrKq) ; by in- 
 verting the function which expresses (see § lxxix.), the normal vector v 
 for the ellipsoid in terms of p, we find 
 
 |0 = (t2 + K2)a/-2V. wk+4(i-k)-2 V. ikS . ikv; 
 
 hence the equation of that other and reciprocal ellipsoid, on which v ter- 
 minates, may be thus wiitten, 
 
 1 = S . vp = (l2 + k2) J^2 _ 2S . iVKV + 4 (i - k:)-2 (S . tKvy ; 
 
 the mean semi-axis oithis reciprocal ellipsoid is 6"i (contrast § lxxxiv.); 
 in. general, the locus of the extremity of the vector of proximity (see 
 § xcix.), for any surface, may be very simply proved to be (as is other- 
 wise known) a surface reciprocal thereto, by shewing that the equations 
 
 S.vp = c, S ,vdp = 0, give S. pv= c, S , pdv=0, . . . . 
 
 Articles 591 to 597; Pages 584 to 588. 
 
 § cii. More close examination of the extension (§ xcviii.) of Taylor's Series to 
 quaternions ; proof that whenever the quaternion function f{q + xr) can 
 be developed, in a finite or infinite series, of the form /o + xfi -f x'^f^ + hc, 
 X being a scalar, we must have d"/g = A"0"/,j, if dq be treated as con- 
 stant, and = r ; other proof of this theorem, under the form that if 
 / (gi + xdq) =/o+ xji + j;3/2 + &c., then nfn = dfn-\ ; proof that if we sup- 
 pose the n first of the successive differentials of the function oifq to be 
 finite, and if x be supposed small of the first order, then the expression *« = 
 
 fiq + xdq^-fq-xdfq-^x^d'^fq- . . - ^— ^ .r"d"/5 is small of an 
 
 order higher than the k"^ ; or that not only the expression s„ itself, but 
 
Ix CONTENTS. 
 
 its n first successive differential coefficients, taken with respect to a;, va- 
 nish with that scalar variable ; it is to be remembered that q and Aq are 
 treated throughout this section (compare § xcviii.) as two arhitrai~y quater- 
 nions ; and that Tdg is not here supposed to be small, although in geome- 
 trical applications it is often convenient to attribute small values to Tdp; 
 example from the equation of the ellipsoid, which may be rigorously de- 
 veloped under the_^wj^e form, =f{p + dp) —fp = Afp + id^/p, dp denoting 
 an arbitrary chordal vector, drawn from the extremity of p, to any other 
 ^oiwf of the surface, Articles 598 to 601; Pages 588 to 592. 
 
 § cm. When dp is thus treated as a finite and chordal vector, the equation 
 
 = d/p + id2/p, or = 2S . vdp + S . dj^dp, 
 
 or the same equation with an additional term S . v&p S . OTdp, where zy 
 is an arbitrary vector, represents an ellipsoid, or other surface of the se- 
 cond order, which osculates in all directions to the given surface /p = 
 const., or has with it complete contact of the second order, at the extre- 
 mity of p ; if (7 be the vector of the centre of the sphere which osculates 
 to the surface in the direction marked by the limiting value of Udp, then 
 
 V dv 
 
 = S -— , the second member being regarded as a function of this va- 
 
 p - (T dp 
 
 lue of Udp ; applied to the ellipsoid, this formula reproduces the known 
 expression Z>^ . P-i, as the value for T (p - er), or for the radius of cur- 
 vature of a normal section of the surface, 
 
 Ai-ticles 602 to 606; Pages 592 to 596. 
 
 § CIV. For any surface, S . SAvAp = S . AvMp, if in forming Sdv we operate only 
 on dp, but not on p itself, as contained in the expression of dj' ; hence it 
 may be inferred that the directions of osculation of the greatest and least 
 spheres, determined by the formula SS . dvdp-i = 0, are also the directions 
 of the lines of curvature, for which consecutive normals intersect, and 
 which have for their differential equation 0_= S . vdvdp ; this latter equa- 
 tion expresses that dpjjv. vdv, and therefore contains one of the theo- 
 rems of Dupin, namely, that the tangent to a line of curvature on any sur- 
 face at any point is perpendicular to its conjugate tangent ; equations of 
 the indicatrix, S . v&p = 0, S . dvdp = constant ; the equation of the lines 
 of curvature may also be thus written, = S . dv^Udp ; or thus, 
 = V . dpdUv ; this last form contains a theorem of Mr. Dickson, that if 
 two surfaces cut along a common line of curvature, they do so at a con- 
 stant angle ; transformation of the equation of § cm., for the curvature 
 of a section of a surface, 
 
 dp2 w — p 
 
 conducting to the theorem of Meusnier ; other general transformation and 
 interpretation of the formula of § cm., for the curvature of a normal sec- 
 tion ; if on the normal plane cpp' to a given surface, containing a given 
 linear element pp', we project the normal to the siu-face at the 7iear point, 
 
CONTENTS. 
 
 Ixi 
 
 p', this projected normal will cross the given normal CP, which is drawn 
 at the given point p, in the centre c of the sphere which osculates to the 
 surface along the eletnent, .... Articles 607 to 612 ; Pages 596 to 601. 
 
 § cv. Considering the vector p, of a variable point on any surface, as a function, 
 = i// (a^, y), of two scalar variables, x and y, which are themselves re- 
 garded as functions of some one independent and scalar variable, we may 
 write, 
 
 dp = p'Ax + p&y ; dp' = p"dar + p'Ay ; dp, = p'Ax + p„dy ; 
 
 d2p = p"da;2 + 2p,'da^dy + p„dy2 + p'^^x + pp?y ; 
 
 p', p,, p", p,', p„ being five, new vectors ; 
 
 it is allowed to write j^ = V. pp^^ because p and p, are tangential, and there- 
 fore the V thus found is normal ; in the expression for S . vd^p, d^ar and 
 d2y disappear; and if we make Uv {a- p)-i = i2-i, so that R is the ra- 
 dius of curvature of a normal section, of which a is the vector of the centre 
 of curvature, we shall have, by § civ., an equation of the form, 
 
 = i2- 1 dp2- S .UvdSp = AAx^ + 2BAxAy + CAy'^ ; 
 
 for a line of curvature, we have 
 
 = AAx + BAy = £&x + Cdy, and therefore AB- C^=0, 
 
 where 
 
 ^ = .fi-ip'2-S.p"Uv, -B = i2-iS.p'p,-S.p;Uj/, C = i?-ip2-S.p„Uj/; 
 
 Ri, R% being the two extreme radii of curvature, the measure of cukva- 
 TURE of the surface may be thus expressed. 
 
 V V \ V j 
 
 example; deduction of the usual formula, (rt — s'^') (l+/)2+g'2)-2; in 
 general if e =— p'2, f= — S . p'p„ g — —p}., so that the square of the length 
 of a linear element of the surface has for expression 
 
 Tdp2 = eda:2 + 2/da:dy + gAy\ 
 
 the recent expression for the measure of curvature is shewn to depend only 
 on the three scalars e, /, g^ on theii" six partial differential coefficients of 
 the first order, and on three of their nine partial differential coeflScients of 
 the second order, taken with respect to x and y ; in this way is reproduced 
 by quaternions a very remarkable theorem of Gauss, namely, that if a sur- 
 face be treated as an infinitely thin and flexible, but inextensible solid, 
 and be then tkansfoemted as such into another surface, such that each 
 LINEAR ELEMENT of the new is equal in length to the corresponding ele- 
 ment of the old one, the measure of curvature at each point will not 
 
 BE ALTERED ly this TRANSFORMATION, 
 
 Articles 613 to 615; Pages 601 to 604. 
 § cvi. If X denote the length of the geodetic line ap, drawn on the surface from a 
 
Ixii CONTENTS. 
 
 fixed point A, and if y denote the angle bap which the variable geodetic 
 AP makes there with a fixed line ab, then 
 
 |0'2 = - 1, S . |0>,= 0, or e = 1, /= 0, 
 and these equations may be diflferentiated ; hence if we make m = '^g = Tp,, 
 the general expression for the measure of curvature reduces itself to the 
 following, which (with a somewhat different notation) was first discovered 
 by Gauss, 
 
 i?i- iRo- 1 = - nimr i ; or, ffr ^Ro- 1 = d2 TSp -~ (dp2 T^p) ; 
 
 treating x and y as functions of the arc s of a new geodetic on the surface, 
 not drawn from the fixed point A, and denoting by v the angle between an 
 element ds or pp' of this new geodetic, and the prolongation of the old geo- 
 detic line AP, the difierential equation of the new geodetic becomes, 
 
 x" = mmy"^^ or v = — my, or &.v = — rnd^y ; 
 
 we may also conveniently write, in a slightly modified notation, 
 
 Sv = - m'Sy, or dv = - dTSp -j- Tip, 
 
 d referring here to motion alonff the original geodetic ap, and S to passage 
 froin that line to a near one ; ddv, or — m'dxSy, is then a symbol for the 
 spheroidical excess (compare § c.) of a little geodetic quadrilateral, of 
 which the area = mdixly ; dividing the excess hy the area, we find the quo- 
 tient = —rd'm-'^ = the measure of curvature of the surface ; but also this 
 measure = Rr^R^'^ = "V . dUj'^Uv -r- V. dpdp = the area of the corres- 
 ponding superficial element of the unit-sphere, divided by the element of 
 area of the given surface, this con-espondence consisting in a parallelism 
 between radii and normals ; hence, as Gauss proved, the total curva- 
 TUKE of any small or large closed figure, on any arbitrary surface, bovmded 
 by geodetic hnes, or the area of the corresponding portion of the surface 
 of the unit-sphere (not generally bounded by great circles), is equal 
 (with a proper choice of units) to the spheroidical, excess oj the figure; 
 singular points are here excluded, and the sign of the element of the sphe- 
 rical area is supposed to change, when we pass from a convexo-convex to 
 a concavo-convex surface, .... Articles 616 to 619; Pages 604 to 609. 
 
 § evil. Many other geometrical applications of difierentials of quaternions might 
 easily be given ; for instance, they serve to express with ease what M. 
 Liouville has called the geodetic curvature of a curve upon any surface ; 
 they may also be employed to calculate the normal and osculating planes, 
 and the evolutes, torsions, &c. of curves of double curvature ; transforma- 
 tions of the symbols <1<1', <1^, where 
 
 ^ id 7'd kd , id jd kd 
 dx dy dz dx dy az 
 
 xyzxyz being six independent and scalar variables; the formula, 
 
 (dt du dv \ 
 Jx^-dy^Tz) 
 
CONTENTS. " Ixiii 
 
 JAv d«\ .(At dv\ /dM &t\ 
 
 +*id^-d;J+^\dr-d^J+Hcb"#)' 
 
 d^v &^ d^v 
 da;2 dy2 ^ dr2 
 
 appear likely to become hereafter important in mathematical physics ; 
 — <i V may represent the flux of heat, if v be the temperature of a body ; 
 or if V be the potential of a system of attracting bodies, then ^ v repre- 
 sents, in amount and ia direction, the accelerating force which they exert 
 at the point xi/z ; in geometry, the vector <1 « is normal to the surface for 
 which the scalar function v = constant ; when operating on such a fimc- 
 
 tion, 
 
 <] = -(S.dp)-id, 
 
 Article 620; Pages 609 to 611. 
 
 CVill. Applications of quaternions to physical astronomy ; the vector fimction, 
 ^a = a-^Ta-i, may be called the tractor of a, because it represents, in 
 length and in direction, the accelerating force of attraction which an unit 
 of mass at the origin exerts on a point placed at the end of the vector of 
 position, a ; by the rules of this calculus, this function may be thus trans- 
 formed, 
 
 fa = dJJa -—Y. ada = (Ua)'-7- V. aa ; 
 
 the differential equation o{ motion of a binary system, d^a = M<padt^, or 
 a!'=M(pa, gives the following integrals of the first order, V. aa'=y, 
 a'= Ary-i(£ — Ua), where y and £ are constant vectors, but a is a varia- 
 ble vector ; the first contains the laws of constant plane and area, and the 
 second contains the law of the circular hodograph ; eliminating the 
 vector of velocity, a', we obtain this equation of the orbit, = Ta + S . ae 
 + iJf-i y2j or r-i =/?-! (1 + e cos v), agreeing with a well-known result 
 respecting the conic-section form of the curve, and /ocaZcAarac^er of that 
 body aboiit which the other is conceived to move ; the varying tangential 
 velocity of this latter body may be decomposed into two parts, both con- 
 stant in amount, and one constant also in direction ; theorem of hodo- 
 GRAPHic isocHRONiSM, Corresponding to Lambert's theorem ; allusion to 
 a conception of Moebius; the difference (a + ^a) — ^a, or A0a, of the 
 tractor function (pa, might perhaps be called the turbator, because it 
 expresses, with Newton's law, the amount and direction of the disturbing 
 force which an unit-mass, supposed to be situated at the common origin 
 B of the two vectors a and a + Aa, exerts on a body a situated at the end 
 of the latter vaiiable vector, to disturb its relative motion about a body c 
 at the end of the former vector ; developement of this disturbing force, 
 under the supposition that TAa < Ta, or that the distance b = ca, of the 
 disturbed body a from the centre c of the relative motion, is less than the 
 distance a = bc of the distm-bing body b from the same centre ; example, 
 where a, b, c denote moon, sun, and earth ; we have the transformation, 
 
 0(/3 + a)=(l+g)-5 (l + 5')-f 0a, ifg = ^a-i, q =^q = a-"^ ji; 
 
Ixiv CONTENTS. 
 
 hence results a developement of the form (j3 -I- a) = 2„, „' 0„, „/, ^n 
 which the law of formation of the terms is assigned ; the sun's disturbing 
 force on the moon is in this way seen to admit of being decomposed into a 
 series of groups of smaller and smaller forces, in the varying plane of the 
 three bodies, represented in amount and in direction by the terms of this 
 developement ; if a, 6 denote the geocentric distances of the sun and moon, 
 and C their geocentric elongation measured from the sun towards the 
 moon in their common great circle in the heavens, then the angle from the 
 sun's geocentric vector — a to the component force 0nj n' is = (n — w') C, 
 and the intensity of the same partial force is = m,,, »' (6a'i)"+"' a"^, >n,j, n' 
 being an assigned and rational numerical coefficient; in the first and 
 principal group, there are two component forces, of which one, (pi, o has its 
 intensity = |&a~3, if the sun's mass be taken for unity, and is directed along 
 the moon's geocentric vector (3 prolonged, or towards the moon's apparent 
 place in the heavens, while the other, <po, i, has an exactly triple intensity, 
 and is directed towards what may be called Si fictitious moon, or to a 
 point which is a sort of reflexion of the moon's place with respect to the 
 sun ; the second group contains three partial forces, which may be said to 
 be directed towards three suns (one real and two fictitious), and the in- 
 tensities of these forces, taken in a suitable order, are exactly proportional 
 to the whole numbers 1, 2, 5 ; these results may be indefinitely extended, 
 and applied to the perturbation of an inferior by a superior planet, &c. ; 
 some of these and other results of the application of quaternions to mecha- 
 nical or physical problems, such as the conditions of equilibrium, the 
 theory of statical couples, and the motion of a system' of mutually attract- 
 ing bodies, were communicated to the Eoyal Irish Academy in 1845 ; the 
 present writer has since made other physical applications of the same prin- 
 ciples, and has published some of them, but is aware that nothing impor- 
 tant in that way is likely to be done, until the more full co-operation of 
 other and better mathematicians shall have been gained, .... 
 
 Articles 621 to 624; Pages 611 to 620. 
 
 § cix. A DEFINITE INTEGRAL in quatemions may be interpreted as a limit of a 
 sum; but, even when the function to be integrated remains ^wiYe between 
 the limits of integration, still if the difierential factor Aq under the sign of 
 integration be itself essentially a quaternion, then a certain degree oiin- 
 
 determination of value of the quaternion integral F(_q, dj) arises from 
 
 the possibility of assuming indefinitely many different laws of dependence 
 of the variable quaternion 5- on a scalar variable t, which latter may be 
 supposed to change from to 1, while q changes from one given quater- 
 nion value qo to another qi ; in this way arises a new sort of variation of 
 a definite integral, depending on the non- commutative character of multi- 
 plication, which may be symbolized by the formula, 
 
 5Q = ^ P^ F(iq, Aq) = {^'{^aFiq. Aq)-d,Fiq, Sq)}; 
 
 Jqo jqo 
 
 for example, 
 
 ^Sfq^q = J {Sfq ■ ^q - Afq . Sq), if Sqo = 0, dqi^O; 
 
CONTENTS. IXV 
 
 more particularly, 
 
 5\ qdq = S\ qt qt'df = \ (Sgtqt - qt'^qi) it, 
 jqo Jo Jo 
 
 the integral relatively to t being interpreted as the limit of a sum ; exam- 
 ples of different functional forms which maybe assumed for g^, and of the 
 
 [qi : 
 
 different quaternion values thereby obtained for the integral qdq ; this 
 
 Jqo 
 sort of variation of a definite integral vanishes, as in the ordinary integral 
 
 calculus, when the function F(jj, Aq") is an exact differential ; for exam- 
 ple, although, between given quaternion limits, the integrals of qdq and 
 &qq are each separately subject to the kind of indetermination above ex- 
 plained, j'et the integral of their sum is fixed, and we may write, defi- 
 nitely, as in algebra, 
 
 Y {qdq + dqq)=q{^-q(?-, 
 analogous remarks would apply to such expressions as 
 
 R=['\^'F{ci,r,Aq,Ar); 
 
 jro^qo 
 
 if the subject of this section shall be hereafter pursued, it will be proper to 
 combine it with the researches of M. Cauchy, respecting definite integrals 
 taken between imaginary limits of the ordinary kind, and respecting that 
 other species of indetermination, which arises from the passage of func- 
 tions through infinity, and not from any supposed absence of tlie commu- 
 tative property of multiplication, . . Articles 625 to 630 ; Pages 620 to 627. 
 
 § ex. Differentiation oi implicit functions, and of radicals ; examples; if/r de- 
 note any scalar function of a scalar variable x, and if dfx =f'xdx, then 
 in passing to quaternions, we have Y.YqYfq = ; if also we suppose VYfq 
 = + VYq, and denote by dq — Sq that part of dq which is a vector per- 
 pendicular to Yq, we shall have, under these conditions, the formula djq 
 =fqdq + TYfq . dVYq, which may be in various ways transformed, and 
 gives the equation, 
 
 Yqdfq + dfqYq=f'q(Yqdq + dqYq); 
 
 connexion of differentials and developements yi\th. equations of the first 
 degree ; to find the differential of the square root of a quaternion r, we 
 are by § xcviii. to resolve the equation qdq + dqq = dr, which is of the 
 same form as the equation bq + qb = c, discussed in § xcvii. ; and a se- 
 ries of equations of this linear form may be employed to develope the 
 square root of a sum, in a quaternion series, of the form 
 
 (6-2+ c)*=6+</l+?3-|-&C., 
 
 Articles 631 to 685 ; Pages 627 to 631. 
 
 § CXI. Quadratic equations in ^Ma^ermoMS (compare § xcvi.) ; an equation of 
 the form q^ = qa + b, or of this connected form, r~ = ar + b, where abqr are 
 
 i 
 
Ixvi CONTENTS. 
 
 quaternions, and q + r = a, qr= — b, has in general six roots, of which 
 two are real, a.nA fow imaginary; the determination of these six quater- 
 nion roots depends on a scalar equation of the sixth degree, which is of 
 cubic form ; the scalar and cubic equation thus obtained has in general 
 one positive and two negative roots ; case in which one root of the cubic 
 vanishes; examples of the above form of a quadratic equation in quater- 
 nions, 
 
 §2 = Sgi + 10/, 52 = qi -)_ J ; 
 
 more general example, ^2 = ga + /3, where a and ^ denote two rectangular 
 vectors, Sa = 0, S/3 = 0, S . rt/3 = ; the six quaternion roots of this last 
 quadratic are given by the three formulae, 
 
 I. g=ia + a-i/3±ia-i(aH4j82)5, 
 II. g=i(l + U/3)[a + (a2+2T/3)*}, 
 III. g=|(l-U;6){a+(a2-2T/3)*}, 
 
 in which it is to be remembered that a/3 = — /3a, so that the ordinai-y rules 
 of algebra are not all applicable here (§§ x., xi., &c.) ; by the peculiar 
 rules of the present calculus, it is easy to shew that the common value of 
 g2 and qa + j3 is, for the first formula, 
 
 ia2 + K«^ + 4/32)^; 
 
 each of the other two formulte may also be shewn, a posteriori, to give 
 equal values for the two quaternions 52 and qa + f3; the third formula 
 gives always two imaginary values for q ; but, according as a^ + 4/32 < or 
 > 0, we shall have two real quaternions from the second formula, and two 
 imaginary vectors from the first, or two real vectors from the first, and 
 two imaginary quaternions from the second expression ; in the former case, 
 the two real quaternion roots of the quadratic equation have a common 
 tensor =VT/3; in the latter case, the two real vector roots have unequal 
 lengths, or tensors, but VT/3 is still the geometrical mean between them ; 
 the distinction between these two cases coiTesponds (compare §lxxvii.) 
 to the imaginariness or reality of the intersections of the sphere, p2= S . ap, 
 and the right line, Y.ap=(3; the imaginary quaternions considered in 
 the present section (compare § xcvi. ) are all reducible to the form, q = q' 
 + 9" V — 1, where q and q" are quaternions of the real and ordinary kind, 
 such as have been hitherto considered in these Lectures,' and V — 1 is the 
 old and ordinary imaginary symbol of common algebra, and is to be 
 treated, in this sort of combination with the pecidiar symbols, {ijk, &c.) 
 of the present calculus, not as a real vector (contrast the earlier use of 
 the same symbol in § xxxv.), but as an imaginary scalar ; an expression 
 of this mixed form, 7' + V — 1 q", is named by the writer a Biquaternion ; 
 the study of them will be found to be important, and indeed essential, in 
 
 the future developement of this calculus, 
 
 Articles 636 to 650 ; Pages 631 to 643. 
 
 § cxii. Application of the foregoing principles, to continued fractions, of the 
 form 
 
CONTENTS. IXVU 
 
 "H^l"' 
 
 where a, b, and c (= mo) are any three given quaternions, and a is a posi- 
 tive whole number ; making 
 
 Vx = (mx + 92) («« + 91)' '» 
 
 we have «^= 92^ voqi'", where g'l, qz are ani/ two roots of the quadratic 
 equation q^ = qa+ b; examples, 
 
 ^h (^r«. mh [^h 
 
 in the two first of these four examples, the continued fraction has gene- 
 rally a period of six values, which may be found at pleasure by emploj'ing 
 the two real quaternion roots of the quadratic equation q^ — qi -\-j, namely, 
 
 q\ = h i.'^ +i'rj - k), qz= l{-l + i-j - k); 
 
 or two conjugate imaginary solutions of that quadratic, such as the pair 
 
 qi=zi- k,qi=z-^i — k, where z = (cos + V— 1 sin) — , V— 1 being the old 
 
 3 
 
 imaginary symbol (compare § cxi.) ; or the other pair of imaginary roots 
 
 of the same quadratic equation, included in the expression, 
 
 q = i(i + k)±i{l-j)V'^; 
 
 or by any other selection of two roots, for instance, by combining one real 
 and one imaginary root ; the six real quaternion terms of the period, found 
 by any of these combinations of roots, agree with those obtained by ac- 
 tually performing the divisions prescribed by the form of the continued 
 fraction ; in the third example above cited, of such a fraction, the value 
 does not circulate, but (geuerally) converges to a limit, so that 
 
 10; \°° 
 c = 2k-i, unless c = 2k — 4.i: 
 
 in this last case, and also in the case when c — 2k — i, that is, when c is a 
 real root of the quadratic c^ + 5ci=10j, the value of the fraction is con- 
 stant ; geometrical interpretations, for the case where c = i>o + ^-^o, x^ and 
 zq being regarded as the coordinates of an assumed point Pq in the plane 
 of ik (or xz') ; successive derivation of other points Pi, P2, &c., according 
 to a law assigned; if the assumed point be placed at either of two fixed 
 points Fi, F3, in the same plane of ik, its position will not be changed by 
 this mode of successive derivation ; but if Po be taken anywhere else in the 
 plane, the derivative points will indefinitely tend to the fixed position F2, 
 60 that we may write 
 
 p^ F2 = 0, p^ — F3, unless Pq = Fi ; 
 
 law of this approach; continual bisection of the quotient, PF2 -f- PFi, of 
 the distances of the variable point p from the two fixed points ; theorem of 
 the two circular segments, on the common base F1F2, and containing the 
 
Ixviii CONTENTS. 
 
 two sets of alternate and derivative points, Pq, P2) P4 • • and Pii P3) ^5 • • 
 to infinity ; verification by co-ordinates ; relation between the two segments ; 
 more general geometrieal theorems of the same kind, obtained as interpre- 
 tations of the results of calculation with quaternions, respecting the fourth 
 example of a continued fraction above mentioned, with the supposition that 
 j8 is a vector perpendicular to a and to po, and under the condition 
 
 a"* + 4/32 > (see again § cxi.) ; 
 
 interpretation of this condition ; when a* + 4/3^ < 0, there is no ten- 
 dency of the variable point to converge to any fixed position ; the quadratic 
 g2 = ga + /3 (of § CXI.) gives 
 
 g4 = 52a2 -f ^2, (2^2 _ a2)2 = a4 + 4^2 ; 
 
 but when a* -i- 4/32 = 0, the biquaternion solutions of the quadratic give, 
 indeed, like the i'eal roots, 
 
 (252 - a2)2 = 0, but not, like them, 2^2 _ a2 = ; 
 
 those solutions give in this case 2^2 — a^ _ 4,QqYq, Yq = p + V— 1 jo", 
 where p and p" denote two real and rectangular and equally long vec- 
 tors ; and the square of such an expression vanishes, without our 
 being allowed to equate the expression itself to zero ; algebraical interpre- 
 tation of the general results at the commencement of this section, divested 
 of quaternion symbols, and connected with 'Afunctional law of derivation 
 of four scalar s from four other scalars arbitrarily assumed, and from 
 eight given and constant scalars ; the indefinite repetition of this process 
 of derivation conducts generally to one ultimate or limiting system, of four 
 derivative scalars, ....... Articles 651 to 668 ; Pages 643 to 664. 
 
 § cxiii. A biquaternion may be considered generally as the sum of a hiscalar and 
 a bivector ; we may also conveniently introduce biconjugates, bitensors, 
 and biversors, and establish general formulse for such functions or combi- 
 nations of biquaternions, which shall be symbolical extensions of earUer 
 results of this calculus ; thus, in any multiplication, the bitensor of a pro- 
 duct can only differ by its sign from the product of the bitensors; there 
 exists an important class of biquaternions, for v.'hich the bitensors vanish; 
 such biquaternions may be called nullific, or nullifieis, because each may 
 be associated (indeed in infinitely many ways), as multiplier or as multi- 
 plicand, with another factor diiferent from zero, so as to make their pro- 
 duct vanish (compare § cxn.) ; general expressions for the reciprocal of a 
 bi(]uaternion ; the reciprocal of a nuUifier is infinite ; a real quaternion has 
 generally a pair of imaginary, as well as a pair of real square roots ; hints 
 respecting the geometrical uLilily of the biquaternions, in transitions (for 
 example) from closed to unclosed surfaces of the second degree, and in 
 other imaginary deformations ; reference to a proposed Appendix to these 
 Lectures, containing a geometrical translation of an investigation so con- 
 ducted, respecting the inscriptioji of gauche polygons, in ellipsoids, and 
 ■ in hyperboloids, Articles 669 to 675 ; Pages 664 to 674. 
 
CONTENTS. Ixix 
 
 § cxiv. Brief outline of the quaternion analysis employed in such researches res- 
 pecting the inscriptions of polj'gons in surfaces (with which are connected 
 other problems respecting the circumscriptions of polyhedra) ; equation of 
 closure, resumed from § lv. ; distinction between the cases of even-sided 
 and odd-sided polygons ; if it be required to inscribe in a given sphere, or 
 other surface of the second order, a gauche polygon with an odd number 
 of sides, passing successively through the same number of given points, 
 there exists in general one real chord of solution, determining two real 
 OR imaginary positions of the initial point of the polygon ; but, if 
 the polygon be erew-sided, there are then (for the sphere, ellipsoid, or dou- 
 ble-sheeted hyperboloid) two real chords of real and imaginary solution; 
 for the single-sheeted hyperboloid (see Appendix), these two chords may 
 themselves become imaginary; in general they are reciprocal polars of 
 each other ; thus there may in general be inscribed, in a surface of the se- 
 cond order, two real or two imaginary gauche polygons, with an oofc? num- 
 ber of sides, passing through as many given and non- superficial points; 
 whereas, if the surface be non-ruled, and if the number of points and sides 
 be even, there may in general be inscribed two real, and two imaginary 
 polygons, which become all four real, or else all four imaginary, when we 
 pass to a ruled surface ; if we conceive that the inscribed gauche polygon 
 PPi . . . p» has n + 1 sides, of which only the first n are obliged to pass 
 through so many given and non-superficial points, Ai, . . . A,„ then the 
 closing side, or final chord, p„p, belongs to a certain system of right lines 
 in space, of which it is interesting to study the arrangement ; quaternion 
 f formulas connected therewith ; when the number n of the given points is 
 even, so that the number w + 1 of the sides of the polygon is odd, the 
 closing chords touch two distinct surfaces of the second order, which have 
 quadruple contact with the original surface, and with each other, and are 
 geometrically related to each other and to the given surface, as are three 
 single-sheeted hyperboloids which have two common pairs of generatrices; 
 when the number of the given points is odd, or of the sides of the polygon 
 even, then the envelope of the closing side consists of a pair of cones, which 
 are imaginary if the given surface be non-ruled, but may become real by 
 imaginary deformation, namely, by passing to the case of inscription in a 
 ruled surface ; in this last case, the lines on the surface, which are analo- 
 gous to lines of curvature, as being those linear loci of the initial point p, 
 which are bases of developable surfaces composed by corresponding sys- 
 tems of positions of the variable chord pp,;, are rectilinear generatrices of 
 the given surface ; these bases become imaginary, when we return to the 
 sphere, ellipsoid, or other non-ruled surface, as that in which the polygon is 
 to be inscribed ; when the number of given points is even, the tangents to 
 the two corresponding curves on the given surface, at any proposed point 
 p, are conjugate, being parallel to two conjugate diameters ; there exist 
 also certain harmonic relations between the lines and planes which enter 
 into this theory of inscription ; references to communications by the pre- 
 sent writer, on this subject, of which some have been already published, 
 (see also Appendix B), Ai-ticles 676, 677; Pages 674 to 678. 
 
l.xx 
 
 CONTENTS. 
 
 § cxv. More full discussion of the signification of an equation, namely, 
 
 Y. pa= pV. p/3, or V. ap = p\. I3p, 
 
 which had presented itself in the foregoing analysis ; this equation repre- 
 sents generally a certain curve of double curvature vfhich. is of the third 
 order, as being cut hy an arbitrary plane in three points, real or imagi- 
 nary ; this curve is the common intersection of a certain system of surfaces 
 of the second order ; it intersects the sphere p2 = _ i jn fj^Q real and two 
 imaginary points, namely, in the initial positions of the first corner of an 
 inscribed and even-sided polygon (§cxrv.), but it may be said also to in- 
 tersect the same sphere in two other imaginary points, at infinity ; if we 
 confine ourselves to reaZ vectors and quaternions, we can express a variety 
 of other geometrical loci by equations of remarkable simplicity ; interpre- 
 tations of the ten equations, 
 
 Vg = 0, Sg = 0, 8^=1, S? = -l, where 2 = (pa- 1)2; 
 
 with the same meaning of g, if /3 _i_ a, the equation Yq = ^ represents a 
 certain hyperbola ; if a^y denote three real and rectangular vectors, the 
 equation (y V. ap)'^ + (y V. ^p)^ = 1 represents a certain ellipse; the equa- 
 tion (S . ap'y-'r (yV. ap'Y^ 1 denotes the system of an ellipse and an hy- 
 perbola, with one common pair of summits, but situated in two rectan- 
 gular planes ; an equally simple equation can be assigned representing a 
 system of two ellipses, in two rectangular planes, having a common pair of 
 summits ; the equation tpicp = pKpi, or V. ipKp = 0, represents a system of 
 two rectangular right lines, bisecting the angles between t, k ; while the 
 equation tp;cp = ptpK, or = V. pV. ipa, represents a system of three rect- 
 angular lines, namely, these two bisectors, and a line perpendicular to 
 their plane ; example from the ellipsoid, equation V. vp = ; general equa- 
 tion of surfaces of the second order ; equation of Fresnel's wave-surface ; 
 general formulae for translating any equation in co-ordinates into an equa- 
 tion in quaternions, 
 
 j; = —iS.ip,y = —jS.jp, z — — kB . kp ; 
 
 other expressions for geometrical loci may be obtained, by regarding p as 
 the vector part of a variable quaternion q, which is obliged to satisfy some 
 given equation, while its scalar part w is variable ; formulae may be as- 
 signed which shall represent, respectively, on this plan, what may be called 
 a. full circle, and full sphere, .... Articles 678, 679 ; Pages 678 to 688. 
 
 § cxvi. Equation of the focal hyperbola, V. ijp .V. p6= (Y. i]9)^, resumed from 
 § Lxxxviii. ; proof of the adequacy of this owe equation to represent that 
 curve ; geometrical illustrations of the significations of the two constant 
 vectors rj and Q ; they are the two oblique co-ordinates of an umbilic of 
 the ellipsoid, referred to the asymptotes of the focal hj'perbola, when di- 
 rections as well as lengths are attended to ; other elementary geometrical 
 illustrations and confirmations of some of the results of earlier sections (es- 
 pecially of §§ Lxxxvi. to LXXXVIII.), chiefly as regards the equations in- 
 
CONTENTS. Ixxi 
 
 volving T], 9 ; additional calculations and interpretations, designed princi- 
 pally as exercises in quaternions ; introduction of the two new vectors, 
 
 Xi = p - 2 (j, + 9)-i S . 6I|0, £ = 2V. nOT {ri + 0)- 1, 
 
 with three other vectors X2, X3, X4, determined in terms of p by expres- 
 sions analogous to that for Xi ; we have the equations, 
 
 T (Xi - £) = 6 + 6- 1 S . ep, T (Xi + = 6 - 6- 1 S . £|0, 
 and therefore T (Xi - e) + T (Xi + t) = 2& ; 
 
 the locus of the extremity of the derived vector Xi is a certain ellipsoid of 
 revolution, with the mean axis 26 of the given ellipsoid for its major axis, 
 and with two foci on that axis of which the vectors are + £ ; if e de- 
 note the linear excentricity of this new ellipsoid, e = Tt, then 
 
 e2 = (a2 _ 62) (62 _ c2) {cfi - 6* + c^)- 1 ; 
 
 the four vectors, Xi, X2, X3, X4 terminate at four points, Li, L2, L3, I4, which 
 are the /bar corners of a quadrilateral, inscribed in a circle, of this de- 
 rived ellipsoid of revolution ; the two opposite sides, LiLj, 1-314, of this 
 plane quadrilateral, are respectively parallel to the two umbilicar diameters 
 of the original ellipsoid abc ; the two other and mutually opposite sides, 
 ■L3L3, L4L1, of the same inscribed quadrilateral, are parallel to the axes of 
 the two cylinders of revolution which can be circumscribed about the same 
 given ellipsoid (or to the asymptotes of the focal hyperbola) ; the former 
 pair of sides of the inscribed but varying quadrilateral intersect in a point 
 E (the termination of the vector p), of which the locus is the given ellip- 
 soid; for this and for other reasons it is proposed to name the new ellip- 
 soid of revolution the mean ellipsoid, and its foci the two medial foci 
 of the given ellipsoid abc, .... Articles 680 to 688 ; Pages 688 to 700. 
 
 § cxvii.* Many other geometrical applications may be made, of the same general 
 principles ; for example, if r be a vector tangential to a line of curvature, 
 then, with the significations of i, k, v in §§ lxxviii., lxxix., we have the 
 equations, 
 
 S . vr = 0, S. vTiTK = 0, giving r = UV. vi + UV. vk; 
 
 this reproduces the known theorem, that the lines of curvature on an ellip- 
 soid bisect at each point the angles between the circular sections; quater- 
 nions may also be employed to prove some theorems elsewhere published 
 by the present writer, respecting the curvature of a spherical conic, . . . 
 
 Article 689 ; Page 700. 
 
 Appendix A (referred to in § cxiii.), Pages 701 to 710. 
 
 Appendix B (respecting the results of § cxiv.), Pages 717 to 730. 
 
 Appendix C (containing some additional account of the analysis by which some 
 
 of those results were obtained), Pages 731 to the end. 
 
 [* The foregoing Analysis of the work into Sections did not occur to the author until it was too 
 late to be incorporated with the text : but it has been printed here, as seeming likely to be useful.] 
 
Ixxii 
 
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 74 
 
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 80 
 
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 81 
 
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 82 
 
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 32 
 
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 83 
 
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 84 
 
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 85 
 
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 35 
 
 
 
 — 
 
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 36 
 
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 t 87 
 
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 i 88 
 
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 89 
 
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 98 
 
 466 
 
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 — 
 
 
 
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 681 
 
 691 
 
ON QUATERNIONS. 
 
 LECTURE I. 
 
 Gentlemen, 
 
 In the preceding Lectures of the present Term, we 
 have taken a rapid view of the chief facts and laws of Astronomy, 
 its leading principles and methods and results. After some gene- 
 ral and preliminary remarks on the connexion between meta- 
 physical and physical science, we have seen how the observation 
 of the elementary phenomena of the Heavens may be assisted, 
 and rendered more precise, by means of astronomical instru- 
 ments, accompanied with astronomical reductions. An outline 
 of Uranography has been given; the laws of Kepler for the Solar 
 System have been stated and illustrated ; with the inductive evi- 
 dence from facts by which their truth may be established. It 
 has been shewn that these laws extend, not only to the Planets 
 known in Kepler's time, namely, Mercury, Venus, Mars, Jupi- 
 ter, and Saturn, with which our Earth must be enumerated, but 
 also to the various other planets since detected : to Uranus, to 
 Ceres, Pallas, Juno, and Vesta; and to those others of more 
 recent date, in the order of human knowledge, of which no fewer 
 than six have been found within the last two years and a half; 
 to Astrsea, Neptune, Hebe, Iris, Flora, and Metis : among which 
 Neptune is remarkable, as having had its existence foreshewn by 
 mathematical calculation, and Metis is interesting to us Irishmen, 
 as having been discovered at an Irish observatory. It has also 
 been shewn you that these celebrated laws of Kepler are them- 
 selves mathematically included in one still greater Law, with 
 which the name of Newton is associated : and that thus, as New- 
 
 B 
 
2 ON QUATERNIONS. 
 
 ton himself demonstrated, in his immortal work, the Principia, 
 the rules of the elliptic motion of the planets are consequences of 
 the principle of universal Gravitation, proportional directly to the 
 mass, and inversely to the square of the distance. With the help 
 of this great principle, or law, of Newton's, combined with proper 
 observations and experiments, — especially, with the Cavendish 
 experiment, as lately repeated by Baily, — not only have the 
 shape and size of the earth which we inhabit, but even (as you 
 have seen explained and illustrated) its very weight has been de- 
 termined ; the number of millions of millions of millions of tons 
 of matter, which this vast globe contains, has been (approx- 
 imately) assigned. And not only have the motions of that Earth 
 of our's around and with its own axis, and round the sun, been 
 established, but that great central body of our system, the Sun, 
 through the persevering application of those faculties which God 
 has given to man, has itself (as you have likewise seen) been 
 measured and weighed, with the line and balance of science. 
 
 2. Such having been our joint contemplations in this place, 
 before the adjournment of these discourses on account of the 
 Examinations for Fellowships, you may remember that it was an- 
 nounced that at our re-assembling we should proceed to the con- 
 sideration of a certain new mathematical Method, or Calculus, 
 which has for some years past occupied a large share of my own 
 attention, but which 1 have hitherto abstained from introducing, 
 except by allusion, to the notice of those who have honoured 
 here my lectures with their attendance. I refer, as you are aware, 
 to what 1 have called the calculus of quaternions, and have 
 applied to the solution of many geometrical and physical pro- 
 blems. However much this new calculus, or method, may natu- 
 rally have interested myself, there has existed, in my mind, until 
 the present time, a fear of seeming egotistical, if 1 should offer 
 to the attention of my hearers in this University an account of 
 such investigations or speculations of my own. Accordingly, 
 with the exception of a short sketch, in the year 1845, of the 
 application to spherical trigonometry of those fundamental con- 
 ceptions and expressions respecting Quaternions, which I had 
 been led to form in 1843, and had in the last mentioned year 
 communicated to the Royal Irish Academy, I have abstained 
 
LECTURE 1. 
 
 from entering on the subject in former courses of Lectures : — 
 unless it be regarded as an exception to this rule, that in the ex- 
 traordinary or supplementary Course which I delivered here, in 
 the winter of 1846, on the occasion of the theoretical discovery 
 of the distant planet Neptune, I ventured to introduce that theory 
 oi Hodographs, which, in the regular course for 1847, I after- 
 wards more fully developed ; and which had been suggested to 
 me as a geometrical interpretation, or construction, of some in- 
 tegrations of equations in physical astronomy whereto I had been 
 conducted by the Method of Quaternions. But since, on the 
 one hand, it has of late been formally announced (as it is stated to 
 me) that the Professor of Mathematics in this University intends 
 to lecture on that Method of mine in the winter of the present 
 year, in connexion, probably, with some of his own original re- 
 searches ; and to make it, or a part of it, one of the subjects of 
 his public Examination of the Candidates for Fellowship in the 
 summer of 1849 ; while, on the other hand, the theory itself has 
 been acquiring, under my own continued study, a wider exten- 
 sion, and perhaps also a firmer consistency : it seems to myself, — 
 and by some mathematical friends, among whom the Professor 
 just referred to is included, I am encouraged to think that it is 
 their opinion too, — that the time has arrived, when instead of its 
 being an obtrusion for me to state here, in the execution of my 
 own professorial office, my views respecting Quaternions, it 
 would, on the contrary, be rather a dereliction of my duty, or a 
 blameable remissness therein, if I were longer to omit to state 
 those views in this place, at least by sketch and outline. 
 
 3. And inasmuch as I am not aware that any one has hi- 
 therto professed to detect error in any of those geometrical and 
 physical theorems to which the Method has conducted me ; while 
 yet I cannot but perceive it to be the feeling of several persons, 
 among my mathematical friends and acquaintances, that in the 
 existing state of the published expositions of my own views upon 
 the subject, some degree of obscurity still hangs over its logical 
 and metaphysical jt?;-mc?/)/e5 ; so that the admitted correctness of 
 the results of this new Calculus may appear, even to candid and 
 not unfriendly lookers-on, to be, in some sense, accidental, rather 
 than necessary, so far as the conceptions and reasonings have 
 
 B 2 
 
4 ON QUATERNIONS. 
 
 hitherto been formally set forth by me : it therefore seems to be, 
 upon the whole, the most expedient plan which can be adopted 
 on the present occasion, that I should state, as distinctly and as 
 fully as my own limited powers of expression, and as your re- 
 maining time in this Course will allow, the Jbntal thoughts, the 
 primal views, the initial attitudes of mind, from which the others 
 flow, and to which they are subordinated. And if, in the fulfil- 
 ment of this purpose, the adoption of a somewhat metaphysical 
 style of expression on some fundamental points may be at least 
 forgiven me, as inevitable, still more may I look to be excused, 
 if not approved of, should I take, even by preference, my illus- 
 trations from Astronotny, in this Supplementary Course of Lec- 
 tures, which, as you know, arises out of, and is appended to a 
 Course more strictly and properly astronomical. 
 
 4. The object which I shall propose to myself, in the Lec- 
 ture of this day, is the statement of the significations, at least the 
 primary significations, which I attach, in the Calculus of Qua- 
 ternions, to the four following familiar marks of combination of 
 symbols, 
 
 + - X ^ 
 
 which marks, or signs, are universally known to correspond, in 
 arithmetic and in ordinary algebra, to the four operations known 
 by the names of Addition, Subtraction, Multiplication, and 
 Division. The new significations of these four signs have a 
 sufficient analogy to the old ones, to make me think it convenient 
 to retain the signs themselves; and yet a sufficient distinction 
 exists, to render a freliminary comment not superfluous t or 
 rather it is indispensable that as clear a definition, or at least ex- 
 position, of the precise force of each of these old marks, used in 
 new senses, should be given, as it is in my power to give. Per- 
 haps, indeed, I may not find it possible, to-day, to speak with 
 what may seem the requisite degree oi fulness of such exposition, 
 of more than the two first of these four signs ; although I hope 
 to touch upon the tvvo last of them also. 
 
 5. First, then, 1 wish to be allowed to say, in general terms 
 (though conscious that they will need to be afterwards particula- 
 rized), that I regard the two connected but contrasted marks or 
 
 ^^^"^' + and -, 
 
LECTURE I. , O 
 
 as being respectively and primarily characteristics of the syn- 
 thesis and ANALYSIS of a state of a Progression^ according as 
 this state is considered as being derived from^ or compared with^ 
 some other state of that progression. And, with the same kind 
 of generality of expression, 1 may observe here that I regard in 
 like manner the other pair of connected and contrasted marks 
 already mentioned, namely, 
 
 X and -^, 
 (when taken in what I look upon as their respectively p'zwary 
 significations), as being signs or characteristics of the correspon- 
 ding SYNTHESIS and ANALYSIS of a step, in any such progression 
 of states, according as that step is considered as derived from, or 
 compared with, some other step in the same progression. But I 
 am aware that this very general and preliminary statement can- 
 not fail to appear vague, and that it is likely to seem also obscure, 
 until it is rendered precise and clear by examples and illustra- 
 tions, which the plan of these Lectures requires that I should 
 select from Geometry, while it allows me to clothe them in an 
 Astronomical garb. And 1 shall begin by endeavouring thus to 
 illustrate and exemplify the view here taken of the sign -, which 
 we may continue to read, as usual, minus, although the opera- 
 tion, of which it is now conceived to direct the performance, is 
 not to be confounded with arithmetical, nor even, in all respects, 
 with common algebraical subtraction. 
 
 6. I have said that 1 regard, primarily, this sign, 
 
 -, or Minus, 
 
 as the mark or characteristic of an analysis of one state of a pro- 
 gression, when considered as compared with a^ioMe/- state of that 
 progression. To illustrate this very general view, which has 
 been here propounded, at first, under a metaphysical rather than 
 a mathematical form, by proceeding to apply it under the limi- 
 tations which the science of geometry suggests, let space be 
 now regarded as the field of the progression which is to be stu- 
 died, and POINTS as the states of that progression. You will 
 then see that in conformity with the general view already enun- 
 ciated, and as its geometrical particularization, I am led to regard 
 the word " Minus," or the mark -, in geometry, as the sign or 
 
b ON QUATERNIONS. 
 
 characteristic of the analysis of one geometrical position (in 
 space), as compared with another (such) position. The compa- 
 rison of one mathematical point with another^ with a view to the 
 determination of what may be called their ordinal relation^ or 
 their relative position in space, is in fact the investigation of the 
 GEOMETRICAL DIFFERENCE of the two points Compared, in that 
 sole respect, namely, position, in which two mathematical points 
 can differ from each other. And even for this reason alone, 
 although I think that other reasons will offer themselves to your 
 own minds, when you shall be more familiar with this whole 
 aspect of the matter, you might already grant it to be not unna- 
 tural to regard, as it has been stated that I do regard, this study 
 or investigation of the relative position of two points in space, as 
 being thsit primary geometrical operation which is analogous to 
 algebraic subtraction, and which I propose accordingly to denote 
 by the usual mark (-) of the well-known operation last men- 
 tioned. Without pretending, however, that 1 have yet exhibited 
 sufficiently conclusive grounds for believing in the existence of 
 such an analogy, I shall now proceed to illustrate, by examples, 
 the modes of symbolical expression to which this belief, or view, 
 conducts. 
 
 7. To illustrate first, by an astronomical example, the con- 
 ception already mentioned, of the analysis of one geometrical 
 position considered with reference to another, I shall here write 
 down, as symbols for the two positions in space which are to be 
 compared among themselves, the astronomical signs, 
 
 © and J ; 
 M'hich represent or denote respectively the sun and earth, and are 
 here supposed to signify, not the masses, nor the longitudes, of 
 those two bodies, nor any other quantities or magnitudes con- 
 nected with them, but simply their situations, or the positions 
 of their centres, regarded as mathematical points in space. To 
 make more manifest to the eye that these astronomical signs are 
 here employed to denote points or positions alone, I shall write 
 under each a dot, and under the dot a Roman capital letter, 
 namely, a for the earth, and b for the sun, as follows : 
 
 © 4 
 
 (Fig. 1.) 
 
LECTURE I. 
 
 and shall suppose that the particular operation of what we have 
 already called analysis, using that word in a very general and 
 rather in a metaphysical than in a mathematical sense, which is 
 now to be performed, consists in the proposed investigation of 
 the position of the sun, b, with respect to the earth, a ; the latter 
 being regarded as comparatively simple and known ; but the 
 former as complex, or at least unknown and undetermined ; and 
 a relation being sought, which shall connect the one with the 
 other. This conceived analytical operation is practically and 
 astronomically performed, to some extent, whenever an observer, 
 as for example, my assistant (or myself), at the Observatory of 
 this University, with that great circular instrument of which you 
 have a model here, directs a telescope to the sun : it is completed, 
 for that particular time of observation, when, after all due micro- 
 metrical measurements and readings, after all reductions and cal- 
 culations, founded in part on astronomical theory, and on facts 
 previously determined, the same observer concludes and records 
 the geocentric right ascension and declination, and (through the 
 semidiameter) the radius vector (or distance) of the sun. In 
 general, we are to conceive the required analysis of the position 
 of the ANALYZAND POINT B, with respect to the analyzer 
 POINT a, to be an operation such that, if it were completely per- 
 formed, it would instruct us not only in what direction the point 
 B is situated with respect to the point a ; but also, at what 
 DISTANCE from the latter the former point is placed. Regarded 
 as a guide, or rule for going (if we could go) from one point to 
 the other, — which rule of transition would, however (according 
 to the general and philosophical, rather than technically mathe- 
 matical distinction between analysis and synthesis, on which this 
 whole exposition is founded), be itself xsither of a synthetic than 
 of an analytic character, — the result of this ordinal analysis 
 might be supposed to tell us in the^r^^ place how we should set 
 out : which conceived geometrical act, oi setting out in a suitable 
 direction, corresponds astronomically to the pointing, or direct- 
 ing of the telescope, in the observation just referred to. And the 
 same synthetic rule, or the same result of a complete analysis, 
 must then be supposed also to tell us, in the second place, how 
 FAR we ought to GO, in order to arrive at the sought point 
 
O ON QUATERNIONS. 
 
 B, after thus setting out from the given point a, in the proper 
 direction of progress (this direction being, of course, here con- 
 ceived to be preserved unaltered) : which latter part of the sup- 
 posed guidance or information corresponds to the astronomical 
 inquiry, how far off\% the sun, or other celestial object, at which 
 we are now looking, with a telescope properly set ? 
 
 8. Now the whole sought result of this (conceived) com- 
 plete analysis, of the position b with respect to the position a, 
 whether it be regarded analytically as an ordinal relation, or 
 synthetically as a rule of transition, is what I propose to denote, 
 
 or signify, by the symbol 
 
 B - a, 
 
 formed by inserting the sign minus between the two separate 
 symbols of the two points compared ; the symbol of the ana- 
 ly Zand point b being written to the left of the mark -, and the 
 symbol of the analyzer 'point a being written to the right of the 
 same mark ; all which I design to illustrate by the following 
 fuller diagram, 
 
 ® B - A i ,_. 
 -= (Fig. 2.) 
 
 B A 
 
 where the arrow indicates the direction in which it would be ne- 
 cessary to set out from the analyzer point, in order to reach the 
 analyzand point; and a straight line is drawn to represent or 
 picture the progression, of which those^om/5 are here conceived 
 to be, respectively, the initial and final states. We may then, 
 as often as we think proper, paraphrase (in this theory) the geo- 
 metrical symbol b - a, by reading it aloud as follows, though it 
 would be tedious always to do so: " b analyzed with respect to 
 A, as regards difference of geometrical position." But for com- 
 mon use it may be sufficient (as already noticed) to retain the 
 shorter and more familiar mode of reading, " b minus a ;" re- 
 membering, however, that (in the present theory) the diffe- 
 rence thus originally ox primarily indicated is one of position, 
 and not of magnitude : which, indeed, the context (so to speak) 
 will always be sufficient to suggest, or to remind us of, when- 
 ever the symbols a and b are recognised as being what they are 
 here supposed to be, namely, signs oi mathematical points. 
 
LECTURE I. 9 
 
 9. Had we chosen to invert the order of the comparison, or 
 of the analysis of these two positions a and b, as related to each 
 other, regarding the sun b as the given or known point, and the 
 earth a as the sought or unknown one; we should have in that 
 case done what in fact astronomers do in those investigations re- 
 specting the solar system, in which the motion of the earth as a 
 planet about the sun, in obedience to Kepler's laws, is treated as 
 an established general fact which it remains to argue from, and 
 to develope into the particular consequences required for some 
 particular question : whenever, in short, they seek rather the 
 heliocentric 'position of the earthy than the geocentric position of 
 the sun ; and so propose to analyze what has been here called a 
 with respect to b, rather than b with respect to a. And it would 
 then have been proper, on the same general plan of notation, to 
 have written the opposite symbol a - b, instead of the former 
 symbol b - a ; and also to have inverted the arrow in the dia- 
 gram (because we now conceive ourselves as going rather from 
 the sun to the earth, than from the earth to the sun) ; which 
 diagram would thus assume the form, 
 
 © a-b 4 
 
 =- (Fig. 3.) 
 
 b a 
 
 Thus b - A and a-b are symbols of two opposite (or mutually 
 inverse) ordinal relations, corresponding to two opposite steps 
 or transitions in space, and mentally discovered, or brought into 
 notice, by these two opposite modes of analyzing the relative po- 
 sition of one cotnmon pair of mathematical points, a and b ; of 
 which two opposite modes of ordinal analysis in space, with the 
 two inverse relations thence resulting, the mutual connexion and 
 contrast may be still more clearly perceived, if we bring them 
 into one view by this diagram : 
 
 © B - A S 
 
 ^ (Fig. 4.) 
 
 B A-B A 
 
 10. Using aform of wouBS, suggested by this mode of sym- 
 bolical notation, 1 should not think it improper, and it would 
 certainly be at least consistent with the manner in which the sub- 
 ject is here viewed, to say that 
 
10 ON QUATERNIONS. 
 
 The Sun's ordinal relation to the Earth in space, 
 or, somewhat move concisely, that what is called in astronomy, 
 « The Suns Geocentric Position" (including distance), 
 is expressed by, and is (in that sense) equivalent, or 
 (with the here proposed use of Minus) symbolically equal to 
 " The Sun's (absolute) Position in space, 
 Minus the Earth's (absolute) Position." 
 And then, of course, we should be allowed, on the same plan, to 
 say, conversely, that 
 
 " The Earth's Heliocentric Position" is equivalent or equal to 
 " The Earth's Position in space, minus the Sun's Position." 
 In the same new mode of speaking, the 
 
 " Position of Venus (in space), minus the Position of the Sun," 
 would be a form of words equivalent to the usual phrase, 
 
 " Heliocentric Position of Venus." 
 And it is evident that examples of this sort might easily be mul- 
 tiplied. 
 
 II. According, then, to the view here taken of the word 
 <' Minus," or of the sign -, if employed, as we propose to employ 
 it, in pure or applied geometry, this word or sign will denote 
 primarily an ordinal analysis in space ; or an analysis (or exa- 
 mination) of the position of a mathematical point, as compared 
 with the position of another such point. And because, according 
 to the foregoing illustrations, this sign or mark (Minus) directs 
 us to DRAW, or to conceive as drawn, a straight line connecting 
 the two points, which are proposed to be compared as to their 
 relative positions, it might, perhaps, on this account be called 
 the SIGN OF TRACTION. If wc wish, however, to diminish, as 
 far as possible, the number of new terms, we may call it still, as 
 usual, the sign of subtraction ; remembering only, that, in the 
 view here proposed, there is no original (nor necessary) reference 
 whatever to any subtraction of one magnitude from another. 
 Indeed, it is well known to every student of the elements of 
 algebra that the word Minus, and the sign -, are, in those ele- 
 ments also, used very frequently to denote an operation which is 
 
LECTURE I. 11 
 
 hy no means identical with the taking away of a partial from a 
 total magnitude, so as to find the remaining part : thus every 
 algebraist is familiar with such results as these, that 
 (Negative Four) Minus (Positive Three) Equals (Negative 
 
 Seven) ; 
 
 where, if mere magnitudes or quantities were attended to, and 
 the adjectives " Positive and Negative" dropped, or neglected, 
 and not replaced by any other equivalent words or marks, the 
 resulting number " seven" would represent the (arithmetical) 
 sum, and not the (arithmetical) difference, of the given numbers 
 " four" and " three." And as, to prevent any risk of such con- 
 fusion with a merely arithmetical difference, or with the result of 
 a merely arithmetical subtraction, it is usual to speak of an alge- 
 braical difference and oi algebraical subtraction ; and thus to say, 
 for example, that " Negative Seven" is the " algebraical diffe- 
 rence" of " Negative Four" and "Positive Three;" oris ob- 
 tained or obtainable by the " algebraical subtraction" of the 
 latter from the former: so may (I think) that other and more 
 geometrical sort of subtraction, which has been illustrated in this 
 day's Lecture, be called, not inconveniently, for the sake of re- 
 cognising a farther distinction or departure from the merely 
 popular use of the word (subtraction), and on account of its con- 
 nexion with a new and enlarged system oi symbols in geometry, 
 the SYMBOLICAL SUBTRACTION of A from B: and the resulting sym- 
 bol of the ordinal relation of the latter point to the former, namely, 
 the symbol b - a, may conveniently be called, in like manner, a 
 SYMBOLICAL DIFFERENCE. It is in fact, as has been already 
 remarked, in this new system of symbols, an expression for what 
 may very naturally be called the geometrical difference of the two 
 points B and a ; that is to say, it is (in this system) a symbol for 
 the difference of the positions of those two mathematical points 
 in space ; this difference being regarded as geometrically con- 
 structed, represented, or pictured, by the straight line drawn 
 from A to B, which line is here considered as having (what it has 
 in fact) not only a determined length, but also a determined direc- 
 tion, when the two points, a and b, themselves, are supposed to 
 have two distinct and determined (or at least determinable) 
 positions. 
 
12 ON QUATERNIONS. 
 
 12. For my own part I cannot conceal that I hold it to be of 
 great and evenjuiidanientalimportance, to regard Pure Mathe- 
 matics as being primarily the science of order (in Time and 
 Space), and tiot primarily the science of magnitude : if we 
 would attain to a perfectly clear and thoroughly self-consistent 
 view of this great and widely-stretching region, namely, the ma- 
 thematical, of human thought and knowledge. In mathematical 
 science the doctrine of magnitude, or of quantity, plays indeed a 
 very important part, but not) as I conceive, the most important 
 one. Its importance is secondary and derivative, 7iot pri- 
 mary and original, according to the view which has long ap- 
 proved itself to my own mind, and in entertaining which I think 
 that I could fortify myself by the sanction of some high autho- 
 rities : although the opposite view is certainly more commonly 
 received. If any one here should regard that opposite view, 
 which refers all to magnitude, as the right one ; and should find 
 it impossible, or think it not worth the effort, to suspend even for 
 a while the habit of such a reference, he may still give for a mo- 
 ment a geometrical interpretation to the symbol b - a, not quite 
 inconsistent with that which has been above proposed, by regard- 
 ing it as an abbreviation for this other symbol bo - ao, where 
 AO and BO are lines, namely, the distances of the two points a 
 and B from another point o, assumed on the same indefinite right 
 line as those two points a, b, and lying beyond a with respect to 
 B, or situate upon the line ba prolonged through a, as in this 
 diagram : 
 
 ©(Bo-Ao) J n 
 
 ^^^ (Fig. 5.) 
 
 B B - A A O 
 
 Here the point o may be conceived, astronomically, to represent 
 a superior planet, for example, Jupiter ("4), in opposition to the 
 Sun (and in the Ecliptic) ; and it is evident that if we knew, for 
 such a configuration, the distance ao in millions of miles, of the 
 Earth from Jupiter, and also the greater distance bo of the Sun 
 from the same superior planet at that time, we should only have 
 to subtract, arithmetically, the former distance ao from the latter 
 distance bo, for the purpose of findi-ng the distance bo -ao, or 
 BA, in millions of miles, between the earth and the sun : which 
 
LECTURE I. 13 
 
 distance, there might thus be some propriety or convenience, on 
 this account, in denoting by the symbol b - a. That symbol, 
 thus viewed, might even be conceived to suggest a reference to 
 direction as well as distance ; because the supposed line oa, pro- 
 longed through A, would in the figure tend to B ; or, in astrono- 
 mical language, the jovicentric place of the Earth, in the 
 configuration supposed, would coincide, on the celestial sphere, 
 with the geocentric -place of the Sun. But I am far indeed from 
 recommending to you to co7nplicate the contemplation of the re- 
 lative position of the two points a and b, at this early stage of the 
 inquiry, by any reference of this sort to any third point o, thus 
 foreign and arbitrarily assumed. On the contrary, I would advise, 
 or even request you, for the present, to abstain from making, in 
 your own minds, such a reference to a.wy foreign point; and to 
 accompany me, for some time longer, in considering only the in- 
 ternal relation of position of the two points, a and b, them- 
 selves : agreeing to regard this internal and ordinal relation of 
 these two mathematical points in space (to whatever extent it 
 may be found useful, or even necessary hereafter, to call in the 
 aid of other points, or lines, or planes, for the purpose of more 
 fully studying, and, above all, of applying that relation), as being 
 sufficiently denoted, at this stage, by one or other of the two 
 symbols, b - a or a - b, according as we choose to regard b or a 
 as the analyzand point, and a or b as the analyzer. 
 
 13. 1 ask you then to concede to me, at least provisionally, 
 and for a while, the privilege of employing this unusual mode of 
 geometrical notation, together with the new mode of geome- 
 trical INTERPRETATION above assigned to it : which modes, after 
 all, do not contradict auythmg previously established in scienti- 
 fic language, nor lead to any real risk of confusion or of ambi- 
 guity, in geometrical science, by attaching any new sense to an old 
 sign : since here the sign itself (b - a), as well as the significa- 
 tion, is new. The component symbol " minus" is indeed old, but 
 it is used here in a new connexion with other elementary sym- 
 bols ; and the new context, hence arising, gives birth to a new 
 COMPLEX SYMBOL, (b - a), in fixing the sense of which we may 
 and must be guided by analogy, and general considerations ; 
 
14 ON QUATERNIONS. 
 
 old usages and received definitions failing to assign any deter- 
 mined signification to the new complex symbol thus produced. 
 The interpretation which I propose does no more than invest 
 with sense, through an explanation which is new, what had 
 seemed before to be devoid of sense. It only gives a meaning, 
 where none had been given before : namely, to a symbolical 
 expression of the form " Point minus Point." This latter^oz-jw 
 of words, and the geometrical notation b - a to which it cor- 
 responds (a and B being still used as signs of mathematical 
 points), had hitherto, according to the received and usual modes 
 of geometrical interpretation, no meaning : but you will, per- 
 haps, admit that these two connected forms of spoken and 
 written expression were, for that very reason, only the more free 
 to receive any new and definitional sense : especially one which 
 you have seen to admit of beng suggested by so simple an ana- 
 logy to subtraction as that which the conception of difference in- 
 volves. It will, however, of course be necessary, for consistency, 
 that we carefully adhere to such new interpretation, when it has 
 once been by definition assigned : unless and until we find rea- 
 sons (if such reasons shall ever be found) which may compel its 
 formal abandonment. 
 
 14. You see, then, to recapitulate briefly the chief part of 
 what has been hitherto said, that I invite you to conceive the 
 RELATIVE rosiTioN of any sought point b of space, when com- 
 pared with any given point a, as being (in what appears to me 
 to be a very easily intelligible and simply symbolizable sense) 
 the geometrical difference of the absolute positions of 
 those two mathematical points: and that I propose to denote it, 
 in this system of symbolical geometry, by writing " the symbol 
 of the sought point, minus the symbol of the given point." Such 
 is, in my view, the analytic aspect of the compound symbol 
 
 B - A, 
 
 if the component symbols a and b be still understood to denote 
 points: such is the primary signification which I attach in geo- 
 metry to the interposed mark -, when it is regarded as being 
 what I have already called, in general terms, a characteristic 
 
 OF ORDINAL ANALYSIS. 
 
LECTURE I. 15 
 
 15. But as you have already also partly seen, the same 
 symbol, 
 
 B - A, 
 
 may be viewed in a synthetic aspect also. It may be thought 
 of, not only as being the result of a past analysis, but also as 
 being the guide to a future synthesis. It may be regarded as 
 not merely answering, or as denoting the answer, to the question : 
 In what Position is the point b situated with respect to the 
 point A ? but also this other, which indeed has been already 
 seen to be only the former question differently viewed: By what 
 Transition may b be reached, if we set oUt from a ? — And to 
 this other question also, or to this other view of the sameyowto/ 
 Question, where, I consider the same symbol, b - a, to be a 
 fit general representation of the Answer : it being reserved for 
 the context to decide, whenever a decision may be necessary, 
 which of these two related although contrasted views is taken 
 at any one time, in any particular investigation. In its synthetic 
 aspect, then, I regard the symbol b - a as denoting " the step 
 to B from A :" namely, that step by making which, fro7n the 
 given point a, we should reach or arrive at the sought point b ; 
 and so determine, generate, mark, or construct that point. 
 This step (which we shall always suppose to be a straight line) 
 may also, in my opinion, be properly called a vector ; or more 
 fully, it may be called " the vector of the point B,from the point 
 a :" because it may be considered as having for its office, func- 
 tion, work, task, or business, to transport or carry (in Latin, 
 vehere) a moveable point, from the given or initial position a, to 
 the sought or final position b. Taking this view, then, of the 
 symbol b - a, or adopting now this synthetic interpretation of it, 
 and of the corresponding form of words, we may say, generally, 
 for any such conceived rectilinear transport of a moveable point 
 in space, that 
 
 " Step equals End of Step, minus Beginning of Step ;" 
 or may write : 
 
 " Vector = (End of Vector) - (Beginning of Vector)." 
 
 16. Thus, in astronomy, whereas, by the mode oi analytic 
 intei'pretation already explained, the phrase, 
 
16 ON QUATERNIONS. 
 
 " Sun's Position minus Earth's Position," 
 has been regarded (in § 10) as equivalent to the more usual form 
 of words, " Sun's Geocentric Position" (including geocentric dis- 
 tance) ; we shall now be led, by the connected mode of synthetic 
 interpretation ]\x%t mentioned, to regard the same spoken phrase, 
 or the written expression, © - J (where the two astronomical 
 marks, © and J , are still supposed to be used to denote the si- 
 tuations alone of the two bodies which they indicate), as being 
 equivalent, in this other view of it, to what may be called the 
 
 " Sun's Geocentric Vector:" 
 which differs from what is called in astronomy the 
 " Geocentric Radius-Vector of the Sun," 
 by its including direction, as well as length, as an element 
 in its complete signification. In like manner, that equally long 
 but opposite line, which may be called, in the same new mode of 
 speaking, the " Earth's Heliocentric Vector," may be denoted 
 by the opposite symbol, S - ®, or expressed by the phrase, 
 *' Earth's Position, minus Sun's Position ;" the Heliocentric Vec- 
 tor of Venus will be, on the same plan, symbolically equal or 
 equivalent to the Position of Venus minus the Position of the 
 Sun: and similarly in other cases. 
 
 17. To' illustrate more fully the distinction which was just 
 now briefly mentioned, between the meanings of the " Vector" 
 and the " Radius Vector" of a point, we may remark that the 
 Radius- Vector, in astronomy, and indeed in geometry also, 
 is usually understood to have only length ; and therefore to be 
 adequately expressed by a single number, denoting the magni- 
 tude {or length) of the straight line which is referred to by this 
 usual name (radius- vector), as compared with the magnitude of 
 some standard line, which has been assumed as the unit of length. 
 Thus, in astronomy, the Geocentric Radius- Vector of the Sun 
 is, in its mean value, nearly equal to ninety-five millions of miles : 
 if, then, a million of miles be assumed as the standard or unit of 
 length, the sun's geocentric radius-vector is equal (nearly) to, 
 or is (approximately) expressible by, the number ninety-Jive : 
 in such a manner that this single number, 95, with the imit here 
 supposed, is (at certain seasons of the year) Si full, complete, and 
 
LECTURE I. 17 
 
 adequate representation or expression for that known radius- 
 vector of the sun. For it is usually the sun itself (or more 
 fully the position of the Sun's centre), and not the Sun's radius- 
 vector, which is regarded as possessing also certain other {polar) 
 co-ordinates of its own, namely, in general, some two angles, 
 such as those which are called the Sun's geocentric right-ascen- 
 sion and declination ; and which are merely associated with the 
 radius-vector, but 7iot inherent therein, nor belonging thereto ; 
 just as the radius-vector is itself in turn, associated with the 
 right ascension and declination, but not includedin them. Those 
 two angular co-ordinates (or some data equivalent to them) are 
 indeed required to assist in the complete determination of the 
 geocentric position of the sun itself : but they are not usually 
 considered as being in any manner necessary for the most com- 
 plete determination, or perfect numerical expression, of the 
 Sun's radius-vector. But in the new mode of speaking which 
 it is here proposed to introduce, and which is guarded from con- 
 fusion with the older mode by the omission of the word " ra- 
 dius," the vector of the sun has (^itself) direction, as well as 
 length. It is, therefore, not sufficiently characterized by any 
 SINGLE number, such as 95 (were this even otherwise rigorous) ; 
 but REQUIRES, for its complete numerical expression, a 
 system of three numbers; such as the usual and well-known 
 rectangular or polar co-ordinates of the Sun or other body or 
 point whose place is to be examined : among which one may 
 6e what is called the radius-wectov, but if so, that radius must 
 (in general) be associated with two other polar co-ordinates, 
 or determining numbers of some kind, before the vector can be 
 numerically expressed. A vector is thus (as you will afterwards 
 more clearly see) a sort of natural triplet (suggested by 
 Geometry) : and accordingly we shall find that quaternions 
 offer an easy mode of symbolically representing every vector by a 
 trinomial form (ix+jy+kz) ; which form brings the conception 
 and expression of such a vector into the closest possible connexion 
 with Cartesian and rectangular co-ordinates. 
 
 18. Denoting, however, for the present, a vector of this sort, 
 or a rectilinear step in space from one point a to another point b, 
 not yet by any such trinomial or triplet form, but simply (for 
 
 c 
 
18 ON QUATERNIONS. 
 
 conciseness) by a single and small Roman letter, such as a ; and 
 proceeding to compai'e, or equate, these two equivalent expres- 
 sions, or equi significant symbols, a and b - a ; we are conducted 
 
 to the EQUATION, 
 
 B - A = a ; 
 which is thus to be regarded as here implying merely that we 
 have chosen to denote, concisely, by the simple symbol, or single 
 letter, a, the same step, or vector, which has also been other- 
 wise denoted, less briefly, but in some respects more fully and 
 expressively, by the complex symbol b - a. Such is, at least, 
 the synthetic aspect under which this equation here presents it- 
 self ; but we may conceive it to occur also, at another time and in 
 another connexion, under an analytic aspect ; namely, as signify- 
 ing that the simple symbol a was used to denote concisely the 
 same ordinal relation of position, which had been more fully 
 denoted by the complex symbol b - a. Or we may imagine the 
 equation offering itself under a mixed (analytic and synthetic) 
 aspect; and as then expressing the perfect correspondence which 
 may be supposed to exist between that relative position of the 
 point B with respect to the point a, which was originally indi- 
 cated by B - A, and that rectilinear transition, or step, from a to 
 B, which we lately supposed to be denoted by a. Between these 
 different modes of interpretation, the context would always be 
 found sufficient to decide, whenever a decision became necessary. 
 But I think that we shall find it more convenient, simple, and 
 clear, during the remainder of the present Lecture, to adhere to 
 the synthetic view of the equation b - a = a; that is, to regard it 
 as signifying that both its members, b - a and a, are symbols for 
 one common step, or vector. And generally I propose to employ, 
 henceforth, the small Roman or Greek letters, a, b, a', &c., or a, j3, 
 a', &c., with or without accents, as symbols of steps, or of vectors. 
 19. But at this stage it is convenient to introduce the employ- 
 ment of another simple notation, which shall more distinctly and 
 expressly recognise and mark that synthetic character which we 
 have thus attributed to a, considered as denoting the step from 
 a to B ; in virtue of which synthetic character we have regarded 
 the latter point b as constructed, generated, determined, or 
 brought into view, by applying to, or performing on, the former 
 
LECTURE I. 19 
 
 point A, that act of vection or of transport, in which the 
 agent or operator is the vector denoted by a. We require a 
 SIGN OF vection: a characteristic of the operation of ordinal 
 synthesis, by which we have conceived a sought position b in 
 space to be constructed, as depending on Sl given position a, with 
 the help of a given vector, or ordinal operator, a, of the kind con- 
 sidered above. And such a characteristic of ordinal syn- 
 thesis, or sign of vection, is, on that general plan which was 
 briefly stated to you early to-day (in art. 5), supplied by the mark 
 +, or by the word Plus, when used in that new sense which has 
 already been referred to in this Lecture, and which may be re- 
 garded as suggested by Algebra, though it cannot (strictly speak- 
 ing) be said to be borrowed from Algebra, at least as Algebra is 
 commonly viewed. For we shall thus be led to write, as another 
 and an equivalent form of the recent equation b - a= a, this other 
 equation, in which Plus is introduced, and which is, in ordinary 
 Algebra also, a transformation of the equation lately written : 
 
 B = a + A ; 
 
 while yet, in conformity with what has been already said, we shall 
 now regard it as being the primary signification of this last equa- 
 tion, or formula, that " the position denoted by b may be 
 REACHED (and, in that sense, constructed), by making the 
 transition denoted by z,, from the position denoted by a." 
 
 20. We shall thus be led to say or to write generally, with 
 this (which is here regarded as being the) primary signification 
 of Plus in Geometry, that for any vector or rectilinear step in 
 space, 
 
 " Step + Beginning of Step = End of Step ;" 
 
 or, " Vector + Beginning of Vector = End of Vector:" 
 
 the mark + being in fact here regarded, by what has been already 
 said, as being primarily the sign of vection, or the characteristic 
 of the application of a step, or of a vector, to a given point con- 
 sidered as the Beginning (of the step, or vector), so as to generate 
 or determine another point considered as the End. In relation 
 to astronomy, this phraseology will allow us to say that 
 
 *' Sun's Position = Sun's Geocentric Vector + Earth's position ;" 
 
 c 2 
 
20 ON QUATERNIONS. 
 
 and the assertion is to be thus interpreted : that if a straight line, 
 agreeing in length and in direction with the line or step in space 
 which we have called in this Lecture the Suns Geocentric Vector, 
 were applied to the position occupied by the Earth, so as to begin 
 there, this line would terminate at the Sun. In exactly the same 
 way, we may say that the " Position of Venus in space" is sym- 
 bolically expressible as the " Heliocentric Vector of Venus, Plus 
 the Position of the Sun in Space ;" or as the " Geocentric Vec- 
 tor of Venus, plus the Position of the Earth ;" and similarly in 
 other cases. 
 
 21. All this, as you perceive, is very simple and intelligible; 
 nor can it ever lead you into any difficulty or obscurity, if you 
 will only consent to use from the outset, and will take pains to 
 remember that you use, the signs in the way which I propose ; 
 although that way may not be, or rather is certainly not, alto- 
 gether the same with that to which you are accustomed. Yet 
 you see that it is not in contradiction to any received and estab- 
 lished use of symbols in Geometry, precisely because no meaning 
 is usually attached to any expression of the form, " Line plus 
 point." (Compare 13). Such an expression would be simply un- 
 meaning, according to common usage ; in short, it would be 
 notisense : but I ask you to allow me to make it sense, by giving 
 to it an INTERPRETATION ; which must indeed remain so far a 
 DEFINITION, as that you may refuse to accompany me in assign- 
 ing to the expression in question the signification here proposed. 
 Yet you see that I have sought at least to present that definition, 
 or that interpretation, as divested of a purely arbitrary character ; 
 by shewing that it may be regarded as the mental and symbolic 
 counterpart of another definitional interpretation, which has al- 
 ready been assigned in this Lecture for another form of spoken 
 and written expression ; namely, for the form, " Point minus 
 Point :" which would, according to common usage, be exactly 
 as unmeaning, not more so, and not less, than the other. If you 
 yield to the reasons, or motives of analogy, which have been already 
 stated, or suggested, for treating the Difference of two Points 
 as a Line, it cannot afterwards appear surprising that you should 
 be called upon to treat the Sum of a Line and Point, as being 
 another Point. 
 
 22. Most fully do I grant, or rather assert and avow, that the 
 
LECTURE I. 21 
 
 'primary signification which I thus, propose for + in Geometry, is 
 altogether distinct from that of denoting the operation of com- 
 bining two partial magnitudes.^ in such a manner as to make up 
 one total magnitude. But surely every student of the elements 
 of Algebra is perfectly ^mz'/mr with another use ofplus^ which 
 is not less distinct from such merely quantitative aggregation, or 
 simple arithmetical addition. When it is granted, as you all 
 know it to be, that " (Negative Seven) + ( Positive Three) = (Ne- 
 gative Four)," where the mark + is still reac? as " Plus ;" and 
 when this operation of combination is commonly called, as you all 
 know that it is called, " Algebraical Addition," and is said to 
 produce an " algebraic sum," although the resulting number Four 
 (if we abstract from the adjectives " positive" and " negative") 
 is the arithmetical difference^ and not the arithmetical sum, of 
 the numbers Seven and Three : there is surely a sufficient depar- 
 ture, thus authorized already by received scientific usage, from 
 the xaQveXy popular meanings of the words " addition," " sum," 
 and " plus," to justify me, or to plead at least my excuse, if I 
 venture on another but scarcely a greater variation from the same 
 first or popular meanings of those words, as indicating (in com- 
 mon language) increase of magnitude ; and if I thus connect 
 them, from the outset of this new symbolical geometry, with 
 
 CHANGE OF POSITION ijl SpaCC. 
 
 23. It seems to me then that it ought not to appear a strange 
 or unpardonable extension of a phraseology which has already 
 been found to require to be extended, in passing from arithmetic 
 to algebra, if I now venture to propose the name of symbolical 
 ADDITION for that operation in Geometry, which you have seen 
 that I denote in writing by the sign + ; and if I thus speak, for 
 example, in the recent case, of the Symbolical Addition of a to a, 
 which operation has been seen to correspond to the composition, 
 or putting together, in thought and in expression, and therefore 
 to the (conceived or spoken or written) synthesis, of the two 
 CONCEPTIONS, of a STEP (a) and the beginning (a) of that step : 
 and not (primarily) to any synthesis or aggregation ofjnagni- 
 tudes. Thus if we now agree to give to the beginning of the 
 step, or to the initial position, the na7ne vehend [punctum ve- 
 hendwn, the point about to be carried), because this is the point 
 
22 ON QUATERNIONS. 
 
 on which we propose to perform the act of vection ; and if in 
 like manner the point which is the end of the step, or the final 
 position (the punctum vectum, the point which in this view is re- 
 garded as having been carried), be shortly called the vectum ; 
 while the step itself has been already named the vector : we 
 may then establish a technical din^ general formula for such sym- 
 bolical addition in geometry, which will serve to characterize and 
 express its nature, by saying that, in general, 
 
 " VECTUM = VECTOR + VEHEND ;" 
 
 while the corresponding general formula for symbolical subtrac- 
 tion in geometry, with the same new names, will be the following : 
 
 '< VECTOR = VECTUM - VEHEND." 
 
 Nor shall I shrink from avowing my own belief that this general, 
 formula, Vectum = Vector + Vehend, may be considered as a 
 TYPE, representing i\iQ,t primary synthesis in Geometry, which, 
 earlier and more than any other, ought to be regarded as ana- 
 logous to addition, in that science, and deserves to be denoted 
 accordingly : namely, the mental and symbolical addition (or 
 application) of a vector to a vehend, not at all as parts of one 
 magnitude, but as elements in one construction, in order to 
 generate as their (mental and symbolical) smn, or as the result 
 of this vection, or transport, a new position in space, which 
 may be thought of as 2i punctum vectum, or carried point ; this 
 VECTUM being simply (as has been seen) the end of that line, or 
 vector, ov carrying path, of which the vehend is the beg iniiing. 
 24. These relations of end and beginning may, of course, be 
 inter cha7iged, while the straight line ab retains not only its 
 length, but even its situation in space, although \t?> direction W\\\ 
 thus come to be reversed: for we may conceive ourselves as re- 
 turning from B to A, after having gone from a to b. l^h\% path 
 of return, this backward step, or reversed journey, considered as 
 having for its office to carry back (revehere) a moveable point 
 from B to a, after that point has been first carried by the former 
 vector from a to b, may naturally be called, by analogy and 
 contrast, a revector ; and then we shall have this general for- 
 mula of revection, 
 
 revector + vectum = vehend : 
 
LECTURE I. 23 
 
 together with this other connected formula : 
 
 VEHEND - VECTUM = REVECTOR. 
 
 The symbol for this revector will thus be a - b, if the vector be 
 still denoted by the symbol b - a ; that is to say, these two oppo- 
 site symbols, 
 
 B - A and A - B, 
 
 which, in their analytic aspect, were formerly regarded by us 
 (see 9) as symbols of two opposite ordinal relations in space, 
 corresponding to two opposite steps, are now, in their synthetic 
 aspect, considered as denoting those two opposite steps them- 
 selves ; namely, the Vector and Revector. With reference to 
 the ACT OF REVECTioN, the point b, which was formerly called 
 the vectum, might now be called the bevehend ; and then the 
 point A, which was the vehend before, would naturally come to 
 receive the name revectum. But I am not anxious that you 
 should take any pains to impress these last names on your me- 
 mory ; though I think that it may have been an assistance, rather 
 than a distraction, to have thus briefly suggested them in passing. 
 25. If in the general formula lately assigned (in 23) for 
 symbolical addition in geometry, namely the formula, vector + 
 vehend = vectum, we substitute for vector its value, or equivalent 
 expression, namely, vectum - vehend, as given by the corres- 
 ponding general formula already assigned (in same art. 23) for 
 symbolical subtraction ; we shall thereby eliminate (or get rid 
 of) the tuord " vector," in the sense that this word will no longer 
 appear in the result of this subtraction ; which result will be the 
 
 equation, 
 
 Vectum - Vehend + Vehend = Vectum. 
 
 In symbols, the corresponding elimination of the letter a, be- 
 tween the two equations, 
 
 B-A = a, a + A = B, (18,19) 
 
 gives, in like manner, the result: b-a + a = b. In ordinary 
 Algebra, not only does the same result hold good, but it is said 
 to be identically true, and the equation which expresses it is 
 called an identity ; and in the present Symbolical Geometry it 
 may still be called by that name : in the sense that its truth does 
 not depend, in any degree, on the positions of the two points, A, b ; 
 
24 ON QUATERNIONS, 
 
 but only on the general comiexion, or contrast, between the two 
 OPERATIONS of ordinal analysis and synthesis, which are here 
 marked by the signs - and +. For the formula b - a + a = b, or 
 more fully, (b-a) + a = b, may be considered as expressing, in 
 the present system of symbols, that if the position a be operated 
 on (synthetically) by what has been called the symbolical ad- 
 dition (or application) of a suitable vector^ namely b - a, it will be 
 changed to the position b ; such suitable operator (b - a) being 
 precisely that vector which is conceived to have been previously 
 discovered (analytically) by what we have called the symbolical 
 subtraction of the proposed vehend a from the vectum b. Until 
 the points A and b are in some degree known, or particularized, 
 the line B - A must also be unknown, or undetermined: yet must 
 this line be such (in virtue of its definition, or of the rule for its 
 construction) as to conduct, or to be capable of conducting, ^om 
 the point a to the point B. We know this, and this is all we 
 know, about that line, in general : and we express it by the ge- 
 neral equation or identity, B - A + A = B. 
 
 26. In like manner, if we eliminate the word " Vectum," or 
 the letter b, between those general equations or formulae of sym- 
 bolical addition and subtraction in geometry which have been 
 already assigned, we arrive at this other identity. 
 
 Vector + Vehend - Vehend = Vector ; 
 or in symbols, 
 
 ai-A-A=a; or more fully, (a + a) - a = a : 
 which must hold good for any vehend a, and any vector a. The 
 same result would evidently be true, and identical, in ordinary 
 Algebra also : but it is here to be interpreted as signifying that 
 if, from any point a, we make any rectilinear step a, and then 
 compare the end a + a of this rectilinear step with the beginning 
 A, we shall be reconducted, by this analysis of the relative posi- 
 tion of these two points, to the consideration and determination 
 of the same straight line a, which is supposed to have been 
 already employed in the previous construction, or syiithesis. 
 You will find hereafter that many other instances occur, on which, 
 however, it will be impossible in these Lectures long to delay, 
 or perhaps often even to notice them at all, where equations or 
 
LECTURE I. 25 
 
 results, that are true in ordinary Algebra, hold good also in this 
 new sort of Symbolical Geometry ; although generally regarded 
 in new lights, and bearing new (if not enlarged) significations. 
 
 27. In all that has yet been said respecting the acts of " vec- 
 tion" and " revection," or the lines " vector" and " revector," 
 we have hitherto had occasion to consider only two points ; 
 namely, those which have been above named the " vehend" (or 
 the revectum) a, and the " vectum" (or revehend) b. Let us 
 now introduce the consideration of a third point, c, which we 
 shall not generally suppose to be situated on the straight line ab, 
 nor on that line either way prolonged ; but rather so that the 
 three points abc may admit (for the sake of greater generality) 
 of being regarded as the three corners of a triangle. And let us 
 conceive that the former act of vection, whereby a moveable 
 point was before imagined to have been carried from the position 
 A to the position b, is now followed by another act of the same 
 kind, that is to say, by an immediately successive vection, which 
 we shall call on that account (from the Latin word provehere) 
 a PROVECTioN : whereby the same moveable point is now car- 
 ried FARTHER, though not (generally) in the same straight line, 
 but along a new and different straight line ; and is in this manner 
 transported from the position b to the position c. We shall thus 
 be led to consider the line c - b as being a new and successive 
 vector, which may conveniently be called, on that account, a 
 PROVECTOR : the point b, which had been named the Vectum^ 
 may now be also named the provehend, with reference to the 
 new act of provection here considered, and which begins where 
 the old act of vection ends : while, with reference to the same 
 new act of transport, or provection, the point c will naturally 
 come to be called (on the same plan) the provectum. And 
 thus we shall have, for any such successive vection, the formulaj 
 
 Provector + Vectum = Provectum ; 
 
 as also the connected formula, 
 
 Provector = Provectum - Vectum. 
 
 It is worth noticing here, that if we substitute, in the first of these 
 two new equations, for the word " Vectum," its value, or equi- 
 
26 ON QUATERNIONS. 
 
 valent expression, namely, " Vector + Vehend" (23), we shall 
 be thereby led to write this oiYier formula ofprovection: 
 Provector + Vector + Vehend = Provectum. 
 28. In symbols, if we write the equation 
 
 c - B = b, 
 so that the small Roman letter b shall here be used as a short 
 symbol for the provector, while a remains, as before, a symbol for 
 the vector, and satisfies still the equation (18), 
 
 B - A = a; 
 we shall then have not only, as before (19), 
 
 B = a + A, 
 
 but also, in like manner, 
 
 c = b + B. 
 
 And then, by eliminating b, we shall have also this other for- 
 mula, 
 
 c = b + a + a; 
 or more fully, 
 
 C =b + (a+ a). 
 We may also write, without introducing the symbols a and b, 
 
 c = (c - b) + {(b - a) + a} ; 
 because the second member of this equation may be reduced (by 
 25) to (c - b) + B, and therefore to c ; or, more concisely, we 
 may write, 
 
 c = (c - b) + (b - a) + A ; 
 
 which gives again, in words, 
 
 Provectum = Provector + Vector + Vehend. 
 The last symbolic formula (with a, b, c) is in common Algebra 
 an identity ; and we see that is here also at least a general equa- 
 tion {of provection), which holds good for any three points of 
 space. A, B, c, independently of the positions of those points, and 
 in virtue merely of the laws of composition and interpretation of 
 the symbols, or in virtue of the relations between the (conceived) 
 operations which the signs denote : so that it may perhaps be 
 called here (compare 25) a geometrical identity. 
 
 29. Astronomically, we may conceive c to denote the position 
 of the centre of a planet ; while a and b denote still the positions 
 
LECTURE r. 27 
 
 of the centres of the earth and sun : and then, while the vector 
 (b - a) is still the geocentric vector of the sun, the provector 
 (c-b) will be the heliocentric vector of the planet. And in a 
 phraseology already explained, we shall not only have as before 
 (20) the equation, 
 
 Sun's position = Sun's geocentric vector + Earth's position, 
 and in like manner, 
 Planet's position = Planet's heliocentric vector + Sun's position, 
 
 but also, by a combination of these two assertions, or phrases, or 
 equations, which combination is effected by substituting in the 
 latter of them the equivalent for the " Sun's position" which is 
 supplied by the former, we shall be able to conclude the correct- 
 ness of the following other assertion (in this general system of 
 expressions) : 
 
 " Planet's position = Planet's Heliocentric Vector 
 + Sun's Geocentric Vector + Earth's Position." 
 30. Instead of thus imagining a moveable point to be carried 
 in succession, first along owe straight line (e - a) from a to b, and 
 then along another straight line (c - b) from b to c, which lines 
 have been supposed to be in general tivo successive sides, ab, bc, 
 of a triangle abc ; we may conceive the moveable point to be 
 CARRIED ACROSS, by the straight line (c - a) or along the third 
 side, or base, ac, of the same triangle, from the original position 
 a to the final position c. And this new act of transport may be 
 called a transvection (from the Latin word transvehere, to carry 
 across) ; while the line c - a, when viewed as such a cross-car- 
 rier, may be called a transvector : and the points a and c, 
 which were before termed the Vehend and the Provectum, will 
 now come to be called, with reference to this new act of trans- 
 port, or transvection, the transvehend and the transvectum, 
 respectively. Comparing then the names of the three points, we 
 shall have the following new equations, or expressions of equiva- 
 lence between them : 
 
 Transvehend = Vehend = a ; "^ 
 Provehend = Vectum = b ; ^ 
 Transvectum = Provectum = c : J 
 
 each corner of the triangle abc being thus regarded in two dif- 
 
28 ON QUATERNIONS. 
 
 ferent views, or presenting itself in two different connexions, and 
 receiving two names in consequence thereof, on account of its 
 relations to some two out of the three different acts^ or operations, 
 of vection, provection, and transvection. And by a suitable se- 
 lection among these names for a and c, the following equation 
 (see 25), 
 
 C = (C - a) + A, 
 
 may now be translated as follows : 
 
 Provectum = Transvector + Vehend. 
 
 31. Combining this result with another recent expression for 
 the Provectum (at end of 27), we see that we may now enun- 
 ciate the equation : 
 
 Pro vector + Vector + Vehend = Transvector + Vehend ; 
 
 each memher of this last equation being an expression for one and 
 the same point, namely the Provectum, or the point c. And 
 when this equation had once been enunciated, under the form 
 just now stated, an instinct of language, which leads to the 
 avoidance of repetition in ordinary expression, and so to the 
 abridgment of discourse, when such abridgment can be attained 
 without loss of clearness or of force, might of itself be sufficient 
 to suggest to us the suppression of the words " plus vehend," 
 which occur at the end of each member of the equation (+ being 
 always read as plus). In this way, then, we may be led to enun- 
 ciate the following shorter formula : 
 
 " Provector + Vector = Transvector ;" 
 
 this latter formula (which we shall find to be a very important 
 one) being thus considered, here, as nothing more than an abbre- 
 viation of that longer equation, from which it is supposed to 
 have been in this way derived. 
 
 32. In symbols, if we write 
 
 c - A = c 
 
 thus making c a symbol of the transvector ; and if we compare 
 the expression hence resulting for c, namely (see 19), 
 
 c = c + A, 
 with the expression already found (in 28), 
 
 c = b + a + A ; 
 
LECTURE I. 29 
 
 we shall thus be led to the equation, 
 
 b+a+A=c+A, 
 which we may (in like manner) be tempted to abridge, by the 
 omission of + a at the end of each of its two members ; and so to 
 reduce it to the shorter form, 
 
 b + a = c, 
 
 which agrees with the recent result, Provector+ Vector = Trans- 
 vector (31); because a, b, c denote here the vector, pro vector, 
 and transvector, respectively. Or, without introducing these 
 symbols a, b, c, if we compare a recent expression for c, namely 
 
 (see 28), 
 
 c = (c - b) + (b - a) + A, 
 
 with this other expression (compare 25), 
 
 c = (c - a) + A, 
 
 and suppress + a in both, as before, we shall thus be conducted 
 to the general equation, or geometrical (as well as algebraical) 
 
 IDENTITY : 
 
 (c - b) + (b - a) = (c - a) ; 
 which again agrees with the result (of 31), 
 
 " Provector + Vector = Transvector." 
 33. In a phraseology suggested by astronomy, and partly em- 
 ployed already in this Lecture, we have on the one hand (as in 
 29), 
 Planet's Position = Planet's Heliocentric Vector 
 
 + Sun's Geocentric Vector + Earth's Position ; 
 
 and on the other hand (see 20), 
 
 Planet's Position = Planet's Geocentric Vector + Earth's Position. 
 Comparing these two different expressions for the position of the 
 planet in space, and suppressing a part which is common to both, 
 namely, the words 
 
 " Plus Earth's Position," 
 
 we shall be led to say that 
 
 " Planet's Heliocentric Vector 
 
 + Sun's Geocentric Vector 
 
 = Planet's Geocentric Vector:" 
 
 where the geocentric vector of the planet is to be regarded as the 
 
 transvector in the triangle, if the planet's heliocentric vector be 
 
30 ON QUATERNIONS. 
 
 the provector, while the geocentric vector of the sun is the origi- 
 nal vector itself. 
 
 34. Since (by 27), 
 
 Provector = Provectum- Vectum, 
 
 while (by 30 and 23), 
 
 Provectum = Transvector+ Vehend, 
 and 
 
 Vectum = Vector + Vehend, 
 
 we have the equation 
 
 Provector = (Transvector -f Vehend) 
 - ( Vector + Vehend) ; 
 which may conveniently be abridged to the following formula : 
 
 " Provector = Transvector - Vector." 
 Thus, in astronomy, we may say that 
 
 " Planet's Heliocentric Vector 
 = Planet's Geocentric Vector 
 -Sun's Geocentric Vector;" 
 regarding the second member of this equation as an abridgment 
 for the following expression : 
 
 (Planet's Geocentric Vector + Earth's Position) 
 - (Sun's Geocentric Vector + Earth's Position) ; 
 which we know to be equivalent, in the phraseology of the pre- 
 sent Lecture, to 
 
 " Planet's Position - Sun's Position ;" 
 and therefore to " Planet's Heliocentric Vector," as above. 
 
 35. In symbols, because (by 28, 32, 19), 
 
 b = C-B, C = C+A, B = a+A, 
 
 we have the equation 
 
 b = (c + a) - (a + a) ; 
 which may be abridged to the following : 
 
 b = c - a . 
 This signification of c - a allows us also to extend to geometry 
 the algebraical identity: 
 
 (c-a)-(b-a) = (c-b); 
 and generally it will be found to prepare for the establishment of 
 a complete agreement between the rules of ordinary Algebra and 
 
LECTURE I. 31 
 
 those of the present Symbolical Geometry, so far as addition and 
 subtraction are concerned. Thus, if we compare the two equa- 
 tions (32, 35), 
 
 c^b+a, ij = c_a, 
 
 we find that generally, for any two co-initial vectors, a, c, we 
 may write (as in ordinary Algebra), 
 
 (c - ^) + a = c ; 
 and that for any two successive vectors, a, b, we have also (as in 
 
 Algebra) : 
 
 (b + a) - a = b ; 
 
 which new geometrical identities are of the same forms as some 
 others that were lately considered (in 25, 26), namely, 
 
 (b - a) + A = B ; (a + a) - A = a. 
 Indeed they have with these a very close connexion, as regards 
 their significations too, arising out of the way in which they have 
 been above obtained ; yet because a, b, c have been used as 
 symbols oi points, buta, b, c as symbols oi lines, it would have 
 been illogical and hazardous to have confounded these two pairs 
 of equations, or identities, with each other; or to have regarded 
 the truth of the one pair as an immediate consequence of the 
 truth of the other pair. 
 
 36. We see, however, that the original view which has been 
 proposed, in the present Lecture, for the primary significations 
 of + and - in geometry, as entering^rs^ into expressions of the 
 (unusual) forms " Line plus Point" and " Point minus Point," 
 conducts, simply enough, when followed out, to interpretations 
 of expressions of the (more common) forms ^^ Line plus Line," 
 and ^^ Line minus Line:" and that thus, from what we have re- 
 garded as the PRIMARY ACTS of synthesis and analysis (of points) 
 in geometry, arise a secondary synthesis and a secondary 
 ANALYSIS (of lines), which correspond to the composition and 
 decomposition of vections (or of motions) ; and which are sym- 
 bolized by the two general formulae already assigned (in 31, 34), 
 
 namely, 
 
 Transvector = Pro vector + Vector, 
 and 
 
 Provector = Transvector - Vector. 
 
 The first formula asserts that of any two successive vectors, 
 
32 ON QUATERNIONS. 
 
 or directed lines (the second or added line being conceived to 
 begin where the first line ends), the geometrical sum is the line 
 drawn from the beginning of the first to the end of the second 
 line. The second formula asserts, that of any two co-initial 
 vectors (or directed lines), the geometrical difference is the 
 line drawn from the end of the subtrahend line to the end of the 
 line from which it is subtracted. The sum and the difference of 
 two directed lines are thus two other lines having direction ; and 
 the geometrical rules for determining them are found to co- 
 incide in this theory, as in several others also, with the rules 
 of composition and decomposition of motions {or of forces). 
 For, although it would be unsuited to the plan and limits of 
 these Lectures to enter deeply, or almost at all, into the history 
 of those speculations to which their subject is allied, yet it seems 
 proper to acknowledge distinctly here, as I am very happy to do, 
 that (whatever may be thought of the foregoing general views 
 respecting + and -), the recognition of an analogy between 
 addition a??c? subtraction of directed lines, on the one hand, 
 and composition and decotnposition o/motions on the other hand, 
 is nothing private or peculiar to myself Indeed, the existence 
 of this fundamentally important analogy has, in different ways, 
 presented itself to several other thinkers, starting from various 
 points of view, in many parts of the world, during the present 
 century : so much so, that it may by this time be well nigh con- 
 sidered to have acquired, in the philosophy of geometrical science, 
 what I cannot doubt its possessing still more fully in time to 
 come, the character of an admitted and established truth, a fixed 
 and settled principle. But of those more novel and hitherto less 
 participated views, respecting the multiplication and division 
 of such directed lines in geometry, on which the theory of qua- 
 ternions is founded, 1 perceive that our time requires that we 
 should postpone the consideration to the next Lecture of this 
 Course : for which, however, I indulge myself meanwhile in 
 hoping, that what has been laid before you to-day will be found 
 to have been an useful, and indeed a necessary preparation. 
 
LECTURE IL 
 
 37. You have had laid before you, Gentlemen, in the fore- 
 going Lecture, a statement or at least a sketch of those general 
 views, respecting the primary significations of the marks 
 
 + and -, 
 or of the words plus and minus, with which views, in the Cal- 
 culus of Quaternions, I connect the two corresponding opera- 
 tions of Addition and Subtraction in Geometry. With me, as 
 you have seen, the primary geometrical operation which has been 
 denoted by the usual mark -, and the one for which I have ven- 
 tured to employ the familia;r name subtraction, though guarded 
 sometimes by the epithet symbolical, consists in a certain ordinal 
 Analysis of the position of a mathematical point in space. This 
 Analysis is performed, as you have seen, thiough the comparison 
 of the position of the point proposed for inquiry, with the posi- 
 tion oi another mathematical point; and it \s pictured, or repre- 
 sented, by the traction (or drawing) of a straight line, from 
 the given to the sought position ; from the analyzer point a, to 
 the analyzand point b : from the one which is regarded as being 
 comparatively simple, familiar, or given, to the other which is 
 (for the purposes of the inquiry) accounted to be comparatively 
 complex, unknown, or sought. In this way, the symbol b - a 
 has come with us to denote the straight line from a ^o b ; the 
 point A being (at first) considered as a known thing, or a datum 
 in some geometrical investigation, and the point b being (by 
 contrast) regarded as a sought \\\\x\g, or a quasitum : while b - a 
 is at first supposed to be a representation of the ordinal relation 
 in space, of the sought point b to the given point a; or of the 
 geometrical difference of those two points, that is to say, the 
 difference of their two positions in space; and this difference is 
 
 D 
 
34 ON QUATERNIONS. 
 
 supposed to be exhibited or constructed by a straight line. Thus, 
 in the astronomical example of earth and sun, the line b - a has 
 been seen to extend/}-o?« the place of observation a (the earth), 
 to the place of the observed body b (the sun) ; and to serve to 
 CONNECT, at least in thought, the latter position with the former. 
 38. Again you have seen that with me the primary geome- 
 trical operation denoted by the mark +, and called by the name 
 addition, or more fully, symbolical Addition, consists in a cer- 
 tain correspondent orrfm«Z synthesis of the position of a mathe- 
 matical point in space. Instead oi co7nparing such a position, b, 
 with another position a, we now regard ourselves as deriving the 
 one position from the other. The point b had been before a 
 punctum analyzandum ; it is now a punctwn cotistructum. It 
 was lately the subject of an analysis ; it is now the result of a 
 synthesis. It was a mark to be aimed at; it is now the endoio. 
 flight, or of a journey. It was a thing to be investigated (ana- 
 lytically) by our studying or examining its position ; it is now a 
 thing which has been produced by our operating (synthetically) 
 on another point a, with the aid of a certain instrument, namely, 
 the straight line B - a, regarded now as a vector, or carrying 
 path, as is expressed by the employment of the sign of vection, 
 +, through the general and identical formula : 
 (b - a) + A = B. 
 
 That other point a, instead of being now a punctum analyzans, 
 comes to be considered and spoken of as a punctum vehendum ; 
 or more briefly, and with phrases of a slightly less foreign form, 
 it was an analyzer, but is now a vehend ; while the point B, 
 which had been an analyzand, has come to be called a vectum, 
 according to the general formula : 
 
 Vector + Vehend = Vectum ; 
 
 where Plus is (as above remarked) the Sign of Vection, or the 
 characteristic of ordinal synthesis. From serving, in the astro- 
 nomical example, as a post of observation, the earth, a, comes 
 to be thought of as the commencement of a transition, b - a, which 
 while thus beginning at the earth is conceived to terminate at the 
 sun ; and conversely the sun, b, is thought of as occupying a 
 situation in space, which is not now proposed to be studied by 
 
LECTURE 11. 35 
 
 observation, but is rather conceived as one which has been reached, 
 or arrived at, by a journey, transition, or transport of some move- 
 able point or body /"rotn the earth, along the geocentric vector of 
 the sun. I think that this brief review, or recapitulation, of some 
 of the chief features or main elements of the view already taken, 
 of the operations of Addition and Subtraction, or of the marks + 
 and -, will be found to have been not useless, as preparatory to 
 our entering now on the consideration of the analogous view 
 which I take of the operations of Multiplication and Division, or 
 of the marks x and -r- in Geometry. 
 
 39. The Analysis and Synthesis, hitherto considered by us, 
 have been of an ordinal kind; but we now proceed to the con- 
 sideration of a different and a more complex sort of analysis and 
 synthesis, which may, by contrast and analogy, be called car- 
 dinal. As we before (analytically) compared a point, b, with 
 a point A, with a view to discover the ordinal relation in space 
 of the one point to the other ; so we shall now go on to co^npare 
 one directed line, or vector, or ray, j3, tuith another ray, a, to 
 discover what (in virtue of the contrast and analogy just now re- 
 ferred to) I shall venture to call the cardinal relation of the one 
 ray to the other, namely, (as will soon be more clearly seen), a 
 certain complex relation of length and of direction. As one 
 among the reasons for the adoption of such a phraseology which 
 may admit of being most easily and familiarly stated, while the 
 statement of it will serve, at the same time, as an initial prepa- 
 ration, or introduction, to questions or cases of greater difficulty 
 or complexity, let me remind you that when the condition j3 = a + a 
 is satisfied, it is then permitted, by ordinary usage, to write also 
 j3 -f- a = 2 ; the quotient of j3, divided by a, being, in this case, 
 equal to the cardinal number, two. Under the same simple con- 
 dition, it is, as you know, allowed by custom to write also j3 = 
 2 X a ; and to say that the multiplication of a, by the same car- 
 dinal number, two, produces j3. Now I think that we may not 
 improperly say that we have here, in the division, cardinally 
 analyzed ^, as a cardinal analyzand, with respect to a, as a car- 
 dinal analyzer ; and that we have obtained the cardinal number, 
 or quotient, 2, as the result of this cardinal analysis ; while, in 
 the converse process of multiplication, we may be said to have 
 
 D 2 
 
36 ON QUATERNIONS. 
 
 employed the same number, two, as a cardinal operator ^ or as the 
 instrument of a cardinal synthesis, which instrument or operator 
 thus serves as a multiplier, or as 2ifactor, to generate or to con- 
 struct j3, as di product or as o. factum, from a as a inultiplicand or 
 faciend. In so simple an instance as this, it might be better, 
 indeed, to abstain from the use of any part of this phraseology 
 which should seem in any degree unusual; but there appears to 
 me to be a convenience in applying the foregoing modes of ex- 
 pression to the much more general case, where it is proposed to 
 compare any one ray, j3, with any other ray, a, with a view to 
 discover the complex relation of length and of direction 
 of the former to the latter ray ; or, conversely, to construct or 
 generate (5 from a, by making use of such a relation. 
 
 40. In adopting, then, from ordinary algebra, as we propose 
 to do, the general and identical formula, 
 
 (5 -f- a X a = ]3, 
 
 we shall now suppose that j3 -r- a denotes generally a certain 
 metrographic relation of the ray |3 to the ray a, including at 
 once, as its metric element, a ratio of length to length, and also, 
 as its graphic element, a relation of direction to direction. The 
 act or process of discovering such a metrographic relation, de- 
 noted by the symbol |3 -^- a, we shall call, generally, the car- 
 dinal analysis of j3, as an analyzand, by a as an analyzer. And 
 the converse act of employing such a cardinal relation, when 
 already found or given, so as to form or to construct (5 by a suit- 
 able operation on a, namely, by altering its length in a given 
 ratio, and by causing its direction to revolve through a given 
 angle, in a given plane, and towards a given hand, we shall call 
 a cardinal synthesis. The cardinal analysis above mentioned, 
 we shall also call the division, or, sometimes more fully, the 
 symbolical division of the ray /3 by the ray a ; and the usual name, 
 quotient, shall be occasionally applied by us to the result of 
 this division, that is, to the metrographic relation denoted above 
 by the symbol j3 -=- a, and supposed to he found by that cardinal 
 analysis, of which the mark -f- is thus the sign, or the charac- 
 teristic. In like manner to that converse cardinal synthesis, of 
 which the characteristic is here supposed to be the mark x, we 
 
LECTURE II. 37 
 
 shall give (from the analogy which it will be found to possess to 
 the operation commonly so called) the name of multiplication, 
 or sometimes, more fully, that of s2/?«6oZ/ca/ multiplication. And 
 when, after writing an equation of the form 
 
 /3 ^ a = ^, 
 we proceed to transform it into this other equation, 
 
 q^a = ^, 
 (by an application of a general formula lately cited), we shall say 
 that q has been multiplied into a, or (sometimes) that a has been 
 multiplied hy q ; avoiding, however, to say, conversely, that q has 
 been multiplied by a, or a into q. Thus q, which had, relatively 
 to the cardinal analysis (-h)? been regarded as a quotient, will 
 come to be regarded, and to be spoken of, with reference to the 
 cardinal synthesis (x), as a multiplier, or as a factor ; while j3 
 may still be called, as above, a product, or a factum : and a 
 may, by contrast, be called a multiplicand, or a faciend. 
 
 41. Without yet entering more minutely into the considera- 
 tion of the precise force, and/w// geometrical signification, of 
 that act or operation which has here been called Multiplication, 
 or FACTION ; it may be seen already that the general type of this 
 process oi cardinal synthesis is, in the present phraseology, con- 
 tained in the following technical statement, ox formula : 
 
 FACTOR X FACIEND = FACTUM ; 
 
 where we shall still read, or translate, the mark x by the word 
 " INTO." It is clear also that the converse process of what has 
 been above called Division, or cardinal analysis, has, in like 
 manner, its general type in the reciprocal formula, 
 
 FACTUM -f- FACIEND = FACTOR ; 
 
 where the mark -^ may still be translated, or read, as equivalent 
 to the word " by." And it is evident that these two general and 
 technical assertions, respecting the kind of (symbolical) Multi- 
 plication and Division in Geometry which we here consider, are 
 closely analogous to the two corresponding formulae, already 
 assigned (in art. 23), as types of those earlier operations in geo- 
 metry which were there called (symbolical) Addition and Sub- 
 traction, namely, the two following : 
 
38 ON QUATERNIONS. 
 
 Vector + Vehend = Vectum ; 
 Vectum - Vehend = Vector. 
 
 42. It is easy to push this analogy farther with clearness and 
 advantage. We have, for instance, the general formula of iden- 
 tity, 
 
 Factum 4- Faciend x Faciend= Factum; 
 
 which corresponds to the identity (of art. 25), 
 
 Vectum- Vehend + Vehend = Vectum. 
 More concisely and symbolically, the written identity (of art. 40), 
 j3 "T- a X a = /3, corresponds exactly to the earlier identical for- 
 mula (of same art. 25), b - a + a = b. Each is to be considered 
 as telling us nothing whatever respecting the points or lines 
 which seem to be compared, and of which the symbols enter into 
 the formulae ; but only as expressing, each in its own way, a 
 general relation, of a metaphysical rather than of a mathematical 
 kind, between the intellectual o^evaXion?,, or mental acts^ oi Syn- 
 thesis and of Analysis. For each of these technical formulae may 
 be regarded as an embodiment, in one or other of two different 
 mathematical forms, of the general and abstract principle, that ij" 
 the KNOWLEDGE previously acquired, by any suitably performed 
 ANALYSIS, be afterwards suitably applied, by the Synthesis an- 
 swering to that Analysis, it will conduct to a suitable result : 
 which result, thus constructed by this synthesis, will be the very 
 SUBJECT (whether point, or line, or other thing, or thought) 
 which had been analyzed before. Or that whatever has been 
 found by Analysis may afterwards be used by Synthesis (or at 
 least may be conceived to be so used) ; and that the thing or 
 thought which is produced (or re-produced) by this synthetic pro- 
 cess, will be the sajne with that which had been examined ox sub- 
 mitted to analysis previously. 
 
 43. Corresponding remarks apply to the written and spoken 
 
 identities, 
 
 q ^ a -T- a = q, 
 and 
 
 Factor x Faciend -t- Faciend = Factor ; 
 
 which are obviously analogous to the identical formulae (of 26), 
 
 a + A - a = a, 
 and 
 
 Vector + Vehend - Vehend = A^ector. 
 
LECTURE II. 39 
 
 In fact these technical formulae may be regarded as being merely 
 so many different mathematical modes of embodying the general 
 and abstract principle, that whatever specific instrument (a or q) 
 of any known sort of synthesis (+ or x), is conceived to have been 
 previoushj used, in operating on a kiiown subject (a or a), may 
 be conceived to be afterwards found, by the converse act of ana- 
 lysis (- or -1-). 
 
 44. After comparing any two rays, a and j3, with each other 
 by cardinal analysis, in one order (/3 with a), we may choose to 
 compare again the same two rays among themselves, but in the 
 opposite order (a with j3) ; exchanging thus the places of the 
 analyzer and analyzand, in the process of the cardinal analysis. 
 The relations, or the quotients, thus obtained, and denoted by 
 the symbols j3 -i- a and a -f- /3, may be called reciprocal cardinal 
 relations, or reciprocal quotients ; as (in art. 9) we called b - a 
 and A - B the symbols of two opposite ordinal relations. Con- 
 sidered as reciprocal operators, or as inverse factors, the same 
 two symbols, j3 -^ a and a -f- /3, may be said to denote, respec- 
 tively, a Factor and its answering refactor; as the two oppo- 
 site steps denoted by b - a and a -b, were called (in art. 24), in 
 respect of each other, by the names of Vector and revector. 
 And in reference to this act of refaction, we might call j3 the 
 REFACiEND, and a the refactum ; as b has been called (in 24) 
 the reverend, and a has been called the revectum. 
 
 45. We shall now proceed to make a further extension of this 
 sort of phraseology ; of which extension the deficiency (what- 
 ever it may be) in elegance will, it is hoped, be compensated by 
 the systematic convenience which will arise from its resemblance 
 or analogy to the language of the former Lecture; and from the 
 consequent illustration which may be thrown on one set of 
 thoughts by their being brought into contact or juxtaposition 
 with another set, which other has been already considered. 1 
 venture, therefore, to propose to you to speak now, or to allow 
 me to speak, of an act ofpROFACTioN as being performed, when, 
 after having constructed a second ray j3, from ?i first ray a, by a 
 first act of faction, or of cardinal synthesis, such as has been al- 
 ready spoken of, we proceed to the construction of a third ray, y, 
 from the second raj, (5, by the performance of a neiv and successive 
 
40 ON QUATERNIONS. 
 
 act of synthesis, of the same general kind as before ; although 
 this new act of faction, by which we pass to y from j3, may not 
 (and generally will not) be a simple continuation^ or a me7'e re- 
 petition, of the first factor act, but 7nay (and generally will) be 
 performed with a quite different factor as its instrument. And 
 then that third act of the same sort, which is able of itself alone 
 to replace, or is singly equivalent to, the system of these two suc- 
 cessive acts of faction and profaction, may be called an act of 
 
 TRANSFACTION. 
 
 46. Writing then the equation, 
 
 7 ^ ^ = r, 
 and, therefore, also (see art. 40), 
 
 7 = /*x/3, 
 we shall call r the profactor, because it is the instrument or 
 agent in the second successive act, above mentioned, of cardinal 
 synthesis, or is the operator of that profaction, by which the ray 
 7 is generated or constructed from the ray j3, after j3 has been 
 already constructed from a by the former act olfaction. And 
 with reference to the same successive faction, or joro-faction, we 
 shall call j3 the profaciend, and 7 the profactum ; in such a 
 manner that we shall be able to enunciate the ioWowm^ formula 
 of profaction : 
 
 Profactor x Profaciend = Profactum ; 
 together with the converse formula, 
 
 Profactum -^Profaciend =Profactor ; 
 as in the foregoing lecture we might have said in speaking of 
 provectlon, 
 
 Provector + Provehend = Provectum ; 
 and 
 
 Provectum - Provehend = Provector. 
 
 47. And inasmuch as the same ray, (5, is here considered and 
 named as the Profaciend, which had before been named, in a 
 different connexion, the Factum, we may substitute for the word 
 " Profaciend," in the first verbal formula of the last article, the 
 word " Factum," so as to obtain this other formula (analogous 
 to one of art. 27), 
 
LECTURE II. 41 
 
 Profactor x Factum = Profactum. 
 We may also proceed to substitute here for " Factum," its value 
 (assigned by art. 41), namely, the equivalent expression, 
 
 Factor x Faciend ; 
 and so obtain this other general formula of pro/action (analogous 
 to the formula of provection at the end of art. 27), 
 
 Profactor x Factor x Faciend = Profactum. 
 In symbols, if, 
 
 (5 ==q X a, and y = rx (5, 
 
 we may write, by elimination of j3, 
 
 y = r X q X a. 
 Or, because q = (5 -i-a, r = y -^ )3, we may write the identical for- 
 mula (analogous to one in art. 28), 
 
 7 - (y -^ /3) X (/3 -^ a) X a. 
 48. Conceiving, in the next place (see end of art. 45), that 
 the two successive acts of faction and profaction are replaced by 
 a single act of the same sort, equivalent to the system, of these 
 tivo ; namely, by a certain act of transfaction, in which the 
 Operator, or the transfactor, shall be (for the present) denoted 
 by the letter s ; we may then write 
 
 y = s X a; y -r-a = s ; 
 and with respect to this act oi transfaction^ may call a the trans- 
 FACiEND, and y the transfactum. We shall thus have the two 
 general and reciprocal formulae, 
 
 Transfactor x Transfaciend = Transfactum ; 
 Transfactum -^ Transfaciend = Transfactor ; 
 
 with two identities, deducible by the comparison of these. And 
 because the ray y is here at once the transfactum and the pro- 
 factum^ according as we consider one or the other of the two 
 operations of which that ray is the result ; while the other ray, 
 namely, a, is at once the faciend and the transfaciend ; we may 
 enunciate this other general formula (compare art. 30), 
 
 Transfactor x Faciend = Profactum ; 
 as, in symbols, we have the identity, 
 
 (7 -J- a) X a = y. 
 
42 ON QUATERNIONS. 
 
 49. Equating then the two expressions for the Profactuni, or 
 for y, found in the two last articles, we have, in symbols (com- 
 pare 32), the formula 
 
 (7 -T- a) X a = (7 -i- /3) X ()3 -f- a) X a ; 
 and in words (compare 31) we have this general enunciation, 
 Transfactor x Faciend = Profactor x Factor x Faciend. 
 
 Hence (compare again the same articles 31 and 32), we may be 
 naturally led to adopt the two following abbreviated forms of 
 assertion, namely, in symbols, 
 
 (7-4-a) = (7-4-/3)x(|3-^-a); 
 and in words, 
 
 TRANSFACTOR = PROFACTOR x FACTOR. 
 
 You see, then, that each of these two last equations (of which 
 the first is true and identical in ordinary algebra also) is here re- 
 garded as an abridged form, which is to be restored (where 
 required) to its complete original significance, or full and deve- 
 loped expression, by restoring the suppressed symbols, x a, or by 
 restoring the suppressed words, " Into Faciend;" exactly as it 
 was supposed (in the articles recently referred to), that the iden- 
 tical equations, 
 
 (c - a) = (c - b) + (b - a), 
 and 
 
 Transvector = Provector + Vector, 
 
 were abridged forms, which were to be interpreted, or restored 
 to their full meanings, by restoring the symbols + a at the right 
 hand of each member of the one equation, or the words " Plus 
 Vehend" after each member of the other. And we see that, on 
 the present plan, as well as in ordinary algebra, whenever we 
 have (as above supposed) 
 
 g' = /3-Ha;r = 7-^/3;5=7-:-a; 
 and when we have, therefore, also the equation (in which each 
 member is = 7, and the ray a is conceived to have some actual 
 length), 
 
 sy.a = ry~qxa; 
 
 we may then abbreviate this last equation to the shorter form, 
 
 s = r y. q. 
 
LECTURE II. 43 
 
 50. In like manner, because, under the conditions recently 
 mentioned, we have 
 
 r = y -~ (i = {s X a) -~ (q X a), 
 or 
 
 Profactor = (Transfactor x Faciend) -i- (Factor x Faciend), 
 
 we may also agree to write, more concisely (compare art. 35), 
 
 r=s-r-q, 
 
 and also to say (compare art. 34), 
 
 PROFACTOR = TRANSFACTOR — FACTOR. 
 
 And thus we shall be conducted (as in ordinary algebra) to the 
 following identical formulae (compare 35), 
 
 (s-^q)xg = s; {rxq)-^q=r; 
 which have, indeed, a very close connexion, both of form and of 
 signification, with the identical equations (of articles 40, 43), 
 
 (j3 -f- a) X a = j3 ; (qx a)-i-a = q', 
 
 yet which are ?iot, in the present system, to be confounded there- 
 with. For a, j3, y, have been supposed to be rays, or directed 
 right lines in tridimensional space ; while q, r, s, are here not 
 (generally) rays, or lines, but certain results of cardinal analysis, 
 or instruments of cardinal synthesis, namely, certain geometrical 
 quotients or factors, the precise nature of which we have pro- 
 posed to ourselves to consider more closely soon, but concerning 
 which we have as yet no right to assume that they must neces- 
 sarily follow, in all respects, the same rules of combination among 
 themselves, as the rays a, j3, 7. (Compare art. 35). 
 
 51. It may be useful here to collect into one tabular view 
 (analogous to that of art. 30) the names above assigned to the 
 three rays, a, j3, 7 ; which names have been the following : 
 
 a = Faciend = Transfaciend ; 
 /3 = Factum = Profaciend ; 
 7 = Profactum = Transfactum. 
 
 ^ach of the three rays, which are here considered and compared, 
 receives thus, as we see, two different names, on account of its 
 being regarded in two different ^;^e^6'.9, as connected with and con- 
 cerned in some two out of the three different (although similar) 
 
44 
 
 ON QUATERNIONS. 
 
 acts of faction, profaction, and transfaction ; exactly as (in art. 30) 
 each of the three points, a, b, c, was formerly tabulated as re- 
 ceiving two names, on account of its connexion with some two 
 of the three acts of vection, provection, and transvection. 
 
 52. To draw still more closely together into one common 
 contemplation, or conspectus, what has thus been separately 
 shewn in the foregoing and in the present lecture, we may now 
 conceive that the three rays, a, j3, 7, are three diverging edges 
 oi a py?'amid, abcd, which has a new point, d, for its vertex, 
 and for the common origin, or initial point, of the three rays ; 
 while the base of this pyramid is the triangle abc (of art. 27), 
 which has the three old points, a, b, c, for its three corners. We 
 may then write, in the notation of the former Lecture, 
 
 a = a-d; /3 = B- D ; 7 = c-d; 
 and shall have also the relations, 
 
 a = B-A = j3-a; 
 b = c -B = 7 -/3; 
 
 c = C — A^-y— a. 
 
 And we may say that while each of the three points, a, b, c, re- 
 ceives two different names, or designations, as belonging at once 
 to two different sides of the triangle of vections, abc, each 
 of the three rays, a, j3, 7, receives, in like manner, two names, 
 as appertaining at once to two different faces of the pyramid of 
 FACTIONS, a[5y ; namely, to some two out of the three faces which 
 may be called, respectively, the^ace of faction (aj3or adb) ; the 
 face of profaction {I5y or bdc) ; and the face of transfaction 
 {ay or adc). 
 
 53. All this may be illustrated by the two following diagrams ; 
 of which one (fig. 6) is designed to represent the triangle ofvec- 
 tions, ABC, while the other (fig. 7) is intended to picture the 
 pyramid of factions, a[5y. 
 
 B = 
 
LECTURE II. 45 
 
 In astronomy we may still conceive, as before, that the three 
 points A, B. c, are situated at the centres of the Earth, Sun, and 
 Venus, respectively; and may then imagine that the fourth point, 
 D, is situated at the centre of the Moon. 
 
 Thus the three diverging edges of the pyramid, or the three 
 rays, a, j3, 7, will coincide, in this astronomical example, with 
 the selenocentric vectors of the Earth, the Sun, and Venus, or 
 with the three rays from the centre of the Moon to the centres 
 of those three other bodies. 
 
 54. And as (in art. 36) we saw that what we had begun by re- 
 garding, in the former Lecture, as ihe primary significations of the 
 marks + and- in geometry, conducted to certain secondary signi- 
 fications oithoseiwo characteristics of operation; so now, from what 
 have been, in the present Lecture, conceived as \he primary sig- 
 nifications of the marks x and -f-, we may observe that we are con- 
 ducted to certain analogous and secondary significations of these 
 two other marks or characteristics. From expressions of the 
 forms, "/me plus point" and ^^ point minus point" we ^vere 
 before led on to the expressions of the forms, *' line plus line," 
 and " line minus line." And, in like manner, from expressions 
 of the forms, '•^factor into ray" and '■'• ray by ray" (where the 
 rays do not diff'er in kind from the lines before considered, and 
 where the words into and by are equivalent to the niarks x and 
 -i-), we have since been conducted to expressions of the forms 
 '* factor into factor," and " factor by factor;" for we have been led 
 to assert that " Profactor, multiplied into Factor, equals Trans- 
 factor" (art. 49), and that " Transfactor, divided by Factor, equals 
 Profactor" (art. 50). It is true that these two last assertions, like 
 the two corresponding enunciations of the preceding Lecture, 
 namely, " Provector /?/^<5 Vector = Transvector" (art. 31), and 
 " Transvector minus Vector = Provector" (art. 34), have, at first, 
 offered themselves to our notice as mere abbreviations of certain 
 other and longer statements, in which the marks + - x -^ had all 
 retained what we have regarded as their primary significations. 
 But as we saw (in art. 36), that the abridged expressions of the 
 forms " line + line," and " line - line," might suggest a certain 
 derivative or secondary ordinal synthesis, and a corresponding 
 derivative or secondary ordinal analysis, which might be called 
 
46 ON QUATERNIONS. 
 
 (as in fact they often are called) " addition and subtraction of 
 lines" and might be interpreted (as in fact they often are inter- 
 preted), as answering to the composition and decomposition of 
 vections (or of motions) ; so we may now see that the newer ab- 
 breviated expressions of the forms " factor x factor" and " factor 
 -r- factor," may suggest a certain derivative or secondary car- 
 dinal SYNTHESIS, and a certain other and correspondent deriva- 
 tive or secondary cardinal analysis, which may be called 
 ^'^Multiplication and Division of Factors " and vi?hich admit of 
 being interpreted as answering to the composition and decom- 
 position of factions, or of operations of the factor kind. 
 
 65. Thus, when (see fig. 6) we assert that the Provector, 
 c- B, from the Sun to Venus, being added geometrically to the 
 Vector, b-a, which extends from the Earth to the Sun, gives, 
 as the geometrical sum, the Transvector, c - a, which goes from 
 the Earth to Venus; we may interpret the assertion (what- 
 ever the original motives for enunciating it may have been), as 
 expressing that to go straight accoss {trans-) from the earth to 
 the planet, if we attend only to the total or final effect of this 
 process, or to the ultimate change of position accomplished by 
 this mode of transport, comes to the same thing., as to go first 
 from the Earth to the Sun, and afterwards from the sun to the 
 planet. And in like manner when we assert (see fig. 7), that the 
 Profactor, 7 -r- 13, being multiplied geometrically into the Factor, 
 ^ _!. a, produces the Transfactor, 7-^0, we may interpret the 
 assertion by saying that to change at once the selenocentric ray 
 or vector of the Earth to the selenocentric vector of Venus, is, 
 as to final effect, the sa7ne thing, as to change j^y^^ that seleno- 
 centric vector of the Earth to the selenocentric vector of the Sun, 
 and afterwards to change this selenocentric vector of the Sun to 
 the selenocentric vector of the Planet. An act ofvection may 
 be compounded W\t\\ &. subsequent act oi pro-wectxon into one sin- 
 gle act of ^;"aw5-vection ; and, in like manner, an act of faction 
 (which changes one ray or vector to another) may be compounded 
 with an act of jo/'O- faction following it, into one single act of 
 ^raw5-faction, which as to its effect, or the ultimate result of its 
 operation, shall be equivalent to the system of those two former 
 acts of the same kind. To move successively along the two sides, 
 
LECTURE II. 47 
 
 AB, BC, of any triangle, abc, is to move, upon the ivhole, from 
 the first point, a, to the last point, c, of the base, ac. To sweep 
 over the face, adc, of the pyramid, abcd, from the edge da, to 
 the edge dc, or from the ray a to the ray y, is an operation 
 which has the same first subject, and the same last result, as to 
 sweep first over the face, adb, from the edge da to the edge 
 DB, or from the ray a to the ray /3, and then over the face bdc, 
 from the edge db to the edge dc, or from the ray j3 to the ray 
 •y. (Compare the commencement of art. 48.) 
 
 bQ. It has been noticed (in art. 54) that there exist two kinds 
 oi secondary analysis, ordinal and cardinal, which answer to the 
 two kinds, recently illustrated, of secondary synthesis: namely, 
 those two modes of analysis which consist, respectively, in the 
 decomposition oi vections, and olfactions. The first or ordinal 
 kind of secondary analysis has been called the subtraction of lines; 
 the second or cardinal kind of secondary analysis has been called 
 the division of factors. The diagrams lately exhibited (figures 
 6 and 7) may serve to illustrate these two processes. Thus we 
 have been led to say (see fig. 6), that the subtraction of the Vec- 
 tor B - a, from the Transvector c - a, gives the Provector c - b 
 as the remainder I or that the subtraction (compare art. 34) of 
 the geocentric vector of the Sun from the geocentric vector of 
 Venus, leaves, as remainder, the heliocentric vector of the planet. 
 And whatever motive of abridgment may have originally led us 
 to enunciate this assertion, while the mark - was still confined 
 by us to what we regarded as its primary signification, we may 
 now be led to interpret the assertion as expressing, that if the 
 act or process of transvection, from the earth a to the planet c, 
 be DECOMPOSED into two successive vections, of which ihe first 
 is the given act of vection from the earth to the sun b, then the 
 second component must be (or be equivalent to) the act oi pro- 
 vection, from the Sun b to Venus c. This, then, is an example 
 of what we have called secondary ordinal analysis, or Analysis 
 OF Vection, arising out of ihdl primary and ordinal analysis, 
 or Analysis of Position, namely, the examination or study 
 of the position of one point b as compared with another point a, 
 which primary sort of analysis in geometry was considered in the 
 former Lecture. And in like manner, from that primary and 
 
48 ON QUATERNIONS. 
 
 cardinal analysis, or Analysis of directed distance, on which, 
 in the present Lecture, we have entered, by comparing one ray 
 |3 with another ray a, we have been conducted to a secondary 
 cardinal analysis^ or to an Analysis of Faction ; that is, to a 
 decomposition of one factor act into two other acts of the same 
 kind, which may be illustrated by figure 7. For we may say that 
 if the act or process of transfaction, from the ray a to the ray y, 
 that is (in our example) from the selenocentric vector of the earth 
 to the selenocentric vector of the planet, be decomposed into two 
 successive acts of the same kind, of which the first is given to be 
 that act olfaction whereby we pass from the ray a to the ray /3, or 
 from the selenocentric vector of the earth to that of the sun, then 
 \}i\Q second h found to be (or to be equivalent to) that other act, of 
 profaction, whereby a passage of the same sort is made (along 
 the remaining face of the pyramid) from the ray j3 to the ray y, 
 or from the selenocentric vector of the Sun to the selenocentric 
 vector of Venus. And thus we may, if we think fit, interpret 
 the assertion, that " the Transfactor divided by the Factor gives 
 the Profactor as the Quotient ;" or in symbols, we may inter- 
 pret thus the formula, 
 
 y-^/3 = (7-a)-0-^a); 
 
 whatever desire of such abbreviation as might be gained by the 
 omission of the twice-recurring signs, x a, or by the suppression 
 of the twice-repeated words, " Multiplied into Faciend," may 
 h&vefrst induced us to adopt the latter usual formula, or the 
 former mode of verbal enunciation, while the mark -^- and the 
 name Division were still, as yet, confined by us to what we re- 
 garded as then primary significations : and were therefore em- 
 ployed to denote only the comparison of one directed dis- 
 tance WITH another. 
 
 57. As examples of such comparison or analysis, which may 
 illustrate what has been already said, we shall here consider a few 
 very simple cases ; in some of which the compared rays shall agree 
 with each other in direction, but differ from each other in length ; 
 while in other cases they shall, on the contrary, agree in length, 
 but differ in direction. 
 
 Supposing then, first, that we have not only (as in the ex- 
 
LECTURE 11. 49 
 
 ample of article 39), /3 = a + a, but also y «= j3 + /3 + j3 ; as is re- 
 presented in this figure, 
 
 Fig. 8. 
 
 We shall then evidently have, not only |3 -r- a = 2 (as in 39), 
 but also 7-f-j3 = 3, and y ^ a = 6. In this case, then, the 
 factor q, the profactor r, and the transfactor s, are respectively 
 equal to the cardinal numbers, 2, 3, 6 ; and the general relation 
 (of art. 49) connecting them, or the formula, s = rxq, becoming 
 here simply 6 = 3x2, is obviously, in this example, consistent 
 with ordinary arithmetic ; as is also the inverse formula (of art. 
 50), r-s -^ q, since it becomes here 3 = 6 -^ 2. Now (compare 
 art. 40), that division of the ray, y, or of the line /3+j3 + j3, or 
 of 6 X a, by the ray or line j3, or 2 x a, which conducts to the quo- 
 tient 3, is what I call a primary cardinal analysis, or is an ex- 
 ample of what I regard as the primary operation of Division in 
 Geometry ; since it leads to an expression for the relative length 
 of a line y, as compared with another line j3; the relation of di- 
 rections being already known to be, in the present case, a relation 
 of sameness, or identity. And on the other hand the division of 
 the number 6 by the number 2 is an example of what I call a se- 
 condary cardinal analysis ; at least when this operation is re- 
 garded as being the comparatively abstract analysis of the act 
 ofsextupling, whereby that act (of transfactiori) is here decom- 
 posed into the given act of doubling (which is in this case the 
 act oi faction), and another act of the same sort (the act oi pro- 
 faction), which \^ here found, by this decomposition, to be the act 
 of tripling, as is expressed by the arithmetical formula 6 -4-2=3, 
 according to the mode of interpretation of such formulse which has 
 been above proposed (in art. 56). In like manner in the synthetic 
 aspect of the question, or of the lines and numbers here compared 
 and combined, I regard as primary that cardinal synthesis by 
 which we construct the ray y, or the line /3 + j3 + j3, by operating 
 on another ray /3 with the number 3 as a multiplier ; and I re- 
 gard as sec07idary that other sort of cardinal synthesis, by which 
 
 E 
 
50 ON QUATERNIONS. 
 
 vte produce the number 6 (the transfactor), by multiplying a num- 
 ber 2 (the factor), by another number 3 (the profactor) ; or by 
 compounding the two successive acts of doubling and of tripling, 
 into a third act of the same sort, namely, the act of sextupling, 
 as is expressed, according to the mode of interpretation above 
 proposed (in art. 55), by writing 6 = 3x2. We may, however, 
 according to another mode of interpretation already mentioned 
 (in 49 and 50), retain theformulce 6 = 3x2, and 6 -^ 2 = 3, with- 
 out introducing the conceptions of such composition and decom- 
 position of factions, provided that we regard these formulae as 
 abhreviations for the fuller assertions 
 
 6xa = 3x2xa, and {^xa) -^ (2 x a) = 3, 
 in which the signs x and -f- are used in what we have called their 
 primary significations in geometry. And similarly in other cases, 
 where the lengths only^ but not the directions, of the rays a, /3, 7, 
 are different ; and when therefore the factor, profactor, and trans- 
 factor, are ordinary numbers^ which, in this class of cases, are al- 
 ways positive or absolute, although they may become fractional 
 or incommensurable. 
 
 58. A slightly different class of cases may here be usefully 
 noticed, as conducting, on the same general plan, to the conside- 
 ration of negative numbers ; and as reproducing the usual rules 
 for the multiplication and division of such numbers: while it will 
 also serve as an useful preparation for those more complex pro- 
 ducts and quotients, of which we shall afterwards have to speak. 
 
 By principles already laid doAvn, the sum of any two opposite 
 lines is a null or evanescent line; for the transvector c-a va- 
 nishes, when the provectum c, becoming a revectum, coincides 
 with the vehend a. In fact it is evident that if we first go, along 
 any line ab, from a to b, and then return along the same line, 
 from B to A, we occupy the same final position as if we had not 
 moved at all. We may then say that 
 
 " REVECTOR + VECTOR = ZERO ;" 
 
 and that conversely, 
 
 " REVECTOR = ZERO - VECTOR ;" 
 
 the word zero, or the symbol 0, being understood to denote a 
 null line, when used in such connexions as these. Thus 
 
lecture ii. 51 
 
 (a - b) + (b - a) = ; 
 and 
 
 (a - b) = - (b - a) ; 
 
 which latter equation may be abridged to the following formula 
 (familiar in ordinary algebra) : 
 
 a - B = - (b - a) ; 
 while, by a similar abridgment of discourse, we may say, in 
 words, that 
 
 REVECTOR = MINUS VECTOR : 
 
 understanding or tsichly supplying the word ze/'o before the word 
 minus, in order to bring this mode of expression into harmony with 
 others which have been already discussed. In like manner, if we 
 conceive the provectum c to coincide with the provehend b (and 
 not now with the vehend a), it will be the provector c - b (in- 
 stead of the transvector c - a), which will vanish, while the trans- 
 vectum and vectum will coincide ; we shall, therefore, have 
 the enunciation : 
 
 VECTOR = ZERO + VECTOR ; 
 
 which may be abridged to the following form : 
 
 VECTOR = PLUS VECTOR ; 
 
 the word zero being still understood. In symbols we have (as in 
 algebra), 
 
 B - A = (b - b)+ (b - a) = + (b - a) ; 
 
 and more concisely, omitting the 0, 
 
 B -A = +(b - a). 
 
 Thus, a being a symbol for a ray, or for a vector, + a comes to be 
 another symbol for the saine rag or vector ; and - a comes to be 
 a symbol for the opposite rag, or for the revector corresponding. 
 In like manner, after agreeing that 2a shall denote concisely the 
 same thing as 2 x a, the symbols + 2a and - 2a come to denote, 
 respectively (as in fact they are often employed to do), the dou- 
 ble of the ray a itself, and the opposite of that doubled ray; and 
 similarly in other instances. 
 
 59. Now, I think, that the clearest way of viewing positive 
 and negative numbers, at least as connected with Geometry (for 
 I endeavoured many years ago to shew thiat such numbers might 
 
 E 2 
 
52 ON QUATERNIONS. 
 
 be regarded as presenting themselves in Algebra, according to 
 the view which 1 took of that science, as results of the division 
 of one step in time hy another), is to regard such numbers as 
 being each the quotient of the division of one step in space, 
 that is, of one ray or vector, by another step in space, which has 
 its direction either exactly similar or else exactly opposite to the 
 former. Thus, the cardinal numbers, " positive two" and " ne- 
 gative two," or + 2 and - 2, would offer themselves in this view 
 as certain geometrical quotients, or at least as quotients of certain 
 geometrical divisions, of that general kind which has been con- 
 sidered in the present Lecture, namely, as quotients of the forms, 
 
 + 2 = + 2a-i-a; — 2 = -2a-r-a; 
 
 where the symbols + 2a and^- 2a are interpreted as in the fore- 
 going article, and do not (here) denote abstract numbers, but 
 certain comparatively concrete conceptions, namely, certain rays, 
 or lines, or steps in space. Observe now this diagram, 
 
 Fig. 9.. 
 
 which is designed to picture the conceptions of the relations, 
 j3 = -2a, 7 = +6a; and you will see that for this set of rays, a, 
 |3, 7, the values of the factor, profactor, and transfactor, are the 
 following negative or positive numbers : 
 
 Factor =^=j3-f-a = -2; 
 
 < Profactor =r=y -r-j3 = -3; 
 Transfactor = 5 = 7 -^ a = + 6. 
 
 You see, then, that the general formula or rule of multiplication 
 assigned in the present Lecture, namely, the rule 
 
 Transfactor = Profactor x Factor, 
 
 gives here, again, as in art. 57, a result agreeing with received 
 principles, namely, with those of elementary algebra, since it 
 gives 
 
 (+6)=(-3)x(-2); 
 
LECTURE II. 53 
 
 or in words, the result, that " Positive Six equals the product of 
 Negative Three into Negative Two." You see, too, that (in 
 consistency with our present views) we may either regard this 
 elementary result as a mere abbreviation of the formula 
 
 (+6)xa = (-3)x(-2)xa, 
 where the sign x may still be considered as being used in what 
 we have called its primary sense ; or we may interpret the same 
 result of multiplication, of the two negative numbers proposed, 
 as signifying that the two successive acts, of negatively doubling 
 and negatively tripling, compound themselves into the single act 
 of positively sextupling. And it is obvious that analogous re- 
 marks apply to the converse formula of division, 
 
 (+6) -4- (-2) = (-3). 
 
 In general, this way of considering the multiplication and divi- 
 sion of positive or negative numbers (whether whole or fractional 
 or incommensurable), reproduces the usual rule of the signs, and 
 is, in all its consequences, consistent with common algebra. 
 
 60. A few words may, however, be said here upon the rule 
 OF THE SIGNS just referred to, in the hope that they may make 
 that rule and the present p?inciples throw light upon each other. 
 Suppose, then, that we have, as in this figure, 
 
 Fig- 10. j =T - 
 
 /3 y 
 
 the relations /3 = - a, 7 = - i3, which give also (as the figure 
 shews) the relation y = +a. We might express these relations 
 under the forms 
 
 /3 = (-l)xa,7=(-I)x/3,y = (+l)xa, 
 
 and so arrive, on the plan of the foregoing article, at the well- 
 known equation of algebra, 
 
 (-l)x (-!) = (+ I). 
 
 But we might also write 
 
 /3 = (-)xa, 7 = (-)x/3, 7 = (+)xa; 
 
 regarding the signs ( + ) and ( - ), when thus employed, as being 
 themselves of the nature of geometrical ^ac^or* or multipliers; 
 
54 ON QUATERNIONS. 
 
 because if they operate at all, they do so on the directions of 
 the rays, or lines, or steps, to the symbols of which they are pre- 
 fixed, with the MARK OF faction X interposed ; so that their opera- 
 tion, whether non-effective or effective, comes to be included 
 under that general head or class of operation to which it has been 
 already stated tliat we apply the name multiplication in geo- 
 metry. And then the general relation of multiplication to divi- 
 sion, or of X to -f-, will enable us to form also, as expressions of 
 the same relations between the three rays a, j3, 7, in fig. 10, 
 combined with the nomenclature of preceding articles, the follow- 
 ing little table : , 
 
 Factor =g' =j3 ^ a = (-) ; 
 
 Profactor =^ = 7 -r- j3=(- ) ; 
 
 Transfactor =,s = 7-f-a = ( + ). 
 
 The general formula " profactor into factor equals transfactor," 
 ov ?' X q = s, becomes, therefore, here, the particular formula, 
 
 (-)x(-)=(+); 
 
 and the converse general formula, " transfactor bt/ factor equals 
 profactor," or s -t- q = r, becomes here, 
 
 The effect of the sign ( - ), when thus used as a factor, being to 
 invert the direction of the ray or step on which it operates (as is 
 exhibited by the arrows in the figure), this factor ( - ) itself may 
 be said to be an inversor; whereas the other sign ( + ), when 
 similarly used as a factor, may be called, by contrast, a non- 
 versor, because its effect is simply to preserve the direction of 
 the ray or step on which it operates, or seems to operate. We 
 may also say (by the introduction of another new but convenient 
 term), that the sign ( + ), as a factor, non-verts the ray, to the 
 symbol of which it is prefixed ; or that its effect is a non-version: 
 whereas the sign ( - ), as before, m- verts, or its effect is an in- 
 version. And thus the formula 
 
 (-)x(-)=(+) 
 may (on our general plan) be interpreted as expressing the re- 
 sult of a certain composition of factions ; that is, here, a composi- 
 
LECTURE II. 65 
 
 tion of versions, or still more precisely, a composition of tivo 
 successive inversions, into a single equivalent operation, namely, 
 a wow- version. It signifies, when translated into ordinary words, 
 that if we twice successively invert, or reverse, the direction of 
 any step, we do what is, upon the whole, equivalent to leaving 
 the step unchanged : since, by this double alteration, we recover, 
 or restore, the original direction of that step. And in like man- 
 ner the converse formula, 
 
 ( + )-(-) = (-), 
 
 may, on the same plan, be interpreted as expressing the decom- 
 position of a non-version into two successive inversions ; or as 
 signifying that if it be required to follow up a first inversion of 
 a step by some second operation, which shall, upon the whole, 
 produce the effect of a non-version, or shall restore the step to 
 the direction which it originally had, this second or successive 
 operation must be itself an inversion, or some operation equiva- 
 lent thereto. Remarks precisely similar apply to all the other 
 formulse of this kind, such as 
 
 ( + )x(-) = (-). (-)^(-) = (+); 
 
 which may all be in like msLnner interpreted, and with this inter- 
 pretation proved, if they be regarded as relating to compositions 
 and decompositions of inversions and nonversions of a ?'ap, or 
 more generally of a step in ani/ proposed progression : the general 
 rule being evidently that any even number of m- versions are equi- 
 valent, on the whole, to a wow- version ; and that, therefore, any 
 odd number of inversions are equivalent to a single inversion ; or 
 produce the sa.me fnal effect, as that single inversion would do. 
 61. It is evident also that if we should prefer to look at these 
 last signs ( + ) and ( - ) in their analytic instead oi then synthetic 
 aspect, or to regard them as quotients rather than d^^ factors, they 
 would then (on the general plan already mentioned) come to be 
 considered respectively as symbols of the relations of simi- 
 larity and opposition between the directions of any two rays 
 or steps. Thus we might write again the formulse, 
 
 /3-^a. = (-), 7-^a=( + ), 
 in connexion with the lines of fig. 10, in order to express that on 
 
56 ON QUATERNIONS. 
 
 analyzing the directions of j3 and -y (as marked by arrows in that 
 figure), considered as analyzands, with respect to the direction of 
 a considered as an analyzer, we should find by this comparison 
 (which we regard as being still a species of cardinal analysis), 
 that the relation of directions between /3 and a is a relation of 
 opposition ; but that the relation of directions between y and a is 
 a relation of similarity. And in this analytic aspect of the signs 
 ( + ) and ( - ) as certain cardinal quotients, the formula (-)x 
 (-) = (+) may be interpreted as expressing that two relations of 
 opposition (of directions) cotnpound themselves into one relation 
 oi similarity ; or that the opposite of the opposite of any direc- 
 tion is the original direction itself: while analogous and equally 
 simple interpretations might be given for all other formulae of 
 this sort, on the plan of the present Lecture. 
 
 62. In the two foregoing articles the three lines a, /3, y, 
 which were compared among themselves, were supposed to have 
 equal lengths, and to differ (so far as they differed at all) in their 
 directions only ; or at most in their situations in space, from which 
 situations, however, we abstract, in the present inquiry or contem- 
 plation. The only operators of the cardinal kind, whether effec- 
 tive or non-effective, which have thus been brought into view by 
 the consideration of the example of art. 60, have been (as we 
 have seen) the factors ( + ) and ( - ), regarded as signs or cha- 
 racteristics of nonversion and of inversion respectively ; and not 
 (when used in this sort of connexion) as marks of addition and 
 subtraction; although it was shewn (in articles 58, &c.) how, in 
 the progress q/" notation those earlier significations of + and - 
 which were connected with addition and subtraction, might gra- 
 dually come to suggest or to permit that other use of them, 
 whereby they are connected with multiplication and division. 
 
 63, On the other hand, in the example of art. 57, the three 
 lines a, j3, 7, which were there compared, had all the same direc- 
 tion, and differed only in their lengths. In that example, there- 
 fore, we had not occasion to consider any kind of turning, or of 
 VERSION ; but we had, on the contrary, occasion to consider what 
 may be called a stretching, or a tension, namely, that other 
 operation of the factor kind, by which we pass from one given 
 length (and not from one given direction) to another. It was on 
 
LECTURE II. 57 
 
 extension (not on direction) in space, that we operated in that 
 earlier example ; the act performed was an act of a metric, and 
 not one of a graphic character. The agents, therefore, or the 
 factors, in those earlier operations of the cardinal kind which 
 were considered in art. 57, may naturally, in consistency with the 
 plan of nomenclature employed in these Lectures, receive the 
 general name of tensors; and we may say, more particularly, 
 that the factor, profactor, and transfactor, were (in the example 
 here referred to) a tensor, protensor, and transtensor respectively. 
 And although these three tensors, in the example of art. 57, being 
 the three cardinal numbers 2, 3, and 6 respectively, were thus 
 each greater than the number one, and so had the effect of ac- 
 tually lengthening the line (a or j3) on which they operated ; yet 
 it seems convenient to enlarge by definition the signification of 
 the new word tensor, so as to render it capable of including also 
 those other cases in which we operate on a line by diminishing 
 instead of increasing its length ; and generally by altering that 
 length in any definite ratio. We shall thus (as was hinted at 
 the end of the article in question) have fractional and even in- 
 commensurable tensors, which will simply be numerical multi- 
 pliers, and will all be positive or (to speak more properly) sign- 
 less NUMBERS, that is, unclothed with the algebraical signs of 
 positive and negative ; because, in the operation here consider- 
 ed, we abstract from the directions (as well as from the situa- 
 tions) of the lines which are compared or operated on. Thus 
 the three acts, of doubling a line, of halving it, and of changing 
 it from the length of a side to the length of a diagonal of a 
 square, shall be regarded as being, all three, acts of tension ; 
 the tensors in these three respective acts being the integral num- 
 ber 2, the fractional number ^, and the incommensurable number 
 V'2. The act of restoring a line to its original length, after 
 that length had been altered by a previous act of tension, might 
 be called an act of re-tension, and the agent in the second 
 operation might be called a re-tensor (compare art. 44) ; thus 
 any tensor and its answering retensor would simply be two 
 numbers of which each is (what is commonly called) the recip- 
 rocal oi i\xe other; or, in their analytic aspect, they would re- 
 present ratios mutually inverse. The number 1 might be called 
 
58 ON QUATERNIONS. 
 
 a NON-TENSOR, because ifc makes no actual alteration in the 
 length of the line which it multiplies; just as the sign (+ ) was 
 lately called a non-versor, because it leaves unchanged the di- 
 rection on which it seems to operate. And the general formula 
 for the multiplication of such signless numbers^ or for the com- 
 position of ratios of lengths (or other magnitudes), will offer itself 
 with these conceptions and denominations, as a particular case of 
 the general multiplication of factors, or of the composition of 
 cardinal relations, under the form (compare art. 49) : 
 
 TRANSTENSOR = PROTENSOR X TENSOR ; 
 
 together with the converse formula of division (compare art. 50): 
 
 PROTENSOR = TRANSTENSOR -f- TENSOR. 
 
 64. As regards the example of art. 59, each act of faction 
 there considered may be said to have been compounded of an act 
 of tension, and an act of inversion or of nonversion, according 
 as the numerical (but not signless) multiplier employed was a 
 negative or a positive number ; and we may express this concep- 
 tion by writing, in reference to that example : 
 
 (-2)=(-)x2; ( + 6)=( + )x6; 
 
 with analogous expressions for all other positive or negative num- 
 bers. It is also evidently allowed to write, with a different ar- 
 rangement of the factors, 
 
 (-2) = 2x(-); ( + 6) = 6x( + ); 
 
 since it comes (for example) to the same thing, whether we first 
 double a step and afterwards reverse its direction, or first reverse 
 and afterwards double. We may agree to give the general name 
 of scALARS to all positive and negative numbers (that is to the 
 REALS of ordinary algebra), on account of the possibility of con- 
 ceiving all such multipliers to be represented, or laid down, on 
 one common but indefinite scale, extending from - oo to + oo , 
 that is, from negative to positive infinity. 
 
 65. Proceeding now to a more general examination of the di- 
 rections of lines, or rays, in space, let us consider a somewhat 
 more complex case of the (analytic) comparison of such directions, 
 or of the (synthetic) composition of versions, than any of those 
 
LECTURE II. 59 
 
 which were discussed in recent articles : and for this purpose let 
 i, j,k, denote three straight lines equally long, but diiferently di- 
 rected ; let it be also supposed that these three different directions 
 are rectangular each to each ; and to fix the conceptions still 
 more precisely, let us conceive that these directions of i, j, k, are 
 respectively southward, westward, and upward (in the present 
 or in some other part of the northern hemisphere of the earth) ; 
 So that i and j are both horizontal, but k is a vertical line. We 
 may further imagine that the common length of these three lines 
 is equal to some assumed unit of length, or more particularly, that 
 it is Sifoot ; so that i is or denotes a southward foot, j is a west- 
 ward foot, and k is an upward foot. Then (by art. 58) + i, + j, + k, 
 will be other symbols for the same three directed lines ; but -i, 
 - j, -k, will denote respectively a northward, an eastward, and 
 a downward foot. This being agreed upon, let the three diverg- 
 ing edges, a, /3, 7, of the pyramid in fig. 7 (of art. 53), be con- 
 ceived to be each a foot long, and to be directed respectively 
 towards the northern point of the horizon, the zenith, and the 
 east point, so that we may write the equations : 
 
 a = -i, /3 = + k, 7 = -j. 
 
 The pyramid being thus constructed, we may next proceed to 
 study the three separate acts of faction, profaction, and trans- 
 faction, by which we may pass respectively from a to j3, from j3 
 to y, and from a to y, by operating on the directions of the rays 
 or lines a and j3, and, therefore, by performing what may be 
 called acts of version, proversion, and transversion : since 
 it is clear that there is, in the present case, no act of tension per- 
 formed, the three lines which are compared being supposed to 
 be all equally long. The agents in the three acts which we are 
 thus to study, may be called respectively the versor, the pro- 
 VERSOR, and the transversor; and we may already enunciate, 
 as a particular case of the general formula of multiplication of 
 factors in art. 49, the relation : 
 
 TRANSVERSOR = PROVERSOR X VERSOR ; 
 
 which must, by the general conceptions and definitions of multi- 
 plication already stated, hold good for eve?'y composition ofver- 
 
60 ON QUATERNIONS. 
 
 sions. We may also, in like manner, as a particular case of the 
 general formula of division of factors in art. 50, enunciate this 
 converse relation, 
 
 PROVERSOR =TRANSVERSOR -^ VERSOR ; 
 
 which is to be regarded as being likewise valid, by the general 
 significations of the terms employed, for every case of decomposi- 
 tion of versions^ or of rotations in geometry. We may also mo- 
 dify the phraseology of former articles, respecting the three lines 
 a, j3, 7, themselves, considered now as the subjects or the results 
 of operations of the versor kind, by naming those three lines as 
 follows (compare the table in art. 51) : 
 
 a = Vertend = Transvertend ; 
 j3 == Versum = Provertend ; 
 y = Proversum = Transversum ; 
 
 in order to mark, by this nomenclature, that we now abstract 
 from the lengths of the lines, or that we treat those three lengths 
 as equal. We shall thus be able to assert generally (compare 
 art. 41), that 
 
 VERSOR X VERTEND == VERSUM, 
 
 and that 
 
 VERSUM -~ VERTEND = versor; 
 
 with other analogous formulse (compare articles 47, 48) for pro- 
 version and transversion respectively. But what the particular 
 acts of version are, for any particular set of lines or rays, as (for 
 example) for the set mentioned at the beginning of the present 
 article, it still remains to consider. 
 
 66. In this consideration or inquiry, we may assist ourselves 
 by remembering the general remarks which were offered at an 
 earlier stage of the present Lecture (in articles 39 and 40). The 
 lengths of the lines which are to be compared being (in the pre- 
 sent question) equal to each other, the metric element of the in- 
 quiry disappears, and only the graphic element remains. We 
 have, therefore, only now to inquire, as concerns the lines a and 
 j3, through what angle, in what plane, and towards which hand, 
 are we to turn the line a as a given vertend, in order to make it 
 
LECTURE II. 61 
 
 attain the proposed direction of the versum^ that is of the line j3? 
 For the answer to this inquiry, when it shall be, in any manner, 
 with suflBcient clearness and fulness assigned, will be, under one 
 form or other of expression, a sufficient description, statement, or 
 particularization of the sought versor, which we have already, by 
 anticipation, denoted by the symbol j3 -f- a, and have called a 
 cardinal quotient. 
 
 67. Now, with the particular directions above assumed or 
 assigned, for the vertend and versum, or for the lines a and j3, 
 namely, those otherwise denoted (in 65) by - i and + k, or the 
 (horizontally) northward and the (vertically) upward directions, 
 it is clear that the angle of version is a right angle ; the plane is 
 meridional; and the axis of right handed rotation, from a to j3, 
 is a right line directed westward. In that little model of a tran- 
 sit instrument which you see here, the line a may be conceived 
 to be the telescope when pointed to a north meridian mark ; and 
 /3 the same telescope, directed towards the zenith. And when 
 I lay my hand on the westward half of the axis in the model, 
 and turn that part right handedly, with a motion of the screwing 
 kind, you see that the northern (or object) end of the tele- 
 scope comes to be elevated^ while the southern {ox eye) end is de- 
 pressed. Continuing this motion of rotation through a quad- 
 rant of altitude, you see that I have erected the telescope in the 
 model, in such a manner as to cause it to attain a vertically 
 upward direction ; and that thus I have, in fact, changed the 
 telescope (that is, its object half) from the direction symbolized 
 by a to the direction symbolized by /3. The required act of ver- 
 sion, symbolized by j3 -r- a, has, therefore, in this case, been 
 actually and practically performed. 
 
 68. And since the (mechanical) agent in producing this (me- 
 chanical) rotation, or in this right-handed (or screwing) act of 
 version, has been an axis or handle directed to the west^ which 
 direction has also been lately supposed (in art. 65) to belong to 
 the line denoted by the symbol +j, I propose now to denote the 
 versor itself, or the conceived agent of the conceived version, or 
 of the purely geometrical rotation from a to /3, by the connected 
 symbol j; availing myself (as you see) of the distinction between 
 the roman and the italic alphabets, to mark, at least temporarily, 
 
62 ON QUATERNIONS. 
 
 the distinction between the two different conceptions of a line, as a 
 turned and as a turning thing ; a versum and a versor ; a subject 
 of operation and an operator. We shall thus have, on the ge- 
 neral plan of notation already stated or sketched for you, the for- 
 mulae : 
 
 i3^a = ( + k)--(-i)=i; 
 
 jxa=jx(-i)=^ = + k; 
 
 and the ^^j- operation" or the operation of multiplying a line 
 by the factor or versor y, is seen to have the effect of elevating a 
 transit telescope from that position in which it is directed to the 
 north point of the horizon, to that other position in which it is 
 directed towards the zenith. The conception of this operation 
 may be illustrated by figure 11, where the axis J h drawn as di- 
 rected to the west, and as ready to operate on the telescope or 
 line a, which line is, before the operation, represented as directed 
 towards the north ; but is to be conceived as taking, after that 
 operation, the direction towards the zenith, represented by j3 in 
 fig. 12 : with which two figures, I shall here, by anticipation, as- 
 sociate a third (fig. 13). 
 
 Fig. 11- Fig. 12. Fig. 13. 
 
 Z E 
 
 A 
 
 I^S N< 
 
 w 
 
 
 I ! j 
 
 I j 
 I 
 
 w 
 
 69. Having thus passed, by the way of rotation, from a to j3, 
 or from - i to + k, there is no difficulty in passing similarly from 
 /3 to y, or from + k to - j. The act of version having been stu- 
 died and symbolized, it becomes easy to study and symbolize, in 
 like manner, the subsequent but analogous act of proversion. 
 We have passed from a northward to an upward position of the 
 telescope ; and we are now to pass from an upward to an east- 
 ward position thereof This cannot, indeed, be done by any such 
 meridional motion as belongs to an ordinary transit telescope ; 
 
LECTURE II. 63 
 
 but it can be done by that other important mode of motion of a 
 telescope, of the ea;^;-«-meridional kind, in the plane of theprme- 
 vertical, which has been used, with great success, in some cele- 
 brated geodetic surveys, and also at some fixed observatories, in 
 Russia and elsewhere. Having already erected the telescope to 
 the zenith in this little model of a transit, you see that I can turn 
 the model through a quadrant of azimuth, so as to cause that 
 axis, or semiaxis, which had been directed westward, to take the 
 southward direction. And if I now lay my hand on the same 
 physical or mechanical semiaxis as before, but in its new and 
 southward direction, you see that the sa77ie sort of screwing mo- 
 tion, as that which was before employed, being continued through 
 the same angular quantity, namely, through a quadrant of rota- 
 tion of the telescope, in the plane of the prime vertical, has the 
 effect of turning that telescope from the upward to the eastward 
 direction, or from the direction of/3 to that of y, that is, from 
 the direction of + k to that of - j. In short, you see that the re- 
 quired act of Proversion is thus effected ; and that I may natu- 
 rally denote the Proversor, or the agent of the proversion, on the 
 plan of the foregoing article, by the symbol i; because, as you 
 may see illustrated by the diagram last referred to (fig. 12), the 
 axis, or handle, of this proversion, is, like the line already de- 
 noted by + i, a line directed towards the south. We are thus led 
 to write the equations : 
 
 ^x/3 = ^■x( + k) = 7=-j; 
 
 by combining which with the equations of the foregoing arti- 
 cle, on the plan of art. 49, we obtain these other formulae ; 
 
 i xj X a = y ', i xj = y -^ a. 
 
 70. Proceeding to consider the transversion, we are next to 
 inquire what one rotation in a single plane would bring the ver- 
 tend a into the direction of the proversum y ; or would cause the 
 telescope to pass, by a single act of turning, from its original and 
 northward, to its final and eastward direction. And it is clear, 
 either from the model before you of the eight-feet Circle, which 
 
64 ON QUATERNIONS. 
 
 belongs to the Observatory of this University, or from the little 
 diagram above drawn (fig. 13), that the plane of this transversion 
 is horizontal ; that its angular quantity is a quadrant; and that, 
 if the rotation be still conceived to be n^/i^-handed, its axis is a 
 line directed vertically upwards : so that the Transversor itself 
 may be denoted (on the plan of recent articles) by the italic let- 
 ter k, because the axis or handle of its operation has the direc- 
 tion of the line which we have above denoted by + k. We shall 
 thus have the formulae : 
 
 7^« = (-J)-^(-0 = ^i 
 ^ X a = ^ X ( -i) = -j. 
 
 And by comparison of the last value of 7 -f- a, with that assigned 
 in the preceding article, or by the general principle that trans- 
 versor = proversor x versor (art. 65), we arrive at the simple but 
 useful equation following : 
 
 ixj^k; 
 
 which may either be interpreted (synthetically) as asserting that 
 the quadrantal rotation^ round a westward axis, being succeeded 
 by another quadrantal rotation i, round a southward axis, produces 
 finally, and upon the whole, the same change of direction as that 
 third quadrantal rotation k would do, which is performed round 
 an upward axis, these three rotations being all supposed to be 
 right-handed ; or (analytically) as expressing a composition of 
 relations of directions in space, which corresponds to this com- 
 position of rotations. 
 
 71. After settling, as above, the significations of the symbols 
 i,j^ k, regarded as certain quadrantal versors, or as symbols denot- 
 ing the conceived agents or operators of certain quadrantal and 
 right-handed rotations in the three rectangular planes of the prime 
 vertical, the meridian, and the horizon, round axes directed res- 
 pectively towards the south, the west, and the zenith ; we may 
 proceed to investigate, on similar principles, and by analogous 
 compositions of rotations, the symbolic values of all the other 
 binary products of these three factors or versors 2, j, k ; and 
 should find for eac/« such product a determinate result, unafi"ected 
 by any change of the line (a) assumed as the original vertend^ 
 
LECTURE II. 65 
 
 which change the general plan of the construction might allow. 
 Thus, in order to find anew the value of the product i xj\ we may 
 indeed vary the vertend a, since we need not assume this line to 
 be (as was supposed in art. 65) ayoo^ directed towards the north. 
 We might assume the line a to denote any longer or sho/^ter line 
 in the same northward direction ; but then we should only alter, 
 in the sa7ne ratio, the lengths of the two other lines j3 and y, 
 without their ceasing to be directed respectively towards the 
 zenith, and the east, so that the geometrical quotient 7 -f- a, or 
 the product i xj\ would still be found equal to k, since the pro- 
 versum y would still be a line of the same length as the ver- 
 tend a, and would still be advanced beyond it by a quadrant of 
 azimuth, while both these lines would still be contained in the 
 same horizontal plane, if they be conceived to radiate from one 
 common origin. We might even assume the vertend a to be a 
 line directed to the south, and not to the north as before; for the 
 only eflfect of this change would be that the versum j3 would take 
 a downward (instead of an upward) direction ; and that the pro- 
 versum y would be directed to the west, instead of being pointed 
 to the east : and on finally comparing the (new) westward direc- 
 tion of y with the (new) southward direction of a, we should find 
 that y was still, as before, more advanced in azimuth than a by 
 a quadrant, both being still in a horizontal plane, so that 7 -^-a 
 would still be found equal to k. It was thus (for example), that 
 in the recent act of version (68), the eye-end of the telescope in 
 the model was depressed from the south to the nadir ; while in 
 the proversion (69), the same eye-end was elevated from the 
 nadir to the west: and the same horizontal transversion (70), 
 which brought the object-end from north to east, brought also, at 
 the same time, the eye-end from south to west. In symbols, re- 
 taining the recent significations of i, j, k, as well as those of i,J, 
 k, we might have assumed, 
 
 a = + i, /3 = -k, 7-+j, 
 
 instead of the values or directions which were assumed for a, j3, 
 7, in art. 65 ; and then we should have had the relations,, 
 
66 ON QUATERNIONS. 
 
 i3-^a=(-k)-( + ^)=i; 
 7-f-/3 = (+j)-(-k) = e; 
 7 -f-a=(+j) ^( + i) = ^; 
 
 whence there would have followed, as before, the equation, 
 
 ixj = k. 
 
 Nor could any variation of this result be obtained by assuming 
 o^Aer positions of a; for the plan of construction requires that 
 this line a should have either a northward or a southward direc- 
 tion, if it is to be used as the vertend in the determination of the 
 product i xj ; since it is to be in the plane of version, that is here 
 in the meridian plane, and is also to be perpendicular to the ver- 
 sum, or provertend, |3 ; which latter line /3 must lie at once in 
 the two planes of version and proversion, or in the planes of the 
 meridian and prime vertical, and must, therefore, be a vertical 
 line, directed either upwards or downwards. 
 
 72. With respect to the other binary products of i,j, k, it is 
 easy to perceive, first, that we have, by an exactly similar com- 
 position of rotations, the formulae, 
 
 j X k = i, and k x i =j ; 
 
 which only differ from the formula i xj=^k, by a cyclical permuta- 
 tion of the symbols, and can, on this account, be easily reinem-- 
 bered. In fact if it were required to determine directly the value 
 of the product^' X /i, on the same plan of construction as before, 
 we should have to assume a direction for the versum j3, which 
 should be contained at once in the two planes of version and pro- 
 version, or be perpendicular at once to the axes of the two suc- 
 cessive rotations; thus j3 must be perpendicular to both k and j, 
 and must, therefore, have one or other of the two opposite direc^ 
 tions denoted by the ambiguous symbol + i ; and by a principle 
 already mentioned, it is unimportant which of these two we select, 
 the choice not affecting the value of the transversor 7 -^ a ; since 
 a change in this choice can only invert both, at 07ice, of the direc- 
 tions to be finally compared. Assuming then j3 = + i, we easily 
 find that we are to assume, at the same time, a = -j, and7 = -k, 
 in order that we may have 
 
LECTURE 11, 67 
 
 k X a = (5 = i, J X (3 =j X [ = y ; 
 
 and thus we find that the required product is 
 
 In like manner, to determine the value of ^ x ^, we may assume 
 
 /3 = + j, a--k, 7 = -i, 
 and we find that 
 
 73. On the other hand, to find the value ofj x i, although 
 we may still suppose, as in the example of articles 65, &c., that 
 the versum j3 is directed vertically upward, we must then vary 
 the directions of a and y from those which were employed in that 
 example; for if we takej3 = + k, we must take a= + j, and -y^ + i, 
 in order that we may have the relations, 
 
 i X a = /3 = + k, J X j3 =7 X ( + k) = y. 
 
 The telescope is now to be conceived as being originally directed 
 to the west ; as being next elevated to the zenith, by a rotation in 
 the plane of the prime vertical, of which the agent or versor is 
 i ; and as being finally depressed to the south point of the horizon, 
 by operating with the proversor j. It has, therefore, in this case, 
 been caused upon the whole to retrograde (and not to advance) 
 in azimuth through a quadrant, since it has been moved from the 
 west to the south. Or we might assume 
 
 « = -J' /3=-k, 7--i, 
 because 
 
 «x(-j)=(-k),ix(-k)=-i; 
 
 that is, we might conceive the telescope to be first depressed by 
 the versor i from the east to the nadir, and then elevated by the 
 proversor _/ from the nadir to the north point; but we should 
 still have, on the whole, a retrogression of a quadrant in azimuth, 
 or a Ze/5^-handed motion (from east to north) through a right an- 
 gle, round an axis directed vertically upwards. Thus, 
 
 >x^ = ( + i)-^( + j) = (-i)--(-j); 
 
 but also (by 72 and 60), 
 
 F 2 
 
68 
 
 ON QUATERNIONS. 
 
 ^x(-j) = ( + i), and(-) x(+i)=(-i); 
 whence it follows that 
 
 (-i) = (-)xAx(-j), (-i)^(-j) = (-)xA, 
 
 and finally that 
 
 jxi = {-)xk. 
 
 In words this comes to substituting for the quadrantal retrogres- 
 sion in azimuth a quadrantal advance, succeeded by an inversion 
 of the telescope. 
 
 74. But we may also conceive the motion from east to north, 
 or from west to south, to be effected by a ri^A^-handed rotation 
 through a quadrant, performed round a downward axis ; and in 
 this view, the transversor in the present question is seen to be a 
 line in the direction of -k, so that it may conveniently be de- 
 noted by the symbol -k, as is exhibited in figure 14. 
 
 Fig. 14. 
 
 We may then write also, 
 
 jxi = -k', 
 
 and in fact this shorter notation is seen to harmonize with the 
 formula recently obtained. It is proper, however, to observe 
 that we have thus been conducted to one important departure 
 (the only one, indeed, that has hitherto offered itself to our atten- 
 tion) y/-om the rules or mechanism of common algebra. For we 
 have been led to conclude the two contrasted results : 
 
 ixj=k ; J xi = -k ; 
 
 which shew that (in the present system) the multiplication of 
 versors among themselves is not generally a commutative ope- 
 ration: or that the order of the^ac^or* is not indifferent to the 
 
LECTURE II. 69 
 
 result- In fact we have been led to express thus a theorem of 
 ROTATION, which is indeed very simple, but is, at the same time, 
 very important, and which there is consequently an advantage 
 in having so short a mode of formulizing : namely, the theorem 
 that two rectangular and quadrantal rotations compound them- 
 selves into a third quadrantal rotation, rectangular to both the 
 components, but having one or other of two opposite directions 
 (or characters, as right-handed or left-handed, round one axis), 
 according as the composition has been effected in one order or iti 
 the other. It is thus that, for example, in figs. 11 , 12, 1 3, if the ro- 
 tation denoted byj he followed hy that denoted by i, the telescope 
 has been seen to be turned upon the whole from north to east, its 
 intermediate position being upward ; whereas the same telescope 
 would (as we also saw) be brought back from the east to the north, 
 through an intermediate and downward direction, if the rotation 
 i were performed y?r5^, and aftenvards the rotation j'; or would 
 be brought, as in fig. 14, from a westward to a southward posi- 
 tion. It is easy to deduce, on the same plan, the analogous equa- 
 tions, 
 
 k xj = — i, ix k = —J, 
 
 which are contrasted respectively, in the same way, with the 
 equations 
 
 J X k=^i, kx i^j; 
 
 and in which -i is a versor with a northward axis of right- 
 handed rotation, and -j is another versor, with an eastward axis 
 of a rotation likewise right-handed. Or we may write (on the 
 plan of the last article) these other and equivalent formulae : 
 
 k^j={-)^i; ixk=(-) xj; 
 
 which would express that the old resultant rotations round south 
 and west (in 72) were now to be succeeded by inversions. 
 
 75. We have not yet considered the squares of the symbols 
 i,j,k, or the products of equal versor s. But we have seen (in 
 73 and 69), that 
 
 «'x (+J) = + ^^> andz x( + k)--j = (-l)xj; 
 by combining which two results it follows that 
 
70 ON QUATERNIONS. 
 
 or that 
 
 2 X 2 = - 1 . 
 
 The same conclusion would have followed, if we had twice suc- 
 cessively operated by i on the line -j, or on either of the two 
 lines + k. In general it is clear that if any line in the prime-ver- 
 tical (or in any other) plane receive two successive and similar 
 quadrantal rotations, its direction is thereby on the whole in- 
 verted or reversed, or multiplied by - 1. For the same reason, 
 we have, in like manner, the values : 
 
 jxj=-l; kxk = ~l. 
 
 We may also write more concisely (compare art. 60), 
 
 i X I =j xj = k X k = {- ) ; 
 
 and may say that these three quadrantal versors i, j, k, together 
 with their own opposites, -i, -j, - k, are semi-inversors, or 
 produce each a semi-inversion. Indeed we see more generally 
 that every other quadrantal versor with any arbitrary axis 
 in space, is, in like manner, a semi-inversor, and may be re- 
 garded as a geometrical square root of negative unity ; or even 
 as a square root of minus, when " minus" is treated as 2l factor : 
 so that every such versor may be considered as included among 
 the interpretations of the symbol V - 1 or ( - )*; at least if we 
 suppose, for the present, each such versor to operate on a line 
 perpendicular to itself, or perpendicular to the axis of that quad- 
 rantal rotation of which the versor is conceived to be the agent. 
 76. It may have been noticed that we have not only the six 
 formulae : 
 
 (ixj = k, jxk = i, kx i =j, 
 '^j X i = - k, k xj = - i, i xk = -j, 
 
 considered as results of the multiplication of versors, or of the 
 composition of rotations, but also the closely analogous formulse, 
 
 zxj=k, yxk = i, ^xi=j, 
 Lyxi = -k, ^x j --i, 2xk = -j, 
 
 considered as the six results of so many single versions, and not 
 of versions compounded among themselves. These two sets of 
 
 {; 
 
LECTURE II. 
 
 71 
 
 results correspond to diiFerent conceptions and constructions, and 
 are not to be confounded with each other. We saw, for instance 
 (in connexion with the figures 11, 12, 13), that the formula ixj 
 = k expressed (as above interpreted) the result of a process, 
 whereby a telescope was first elevated from a northward to a 
 vertical position, and then depressed to an eastward one, being 
 thereby qaused upon the whole to advance through a quadrant of 
 azimuth. But the formula 2xj =k (which occurred in art. 73, 
 the line j being there denoted by a), expressed, at least according 
 to the interpretation already given, that a telescope originally 
 directed towards the west would be elevated to the zenith, if it 
 were caused to revolve right-handedly through a quadrant round 
 an axis directed to the south (as in the first part of figure 14). 
 The signification of the one formula {ixj = k) has thus been 
 made to depend on the consideration oi three quadrantal rota- 
 tions, in three rectangular planes ; whereas the signification ot 
 the other formula (zxj=k) has been made to depend on the con- 
 sideration of a single rotation of this sort. Yet the two results 
 are by no means unconnected geometrically, nor is it accidental 
 that their symbolic expressions have so close a resemblance to 
 each other ; for this symbolical analogy arises from, and em- 
 bodies, a general theorem of rotation. And 1 conceive that we 
 may now legitimately, and with advantage, avail ourselves of the 
 same analogy, or of the theorem to which it corresponds, to dis- 
 pense with that symbolic distinction which has been above ob- 
 served, between the three quadrantal versors i,j, k, and the three 
 lines, i, j, k, which have respectively the directions of their three 
 axes. Dismissing, therefore, or suspending, the use of the ro- 
 man letters i, j, k, I propose now to regard the formula i xj = k, 
 as being the common expression of the two connected results rela- 
 tive to rotation, of which one was illustrated by the three figures 
 11, 12, 13, and the other by the first part of figure 14. And in like 
 manner, each of the five other formulse of the same sort, respect- 
 ing the binary products of i, j, k, as for example, the formula^ x k 
 = i, will come to be regarded as the commo7i expression of two 
 distinct but connected results; one relative to a certain composi- 
 tion of versions, and the other relative to a single rotation. It is 
 clear that similar remarks apply to the comparison of such results 
 
72 ON QUATERNIONS. 
 
 of division of rays, and of decomposition of versions, as are ex- 
 pressed by the following formulae : 
 
 z = k -^ j ; 1 = 1^ ^j; 
 
 and by others analogous thereto. 
 
 77. In this manner we may be led to regard the three italic 
 letters «,j', k^ as symbols of the same three lines which were 
 lately denoted by the three roman letters i, j, k. Ox rather, for 
 the sake of a somewhat greater generality, in future applications, 
 we shall now say that i,j, k, may be regarded as symbols of 
 
 ANY THREE MUTUALLY RECTANGULAR AND EQUALLY LONG 
 
 LINES, whose common length is still supposed to be the unit of 
 LENGTH ; while the rotation, i'ound the first (i), from the se- 
 cond (j), to the third (/t), is positive ; that is (as we shall still 
 suppose) right-handed: these last suppositions being a little 
 more general than those of art. ^b^ in virtue of which the three 
 lines i, j, k, were respectively a southward, a westward, and an 
 upward foot. And, on the other hand, we are conducted to regard 
 each of these three right lines, i,j, k, and similarly every other 
 unit line in space, as being a quadrantal versor ; whose 
 operation, on any right line in a plane perpendicular to itself, 
 has the effect of turning this latter line through a right 
 angle, towards the right hand, in the same perpendicular 
 
 PLANE. 
 
 78. Indeed this view of the directional or graphic opera- 
 tion of one right line on another line perpe7idicular thereto, 
 whereby that operation is considered as producing or determin- 
 ino-, by a rotation towards a given hand, a third line perpendi- 
 cular to both, appears to be so simple in itself, and so intimately 
 connected with whatever is most characteristic in the whole 
 conception of tridimensional space, that we might have been 
 pardoned if we had chosen to set out with it, and to define that 
 such should be regarded, in our system, as the operation of mul- 
 tiplying one of two rectangular lines by another, when direc- 
 tions alone were attended to. And then the contrast between 
 the two formulae, 
 
 i^j^k, j xi = -k, 
 or the non-commutative character of this sort of geometrical mul- 
 
LECTURE II. 73 
 
 tiplication, would have oflFered itself to our notice, even more 
 simply than in art. 74 ; as expressing, for example, that if a west- 
 ward line be turned right-handedly through a right angle, round 
 a southward axis, it is elevated to the zenith ; but that if (by 
 an interchange of operator and operand) a southward line be 
 turned, in like manner, round a westward axis, through a quad- 
 rant, and towards the right-hand, it is, on the contrary, de- 
 pressed to the nadir. And so many other consequences could 
 be drawn from the same simple conception of this directional 
 operation of line on line, that it might not be too much to say, 
 that the whole Theory of Quaternions, or that all the symbo- 
 lical and geometrical properties of quadrinomial expressions of 
 the form w + ix+jy + kz, where w, x, y, z are any four scalar 
 constituents (four positive or negative numbers), while ?', j, k 
 are three rectangular vector units, would admit of being 
 systematically developed from the supposed definition, above 
 mentioned, of this case of the geometrical and graphic multipli' 
 cation of lines ; at least if this were combined with those other 
 and earlier definitions of geometrical addition and subtraction, 
 which other definitions (as was noticed in art. 36) are not pecu- 
 liar to quaternions, but are common to several systems of appli- 
 cation of symbols to geometry. But it has seemed to me that 
 the subject allowed of its being presented to you under a still 
 clearer light, and with a still closer philosophic unity, by the 
 adoption of the plan on which these Lectures have hitherto 
 been framed, and on which it is my purpose to pursue them, if 
 favoured for some time longer with your attention. 
 
LECTURE III. 
 
 79. The two preceding Lectures, Gentlemen, will be found, 
 I think, to have advanced us, in no inconsiderable degree, towards 
 a correct and clear understanding of the principles of the Calcu- 
 lus of Quaternions : since they have contained an exposition of 
 the primary (and of some of the chief derivative) significations 
 attached, in that Calculus, to the four elementary signs + - x -r-, 
 or to the four fundamental operations of Addition, Subtraction, 
 Multiplication, and Division, when viewed in connexion with 
 Geometry. Those primary significations (in the view thus taken 
 of them) have indeed been stated, at first, in a very general and 
 somewhat metaphysical manner ; but they have since been illus- 
 trated by so many and such simple examples^ geometrical or 
 astronomical, combined with the exhibition, in some cases, of ap- 
 propriate models and diagrams, that the seeming vagueness or 
 obscurity, whatever it may have been, of those early statements 
 (in art. 5), may be hoped to have been, by this time, sufficiently 
 done away. We must, however, now proceed to develope still 
 farther the same principles, and to apply them to new questions, 
 in order to render still more manifest their geometrical meanincr 
 and utility. We may not indeed be obliged to enlarge, except 
 in a few instances, the nomenclature or vocabulary of the 
 science, which some may think already too copious; but its no- 
 tation will require to be extended and illustrated by new defi- 
 nitions and examples. The conceptions themselves must be 
 still further unfolded and combined; and the symbols by which 
 they are to be embodied and expressed must be shewn to be the 
 elements of a Calculus, possessing, on several important points, 
 its own appropriate rules ; although aiming in many other res- 
 pects, and indeed wherever this can be done without sacrifice of 
 
LECTURE III. 75 
 
 its peculiar features, to render available, in conjunction with its 
 own new usages, the results and habits of Algebra. More ge- 
 neral processes for geometrical Multiplication and Division must 
 be exhibited, than have been given in the foregoing Lecture ; 
 and these must be combined with those already stated, for geo- 
 metrical Addition and Subtraction. And above all, it will be 
 indispensably required by the plan of the present Course, that 
 we should soon proceed to consider more closely than we have 
 hitherto done, the questions, What is, in this System, a Quater- 
 nion ? and On what grounds is if so called? 
 
 80. The general notion of multiplication, or of faction, in 
 geometry, proposed in the foregoing Lecture, has been, that it is 
 an act or process which operates 1st, on the length of a ray; or 
 2nd, on its direction; or 3rd, on both length and direction at once. 
 The multiplier or factor has been conceived to be the agent or 
 the operator in this act or process; and the multiplication of any 
 two factors among thetnselves, in any assigned order, has been 
 conceived to correspond to the composition oi two successive acts 
 of faction, and to the determination of the agent in the resulting 
 act oi transf action. And the mai^h or characteristic of such fac- 
 tion, or of such composition of factions, has been with us the 
 familiar sign x, pronounced or read, as usual, by the word into. 
 As examples of such factors in geometry, we have as yet con- 
 sidered only t\\Qfour following classes : L tensors or signless 
 numbers, such as 2, 3, 6, |, -v/2, which operate only metrically 
 on the lengths of the lines which they multiply, and which are 
 to be combined among themselves, as factors, by arithmetical 
 multiplication, or by the laws of the composition ofratios; IL 
 SIGNS, namely (+) and (-), regarded as marks of nonversion and 
 inversion, which operate (as such) only to preserve or to reverse 
 tbe direction of a line, and are combined among themselves ac- 
 cording to the usual rule of the signs; IIL scalars, or sign- 
 bearing numbers, such as - 2 or + 6, which are simply the reals 
 of ordinary algebra, and are combined with each other as factors 
 according to the known rules of algebraic multiplication, while 
 each may be regarded as being itself the product of a tensor and 
 a sign, and may at once alter the length of a line in a given ratio, 
 and also nonvert or invert its direction; IV. vector-units, 
 
76 ON QUATERNIONS. 
 
 or quadrantal versors, such as i, j, k, and their opposites - i, -j\ 
 -k, of which each is a purely graphic operator, having the effect 
 of turning a line, in a plane perpendicular to itself, right-handedly 
 through a quadrant, but having no power to alter the length of 
 the line whereon it thus operates. As yet, therefore, we have 
 not considered, V. how to multiply one of two rectangular lines 
 by another perpendicular thereto, when the multiplier-line has a 
 length different from that which has been assumed as the unit of 
 length; nor VI. how to multiply a scalar hy a vector; nor 
 VII. have we considered the product of two parallel lines ; 
 much less have we shewn, VIII. how to multiply generally 
 any one vector by any other, and thereby obtain a Quaternion as 
 the product; nor IX. how to multiply any one such quater- 
 nion, as a factor, by any other quaternion. It is obvious that 
 there must remain questions of the same sort to be considered 
 with respect to the division of lines and of quaternions. But I 
 think that before entering on these new problems, it will be use- 
 ful to suggest still another mode of elementary illustration (be- 
 sides those given in the last Lecture) of the multiplications of 
 the I Vth class enumerated above ; because the smallest degree of 
 obscurity, existing with respect to these, would be fatal to our 
 subsequent success, or at least would materially interfere with 
 the facility and clearness of our future investigations. 
 
 81. Conceive then that there are two clock faces or dial-plates, 
 one facing the south, as represented in fig. 15, and the other fac- 
 ing the west, as indicated in fig. 16 : where the letters Z, W, E, 
 N, S, denote, as in some earlier diagrams, the zenith (or highest 
 point), and the west, east, north, and south, respectively. Then 
 the former of these two figures may become a sort of picture of 
 the " ^-operation," and the latter figure of the "J-operation," if 
 we proceed to interpret them as follows. In fig. 15, with the 
 clock face south, the 2-operation, or the multiplication by the fac- 
 tor i, has the effect of advancing the hour-hand by three hours, 
 or of putting the minute-hand forward fifteen minutes, or a quarter 
 of an hour. And in like manner, in fig. 16, where the face is 
 supposed to be turned towards the west, an exactly similar ad- 
 vance of either clock-hand (through a quadrant) is effected by 
 the^-operation, or by a multiplication by the factory. Conceiv- 
 
LECTURE III. 
 
 77 
 
 ii\g, therefore, that we watch the motion of the hour-hand from 
 IX. to XII. on the dial-plate with face to the south (fig. 15), 
 and again from III. to VI. on that other dial-plate which faces 
 the west (fig. 16), we may suppose ourselves to see upon these 
 
 E NW- 
 
 Face South. 
 
 Face West. 
 
 dials, or clock-faces, that the hour-hand is brought up from +j to 
 + A, by the e'-operation, but that it is, on the contrary, brought 
 down from + e to -k, by the_;"-operation, as marked by the curved 
 arrows in the figures : and thus, or by watching the motions of 
 the minute-hand on the same two faces, during the fourth and 
 second quarters of an hour, we might in a new way exhibit to 
 ourselves the truth and contrast of the two important formulae : 
 
 i xj = k, jxi--k; 
 
 at least if (to fix our conceptions) we retain, for some time lon- 
 ger, that particular choice of the directions of the lines i, j, k, 
 which is suggested by the examples given in the foregoing Lec- 
 ture. The figure 15 may, on the same plan, illustrate the for- 
 mulae : 
 
 ixk = -j, ix{-j) = -k, 
 and, therefore, also the resulting formulae, 
 
 i X I X k = - k, i X i = — I ; 
 which last result may be considered as here expressing, that if 
 
78 ON QUATERNIONS. 
 
 the minute-hand be advanced upon the southward dial-plate, 
 through two successive quarters of an hour, it is brought from 
 pointing up to pointing doivn, or is otherwise reversed in direc- 
 tion. In like manner, figure 16 exhibits the results, that 
 
 jyk = i, jxi = -k, 
 
 and that consequently, 
 
 jxjxk=^-k, jxj = -\; 
 
 while the analogous results respecting the ^-operation, or multi- 
 plication by the factor k, may be illustrated by simply laying a 
 watch upon a table, with its face upward. 
 
 82. Assuming then that we are by this time quite familiar 
 (compare 80, IV.) with the effect of a vector-unit, such as ^, or 
 j, or k, when thus operating as a graphic factor on any line per- 
 pendicular to itself, let us consider, in the next place, what our 
 principles oblige us to regard as being the product obtained by 
 the multiplication of a line by another perpendicular thereto, 
 when (see 80, V.) the multiplier line has a length different from 
 that which has been chosen for the unit of length. Suppose, for 
 instance, that it is required to multiply the line 3j by the line 2i; 
 which latter line (by art. 58) is the same with the product 2 x i. 
 To adapt to this particular question the principles of the forego- 
 ing Lecture, we have only to assume that 3/ is the faciend ; i the 
 factor; ix 3/ the factum, or the profaciend; 2 the profactor; and 
 therefore 2i, the transfactor ; and to seek what line the trans- 
 factum, or the profactum, is: for (by articles 39, 40, 41, 46, 47, 
 48, 49) the line thus found will be the product required, since it 
 will be the result of the multiplication, Transfactor into Faciend. 
 ISIow the z-operation, or the multiplication by the versori, being 
 performed on the line 3j,according to the rules which we already 
 know, has simply the effect of turning that proposed line 3; into 
 the new position 3^, without any change in its length ; hence 
 3k is, in this case, the factum, and we may write the equation, 
 
 i X 3j= 3k. 
 
 Operating next on this factum 3^, regarded as a profaciend, by 
 
LECTURE III. 79 
 
 the profactor 2, which belongs to the class of tensors, we now do 
 not turn at all the line which we thus multiply, but we stretch it 
 so as to double its length, and change it to the line Qk; which 
 consequently is the required profactum, or transfactum, or final 
 product ; so that we have the equations, 
 
 2^■ X 37 = 2 X ^■ X 3;" - 2 X 3/^ = 6^. 
 
 In like manner we should find that 
 
 ?,j x2i = 3 xj X 2/ = 3 X (- 2k) = -6k; 
 -2ix3j=-2xix3j=-2xSk = - 6k, &c. ; 
 
 and generally we see that (as in algebra), 
 
 ai X bK = ah X ttc, 
 
 if « and b be any two tensors, or scalars, while i and k are any two 
 rectangular vector units. We have then this Theorem, as a neces- 
 sary and important consequence of the principles of the present 
 System of Symbolical Geometry : theproduct o/'any tvv^o rect- 
 angular LINES is a THIRD LINE perpendicular to both; its 
 LENGTH being the product of their lengths (or bearing to the unit 
 of length the same ratio which the rectangle under the factors 
 bears to the unit of area); and the rotation round the multi- 
 plier line, from the multiplicand line to the product line, being 
 POSITIVE (that is, as we continue to suppose, right-handed). 
 But we see, at the same time, that this product line assumes ^ewe- 
 r«//2/ one or other of two opposite directions, according as 
 the two rectano'ular factor lines are taken in one or in the other 
 ORDER ; just as we found more particularly before, that the lines 
 (+A), represented by the two products i xj andj x i, were oppo- 
 site ; so that we may now write, generally, the following equa- 
 tion OF perpendicularity : 
 
 afi = -(ia, if j3 _La ; 
 
 where _L is the usual sign for one line being at right angles to 
 another; and, in the symbols of the two products a/3 and [3a, 
 the mark of multiplication is omitted. 
 
 83. It will now be easy to fix the signification which should 
 be attached to the product of a number multiplied by a line (see 
 
80 ON QUATERNIONS. 
 
 80, VI.), or of a vector into a scalar. Suppose that it Is required, 
 for example, to multiply the scalar - 2 by the vector z; or to 
 find the value of the product ix-2. For this purpose we may 
 assume any line perpendicular to i, suppose the line 3/', as a fa- 
 ciend ; operate^rs^ on this line by the factor - 2, which will give 
 the factum - 6;'; operate next on this factum, or profaciend, -6/, 
 by the profactor i, which will give the profactum - 6A ; and 
 finally inquire what one transfactor, operating on the assumed 
 faciend or transfaciend 2>j, would conduct to this profactum, or 
 transfactum, namely, to the line - 6A : for this transfactor, so 
 found, will (by 49) be the sought product of profactor into factor. 
 In this way (since -2ix 3/ = - 6^) we find, in this example, that 
 
 e X - 2 - - 6)(; -^- 3/" = - 2« ; 
 
 and generally we may conclude, by a process of the same sort, 
 that 
 
 axaxj3 = «xaxj3, 
 
 if a be any scalar, and j3 any line perpendicular to a ; whence we 
 infer (see 49) that 
 
 ay^a = ay. a, 
 
 or that the product of a scalar and a vector is independent of the 
 order of the factors. But we know how to interpret this pro- 
 duct as a line, when the vector a is multiplied by the scalar a 
 (see art. 59) ; we are led, therefore, to interpret the product as 
 denoting the same line, when the scalar a is multiplied by the 
 vector a : and omitting the mark x, we may denote this product- 
 line indifferently by either of the two symbols aa or aa. 
 
 84. We have not yet fixed generally (see 80, VII.) the in- 
 terpretation which should be attached to the product of two 
 2)arallel lines, or to the square of a vector, in this system of sym- 
 bolical geometry. However we saw (in art. 75) that the squares 
 of the three vector-units i, j, k, and generally that the squares of 
 all quadrantal versors, such as (by art. 77) all vector-units are, 
 have negative unity for their common value. And if we wish to 
 determine generally the product of any two vectors, such as ia 
 and icV, which are parallel to one common line (the factors a and 
 X being here supposed to be scalars), and which may, therefore. 
 
LECTURE III. 81 
 
 be said to be themselves parallel lines, even if they should hap- 
 pen to be situated as parts of one common and indefinite axis, 
 we have only to assume some perpendicular line such as^y for 
 the faciend ; to deduce hence the factum, namely, ix^jy-xyk, 
 by the rule in art. 82 ; and then (by the same rule in 82), to 
 calculate an expression for the profactum, namely, 
 
 ia X xyk = axy xik = - axyj = - ax xjy ; 
 
 for thus we find that the transfactor is -ax, or that the product 
 required is 
 
 ia X ia; = - ax. 
 
 In general this mode of proceeding shews that the product o/any 
 TWO PARALLEL VECTORS is (in the present theory) a scalar ; 
 namely, the number which expresses the product of the 
 LENGTHS of the two factor lines, ^A25 number being taken nega- 
 tively or POSITIVELY, according as those two parallel factor- 
 lines agree or DIFFER iu direction. 
 
 85. For example, the square o/every vector is a nega- 
 tive scalar, of which the positive opposite expresses the square 
 of the length of the vector ; thus 
 
 or using the exponent 2, we have the equation 
 
 {ixy = - X-. 
 
 If this result appear at all surprising, it is to be remembered, on 
 the one hand, that we had already (by 75) the values 
 
 and it may be remarked, on the other hand, that the general rule 
 recently deduced (in 84) for the multiplication of parallel lines, 
 gives the following equation of parallelism : 
 
 a/3 =+ /3a, if /3 II a ; 
 
 where H is used as the known sign of paralleHsm, and lines are 
 still regarded as parallel to each other, if they be parallel to one 
 common line ; and that this last equation not only agrees (so far 
 
 G 
 
82 ON QUATERNIONS. 
 
 as it goes) with ordinary algebra, but also contrasts, in a strik- 
 ing and (as it will be found) useful way, with the lately deduced 
 equation of perpendicularity (namely, aj3 =-j3a, in art. 82). It 
 may be added that there appears to be something convenient, and 
 even natural, in the (symbolical) distinction thus sharply drawn 
 in the Calculus of Quaternions, between the two (mentally dis- 
 tinct) conceptions of line and number; every w^ctob,, or directed 
 right line in tridimensional space, having (as above shewn) a 
 NEGATIVE square; while fuerz/ SCALAR, whether it he itself a. 
 positive or a negative number, has, on the contrary, in this sys- 
 tem as in algebra, a positive square. But whatever may be 
 thought, at this stage, of the convenience or advantage of this 
 distinction, it may be already clearly seen, that the distinction 
 itself is a necessary part of the present Theory, indispensable to 
 its self-coherence, and required by its internal unity. To reject 
 this result, while other essential elements of the system were re- 
 tained, would be equivalent to the absurdity of asserting, that 
 two quadrantal and similarly directed rotations, in one common 
 plane, did not invert the direction of the revolving line; or that 
 two quadrants did not make one semicircle. 
 
 86., By a slight extension of the recent use of an exponent, it 
 is easy to give a clear and definite signification to such symbols 
 as i3,fl, M, &c., and to shew that these symbols also may repre- 
 sent versors, a though not quadrantal versors. The symbol i^ 
 has been already seen to represent an inversor, namely, - or - 1 
 (see articles 75, 85), because it represents an operator or factor 
 which produces two semz'-inversions in one plane. In like man- 
 ner, the symbol is may now naturally represent an operator 
 which produces, in the plane perpendicular to i, the third part 
 of a semi-inversion, or the third part of a quadrantal rotation. 
 This operator would, therefore, cause a telescope, in the plane 
 of the prime vertical, to advance through thirty degrees in a 
 right-handed rotation round a southward axis; or in fig. 15, it 
 would have the effect of making the hour-hand advance from IX. 
 to X., or generally from one hour to the next, on a dial-plate 
 facing the south. Again, the operator jf is another versor, which 
 would cause the minute-hand, in fig. 16, to advance through 
 eight-fifths of a quadrant, or would push this hand forward by 
 
LECTURE III. 83 
 
 an interval, upon this westward dial, corresponding to twenty- 
 four minutes of time. Considered as operating on a transit teles- 
 cope, this versor would not merely elevate that telescope from a 
 horizontal and northward to a vertical and upward direction, as 
 supposed in art. 68, but would carry the same telescope still ^r- 
 ther, in the same direction of rotation, through three-Jifths of 
 another quadrant, till it should come to have a zenith distance of 
 54°, or an altitude of 36° above the south point of the horizon ; 
 or in other words till it were brought into a position for observing 
 the transit of an eqiiatoreal star over the meridian, if the north- 
 ern colatitude of the place of observation were 36°: or (in fig. 
 17, art. 87) from the position on to the position oq. And finally, 
 the versor k^ would cause the telescope of a theodolite to advance 
 through half a quadrant, that is, through 45° of azimuth ; or 
 would push on through an hour and a half (that is, through the 
 half of three hours) the hour-hand of a watch which should be 
 laid with its face upward on a table. In general, if t denote any 
 vector-unit, and if t be any scalar exponent, the symbol l* de- 
 notes, on this plan, a versor, which would cause any right line, 
 in a plane perpendicular to t, to revolve in that plane through t 
 quadrants, or through an arc =if x 90°; right-handedly round t, if 
 t be positive, but left-handedly, if t be negative. Thus every such 
 POWER, of every unit-vector, comes with us to be interpreted 
 as a versor (not generally quadrantal); and reciprocally every 
 versor may be regarded as such a power : the base of this power 
 being the unit-line in the direction of the axis of the versor; and 
 the scalar exponent expressing the ratio which the angle (or 
 amplitude) of the same versor bears to a quadrant ; while this 
 scalar is positive or negative, according as that rotation round 
 the axis, in a plane perpendicular thereto (in producing which 
 rotation round this axis and through this angle, the versor is con- 
 ceived to be the agent), is directed towards the ?'ight hand, or 
 towards the left. We know then how to interpret the symbol 
 i*K, iff be thus an unit-line, and /ca vector perpendicular thereto; 
 namely, as denoting a third line A, which is likewise perpendi- 
 cular to I, and has the same length as k, but is inclined thereto, 
 at a determined side thereof, by an angle =tx 90°. 
 
 87. Proceeding to the consideration (see 80, VIII.) of the 
 
 G 2 
 
84 ON QUATERNIONS. 
 
 multiplication of one line by another, which is neither parallel 
 nor perpendicular thereto, let us at first suppose, for simplicity, 
 that each factor is a vector-unit ; let one of them be imagined to 
 have a vertically upward direction, so that it may be denoted (as 
 before) by the letter k ; let the other be supposed to be directed 
 to the north pole in a northern latitude of 54°; let this latter 
 unit-line be denoted, for the present, by p; and to fix the order 
 of the factors, let this line jo be taken for the multiplier, while 
 the other unit-line k shall be regarded as the multiplicand. We 
 are, therefore, to seek the value (or the interpretation) of the pro- 
 duct p X k, or pk, by the principle (see art. 49) that pk ^pka -r- a ; 
 or that 
 
 ph=j-^a, ifj3 = Aa, 7=/'j3, 
 
 where a [5 y are three lines, or rays, which it remains to assume 
 so as to satisfy these last equations. Now, because j3 = Aa, we 
 know (compare articles 70, 71) that a and j3 must be two hori- 
 zontal and equally long lines, of which j3 is more advanced by a 
 quadrant in azimuth than a; and because 7=pj3, we know that 
 /3 and y are two equally long lines in the plane of the equator 
 (perpendicular to the polar axis p), and such that y is more ad- 
 vanced by a quadrant towards the right hand, or in the order of 
 the diurnal rotation of the heavens, than j3, or has an hour-angle 
 greater by an amount which answers to six hours of such rota- 
 tion. We must, therefore, on the present jo^aw of construction, 
 conceive jd to be directed towards either the east or the west 
 point of the horizon, and may suppose its direction to be to the 
 east flfor (compare art. 71), an inversion of j3 would only invert 
 both^of the two other lines a and-y at once, and would, therefore, 
 not affect their quotient : we may also assume that the common 
 length of these three lines is unity. Making then ^=-j, we 
 find that a = -/, or that the line a is directed towards the north; 
 we find also that the line 7 is directed towards the culminating 
 point Q of the equator, or that it has the position OQ lately con- 
 sidered (in art. 86), |which was seen to be derived from a north- 
 ward line ON, by operating with the versor, or graphic factor, de- 
 noted by the power Jl. Thus, in the present question, the required 
 product is known, for we find the equations, 
 
LECTURE III. 
 
 85 
 
 The product px k is, therefore, a versor^ of which the unit-axis 
 is the westward line j>, while its angle, or amplitude, is = f x 90° 
 = 144° ; that is to say, the sup- 
 plement (to two right angles) 
 of the angle of 36°, which has 
 been supposed to be the north- 
 ern co-latitude qos of the place 
 of observation, or the north po- 
 lar distance poz of the zenith ; 
 whilethe rotation (of 36°), /rom -^l 
 the multiplier p to the multi- 
 plicand k, is right-handed, round the (westward) axis of the pro- 
 duct. All this may be illustrated by the annexed diagram (Fig. 
 17), to which reference has already been made. 
 
 88. It is easy now to see that this mode of constructing the 
 product of two unit-lines may be applied to all other cases of 
 such products; and that if the factor lines were different in their 
 lengths from unity, we should only (by 82) be obliged to combine 
 with the foregoing composition of versions a certain composition 
 of tensions, or to multiply the resulting versor by (or into) a 
 tensor, which would simply be the number that expressed the 
 product of the lengths of the two factor lines, or the area of the 
 rectangle under them. We have, therefore, this theorem, which 
 includes several of those already given: " The product kA, of any 
 TWO VECTORS K and A, is in general equal to the product of a 
 Tensor and a Versor; whereof the tensor is the numerical pro- 
 duct he, lib and c be numbers expressing the lengths of the fac- 
 tor lines, or their ratios to an assumed unit of length; while the 
 versor is the power r"* of the vector-unit i, this unit-line i having 
 the direction of the axis of right-handed rotation y}-o»z the mul- 
 tiplier-line fc to the multiplicand-line A ; and the supplement t, 
 of the exponent 2-^tothe constant number 2, expressing the 
 ratio of the angle of this last rotation to a right angle." In short, 
 with the foregoing significations of the symbols, we shall have 
 the two following connected expressions: 
 
 A -i- (c = 7 f' ; KX = bci^'*\ 
 
86 ON QUATERNIONS. 
 
 where - is, as usual, a symbol equivalent to c -^ h. In the ex- 
 ample of the foregoing article, the particular values of these sym- 
 bols were ; 
 
 I =j ; K=p ; X = k ; b = c= I ; t = 
 
 89. As another example, let i = -j, K = k, X=/?, where/? shall 
 be supposed to retain its recent meaning ; so that we shall have 
 still b^c= 1, and ^ = f. The general theorem of the last article, 
 gives now the expression, 
 
 as the value of the product k into p, which differs only by the 
 order of its factors from that considered in art. 87, and represents 
 a versor whose an^le is still = f x 90°, but whose axis is now 
 directed to the east, instead of being directed to the west point 
 of the horizon. In fact, if we had immediately sought to deter- 
 mine this new product kp as the value of kpa H-a, we might have 
 conveniently taken for a the line which was lately y, or the position 
 of a telescope OQ directed towards the culminating point Q of the 
 equator; and then we should have found pa =y, 'dnd kpa = kj = -i, 
 so that the new product Ap, regarded as a transfactor (49), would 
 be seen to have the effect of turning the telescope from the position 
 just now mentioned, through 144°, right-handedly round an east- 
 ward axis, till it pointed horizontally towards the north. We see 
 in this example what the theorem of the preceding article proves 
 to be generally true, that the two products (in this ease pk and 
 kp) of any two unit-lines, taken in two opposite orders, are mu- 
 tually inverse or reciprocal as to their effects as versors, one un- 
 doing what the other does ; because their axes (of right-handed 
 rotation) are opposite, while their angles (of such rotation) are 
 equal. They might, therefore, be called, with respect to each 
 other (compare art. 44), by the names of Versor and reversor. 
 They may also conveniently be said to be conjugate versors: 
 and I am accustomed to denote this relation between them, or 
 to form a symbol oi one such versor from the symbol of the 
 other, by prefixing the capital letter K, as the characteristic 
 OF conjugation : thus with the recent significations oi k and p, 
 as certain unit-lines, I should write the equations, 
 
LECTURE III. 87 
 
 K.pk = kp; }L.kp=pk. 
 
 And because it is the same thing, whether we turn a telescope 
 right-handedly, round an east-ward axis, or /e/Jf-handedly round 
 a i^^e^^-ward axis, through any given angle, such as that of 144°, 
 we may, in the recent example, write an expression with a ne- 
 gative exponent, namely, 
 
 kp ==J-i, 
 
 instead of that other expression which was lately given for this 
 product kp (near the beginning of the present article). The 
 powers Jl and j"f, with one common unit-line J for base, but with 
 opposite scalar exponents, are, therefore, covjugate versors ; the 
 former power being a value for pk (by 87), and the latter being 
 a value for kp. Thus we are led to write, 
 
 and generally for aiig unit-vector l as base, and any scalar t as 
 exponent, we have the formula. 
 
 More generally kX and X/c may be said (by analogy) to be con- 
 jugate PRODUCTS, whether the lines denoted by k and X have 
 their lengths equal to unity, or different therefrom ; using then 
 still the same characteristic of conjugation K, we may agree to 
 write, in this more general case, 
 
 K . kX = Xfc ; K . Xk = jcX. 
 
 90. Since every geometrical product, of any one of the classes 
 hitherto considered, is also at the same time a certain geometrical 
 quotient, or is equal to the quotient of some one directed line 
 divided by another, according to the general notion of such divi- 
 sion, which has been given above ; and because it may thus be 
 used as a factor, or multiplier, to generate or produce the divi- 
 dend line of this quotient as a factum, or as a product, from the 
 divisor line as a faciend or multiplicand; while every such act 
 of faction, or of multiplication, may be resolved into a metric 
 and a graphic element, namely, into two factor acts oi tension 
 
80 ON QUATERNIONS. 
 
 and of version : we may already see that it must be useful to 
 possess signs, or marks, for expressing this general resolution of 
 any geometrical factor into these two important elements, or for 
 denoting separately, in each particular case, on one general plan 
 of notation, the particular tensor, and the particular versor, by 
 whose multiplication among themselves the proposed factor may 
 be conceived to have been produced. Accordingly I employ, with 
 this view, the two capital letters T and U, as characteristics 
 of the two operations which I call taking the tensor, and 
 TAKING THE VERSOR respectively; that is to say, the operations 
 of obtaining, hy a general mode of decomposition thus denoted, 
 from any proposed geometrical multiplier, q, or from any pro- 
 posed product or quotient of lines or numbers, regarded as such 
 a multiplier, the tivo separate factors, or factor-elements, Tq 
 and \]q, whereof the former is a tensor, and the latter is a versor, 
 and which satisfy the two following general equations, or sym- 
 bolical identities (in the present system of symbols) : 
 
 q = Tq X \]q ; q = \Jq x Tq : 
 
 implying that we may either first turn, and then stretch, or else, 
 at pleasure, ^?'S^ stretch, and then turn a line. 
 
 And these two new characteristics, T and U (in conjunction 
 with K, and with a few others to be hereafter mentioned), are 
 among the main elements of that Calculus to which these Lec- 
 tures relate, so far as its notation is concerned. It will readily be 
 understood that if, instead of a single letter, such as q, we have 
 any more complex symbol, such as X -^ jc, or kX, denoting the 
 subject of these two new operations, it may then become neces- 
 sary, for distinctness, to enclose this symbol in parentheses, or to 
 interpose a jyoitit between it and the prefixed characteristic T or 
 U. Thus the equations of art. 88 give 
 
 T.k\ = be; U.KX = f2-*. 
 
 In words we may agree to call Tq the tensor of q, and similarly 
 may say that Uq is the versor of q. And because a versor 
 
LECTURE III. 89 
 
 does not stretchy while a tensor does not turn, we may write ge- 
 nerally, 
 
 T.Ug=l; U.T^=+; 
 
 the tensor-element of any versor, such as l^^-, being properly a 
 non-tensor, namely, unity, or the factor 1 (see art, 63) ; and the 
 versor -element of any tensor, such as Tg-, being in like manner a 
 non-versor, namely, the /?os?7?ue sz'^w + (compare art. 60). On 
 the other hand, we have also, with equal generality, the two for- 
 mulae: 
 
 T.T^=T^; U.U^=U^; 
 
 because the tensor-element of a tensor is simply that tensor itself; 
 while, in like manner, a versor is its own versor-element. 
 
 91. The factor T^' is always a number, commensurable or in- 
 commensurable with unity (see art. 63) ; and the other factor \Jq 
 admits (by 86) of being expressed under the form of a power 
 such as i*, where the exponent t is another number, positive or 
 negative, and the base i is an unit-line with some determined di- 
 rection in space. Now, for the complete numerical expression 
 or determination of this direction, two other numbers are, in ge- 
 geral, required ; for if we conceive the line i to be (at some given 
 moment of sidereal time, and some given place of observation) a 
 telescope pointed to a star, then in order to express numerically 
 the position or direction of this telescope, and thereby to distin- 
 guish this from other directions, we must know so7}ie two astro- 
 nomical coordinates of the star, such as its right-ascension and 
 declination, or its longitude and latitude, which would suffice to 
 identify the star on a globe or chart, or to fix its place in a cata- 
 logue. We see, then, that the power l\ or the versor U^-, de- 
 pends upon, and implicitly involves thkee numerical ele- 
 ments, the knowledge oi all of which is generally necessary for 
 its complete numerical identification. In fact to know completely 
 WHICH versor among all possible versors is denoted in any 
 particular investigation by such a symbol as U^", we ought to 
 know through what angle the corresponding version is per- 
 formed, and round what axis of right-handed rotation ; but in 
 order to adjust this axis properly, or to set a telescope in its di- 
 
90 ON QUATERNIONS. 
 
 rection, two motions, measured by two other angles, would 
 in general be required to be performed. The perfect knowledge 
 of any one Versor, such as \Jq, includes, therefore, generally, the 
 knowledge of the values of three angles, expressed, or at least 
 expressible, by a system of three numbers. And because the 
 Tensor Tq is itself another number, we find, upon the whole, 
 that the geometrical factor, or quotient, or product, which 
 has been above denoted by q, and which has been seen to be 
 equal to the product of its own tensor Tq, and of its own versor 
 \Jq, is generally a Quaternion : in the sense that it is found by 
 this (and by every other) mode of analysis, or of decomposition, 
 to depend upon, and conversely to include within itself, a System 
 of Four Numbers. 
 
 92. This conclusion is so important (we might almost say so 
 fundamental), with reference to the subject of the present Lec- 
 tures, that it may be worth while to confirm it by at least one 
 other mode of illustration, or of derivation, here; although we shall 
 meet afterwards with other confirmations and illustrations of the 
 same conclusion. 
 
 We have lately been considering what has been above de- 
 noted by the symbol q, in a synthetic, rather than in an analytic 
 point of view. We have (upon the whole), in the two last ar- 
 ticles, regarded this q as o. factor, rather than as a quotient; 
 although this latter view of q has also, in those articles, been 
 mentioned or alluded to. While decomposing this geometrical 
 multiplier q, as such a factor, into its own two component ia.ctox% 
 of the tensor and versor classes, denoted respectively by the sym- 
 bols Tq and \Jq, we have thought of q itself rather as operating 
 on a faciend ray a to produce a factum j3, then as hemg foundhy 
 our comparing the latter ray j3, as a dividend, with the former 
 ray a, as a divisor. In short, we have recently been studying the 
 composition of q, as an agent, rather than as a relation; or 
 as satisfying the equation, 
 
 qy- a = fi, 
 
 rather than as determined by the inverse equation, 
 
 g = /3-f-a. 
 
LECTURE III. 91 
 
 which is, indeed, intrinsically, the same, but presents itself un- 
 der a different foryn. But we propose to vary our modes of illus- 
 tration of the subject by taking now, for a while, in preference, 
 this latter view. Instead of studying the (synthetic) operation 
 denoted by the symbol q y^a, we shall aim now to study, unfold, 
 represent, construct, and picture, as clearly but also as briefly 
 as the subject may allow, the converse (analytic) conception of 
 what has already been denoted by the symbol j3 -r- a; and was 
 spoken of (perhaps inelegantly) at an early stage of the foregoing 
 Lecture (see art. 40), as being a certain metrographic rela- 
 tion of the ray j3, to the ray a: involving partly, as was there 
 remarked, a (metric) relation of length to length, and partly also 
 a (graphic) relation of direction to direction. Fixing, then, our at- 
 tention, for the present, on this metrographic relation, or on this 
 quotient of two rays, we are now to seek for some simple construc- 
 tion, diagram, ox fgure, which may represent ox picture this con- 
 ception, and may thereby be analogous to the construction or 
 representation given in the first Lecture, for the corresponding 
 conception of the difference of two points. 
 
 93. Resuming, then, the expression of art. 40 for q, namely, 
 
 q = ^ -^ a, 
 
 where a and j3 denote two rays or directed right lines in space ; 
 and comparing it with the expression of art. 18, for a rectilinear 
 step or vector a, namely 
 
 a = B - A, 
 
 where A and B denote two points, namely, the beginning and end 
 of the step ; we see that as this vector a, regarded as a geome- 
 trical DIFFERENCE, B - A, has been already constructed (in fig. 
 2 of art. 8, or in fig. 6 of art. 53) by a straight line ab, with a 
 straight arrow attached, so the factor q, when regarded as a 
 GEOMETRICAL QUOTIENT, j3 -^ a, may naturally be pictured by a 
 PAIR OF RAYS, or of right lines diverging from an origin or com- 
 mon point, with a curved arrow inserted between them: as has 
 indeed been done in fig. 7 (of same art. 53), where the angle adb 
 (for example), between the two rays da and db, or a and j3, being 
 one of three angles (adb, bdc, adc) at the vertex d of thetrian- 
 
92 ON QUATERNIONS. 
 
 gular pyramid abcd, has a curved arrow thus drawn within it, 
 while the word Factor is written above this arrow, and the letter 
 q below; the arrow being- directed //'om the faciend, da or a, to 
 the factum, db or /3. k figure constructed in this manner, such 
 as the figure adb just mentioned, may be called a Biradial : it 
 differs from the ordinary plane triangle adb, by not expressly in- 
 volving, in its conception or description, the third or closing side 
 AB ; and it differs also from the ordinary plane angle adb, by its 
 essentially involving- the conception of the relative length, and 
 indeed by its depending also on the order and plane of the two 
 lines or rays, da and db, which enclose it. It might, therefore, 
 be otherwise called an unclosed triangle ; or an angle with finite 
 legs: but the recent name biradial appears to be more convenient 
 and expressive than either. The point d, from which the two 
 rays diverge, may be said to be the vertex of this biradial ; the 
 divisor line (or faciend) da may be called the initial ray ; and 
 the dividend line (or factum) db may be called, on the same plan, 
 the final ray of the same biradial figure adb. A biradial has, 
 in general, a shape, or species, depending on the ratio which 
 the length of the final ray bears to the length of the initial, and 
 also on the angle at which the final is inclined to the initial ray; 
 this shape of the biradial determining thus the shape or species of 
 the triangle, which is formed by closing the figure, or by drawing 
 a straight line from the end of the initial to the end of the final 
 ray: and two biradials which have, in this sense, the same shape, 
 by their ratios and angles being equal, may be said to be similar 
 biradials. a biradial has also a plane and an aspect, depend- 
 ing on the star or region of infinite space, towards which its plane 
 may be conceived to face ; this region being distinguished from 
 that other which is diametrically opposite thereto, by the direc- 
 tion of the curved arrow in the figure, or by the condition that 
 if the biradial were looked at by a beholder situated in the proper 
 (or positive) region, the rotation indicated by that arrow, from 
 the initial to the final ray, would appear to be right-handed, like 
 the motion of the hands of a watch ; whereas, if viewed from the 
 opposite (or relatively negative) region, this rotation would seem 
 to be /e/f-handed, or contrary to the motion of a watch-hand. 
 When two biradials have, in the sense just now explained, the 
 
LECTURE III. 
 
 93 
 
 same aspect, their planes both facing at the same moment the 
 sajne star, they may be said to be condirectional biradials. 
 When, on the other hand, they face in exactly contrary ways, 
 and, therefore, have opposite aspects, they may be called con- 
 tradirectional, or sometimes simply opposite biradials. 
 Both these two latter classes may be included under the common 
 name of unidirectional or (somewhat more shortly) parallel 
 biradials, so that the planes of any two parallel biradials are 
 either coincident or parallel. And finally, when two biradials are 
 at once similar and condirectional, we shall say that they are 
 Equivalent Biradials. 
 
 94. For example, if abc (in fig. 18) be an equilateral tri- 
 angle, and if d, e, r be respectively the points of bisection of 
 the sides opposite to the corners 
 A, b, c,then the six biradials, dba, 
 ECB, FAC, and fbc, dca, eab, are 
 all similar to each other, the angle 
 in each being = 60°, and the final 
 ray in each being twice as long as 
 
 the initial, ba = 2bd, &c. But 
 while the aspect of each of the 
 three first of these six biradials 
 is upward, if the figure be laid 
 upon a table, because when we 
 look, for instance, at the biradial 
 DBA in the figure 18 so laid, the rotation from bd to ba resembles 
 the motion of the hands of a watch, yet the aspect of each of the 
 three last of the same six biradials is downward, since we should 
 be obliged to look from below the table, or from below a horizon- 
 tal sheet of paper on which the same figure might be traced, in 
 order to see (for example), in the biradial fbc, the rotation from 
 BF to BC resemble the motion of those hands, to which motion 
 this last mentioned rotation appears contrary, when we look on 
 the figure from above. Thus the three first of these six biradials 
 are con-directional, if they be compared with each other, and so 
 likewise are the three last of them, if they too be compared among 
 themselves: consequently the three former biradials, namely, dba, 
 
94 , ON QUATERNIONS, 
 
 ECB, FAC, are hexe equivalent biradials ; and the three latter bira- 
 dials, namely, fbc, dca, eab, are, in like manner, mutuallt/ eqm- 
 valent. But the conditions of equivalence are not satisfied when 
 we compare any one of the first set with any one of the second 
 set of these biradials, because we then find an opposition in the 
 characters of the rotation as right-handed and left-handed in one 
 plane ; and the two biradials thus compared, for example, dba 
 and FBC, as the arrows in the diagram indicate, are now contra- 
 directional biradials, and consequently are not equivalent. 
 
 As additional illustrations of these conceptions and expres- 
 sions, it may be noted that if, in the same figure 18, we let fall 
 from E two perpendiculars, eh and ek. on af and cf, the new 
 biradial hae is equivalent to the removed biradial kec, to the en- 
 larged biradial fac, and to the revolved biradial dba ; the aspect 
 of each being upward, while the angle of each is sixty degrees, 
 and the ratio of the final to the initial ray in each is that of two 
 to one. 
 
 95. The very object and purpose of introducing such bira- 
 dial figures as the above, being to make each of them serve as a 
 representation of what we have already several times spoken of 
 as a geometrical quotient, namely, the quotient of a final ray j3 
 divided by an initial ray a, it is clear that we ought now to con- 
 sider and determine ivhat degree of variety may be allowed in the 
 construction of the particular biradial which is to represent any 
 proposed or particular quotient j3 -^ a, or a quotient e^-i^a^ thereto. 
 For until we shall have thus settled the changes that a biradial 
 figure may undergo, without ceasing to represent the same quotient 
 or equal quotients, we shall not be prepared to decide, by the con- 
 sideration of this mode of representation, m how many distinct 
 ways a biradial may be changed, so as to make it represent new 
 and unequal quotients, or new and varied relations of the metro- 
 graphic kind, of one ray to another. And the number of distinct 
 ways of varying this last sort of relation must be investigated in 
 order to confirm (as we proposed at the commencement of art. 92), 
 or else to correct (if correction shall be found to be necessary), 
 that conclusion of article 91, in virtue of which we have been led 
 to regard such a quotient, or such a relation, or at least the geo- 
 metrical factor which synthetically corresponds thereto, as in 
 
LECTURE III. 95 
 
 general depending essentially on fow^ distinct riumerical elements, 
 and as being, in that sense, a Quaternion. In short, we are 
 led to seek now to determine the conditions of equality of two 
 quotients^ or the degree of restriction imposed on the four rays 
 a /3 7 S, or on any one or more of them, and also the degree of 
 liberty allowed to them, when an equation such as 
 
 is given ; in order that we may afterwards enumei'ate the modes 
 OF inequality of any two such quotients, or the ways in which 
 one quotient, S -=- 7, may differ from another quotient, j3-^ a: and 
 in this determination and enumeration, it is a part of our present 
 plan that we should assist ourselves by the conception and con- 
 struction of those biradial figures, of which the nature has been 
 already explained. 
 
 96. As preliminary and analogous, but easier and less complex 
 investigations, we may here inquire, first, what are the conditions 
 of equality of two geometrical differences of 'points; and secondly, 
 how many are the distinct modes of inequality, which may subsist 
 between one such difference and another? And because these 
 differences of points have been already represented ox constructed 
 by straight lines, or vectors, we may now propose also two other, 
 but closely connected questions respecting such lines, which shall 
 bear a still more strict analogy than the questions just now men- 
 tioned, to those inquiries respecting i/ra^m/s that were suggested 
 in the foregoing article : namely, 1. How may we change a line, 
 or vector, such as that above denoted by the symbol a, without 
 its ceasing to represent a given or particular difference, such as 
 B - a ; or at least some difference of the same general kind, such 
 as D - c, which shall be equal to the given difference b - a ? and 
 II. How many distinct modes of change of a line, or vector, cor- 
 respond to real (and not merely apparent) alterations, in such 
 a geometrical difference of points ; so that the varied lines shall 
 represent unequal differences, or varied relations between points 
 in space, belonging to what we have already called the ordinal 
 class? These questions might indeed have been proposed and 
 resolved, so early as in the first of these Lectures on Quater- 
 nions, if it had not seemed convenient to reserve them for the 
 
96 ON QUATERNIONS. 
 
 present portion of the Course, at which their signification and 
 importance may be more fully felt than it might then have been. 
 For we may now see, that by their leading to the determination 
 of the NUMBER (namely three) of distinct numerical elements^ 
 which are involved in the conception of an ordinal relation be- 
 tween two points, when that conception is closely enough con- 
 sidered, and unfolded fully enough, they are adapted to assist us 
 to determine also the number {ndimely four) of those other dis- 
 tinct numerical elements, which enter into, or are essentially 
 included in, the conception of a cardinal relation between two 
 rays, when the notion of this cardinal relation is likewise suffi- 
 ciently developed. By confirming in a new way the conclusion 
 of art. 17, that a Vector is a natural Triplet, they may pre- 
 pare for confirming also the conclusion, more lately proposed for 
 discussion, that a Biradial represents a Quaternion. 
 
 97. Of the problems (if they may be so called), which were 
 proposed in the foregoing article, the first related to the determi- 
 nation of the conditions of equality of two geometrical differences 
 of points, such as b - a and o - c. In other words, we were to 
 determine the degree of restriction imposed on any one or more 
 of the four points a b c d, and also the degree of liberty allowed 
 them, when the equation 
 
 D - c= B - A 
 
 is given. It resulted, however, from what was remarked in the 
 same article, that this problem admits also of being proposed 
 under the following other but connected form : To assign the 
 various modes of changing one line, a, into another line, b, so 
 that these two different lines, a and b, may represent equal dif- 
 ferences of points ; or may satisfy the two equations, 
 
 a=B-A, b=D-c, 
 
 when the difference d - c is still supposed to be equal to b - a ; 
 or when the ordinal relation in space, of the point d to the point 
 c, is the same relation with that of the point b to the point a : 
 although the two points themselves of the one pair have not (in 
 general) the same positions as the points of the other pair. Now 
 a little consideration suffices to shew, that this sameness ofordi- 
 
/'F 
 
 LECTURE III. 97 
 
 nal relations between two pairs of points, ab and cd, which is 
 denoted as above by the equation d - c = b - a, may and ought 
 to be considered as holding good, when the four points taken in 
 the order a b d c, are, in this order, the four successive corners 
 of a parallelogram , as in the diagram annexed (figure 19). For 
 when the four points are so arranged, then whatever is the dis- 
 i?«wceofBfromAwill p;g 19 
 
 also be (in length, ^, -,.. 
 
 magnitude, or quan- 
 tity) the distance of 
 D from c ; and what- 
 ever is the direction 
 of the one distance, 
 will also be the di- 
 rection of the other. 
 But if, after once 
 constructing such a parallelogram, a b d c, we were to alter any 
 one alone of its four corners, for example, the corner d, we should 
 thereby violate at least one, if not both, of the two foregoing 
 conditions for the identity of the two ordinal relations, of d 
 to c, and of b to a. If, for instance, we prolonged cd to e, 
 the point e would be more distant from c than b is from a ; it 
 would not therefore have, in a sense so full as that which we are 
 entitled to demand that it should have, the same ordinal rela- 
 tion to c as that which b has to a ; and therefore the equation 
 E - c =B - A would not hold good, in the sense of expressing a 
 complete agreement between two ordinal relations. Again, if, 
 with c for centre, we were to describe, in the plane of abc, an 
 arc of a circle from d to f, and then to join cf, this joining line 
 would indeed be as long as cd or as ab, but its direction would 
 be different ; including then, as we do, the conception of direc- 
 tion of distance, in the conception of the ordinal relation of one 
 point to another, we cannot say that the new point f is ordinally 
 related to c as b is to a ; and must not assert the equation f - c 
 = B - A.. Still less should we be permitted to assert the equation 
 G -c = B - A, if the point g were obtained by prolonging cf, or 
 by causing CE to revolve round c ; for now both the length and 
 direction of the line cg would differ from those of the line ab, 
 
 H 
 
98 ON QUATERNIONS. 
 
 and, therefore, in both of these two respects, the ordinal relation 
 of G to c would be different from the ordinal relation of b to a. 
 And a point h, if assumed out of the 'plane of the parallelogram 
 (and consequently out of the plane of the figure), might be re- 
 garded as being, if possible, still more unfit to be substituted for 
 D in the equation d - c = b - a ; because the directional relation 
 of this point h to c would be still more unlike to that of b to a; 
 or at least would be unlike in another and in a somewhat less ele- 
 mentary way, since the passage from the direction of cd to that 
 of CH would be made by a rotation which was not even contained 
 in the given plane of abc. If, then, the three points abc be not 
 all situated upon one common right line, we can always find one 
 definitepoint d, and 07ily one, which shall (in the full sense above 
 considered) be ordinally related to c as b is to a, or which shall 
 satisfy the above written equation between differences, 
 
 D— c = B - a; 
 
 namely, the corner opposite to a, in the parallelogram of which 
 two adjacent sides are the lines ab and ac. And the only other 
 case in which, with the foregoing general view of an ordinal re- 
 lation of point to point in space, the required sameness of rela- 
 tions can ever exist, or in which the lately written equation can 
 be satisfied by any two distinct pairs of points ab and cd, is when 
 these ^wr points are on one common right line ; d being also as 
 far removed from c upon that line, as b is from a, and towards 
 the same (infinitely distant) parts of space, but not in the oppo- 
 site direction, as is represented in the subjoined diagram : 
 
 Fig. 20. 
 A B C D 
 
 a b 
 O *■ • • ^ • 
 
 In this remaining case, then, also (which case may indeed be re- 
 garded as a limit of the more general case of the parallelogram, 
 the altitude thereof being conceived to diminish indefinitely in 
 passing from the one figure to the other), the position of the 
 fourth point d is entirely fixed, when it is obliged to satisfy 
 the equation already several times written, and when the other 
 three points abc have given or fixed positions. The geometrical 
 
LECTURE III. 
 
 99 
 
 SIGNIFICATION of this equation, at least as thus interpreted, is, 
 therefore, itself perfectly determinate : for it suffices to fix the 
 position of D, and, in like manner to determine the position of any 
 one of the/our points a, b, c, d, when the positions of the three 
 other points are known. It is evident, from inspection of the two 
 last figures, that this equation, 
 
 D - c = B - A, 
 
 interpreted as above, gives, as a necessary consequence of its sig- 
 nification, the inverse equation, 
 
 C - D = A-B ; 
 
 and also the alternate equation, 
 
 D - B = c - A. 
 
 98. Such being the restriction imposed on the four points by 
 the lately written equation, in virtue of which no one of those four 
 points, taken separately, can vary its position in space, we see, at 
 the same time, as regards the liberty allowed them, that any two 
 of the same four points may vary their positions together, and even 
 that they may do this in indefinitely many ways, though all in- 
 cluded in one common class. For while the two first of the four 
 points remain ^icec? at a and b, the third point may be removed 
 from its original position c to any other position e, provided that 
 ih^ fourth point is, at the same time, removed to a certain corres- 
 ponding position F, as in the annexed figure 21. 
 And it is clear that the condition ovlaw of this b 
 correspondence, or connexion, between the two 
 new and variable points, e and f, which are ^ 
 thus substituted for the two old and fixed points, a 
 c and D, is that the ordinal relation f - e of the > 
 two points of the new pair ef, should be the 
 same with the ordinal relation d - c of the two K 
 points of the old pair cd, or that the equation 
 
 F - E = D - c 
 
 should be satisfied. For then, as in ordinary algebra, the two 
 equations, 
 
 F-E=D-C, d-c = b-a, 
 H 2 
 
 Fig. 21. 
 
100 ON QUATERNIONS. 
 
 will conduct to the required equation, 
 
 F -E = B - a; 
 
 because two ordinal relations, which coincide each with the same 
 third ordinal relation, as here with d - c, must also coincide with 
 each other. In fact, it is proved in Euclid's Elements (Book xi. 
 Prop. 9), that if two straight lines, as here ab and ef, be both 
 parallel to any third straight line, as here cd, then, although 
 they be not contained in any one common plane with that third 
 line, they will be parallel to each other; the three lines (if 
 equally long) being edges of a triangular prism. We may enunciate 
 otherwise this principle of the elimination of an ordinal rela- 
 tion D -c between two equations into which it enters as above, 
 by saying that "if any two vectors (as a and c in fig. 21) be equal 
 to the same third vector (as in that figure to b), they are also equal 
 to each other f at least if we now adopt, as the considerations of 
 the preceding article lead us to do, the conclusion, or the defini- 
 tion^ that two VECTORS are equal (as representing equal differ- 
 ences of points), when, and only when, they are opposite (but 
 similarly and not oppositely directed) sides of a parallelogram, 
 or else are equally long and similarly directed portions of one 
 common indefinite right line (the latter case being a limit of the 
 former). Indeed this use of the joara//e/o^;'a»2 to construct the 
 relation oi equality between directed lines, is one of those elements 
 of the present theory which it shares with several others. We 
 may also say that a line, a, may be changed to another line b, as 
 in figures 19, 20, 21, without ceasing to represent the same ordi- 
 nal relation, or the same difference of points as before, or at least 
 an equal difference, if it be merely made to move, or to change its 
 situation in space, without change of length or of direction : and 
 thus another of the questions lately proposed is simply and fully 
 answered. In fact, we may be considered to have already adopted, 
 at least tacitly, this view oi equal vectors, when, in the foregoing 
 Lecture, we abstracted from the situation of a line, or treated 
 that situation as unimportant, while comparing length with 
 length, and direction with direction. 
 
 99. An easy consequence or two of this conception of equa- 
 lity of vectors may be conveniently here mentioned. Thus hav- 
 
LECTURE III. 101 
 
 ing once established (with the signification already explained) 
 the equation D - c = B - a, we may naturally be led, by the known 
 analogies of algebraical notation, to write also (under the same 
 conditions of relative position of the four points compared) this 
 other form of the same equation, 
 
 D = (b - a) + c ; 
 
 or even this slightly simpler form (omitting the parentheses), 
 
 D =B - A+ c. 
 
 And then, returning from notations to conceptions, from signs 
 to thoughts, from symbolical expressions to geometrical inter- 
 pretations, we may regard ourselves as having thus been led to 
 enlarge that notion of the addition of a line to a point, which 
 was proposed in the first of these Lectures. For whereas we 
 there employed only the identity b = b - a + a, or considered only 
 that primary case of addition of a vector b - a to a vehend a, in 
 which this ^^ punctum vehendum" a, via.% already given as th.Q ini- 
 tial point of that " linea vector" b - a, which was to be applied 
 or (in the language of these Lectures) added to it; and regarded 
 ourselves as thus obtaining the fnal point b of the proposed line, 
 as (what we called) the sum, or as the geometrical result of this 
 conceived addition : we now, on the contrary, employ the equa- 
 tion above written, namely, d = b - a + c, and thereby enlarge 
 our view, so as to include the more general case, where the pro- 
 posed line B-A does 7iot already begin at the proposed point c, to 
 which it is to be added or applied, but is made to move, without 
 change of length or of direction, until, in its new and altered 
 situation, denoted by d - c, it comes to begin there ; the point d, 
 in which it thus comes to end, being now the result of this pro- 
 cess, or the geometrical sum required. From the remark made 
 at the end of article 97, it is clear that with this notation, thus 
 interpreted, we shall have also, by alternation, for the same sup- 
 posed arrangement of the points, this other connected equation, 
 
 D = c- A + B ; 
 
 and, therefore, that for any three points of space, a b c, we may 
 write (as in algebra) the identity. 
 
102 ON QUATERNIONS. 
 
 C-A + B = B-A + C, 
 
 each member being a symbol for one common /ourth point d. 
 
 100. The same conception of equal vectors conducts also to 
 several useful results respecting the addition of directed lines. 
 Thus, in connexion with fig. 21, we may write 
 
 D - A = (d - c) + (c - a) = (b - a) + (c - a) ; 
 
 and again, by the last formula of art. 97, or by the principle of 
 alternation of an equation between differences of points, we have 
 
 D - A = (d - b) + (b - a) = (c - a) + (b - a) ; 
 
 the sum, therefore, of two directed and coinitial lines, such as 
 the vectors b - a and c - a, is the intermediate and coinitial 
 diagonal, d - a, of the parallelogram abdc, described with those 
 two lines as sides ; as, in several other modern systems (resem- 
 bling so far the present theory), it has been inferred or defined to 
 be. And we see that this sum of two vectors is independent of 
 the order of the summands, so that we may write, generally, as 
 in algebra, 
 
 a + /3 = /3 + a; 
 
 and may say that the Addition of Vectors is always a commuta- 
 tive operation. It is also an associative operation ; that is to 
 say, we may write, generally, 
 
 (y + j3) + a = 7 + (j3 + a). 
 
 For if we make, in connexion with the same figure 21, 
 
 a=a=B-A=D-c=r-E; 
 
 /3 = c - A = D - B ; 7 = E - c = F - D ; 
 
 we shall then have the two partial sums, 
 
 /3 + a = D-A;y + j3 = E-A = r-B; 
 
 and the total sum of the three successive vectors a (3 j, whether 
 they be associated (or grouped) in one way, by adding y to /3 + a, 
 or in another way by adding 7 +/3 to a, is still, in each case, the 
 same fnal vector, T - A. ; since 
 
LECTURE III. 103 
 
 7 + (j3 + a) = (f - d) + (d - a) = F - A, 
 and 
 
 (7 + /3) + a = (f - b) + (b - a) = F - A. 
 
 We may therefore omit the parentheses^ and write simply, here, 
 the equation 
 
 7 + /3 + a = F-A. 
 
 Or if we attend only to the gauche quadrilateral acef, with 
 /S, 7, a for three of its successive sides, and with ae for one 
 diagonal, and cf (not marked in fig. 21) for the other, we shall 
 have 
 
 7 + j3=E-A, a + 7 = F-c; 
 
 and therefore, without introducing the points b and d, 
 
 a + (7 + j3) = (f - e) + (e - a) = F - A ; 
 (a + 7) + j3 = (f - c) + (c - a) == F - A ; 
 
 so that the associative principle of addition is again seen to hold 
 good, and we may write 
 
 (a + 7) + j3 = a + (7 + j3) = a + 7 + /3. 
 We see, at the same time, that 
 
 a + 7 + /3 = 7 + /3 + a, 
 
 the common value of these two sums being the vector f-a; and 
 generally it is clear, from considerations such as the above, that 
 in the addition of any number of directed lines in space, those 
 summand lines may he in any manner grouped and transposed^ 
 without altering the final result, provided that no one of the given 
 lines is changed in length or in direction ; and also that this sum 
 of any set of vectors is simply that one resultant vector which 
 represents or is the instrument of a vection or motion in space, 
 equivalent, as to its total or final effect, to all the proposed com- 
 ponent or partial motions, simultaneously or successively per- 
 formed. In short, the addition of vectors still answers to the 
 composition ofvections. 
 
 101. We have now completely resolved the first problem of 
 article 96, under the two aspects of the question which were 
 mentioned near the commencement of art. 97 ; the restriction. 
 
104 ON QUATERNIONS. 
 
 there spoken of, having since been pictured by a parallelogram, 
 and the liberty having been constructed by 2i prism. And there 
 can now be no difficulty in resolving also the second problem of 
 art. 96, with the help of the remarks which have been made in 
 art. 97, in connexion with figure 19. For, after constructing, as 
 in that figure, the parallelogram abdc, to represent (as above) 
 the equality 
 
 D-C = B - A, 
 
 we see, by the remarks just now referred to, that we shall ((really) 
 change the value of one of the two equated vectors, or make it 
 (really and not merely in appearance) cease to he equal to the 
 other vector, if, by any one of three distinct sorts of changes of 
 the position of the sought point d (the three other points abc re- 
 maining ^a;ec?), we either _^r5^, lengthen (or shorten) the line cd, 
 as by removing d to e ; or, secondly, turn that line cd, in the 
 plane of abc, as by changing d to f ; or else, and thirdly, turn 
 that line cd out of the plane abc, into some other position, which 
 is not represented in the figure. Conversely these three distinct 
 and elementally modes, of change of the vector d - c, exhaust all 
 the possible varieties of real alteration of that^vector. For what- 
 ever position in space may be denoted by the letter h, we may 
 always conceive that the point d comes to be removed to this 
 new position h, and that the vector cd is thereby changed to the 
 vector CH, or that the difference d - c is changed to h - c, by three 
 successive and component alterations of the kinds enumerated 
 above: namely, by first lengthening (or shortening) cd to ce; 
 then turning ce, in the plane abc, till it becomes cg (in fig. 19) ; 
 and finally causing cg to revolve, in a plane perpendicular to 
 the plane of the figure, till it takes the position ch. In fact we 
 could always, by an opposite rotation, in such a perpendicular 
 plane, bring ch to, coincide with some such line as cg, in the 
 given plane of abc; then, in that plane, turn cg till it became, 
 like ce, a line in the same direction as cd ; and finally shorten 
 (or lengthen) ce, till it became the line cd itself But each of 
 these three operations would make a real change in the vector on 
 which the operation was performed, since it would alter either 
 the direction (in one or other of two different ways), or else the 
 
LECTURE III. 105 
 
 length of that line; and to these three distinct modes of change 
 of a vector d-c, we see that all others may he reduced. A 
 Vector, such as h-c, is therefore, in this sense, a Triplet, 
 since it depends upon three distinct elements, which admit of 
 being expressed numerically ; namely one to tell us in what ratio 
 the length of cd has been changed, in order to make it become 
 CE (in the foregoing process) ; another^ to express, in degrees 
 or quadrants, &c., the angle ecg, through which the line ce has 
 been turned, in the given plane abc ; and finally a third number, 
 to record the magnitude of that other angle gch, through which 
 CG has been caused to revolve, in a new and perpendicular plane, 
 that it might take the position ch. In astronomical language, 
 if ABC be the plane of the horizon ; and if cd be a line whose 
 length is unity, directed towards the south, while c is some 
 known origin or post of observation ; then the vector ch (or the 
 position H of its extremity) will be entirely known, if we know, 
 first, its length, or the number of linear units, such as the lengthen, 
 which are contained in what is often spoken of, and tabulated, as 
 the radius-vector of the point (or celestial body) h ; secondly, 
 the azimuth, ecg, of that point or body ; and thirdly, the alti- 
 tude, GCH : but the knowledge of any tico of these three data 
 cannot, in general, dispense with knowing the third. All must 
 be known, if we would fully know what particular vector the 
 line CH is, or where in space the point or body h is situated ; 
 unless we should employ the aid of dataof some o^Aer Amc?, which 
 would however always be found to furnish, when sufficiently 
 discussed, a triple variety, and one not more than triple, as 
 answering, in fact, to the tridimensional character of space. 
 Indeed we have of late been merely reproducing, under a some- 
 what different aspect, and in a somewhat greater detail, con- 
 siderations which were briefly stated, or suggested, in article 17 
 of the first of these Lectures on Quaternions ; and there can now 
 be no difficulty in distinctly seeing that (as was stated by antici- 
 pation in that earlier article) any vector whatever may be re- 
 presented by the trinomial form, 
 
 Q = ix+jy + kz; 
 
 where ijk retain their significations as unit lines, while the scalars 
 X y z are simply Cartesian co-ordinates. 
 
106 ON QUATERNIONS. 
 
 102. Resuming now the consideration of the questions pro- 
 posed in art. 95, it is easy to see that equal quotients are 
 represented by equivalent biradials ; and conversely, that 
 whatever change of a ray disturbs the latter equivalence, dis- 
 turbs also the former equality ; whereas, so long as the equiva- 
 lence of the biradials remains, on equation between the quotients 
 holds good. Thus, for example, in fig. 18, art. 94, the five bira- 
 dials HAE, kec, fac, DBA, ECB, havc been seen to be all van- 
 tually equivalent, in the sense defined in art. 93; and accordingly, 
 if the final ray of any one of these five biradials be divided by the 
 initial ray, as for instance ae by ah, or e - a by h - a, the quo- 
 tient is, for each of these five divisions, expressed by one common 
 symbol, namely by 2k^, if the figure be conceived to be laid 
 upon a table, and looked at from above. That is to say, we 
 have the five following formulae, to be interpreted on the plan of 
 art. 86, in connexion with figure 18 : 
 
 (e - a) -^ (h - a) = 2k^ ; 
 (c - e) -^ (k - e) = 20 ; 
 (c - a) H- (f - a) = 2k^ ; 
 (a - b) -j- (d - b) = 2P ; 
 (b - c) -T- (e - c) = 2ki. 
 
 And again, whereas the three other biradials fbc, dca, eab, 
 were seen (in art. 94) to be indeed similar to the five biradials 
 just now mentioned, but not equivalent to them, because the di- 
 rection of the rotation from one ray to another is reversed, or 
 because the aspects are opposite ; while yet the three biradials 
 last named are at least equivalent to each other : we have ac- 
 cordingly, for them, these three other formulae, in which the sign 
 alone of the exponent | is changed from what it was in the five 
 formulae last written : 
 
 (C-B) 
 (A-C) 
 (B-A) 
 
 (f-b) = 2&'§; 
 (d-c) = 2^'*; 
 (e-a) = 2A'«. 
 
 103. The same conception oi equality of quotients maybe 
 illustrated by the following simpler figure (fig. 22) ; in which 
 AOB and cod are halves of equilateral triangles, if the closing 
 
A 
 
 LECTURE III. 107 
 
 lines AB, CD be drawn, but may also be conceived to be two bi- 
 radial figures, with a common vertex at o, and with one common 
 upward aspect, and one Fig. 22. 
 
 common shape ; the se- IB C _p 
 
 cond biradial being ob- 
 tained from the first, by 
 first causing it to revolve 
 through a certain amount ,/' 
 (in the figure, a quadrant) 
 of right-handed rotation, in its own plane, round its own vertex, 
 till it takes the position eof, and by then increasing the length 
 of each of the two rays oe and of, in one common ratio (namely, 
 in the figure, the ratio of -y/ 3 to 1) : the pair of rays a, /3, being 
 thus changed to a new pair of rays, y, S, but so that the quotient 
 of the new pair is equal to the quotient of the old pair (each 
 being still, in this case ^2k^), and that thus the equation of art. 
 95 is satisfied, namely 
 
 In fact, when a biradial is thus merely turned round in its plane, 
 and when its legs are altered proportionally, so that it is, in its 
 new state, equivalent, as a biradial, to what it was in its old 
 state, according to the definition of such equivalence in art. 93, 
 it is clear that neither the relative length, nor yet the relative 
 direction, of the second ray of the pair to the first ray of the 
 same pair, is altered ; but (by art. 40 of the second Lecture) 
 the QUOTIENT of the division of the second ray by the first ray 
 depends only on this relative length, and upon this relative direc- 
 tion : the quotient itself therefore remains unaltered, during these 
 changes of the rays which are compared. 
 
 104. It might, at first sight, appear to be enough, in estima- 
 ting the relative direction of two rays, to attend simply to the 
 angle between them, considered as to its magnitude or quantity, 
 and without any attention being paid to its plane. But a little 
 reflection will suffice to show that this would not be sufficient, in 
 the study and comparison of directed lines in space. For if, for 
 example, in fig. 22, after multiplying the length of the ray a by 
 ■v/3, and causing it to revolve right-handedly through a quadrant 
 
108 ON QUATERNIONS. 
 
 in the plane of a and /3, so as to make it take the length and di- 
 rection of y, we were to imagine that it was enough to multiply- 
 in like manner the length of |3 by the same incommensurable 
 tensor \/3; and then simply to set off so7)ie fourth line d, with a 
 length thus obtained, at an angle of sixty degrees to y, such 
 having been the angle of inclination of j3 to a; and if we were to 
 suppose that thus we should satisfy the condition of the equality 
 of quotients, or the equation 
 
 8-r-7 = /3 -^ a ; 
 
 the consequence would be that we should find, for the ray S, no 
 ONE determined direction^ but merely a conical locus, even if 
 its initial point or origin o, were regarded as given and fixed : 
 namely that right cone, or cone of revolution^ which would be 
 described round the ray y, or round the line oc as axis, with the 
 point o for vertex, and with a semi-angle of sixty degrees. We 
 should therefore be led into a vagueness, and an indetermina- 
 tioUi which it is very desirable to avoid, if it be possible to do 
 so ; and which indeed, it would be inexcusable to introduce, or 
 'tolerate, if by a better choice of definitions we can avoid it: as 
 we can, in fact, avoid it, by taking plane and hand into ac- 
 count. Neglecting these, and attending merely to the magni- 
 tude of the angle, we could no longer say, definitely, that the 
 identity 
 
 (j3 -i- a) X a = j3 
 held good ; we could only say that the simple symbol in the se- 
 cond or right hand member, namely j3, denoted one among the 
 infinitely many values of the complex symbol in the first or left 
 hand member, namely (/3 -=- a) x a ; that is, geometrically speak- 
 ing, jS would denote one of the infinitely many directions of the 
 sides of a certain right cone, all which directions would be in- 
 cluded among the meanings of the (on this plan) comparatively 
 indeterminate symbol (j3 -^ a) x a. But when plane and hand 
 are attended to (by our considering towards which hand and in 
 WHAT plane the rotation is to be performed), this indetertmina- 
 TioN entirely disappears. There is, therefore, a good and suffi- 
 cient reason for our taking them into account, as we have done, 
 and as we shall continue to do. 
 
LECTURE III. 109 
 
 105. On the other hand, if any one were to deny to us the li- 
 berty of turning the proposed angle about, even in its oivnplane; 
 or were to require that we should 7iot alter, even pi'oportionally^ 
 the lengths of its legs at all ; if, in short, conceding that when 
 the quotients are equal, the biradials must be equivalent, he were 
 to refuse to admit, conversely, that equivalent biradials represent, 
 in all cases, equal quotients: we might remind this supposed ob- 
 jector, that in studying the quotient of two rays we have (in 
 art. 40.) proposed to study only a certain complex relation, of 
 (what we called) the metrographic kind : not lengths themselves, 
 nor directions themselves, as his objection would require us to do, 
 but a relation between lengths, combined with a relation between 
 directions. We must, therefore, not forego the liberty above de- 
 scribed, while we submit to the restrictions which accompany it. 
 Indeed, before the invention of the quaternions, the same inter- 
 pretation of the equation S-i-7 = j3-^a, as expressing a pro- 
 portionality of lengths, and an equality of angles, directed towards 
 one hand in one fixed plane, had been published by other writers 
 with whom I am happy so far to agree: although my view of 
 either of the two equated quotients, separately taken, appears to 
 be in many respects peculiar to myself; as also does my mode of 
 passing//'0W2 plane to plane. 
 
 106. Having thus come to understand fully the conditions of 
 equality of two quotients, j3 -f- a and S -^ y, we are next to enu- 
 merate their modes of inequality, as, towards the end of article 
 95, it was proposed to do. And this enumeration is easy : for if 
 we regard the rays a and j3 as given and fixed, and retain also y, 
 at first, as an unaltered vector, we know, by the discussion in 
 article 101, that the remaining vector S may be changed in three 
 distinct ways, or admits of a triple variety. And if we next con- 
 ceive the new biradial, whose rays are the old y and the new S, 
 to turn (not in but) with its own plane, preserving its new incli- 
 nation to the old plane of a and j3 unchanged; we shall thereby 
 alter,in a new and fourth way, thebiradial (y, S), or thequotient 
 S-T-y; because we shall alter its plane. You see this little, 
 moveable, reading-desk, upon the table before us: the line ox edge 
 where its slope meets the table is, at this moment, in a meridional 
 direction, or in the line of north and south; but it is obvious that 
 
110 
 
 ON QUATERNIONS. 
 
 I can move it, as I now do, by making the desk turn, while it 
 still rests wpon the table, till the same edge comes to be inclined, 
 or (if I choose) perpendicular to the meridian. (See figure 23, 
 where two positions of a prismatic desk abcuef on a rectan- 
 gular table GHiK are represented.) 
 
 Fig. 23. 
 
 And thus I have altered the aspect of thedesk, and therefore 
 (by art. 93) the value of any biradial, which might have pre- 
 viously been traced upon it ; the new biradial, after such a turn- 
 ing OF and WITH its own plane, being no longer equivalent to 
 the old one. In astronomical language, it is not enough that we 
 know the perihelion distance of a comet, the distance oi perihe- 
 lion from node, and the inclination of the orbit to the ecliptic ; 
 the ORBIT, as di plane, remains in part unknown, until we know 
 also the longitude of the node, or the line in which it intersects 
 the ecliptic. The required enumeration of elements has 
 therefore been effected ; and we become aware that the quotient 
 OF TWO rays involves, when thus geometrically and numerically 
 analyzed, a quadruple variety : it is, therefore, found again^ 
 by this way of examination, as well as by the method of article 
 91, to include within itself a system of four numbers, and to 
 be, in that sense, a Quaternion. 
 
 107. The following additional remarks on this important con- 
 clusion may not be wholly useless. If the situations of the two 
 extreme points a and b, of the vector b-a, were attended to, 
 that vector would depend on six distinct numerical elements 
 (such as the six co-ordinates of the two points) ; because the 
 situation of each poiiit, in particular, depends on, and involves, 
 
LECTURE III. Ill 
 
 three numbers, by the tridimensional character of space. Again, 
 if a quotient of two such vectors, expressed under the form 
 (d-c) -^(b-a), depended essentially on the situations of the 
 four points a b c d, it would, for the same reason, involve no 
 fewer than tw'^elve numerical elements; namely three for each 
 of these four points. But because the vector, denoted by the 
 symbol b - a, is conceived to depend, essentially, only on the 
 RELATIVE and not on the absolute positions of the points a and 
 B, we are allowed, in examining the degree of essential variety 
 of which a vector, so regarded, is capable, to abstract from all 
 that seeming or merely apparent variety, which the mere change 
 of SITUATION oi the pair of points can produce. We may, there- 
 fore, conceive the initial point a asfxed, and attend only to the 
 change of the position of the fnal point b ; and then we find 
 that the vector b - a depends essentially upon three numbers 
 only, and is, in that sense, a triplet. And here we might 
 already see that the quotient of two vectors such as 
 
 (b - c) H- (b - a), 
 
 may be put under the form 
 
 (e - a) -H (b - a), 
 
 by shifting merely the situation of the line cd, till it comes to 
 coincide with a new line ae, commencing at, or radiating from, 
 the point a, without its length or its direction having been al- 
 tered, so that the equation 
 
 e - a= d-c 
 
 shall be satisfied. And thus, by treating a as a known and fixed 
 point, or origin of vectors, we should, in studying the amount 
 of possible variety of a quotient of the kind above considered, be 
 only obliged, at most, to consider that degree of variety which 
 might arise from changes of the two points b and e; so that the 
 Quotient in question could not involve more than six distinct 
 numerical elements. Considering, next, that it is not on the 
 actual or absolute lengths of the two vectors that their quotient 
 depends, but rather on their relative length, or on the ratio of 
 the one length to the other, we see that the divisor-line b - a 
 
112 ON QUATERNIONS. 
 
 may be treated as having its length equal to some one fixed 
 standard, or unit, provided that we suitably, that is to say pro- 
 portionally^ change the length of the dividend-line e - a ; and 
 thus the NUMBER of distinct numerical elements, in the concep- 
 tion of the quotient, is reduced at least as low as five ; because 
 the point b may be conceived to be situated upon the surface of 
 a sphere, with its radius equal to the unit of length, described 
 about the fixed point a as centre : so that ils degree of possible 
 variety is reduced from a dependence on three numbers to a de- 
 pendence on TWO only, while the other variable point e continues 
 to furnish only three numbers. But again, it is not absolute, but 
 relative directions with which we have to deal; we must there- 
 fore allow the angle bae to turn in its own plane, round its own 
 vertex a, and must exclude, as merely apparent, whatever dis- 
 tinction or variety seetns to result, from the comparison of any 07ie 
 such position of the angle (or biradial) so revolving, with another 
 position thereof. We may then conceive the unit-vector ab to 
 be brought, by this sort of rotation, into one fixed plane, such as 
 the horizontal plane drawn through the fixed point a ; and then, 
 although the possible variety of the point e will still remain nu- 
 merically triple, yet the variety allowed to the point b will be re- 
 duced to a dependence upon a single number, such as that which 
 would express the azimuth of this point b, or generally a single 
 angle in the horizontal plane. The whole possible variety 
 OF the quotient of two vectors, or of one directed line in 
 space divided by another, is found, therefore, by this mode of 
 examination or analysis, to involve a dependence upon not more 
 than Four distinct numerical elements. And that it in- 
 volves not fewer than Four such elements appears from con- 
 siderations stated above. It may therefore be properly called (as 
 in fact I do call it) a Quaternion. In short, when such a 
 quotient is pictured by a biradial, it is found to involve two nu- 
 merical elements for species, and two others for aspect ; or more 
 concisely, two for shape, and two for plane : but two and 
 two make Four. 
 
 108. It is easy now to answer the last of the questions (80, 
 IX.), which were proposed at the commencement of this Lecture; 
 or to shew, generally, what ought to be understood by the mul- 
 
LECTURE III. 113 
 
 tiplication o/one Quaternion hy another. For we need only 
 conceive the two factor quaternions as being represented or con- 
 structed by two biradial figures^ having, for greater simplicity, 
 one commo)i vertex; to inquire next in ivhat line ^ the platies of 
 these two figures intersect each other ; to determine thence two 
 other lines a and y, so that the quotient j3 -^ a may be equal to the 
 multiplicand quaternion, and that 7 -4- ]3 may be in like manner 
 equal to the multiplier^ according to the notion of equality be- 
 tween quotients, which has been already fully explained; and finally 
 to determine the product quaternion, namely, the new quotient 
 y -=r a, according to the identity in art. 49, by completing a tri- 
 angular pyramid, or at least by closing a. trihedral angle. That 
 the process, thus sketched out, is an absolutely definite one, 
 and altogether free from vagueness, you may already see. 
 You cannot, therefore, be surprised to have it shewn to you, as 
 I hope in the next Lecture to shew it, that the results of such 
 multiplication of quaternions constitute, in many remark- 
 able instances, or classes of cases, connected with useful geo- 
 metrical interpretations and applications, the subject-matter of 
 theorems. 
 
 For example, the associative principle of the multiplication of 
 quaternions, or the equation 
 
 q" q .q^q . q q, 
 
 (where the point is used as a mark of multiplication), will be 
 found to be such a theorem. It will be shewn to be a truth, but 
 not a truism ; corresponding, in this system of symbolical geo- 
 metry, to certain properties of spherical figures, which are indeed 
 important, but are not obvious: and which cannot probably be in 
 any other way so simply expressed. 
 
 109. But while thus reserving for another occasion any such 
 investigations as these, respecting the theory of Operations on 
 Quaternions, with the geometrical constructions and conse- 
 quences that pertain to them, a few remarks may usefully be 
 added here as illustrations of, or corollaries from, some things 
 which have been already stated in the present Lecture, respect- 
 ing operations on lines and numbers. Thus, without entering 
 yet on the general operation of taking the tensor, we may at 
 
 I 
 
114 ON QUATERNIONS. 
 
 least consider here the two particular but useful cases, where the 
 general quaternion, on which it is proposed to operate, reduces 
 itself, first, to a number, and second, to a line: and so may 'at 
 present inquire only, in the first place, what is the tensor of a 
 scalar: and, in the second place, what is the tensor of a vector? 
 And then we may observe, that whereas every tensor is (by art. 
 63) to be regarded as a signless number, which denotes gene- 
 rally (by 90) the metric element of a factor, the former of 
 the two tensors just now mentioned expresses that factor-element 
 of the scalar, namely, its absolute value, or arithmetical magni- 
 tude, which is independent of algebraical sign ; while the latter 
 of the same two tensors expresses that analogous factor-element 
 of the vector, namely, its length or geometrical magnitude, which 
 in itidepe?ident of geometrical direction. As examples of such 
 tensors of scalars, we have the values, 
 
 T(±3) = 3; T(± v/2)=V2; 
 
 and as examples of such tensors of vectors, we have the equa- 
 tions, 
 
 Tz = T;'-T/t=l. 
 
 110. In fact, by prefixing the characteristic T to any sym- 
 bol |0 of a vector, or directed line in space, regarded as being itself 
 a geometrical factor (on the plan of art. 82), we imply (see art. 
 DO) that we ais^/'ac^ from the graphic operation of this ^c^o/- 
 line, and attend only to its metric effect ; which comes to 
 abstracting from the direction of the line p, and attending only 
 to its length. This length of any vector p may hence be de- 
 noted by the symbol Tp, and may be called, as above, on the 
 general plan of these Lectures (see in particular the latter part 
 of art. 90), the tensor of that vector p. In other words, the num- 
 ber Tp is to be conceived to denote the answer to the question, 
 How many linear units (of a length previously assumed as the 
 standard of length) are contained in the line p? For when the 
 tensor Tp is considered (on the plan of same art. 90) as one ele- 
 ment of the factor p {l\\e other factor-element being the versor 
 Up), it must be supposed to answer this other but connected 
 question : In what ratio does the proposed vector p, regarded .\s 
 
LECTURE III. 115 
 
 a MULTIPLIER-LINE, alter the length of any other vector o-, 
 perpendicular to itself, on which it operates, in the way explain- 
 ed in the eighty-second article ? — that is to say (a being still sup- 
 posed perpendicular to p), What is the ratio of the length of the 
 product-line ^a to the length of the multi'plicand-line o- ? On 
 the one hand, by art, 90, this ratio must be that of Tp to 1, 
 because it is, in general, the ratio of Tq to 1, if 5' be the factor 
 of the multiplication, whatever that factor maybe: while, on 
 the other hand, by art. 82, the same ratio is expressed by the 
 number of linear units in p, because the length of the product- 
 line per was found, in that article, to be the -product oi\k\Q lengths 
 of the two factor-lines, in the sense that the number denoting the 
 length of po- is the product of those which denote the lengths of 
 p and (T. We must, therefore, conclude, as before, that the num- 
 ber Tp expresses the length of the line p ; or that " the tensor 
 of a vector is the number denoting its length." 
 
 With this signification of a symbol such as Tp, it is clear that 
 the equations of art. 90, 
 
 T . kX = be, T(A -7- k) = c-^b, 
 
 may be written as identities thus, 
 
 T.KA = Tfc.TA, T(A--k)-TX^Tk; 
 
 where k and X are symbols oiany two vectors: and indeed it will 
 be found that analogous identities exist, for the more general 
 case where those symbols under the characteristic T are supposed 
 to represent two quaternions. 
 
 111. There is, however, another mode of expressing the length 
 of a line p, on the principles of the present theory, without em- 
 ploying the characteristic T, which mode it may be proper here 
 to mention, and which depends on the principle enunciated at 
 the beginning of art. 85. It was there shewn, as a particular 
 case of the multiplication of parallel vectors, that the square of 
 every vector is a 7iegative scalar, of which the positive opposite 
 expresses the square of the length of the vector ; that is, the 
 square of the number which denotes that length, by denoting (as 
 usual) the number of linear units contained in it. Hence, for 
 
 I 2 
 
116 ON QUATERNIONS. 
 
 example, if?' be the number which thus denotes the length of 
 the vector p, we shall have the equations, 
 
 p2 = - ?-2j p2 + r'3 = ; 
 
 which give also these others, 
 
 the expression - p^, under this last radical sign, being here a posi- 
 tive number, because the square p^ of the vector p is itself {hy 
 the lately cited article) a negative number. The radical \J (- p^) is 
 therefore, in this theory, another symbol Jbr the length q/^^e 
 line p ; and by comparing the results of the present and of the 
 foregoing article, we arrive at this important symbolical equality, 
 where p may represent any vector, 
 
 Tp=v(-P^); 
 
 giving also this equation freed from radicals, 
 
 (Tp)2+p3 = 0. 
 
 1( w be a. scalar, then, by what was shewn in art. 109, its tensor 
 is, on the other hand, 
 
 where the positive or absolute value of the radical is to be taken ; 
 and we may just mention by anticipation here, that when a qua- 
 ternion q shall have been put under the general form already 
 referred to in art. 78, namely, 
 
 q = w + ix +jy + hz, 
 or, more concisely, 
 
 q = w + Q, 
 
 where iv is a scalar, and p is a vector, the tensor of this quater- 
 nion will be found to admit of being so expressed as to include 
 the two radical forms lately written; namely, in the following 
 way : 
 
 T^ = T{ic + p) = V (^^^ - p')- 
 
 112. It may be instructive here to remark, that because when 
 
LECTURE III. 117 
 
 p and o- are any two perpendicular lines, their product per is itself 
 another line, the tensor of this product may, by the last article, 
 be thus expressed : 
 
 T . per = -v/ (- (pff)3), if (7 -L p. 
 
 And because the length of this product line po- is the product of 
 the lengths of the two factor lines p and a, we have also (com- 
 pare art. 110), 
 
 T . p(T = Tp . To-. 
 
 Eliminating, therefore, the characteristic T, by the principles of 
 the preceding article, we arrive at the equation, 
 
 V (- (p^Y) = V (- p') V (- a^), if C7 -L p ; 
 
 which must no doubt seem strange to those who are accustomed 
 only to the expressions of ordinary or commutative Algebra. But 
 in the present Geometrical Calculus, by the equation of perpen- 
 dicularity assigned in art. 82, the formula last written, when 
 cleared of radicals, expresses simply that 
 
 — per . per = pp . (7(T, if - (xp = + po- ; 
 
 and since this last condition gives evidently, 
 
 — p . op . o = + p . po . O, 
 
 we see that we have only to remove the points, regarded as marks 
 of multiplication, which serve to groupe (and, at the same time, 
 to separate) the factors, in order to arrive at the expression of 
 the equality asserted in the formula. Now such removal of 
 POINTS, or of other separating and associating marks inserted be- 
 tween factor-symbols, is precisely what is allowed by that Asso- 
 ciative Principle of multiplication, which was stated, in art. 
 108, to hold good for quaternions generally. We have, therefore, 
 not only explained what might for a moment appear a difficulty, 
 but also have verified, in one useful case of application, that ge- 
 neral associative pi'inciple, which will be found to be among the 
 most important links of connexion between Algebra and the 
 Calculus of Quaternions. 
 
118 ON QUATERNIONS. 
 
 113. The versor of a scalar is simply the sign +, if the scalar 
 be positive, or the sign -, if the scalar be negative ; but because 
 these SIGNS, regarded 2& factors, have respectively the same ef- 
 fects as the factors + 1 and - 1, we may write for any scalar w^ 
 the formula, 
 
 \}w = ± I, according as w 0. 
 
 For example, 
 
 U (+3) = + = +1; 
 U(-V2) = - = -l. 
 
 The versor of a vector p is the vector-unit in the direction of 
 that vector ; for such is the other factor of p, in the identity 
 
 p = Tp . Up ; 
 
 the factor Tp having been seen (in art. 110) to be the number 
 which denotes the length of the line p, so that on dividing the 
 line by this number, the quotient 
 
 Up = p -^ Tp 
 
 must be in general a neiv line, with the sa)ne direction as p, but 
 with its length reduced to unity. For example 
 
 U(30=/; U(-i V2) = -i. 
 
 We may also write (in virtue of the value of Tp, assigned in art. 
 Ill) this general expression, 
 
 Up = p-- V(-P^), 
 
 where p may denote any vector ; and we shall have, with the 
 same generality, the equation (compare arts. 75, 77), 
 
 (Up)3 = -1. 
 
 The versor of zero must be regarded as indeterminate, unless the 
 zero be supposed to be the limit of some known process, in which 
 case we may be induced to treat it as an infinitesimal scalar with 
 known sign, or (according to the case) as an infinitesimal vector 
 with a known direction ; and then this sign, or this direction, 
 
LECTURE III. 119 
 
 may be considered as the particular value of the sytnbol UO, for 
 that particular question. And for the same reason that + 1 or- 1 
 may be substituted for + or -, as the value of the versor of any 
 scalar different from zero, we may also, whenever we think fit, 
 equate a tensor to a positive scalar, although it was seen (in art. 
 63) to be more properly a signless number, or one unaccompanied 
 with algebraic sign. 
 
 1 14. The conjugate of a scalar is simply that scalar itself; but 
 the conjugate of a vector is the vector reversed, or taken with a 
 direction opposite to the original, without any change of length; 
 because in general (by art. 89) conjugate factors produce the 
 same effects in the way of tension, but produce opj^osite effects 
 in the way oi version : and opposite lines (by same art. 89) pro- 
 duce such opposite effects, when used as axes of right-handed 
 rotation, to operate on any other line to which they are both per- 
 pendicular. Thus with the recent significations oi w and p, and 
 with the characteristic of conjugation K, we have generally, 
 
 Kw ■= + iv; Kp = - p ; 
 
 and it may be stated by anticipation, that when any quaternion 
 q is put under the form (see art. Ill) q = iu + Q, its conjugate is 
 
 Kq = Kiw + p) = ic - p. 
 
 1 15. Finally, as regards powers of lines, with positive or ne- 
 gative numbers for their exponents, it is easy to give a clear and 
 simple interpretation to any symbol of such a power, by an ob- 
 vious extension of what was shown in art. 86, respecting powers 
 oi' unit-vectors. We saw, when considering such powers, that 
 whereas the unit-line k, for example, if regarded as a factor, 
 would have the effect of tui'ning any horizontal vector on which 
 it operates, horizontally and right-handedly through a quadrant, 
 or of causing this multiplicand vector to advance through 90° of 
 azimuth, the power k^ with the fraction \ for its exponent, would 
 only cause the vector to turn, in the same plane and towards the 
 same hand, through half a quadrant, or would make it advance 
 through 45° of azimuth. The operation of which the factor k^ is 
 the agent, is therefore half of that other operation, of which the 
 agent is the factor k itself; in the sense that two operations of 
 
120 ON QUATERNIONS. 
 
 the one kind are equivalent to one of the other. In symbols we 
 have, therefore, here, as in common algebra, the equation or 
 idenlit}^, 
 
 h^ k^ = k. 
 
 Suppose now that p is some other upward vector, 
 
 where 2 is a positive number different from unity; for instance 
 let 
 
 z = 2V 2, p = A ^/8. 
 
 To interpret, then, the symbol p^, we have only to combine, with 
 the recent act of version through half a quadrant, an act of ten- 
 sion, which shall, in like manner, produce /*«//' ^Ae effect of mul- 
 tiplying by the number z: in other words we are to multiply the 
 square-root ki of the given versor k, by the square-root 2* of the 
 given tensor z. For the product thus found, namely, 
 
 where 8* has its usual arithmetical signification, is a symbol satis- 
 fying the analogous identity, 
 
 p4 p5 = p ; 
 
 and the symbol pi, when thus interpreted, represents a factor 
 which is the agent of a certain complex operation, on length and 
 on direction, whereof the metric and the graphic elements are 
 respectively, as operations, the halves of the corresponding ope- 
 rations of tension and version, which are the elements of that 
 other operation, whereof the given factor p is the agent. In fact, 
 if we twice successively multiply the length of any proposed hori- 
 rizontal line by the new incommensurable tensor -v/ -y/ S, we shall 
 thereby, upon the whole, have multiplied that length by the ori- 
 ginal number V 8 or 2: ; that is, by the proposed tensor of p. And 
 if, in like manner, we twice successively operate on the direction 
 of the same horizontal line, by the versor A=, regarded as a gra- 
 phic factor, we shall, on the whole, have caused the line to 
 advance through ^«ro octants, or through one quadrant ofazi- 
 
LECTURE III. 121 
 
 muth, which is precisely the eflfect of operating once by the pro- 
 posed versor k of the factor p itself. Again, with the same base 
 p =^ V' 8, but with the fraction ^ for the exponent, we obtain on 
 the same plan the power, 
 
 pi = Ai y' 2, 
 
 which satisfies the identity, 
 
 pa pa^ p? = p ; 
 
 and, as a factor, has the effect of turning any horizontal line on 
 which it operates through 30° of azimuth, and of increasing the 
 length of that line in the ratio of the diagonal to the side of a 
 square, or in the ratio of the cube root of the number z to unity. 
 And the power 
 
 pS = 2At, 
 
 when used as a factor, changes the half base to an adjacent side 
 of a horizontal and equilateral triangle, in such a manner that 
 this last-mentioned power of p coincides with that quaternion 
 which has been already considered in articles 102, 103 of the pre- 
 sent Lecture, and is represented or constructed by any one of 
 the five equivalent biradials dba, &c., of the figure 18, or by any 
 one of the three other equivalent biradials, aob, cod, eof of fig. 
 22. 
 
 1 16. More generally, for the same base p, and for any nume- 
 rical exponent t, we may write, as in ordinary algebra, the fol- 
 lowing expression for the power : 
 
 p« = {kzy - k< zK 
 
 That is to say, the tensor z\ of the power p', is the corresponding 
 power of the tensor z ; and the versor M of the same power p*, is 
 the power of the versor k. It is evident that analogous results 
 must hold good for the powers of all other vectors, and that we 
 may write generally, for any such power, with a vector for base, 
 and a scalar for exponent, the formulae, 
 
 T.p' = (Tpy; 
 
 U.p'=.(Upy. 
 
122 ON QUATERNIONS. 
 
 A POWER of this sort is, therefore, in general a quaternion, 
 of which the tensor and the versor can be assigned by the fore- 
 going rules : but this quaternion may, in certain particular cases, 
 degenerate into a line or a number. In fact, since, with the in- 
 terpretation assigned above, the power p*, regarded as a factor, 
 has, in general, the effect of causing any line a, perpendicular to 
 the base-line p, to revolve round that base through an angle =t 
 X 90°; while it multiplies the length of the same multiplicand 
 line by the i''' power of the number Tp, which expresses the length 
 of the base ; we see that in the equations, 
 
 pV = T, p* = r -T- O", 
 
 where t denotes the product-line, or the result of the multipli- 
 cation thus conceived, this line r will not only be perpendicular 
 to p, but also to (T, if the exponent t be any odd whole number ; 
 in this case, therefore, the power p*, being equal to the quotient 
 of two rectangular lines, will be itself a line or vector. For ex- 
 ample, the power p^ is evidently the base-line p itself. On the 
 other hand, if the exponent t be zero, or any positive or negative 
 multiple of 4, the direction of the product line r coincides with 
 that of the multiplicand line o-, and the power p^ regarded as the 
 quotient r -f- a-, is seen to be a positive number ; for example, we 
 have, as in algebra, the value 
 
 pO= 1. 
 
 But if the exponent t be any positive or negative multiple of 2, 
 without being a multiple of 4, then the direction of r is opposite 
 to that of (T, and the power p^ hanegative number : and, in fact, 
 we saw, for example, that the square p^ of every vector p is equal 
 to a negative scalar, or that (by arts. 85, 111), 
 
 p2 = -(Tp)2. 
 
 117. Another useful though particular case, in this theory of 
 powers of lines, is the power with negative unity for exponent. 
 This power p~i is itself, by the last article, a line, because the 
 exponent is an odd whole number ; and this new line may be 
 called the reciprocal of the old or given line p, on account of 
 the relation 
 
LECTURE III. 
 
 123 
 
 pp-i = pi-i = p0^1. 
 
 which is included in the more general formula (common to alge- 
 bra and to quaternions), 
 
 pm p« ^ ^vi.n^ 
 
 where m and n are any scalar exponents. The tensor of the re- 
 ciprocal of any vector is evidently the reciprocal of the tensor of 
 that vector ; and, in like manner, the versor of the reciprocal is 
 the reciprocal of the versor. The factor p"i has, therefore, the 
 effect of dividing by Tp the length of any line a perpendicular to 
 p, on which it is conceived to operate, and also of turning that 
 line o- left-liandedly through a quadrant round the direction of 
 H: p, or right-handedly through a quadrant round the opposite di- 
 rection of-p as an axis. We may then write 
 
 U(p-i)=(Up)-i==-Up; 
 
 which result evidently agrees with the formula of art. 113, 
 
 (Up)--i; 
 
 and gives the general expression 
 
 Tp-^ 
 
 Ih 
 
 Fig. 24. 
 
 Any two reciprocal vectors, such as p and p"i, have, therefore, 
 their directions opposite^ and their lengths reciprocal ; in such a 
 manner that the rectangle con- 
 structed with those lengths fov 
 its sides is equal in area to the 
 square described upon the unit 
 of length. For example, if 
 AOB, in fig. 24, be a diameter 
 of a circle, and if the ordinate / 
 or half chord oc or od, per-^ 
 pendicular to that diameter, be 
 taken for the unit of length, 
 then the two oppositely direct- 
 ed segments of that or of any 
 other chord through o, for in- 
 stance the two opposite parts or segments e - o and f - o of the 
 
124 ON QUATERNIONS. 
 
 chord EOF, are, in the sense above explained, reciprocal vectors, 
 
 so that 
 
 if E - o = p, then f - o = p"^. 
 
 118. If we combine this notion of a reciprocal with the rule 
 for forming generally the product of any two vectors, which rule 
 was deduced in art. 88, we shall infer easily that " to divide one 
 vector /3 hy another vector a, and to multiply the former vec- 
 tor j3 into the reciprocal a'^ of the latter, are operations which 
 give generally one common quaternion as their result:" or that 
 we may write (in quaternions as in algebra), 
 
 |3^a = i3xa-i. 
 
 In fact, the quotient in the one member, and the product in the 
 other, have one common tensor^ namely Tj3 -^ Ta, or the quo- 
 tient of the length of j3 divided by the length of a. Again, the 
 axis of the versor of the quotient j3 ^ a, regarded as a graphic 
 operator, is perpendicular to the plane which contains both a and 
 j3, or to which they both are parallel ; and the rotation round this 
 axis from the divisor a to the dividend /3, is (by our general con- 
 ception of a geometrical quotient) right-handed ; such then is also 
 the character of the rotation round the same line, from /3 to - a, 
 or from j3 to a"S and, therefore (by 87, 88), this hne is also the 
 axis of the versor of the product, /3 x a" ', or /3a"^. And finally, 
 the angles of rotation are the same ; for the angle of the quotient, 
 |3 _i_ a, which angle may be thus denoted, 
 
 Z (i3 -^ a), 
 
 is simply the angle between the directions of a and j3 ; while (by 
 the same arts. 87, 88) the angle of the product, j3xa'^ which 
 may, on the same plan, be denoted thus, 
 
 ^(jSxa-^, 
 
 is the supplement of the angle between |3 and a"^ or between /3 
 and -a, or is equal to the angle between the directions of a and 
 |3 themselves. We may also agree to denote occasionally the 
 
 reciprocal vector a"^ by ihe fractional symbol - ; and to repre- 
 
 \ 
 
LECTURE III. 
 
 125 
 
 sent the quotient j3-4-a, or the product j3a"^, by the analogous 
 
 symbol — . 
 a 
 
 119. Those who are acquainted with the properties of loga- 
 
 KiTHMic SPIRALS may employ them with advantage to illustrate 
 
 the whole preceding theory oi powers of lines. In figure 25, let 
 
 ABCDEFG be one half-spire of such a curve, subtending two right 
 
 angles at the pole o; while another half spire, proceeding in the 
 
 opposite direction from a, passes through the points uvwxyz. 
 
 Fig. 25. 
 
 Let the six transversals through the pole, aozg, boy, cox, dow, 
 Eov, Fou, be conceived to succeed each other at equal angular 
 intervals of thirty degrees each ; and of the two rectangular rays, 
 or vectors from the pole to the curve, oa and od, let it be sup- 
 posed that the latter is to the former in the ratio of -y/S to 1. 
 Then if the figure be laid upon a table, with its face upwards, 
 the quotient of the ray od, divided by the ray oa, will be (by 
 principles already explained) the same upward vector, p=A-v/8, 
 which was considered in a recent article (1 15) ; and, in general, the 
 power p* of this vector or base-line p, with the scalar exponent t, 
 will be equal to the quotient of some one ray r of this spiral, di- 
 vided by another a; the condition being that r shall be wore ad- 
 vanced than (7, in the order of progression from a to g, by an 
 angle at the pole o, which shall be =t x 90°, if the scalar t be 
 positive ; or else that r shall be less advanced than a-, in the same 
 order of rotation, by the amount so expressed, if the exponent t 
 
126 
 
 ON QUATERNIONS. 
 
 be negative. Thus we may form, for some of the positive powers 
 of p, the table : 
 
 (a-o)--(a-o) = p°==1; 
 (b - o) -f- (a - o) = p^ = ^3 -y/ 2 ; 
 (c - o) -^- (a - o) = p* = 2ki ; 
 (d - o) -^ ( A - o) = pi = A \/ 8 ; 
 
 (e - o) -^ ( A - o) = p^ = 4^3 ; 
 
 (F - o) ~ (a - o) = p3 = 4^* V 2 ; 
 
 (G-o) ^(A-o) = p3=-8; 
 
 with this other table of negative powers : 
 
 (u-o) 
 (v-o) 
 
 (A-o) = p-3 =^-3 Y^i; 
 
 (A-o) = p-t-^^-^; 
 
 (w-o)^(A-o)=p-i = y^-i V^ = ^; 
 
 (X-O) ^ (A-o)-p-^ = ^A-t; 
 
 (y - o) -f- (A-o) = p-* = i A;- 3 Vi ; 
 
 (Z-O) ^(A-0)=p- 
 
 4 
 .-2 =_ J 
 
 The equation of the spiral may, therefore, be said to be the fol- 
 
 lowing 
 
 (7 = p a, 
 
 if a be some fixed ray, such as a - o, while o- is a variable ray 
 (from pole to spiral), and ^ is a variable scalar. If 
 
 r =p 
 
 be the analogous expression for another variable ray of the same 
 spiral, and if, while the exponents ^ and h + t both vary, their 
 difference h xem'cxm^ Jixed, the quotient of the two variable rays, 
 namely, 
 
 T -- ff=p% 
 
 will then remain also fixed, being equal to one constant quater- 
 nion : and the triatigle, whose sides are the two rays a and t and 
 the chord r-o-, will be of a constant species, depending on the 
 length of the base-line p, and on the scalar exponent h. Thus, in 
 fig. 25, making h =f, or conceiving r to be more advanced than 
 
LECTURE III. 127 
 
 o- by 60° of rotation, that is, by two-thirds of a quadrant, we find 
 the fixed quaternion quotient p'' = 20; and the triangle, as for 
 example aoc, or bod, &c., becomes, in this ease, the half of an 
 equilateral triangle. If the difference h of exponents be chosen 
 continually less and less, so as to tend to zero, the vertical angle 
 of the triangle tends to vanish ; and its base-angles tend to be- 
 come the constant acute and obtuse angles which a variable ray 
 (from the pole) makes with the spiral. In the case of fig. 25, this 
 acute angle between ray and curve, which may be called the an- 
 gle of the spiral, suppose the mixtilinear angle at g, is nearly 
 = 56°^ ; and in general it can be computed without difficulty, 
 either by the theory (not yet stated) of differentials of qua- 
 ternions, or by methods otherwise known. 
 
 120. I shall conclude this Lecture, which has already ex- 
 tended to a greater length than I could wish, by observing that 
 (if we set aside, for a moment, the case of numerical quotients or 
 parallel lines), every quotient of two rays may be regarded 
 as a power of a vector, with a scalar for the exponent of this 
 power; and even that we are at liberty to assume that this scalar 
 exponent is confined between the limits and 2 ; so that we may 
 write generally, as an expression for any such geometrical quo- 
 tient, the formula, 
 
 just as the particular quotient 2lc^, which presented itself in some 
 former articles of this Lecture, has been seen to admit of being 
 put under the form pf, where Q = k ^J^. In fact, any given bi- 
 radial, such as aoc in fig. 25, with any actual angle, whether 
 acute, or right, or obtuse, may always be conceived to be in- 
 scribed in a definite spiral (of the logarithmic kind), in such a 
 way that the vertex of the given biradial shall be the pole of the 
 spiral, and that the two given legs or rays of the biradial shall 
 also be two rays of the same spiral, while the arc intercepted be- 
 tween them shall be less than a semi-spire. And, then, by tak- 
 ing any two rectangular rays of the spiral, including between 
 them w^hat may be called a quarter- spire, we shall form a new 
 and quadrantal biradial, such as aod in the same figure 25, 
 whereof the second ray, divided by the first, shall give, as the 
 
128 ON QUATERNIONS. 
 
 quotient, a certain vector p, perpendicular to the plane of the 
 curve, which vector is to be taken as the base of the sought 
 power p* ; while the exponent of that power is simply the num- 
 ber obtained by dividing the angle of the biradial by a quadrant, 
 and therefore is (on this plan of construction or representation) 
 greater than zero, but less than two. Or, without thinking of 
 spirals, we may conceive that after determining, by the last-men- 
 tioned division, the numerical exponent t of the power p*, which 
 power is to be made equal to the given quotient j3 -7- a; and after 
 fixing the direction of the hase-line p, by the condition that it is 
 perpendicular to the plane of the two given rays a and /3, and 
 that the rotation round this base-line p, from the divisor-line a to 
 the dividend-line )3, h positive, or right-handed: we then proceed 
 to determine the length of the same base p, or the number Tp, 
 which expresses this length, by the condition that the t^^ power 
 of this sought number Tp shall be equal to the quotient T/3 -f- 
 Ta, which is obtained by dividing the length of the ray j3 by the 
 length of the other given ray a. At the limit ^ = 0, this process 
 may be said to fail, for it would require us then to take an infi- 
 nitely high power of a number which would generally differ from 
 unity ; but at this limit the a7igle of the biradial vanishes, and 
 the quotient j3 -f- a becomes simply 'a. positive number. And, on 
 the other hand, at the limit ^ = 2, although the process cannot 
 precisely be said io fail, since it still allows di possible construc- 
 tion, yet this construction becomes now partially vague, for it 
 conducts to a semi-spire, in an indeterminate plane ; and the quo- 
 tient is, in this case, a negative number, which is indeed the 
 square of a vector, but of a vector with an indetermifiate direc- 
 tion. But whenever the quotient of the two rays does 7iot thus 
 reduce itself to a scalar, that is, whenever (as above said) the two 
 rays contain between them any actual angle, whether acute, or 
 right, or obtuse, the process then does not merely succeed, but 
 gives a perfectly determinate result; at least if, for the sake of 
 simplicity and definiteness, we still exclude the supposition of a 
 rotation through any greater angle. We may then regard the 
 expression assigned above, namely, the scalar power p*, or 
 more fully, the power, with scalar exponent, of a vector base, 
 as a general expression for the quotient of one ray divided by ana- 
 
LECTURE III. 129 
 
 ther, at least if the two rays do not happen to have one common 
 direction. And because the base p, being a vector, depends (by 
 arts. 17, 101), on a system of three numbers, serving here to 
 fix the aspect and angle of the spiral; while the exponent t is 
 itself ANOTHER NUMBER, Serving to mark ihe fraction of a quar- 
 ter-spire; we are thus conducted anew to that important and 
 fundamental conclusion, from which the present Calculus may 
 be said to derive its name. For we thus are led to conclude again, 
 that the Quotient of two Rays, when directions in space, as 
 well as lengths of lines, are attended to, depends generally on a 
 System of Four Numbers, which result confirms, in a new 
 way, the propriety of our calling such a ^Mo^^ew^ a Quaternion. 
 But the general theory of Operations on such Quaternions must 
 be reserved for the following Lecture. 
 
LECTURE IV. 
 
 121. Although the last long Lecture, Gentlemen, has gone 
 far towards a statement of the chiei notations of that Calculus to 
 which the present Course relates, yet a few other general signs, 
 or characteristics of operation, require to be still explained. And 
 although the chief operations on lines, regarded as having direc- 
 tions (as well as lengths) in tridimensional space, and called 
 sometimes by us, for that reason, rays, or vectors, have been 
 considered, and some leading problemsrespecting them resolved, 
 at least for the cases in which not more than two lines at any 
 one time were to be combined among themselves in the way of 
 multiplication or division, yet even for lines it has not hitherto 
 been distinctly shewn how to combine, in that way, even so many 
 as three with each other. The quotient of any two such rays has 
 been proved to be in general a Quaternion ; and so have also 
 the product of any tivo rays, and the power of any one ray or 
 vector, with any scalar or numerical exponent ; in the sense that 
 each such quotient, or product, or power, denoted by any one of 
 the three symbols, 
 
 /3-f-a, kX, p\ 
 
 and interpreted on the principles of the present system, has been 
 found (in the last Lecture) to involve generally a dependence on 
 a system oifour distinct and numerical elements ; but we have 
 done little more than hint, as yet, at the methods of combining 
 such quaternions among themselves by operations of 07ie on ano- 
 ther. The operation of such a quaternion, as o. factor, on a line, 
 has indeed been seen to involve generally a metric and a graphic 
 element ; a stretching and a turning of the line thus operated 
 upon ; or in other words a tension and a version: to denote ^h.\ch. 
 elements separately we have introduced (in art. 90) the two cha- 
 
LECTURE IV. 131 
 
 racteristic letters T and U, as signs of the operations of what 
 we have called taking the tensor and taking the versor respectively. 
 But while thus decomposing generally a quaternion intoj^c^or^, 
 or into elements to be combined by multiplication, we have as 
 yet proved nothing respecting the equally general and equally 
 important decomposition of a quaternion into parts, or sum- 
 mands, to be combined with each other by addition ; and in par- 
 ticular we have only alluded, by anticipation, to the separation 
 of the scalar and vector parts, such as the parts w and p in the 
 expression 
 
 q = w + p, 
 
 of articles 111, 114; to denote generally which new sort of de- 
 composition of a quaternion, it will be necessary to introduce (as 
 above hinted) two new signs, such as the two new characteris- 
 tic letters S and V, not yet submitted to your notice, for the 
 purpose of indicating the operations of taking the scalar, and 
 taking the vector, respectively, of any proposed quaternion. To 
 express that in passing according to a certain law from one pro- 
 duct of lines or from one quaternion to another, we have con- 
 ceived or found (as for example in passing from kX to Ak), the 
 tensor element of the quateirnion, as a factor, to remain unchanged, 
 but the versor element to be reversed in its effect (114), or to be 
 made to turn the line whereon it operates in a direction contrary 
 to that in which it turned the line before, but through an equal 
 amount of rotation, and in one common plane^ we have introduced 
 (in art. 89) the denomination of conjugate products, or factors, or 
 quaternions, and have employed the letter K as the sign of such 
 conjugation, or as the characteristic of the operation of taking 
 the conjugate of a quaternion ; but we have as yet said nothing 
 respecting the conjugate of a product of quaternions: and no- 
 thing has yet been proved respecting the tensor or the versor 
 of such a product. The outline of a general construction for the 
 multiplication of any two quaternions, by means of a trihedral 
 angle, has indeed been given (in art. 108); and the correspond- 
 ing construction for the division of quaternions may have easily 
 thence suggested itself: but the simplifications and transforma- 
 tions of the constructions, which spherical geometry affords, have 
 
 k2 
 
132 ON QUATERNIONS. 
 
 not yet been touched upon. The multiplication oi lines among 
 themselves has been shewn to give different results^ according as 
 the factors have been taken in one or in another ordei' ; from 
 which it follows, by still stronger reason, that the multiplication 
 of quaternions is not generally a commutative operation ; but it 
 has hitherto been only stated, and not generally proved, that the 
 same new and enlarged operation agrees with the process of the 
 same name in ordinary arithmetic and algebra, by its possessing 
 another general property, which is at least equally important, 
 namely, by its being an associative operation (108) ; much less have 
 the geometrical significations of this general result been brought 
 as yet before your notice. Another great link of connexion be- 
 tween quaternions and ordinary algebra, I allude to the distribu- 
 tive property of multiplication, has not hitherto been so much as 
 mentioned in these Lectures. And while the product or the quo- 
 tient of two rectangular lines has been represented or constructed 
 by a tliird line rectangular to both, yet it may be admitted that 
 the motives for adopting such a representation or construction, 
 which were suggested towards the close of the second Lecture of 
 this Course, even when combined with the degree of success 
 which may be supposed to have been since attained in unfolding 
 the consequences of this geometrical construction or conception, 
 may still leave room for a not unreasonable demand, on the part 
 of a severely logical inquirer, that some new and more stringent 
 TEST should be applied, as a check on the consistency of this 
 view, respecting perpendicular lines, with principles which have 
 been judged, in these Lectures themselves, to possess a character 
 still simpler, earlier, and more fundamental. 
 
 122. To examine then, first, in a new way, the views already 
 propounded respecting the multiplication and division of perpen- 
 dicular lines, as regards the consistency of those views with each 
 other and with still more general principles, let me once more 
 remind you that the quotient (5 -^ a of any two rays in space has 
 been found to be, generally, in our system of interpretation, a 
 Quaternion (see articles 91, 106, 120) : this being indeed that 
 main and fundamental conclusion, from which the present Cal- 
 culus derives its name. But we have also seen that this gene- 
 ral quaternion may, in certain particular cases of relative direc- 
 
LECTURE IV. 133 
 
 tion of the two rays, degenerate into a scalar or into a vector^ that 
 is, into a number or a line : namely into a scalar (by articles 59, 
 64), when j3 |1 a, that is when the two rays compared -dre parallel 
 to each other, or to any common line ; and into a vector (by art. 
 82), when /3 _L a, that is when the two rays are perpendicular 
 to each other; so that numbers and lines are both inclu- 
 ded in the conception o/"quaternions, and a complete theory 
 of the latter must consequently include the theories of both the 
 former. As an example of a quaternion thus degenerating into 
 a vector, we had, in article 83, the equation 
 
 -6^-h3;" = -22; 
 
 and other examples, where the quotient of two rectangular linos 
 has been already treated as a third line rectangular to both, cannot 
 fail to have been observed by you. In fact it was shewn generally, 
 in art. 82, that the product aj3 of any two perpendicular lines is 
 equal (in our system) to a third line; namely, to one which is 
 perpendicular to both the factors, having also its length equal to 
 the product of their lengths, and having its direction distinguished 
 from its own opposite, by a simple rule of rotation, assigned in 
 the last quoted article; a conclusion which is also deducible (by 
 making ^=1) from the more general theorem of art. 88, respect- 
 ing the multiplication of ant/ two lines. Hence, by the general 
 relation of multiplication to division, or immediately by the same 
 art. 88, we may write an equation of the form, 
 
 A-HK = jU, ifX_LK:; 
 
 the new vector jj. being so chosen, as to satisfy the connected 
 equation, 
 
 X = jU X ic, 
 
 with the signification already referred to. That is to say, the 
 length of the quotient-line fx is to be equal to the quotient of the 
 lengths of the two given lines X and k, with the usual reference 
 to an assumed unit of length ; or in symbols (compare art. 110), 
 
 Tfx = TX -r Tfc. 
 
 The direction of the quotient line ^u is to be perpendicular (as 
 
134 ON QUATERNIONS. 
 
 above noticed) both to the dividend-line X and to the divisor-line 
 K ; or in symbols, 
 
 And finally this perpendicular direction of the quotient line is 
 distinguished from its oivn opposite, by the rule that the rotation 
 round ju from k to X h positive ; or more fully, that the rotation 
 round the quotient-line, from the divisor-line to the dividend-line, 
 is right handed. In short a quadrantal quaternion, or a 
 quaternion with a quadrantal versor, is in our system constructed 
 by a LINE, which is drawn in the direction of the axis of the ver- 
 sor, and of which the length represetits the tensor of the quater- 
 nion. All this may indeed have been collected from what was 
 said in former Lectures, but it seemed worth while to state it for- 
 mally and explicitly here : since it is in fact one of the chief fea- 
 tures or main elements of this Calculus, as regards geometrical 
 interpretation. 
 
 123. Conceive now, as an application oi i\ie ioregom^ x\x\e 
 for constructing the quotient of two rectangular lines, that a line 
 £ is drawn from the point o of figure 22 (art. 103), perpendicular 
 to the plane of that figure ; and more particularly, let this new 
 line E be directed \ex\ACs\\y upwards, if the figure be laid horizon- 
 tally with its face upwards on a table. Let the length of this 
 upward line s be equal to the length of the half base oa of the 
 equilateral triangle of which ob is a side ; and let the altitude 
 ab of that triangle be assumed as the unit of length. Then, by 
 the general process of construction above explained, if this new 
 and vertical line £ be employed as a divisor, and if the horizontal 
 ray a or oa of the figure be taken as a dividend, the quotient will 
 be the ray 7 or oc of the same figure; and we may write the 
 
 equation 
 
 a ^ £ = 7. 
 
 For the tensor of the quadrantal quaternion a -^ t will here be 
 equal to unity, on account of the equality of lengths subsisting 
 between the divisor and the dividend ; and the length of the line 
 oc is the same as that of ab, which has been taken as the unit of 
 length, so that we have, in conformity with the first part of the 
 general rule in art. 122, 
 
 T7 = T«--T£=:1. 
 
LECTURE IV. 135 
 
 Again the (horizontal) direction of -y is perpendicular to the (ver- 
 tical) plane of a and e, so that we have here 
 
 7 JL a, 7 _L £, 
 
 as is required by another part of the same general rule for the 
 construction of the quotient-line. And finally the only remain- 
 ing part of the same rule is also satisfied ; for the rotation round 
 7 from £ to a is right handed. In an exactly similar way we shall 
 find that, with reference to the same figure 22, and with the sig- 
 nifications of j3 and S in that figure, as denoting the rays ob and 
 OD, while £ denotes the same upward vector as before, we may 
 write the equation 
 
 i3H-£=g; 
 
 for now the dividend-line j3 is in length double the divisor-line t, 
 and the length of the line § is double of the assumed unit of 
 length, so that 
 
 T/3-^T£=TS=2; 
 we have also the perpendicularities, 
 
 Sj_|3, S±e; 
 
 and the rotation round 8 from e to j3 is positive. 
 
 124. To test now the consistency of these results with other 
 principles, which we regard as being even more essential, and 
 which had in fact been laid down in the Second Lecture, as go- 
 verning generally the composition and tkcomposition of factions^ 
 before we proceeded to consider specially the case of rectangular 
 lines, let us resume the general conclusion of articles 50 and b^^ 
 namely, that in every such "analysis of faction," the " transfac- 
 tor divided by the factor gives the profactor as the quotient ;" or 
 in symbols, the formula, 
 
 7 -^ /3 = (7 -r- a) ^ (/3 -^ a), 
 
 where a, |3, 7 may denote any three rays in space. The identity 
 last written gives evidently this other equation of the same form, 
 
 (/3-^£)4-(a-^£)=/3-^a; 
 
 where a, /3, £ may be supposed to have the significations which 
 
136 ON QUATERNIONS. 
 
 were assigned to them in the foregoing article (123). But it 
 was shewn there that our plan for constructing the quotient of 
 two rectangular lines conducts to the two equations, 
 
 Substituting then these values for these two quotients in the 
 identity written above, we eliminate the symbol e, but introduce 
 -y and S instead, and arrive thus at this other equation, which 
 also ought to be true, 
 
 S -7- 7 = j3 -7- a. 
 
 Here then is a test whereby to judge of the consistency of our 
 principles, notations, and rules ; for we know by the Third Lec- 
 ture how to interpret an equation hetiveen quotients, such as the 
 one just now obtained ; and indeed that particular interpretation 
 had been perceived by others, or at least one partially agreeing 
 therewith had been so, before the quaternions were thought of. 
 And accordingly the test is home; for this very equation §-^ y = 
 j3 -^ a was shewn, in art. 103, to hold good, with reference to 
 figure 22, in the sense that the biradial (7, S) may be formed 
 from the biradial (a, j3) by merely turning the latter biradial 
 round in its own plane, and altering the lengths of its two legs 
 proportionally. 
 
 125. There are therefore at least tv/o essentially distinct 
 interpretations (without counting the distinction between ana- 
 lytic and synthetic views), which may thus be given, on our 
 principles, to the equation, 
 
 S -^- 7 = /3 -r a, 
 
 taken in connexion with the figure 22 of article 103 ; and which- 
 ever of these two we adopt, that equation is found to be true. 
 According to the interpretation which was given in that former 
 article itself, we analyze the lengths and directions of j3 and §, 
 by comparing them respectively with those of a and 7 ; we find 
 thus that while the line j3 is twice as long as a, S is at the same 
 time twice as long as 7 ; and that while j3 is advanced beyond a 
 by sixty degrees of azimuth, S is also advanced beyond 7 by the 
 same amount of rotation, in the same horizontal plane ; and 
 
LECTURE IV. 137 
 
 hence we infer that the quotients j3 -r- a and S ^ 7 are equal, be- 
 cause they correspond to one common relation of lengths, and to 
 one common relation of directions. Or if we regard the quater- 
 nions j3 -^a and 8-^-7 d& factors, then these two quaternions are 
 equal, because they have equal tensors and equal versors; namely, 
 in symbols, in the present example, 
 
 T(g-7) = T(/3-f-a)=2, 
 and 
 
 U(g^7) = U(/3--a)=At; 
 
 so that they answer to precisely similar acts of tension and of 
 version, performed respectively on o and on 7, in order to joro- 
 duce the rays j3 and 8. This is the frst interpretation (analytic 
 or synthetic) of the equation between the quotients j3-i-a and 
 S -i-7 ; it is the one which agrees most closely with views already 
 published, and which flows most naturally from the principles of 
 the foregoing Lecture ; and in adopting it, we have at the same 
 time (by the conception of a quaternion) an interpretation for 
 each quotient separately, which was alluded to at the close of ar- 
 ticle 105, and which involves only the consideration of a single 
 version (or angle), combined with that of a single tension (or 
 ratio), or the comparison of two rays yvhh each other. 
 
 126. But there is also a second interpretatioti of the equation 
 8 -r- 7 = /3 -7- a, or of the quotient S -7- 7 itself, which is suggested 
 by the process in art. 124, and is derived from general principles 
 respecting decompositions of factions, or of acts of tension and 
 version, combined with the construction in art. 122 for the quo- 
 tient of two rectangular lines, or with the earlier construction in 
 art. 82 for the product of any two such lines, as being itself 
 another line. According to this other interpretation, we consi- 
 der 7 and S as being themselves quaternions, namely quadrantal 
 ones, equivalent respectively to the two quotients a -^ £ and j3 -^ £ 
 of article 123 ; and then the act of dividing the line S by the line 
 7 comes to be considered as a particular case of the general ope- 
 ration of dividing one quaternion by another. In this view 7 is 
 2l factor, which operates on the line a as on what was called in 
 the Second Lecture afaciend, to produce what was there called 
 a factum, namely (at present) the line a ; j3 -r a is the profactor, 
 
138 ON QUATERNIONS. 
 
 which operates anew on a, as on a pi'ofaciend^ to produce j3 as a 
 profactum ; and ^ is the transfactor, which operates on the ori- 
 ginal subject £, as on a transfaciend, to produce immediately, by 
 a sort of short cut, or (technically speaking) by an act of trans- 
 faction, the same final result, namely the line j3, regarded now as 
 a transfactum. And then the result that /3 -f- a 2S thus the pro- 
 factor, or is found to be the agent in that successive act of faction 
 which, hy following the operation of 7 as a factor, produces, on 
 the whole, the same effect as that which is produced by S as a 
 transfactor, is precisely the result expressed by the equation 
 
 8 -^ 7 = /3 -^ a, 
 
 according to the second mode of interpretation above alluded to. 
 But we see that (even if we abstract for the moment from any 
 comparison of the acts of tension among themselves) this latter 
 interpretation of the division indicated by the symbol § -f. y in- 
 volves not merely (as at the close of article 125) the considera- 
 tion of a single version, namely the rotation from the ray y to 
 the ray 8, but the consideration and comparison oi three different 
 versions, or rotations, performed in three different planes ; 
 namely the version from £ to a; the pr over sion from a to j3 ; and 
 the transversion from e to /3. Yet we see that the results of these 
 two distinct interpretations harmonize, in the sense that each 
 conducts to one common quaternion, as the value of the quotient 
 S H- 7 ; and also that each conducts to the equation § -f- 7 = j3 -f- a, 
 under the conditions already supposed. All this may be illus- 
 trated by what was said in art. 76, respecting the double signifi- 
 cation of the equation 
 
 i xj = k, 
 
 as being the common expression for two distinct but connected 
 results. It may also be usefully compared with the still earlier 
 and more elementary remarks in article 57, respecting the double 
 view which may be taken of the arithmetical formula 
 
 6 --2 = 3; 
 
 as expressing at one time that on measuring a line = 6 a, suppose 
 a fathom, by another line = 2 a, suppose by a two foot rule, or on 
 
 \ 
 
LECTURE IV. 139 
 
 measuring any other concrete magnitude called 6, by a magni- 
 tude of the same kind, called 2, we find the number 3 as the re- 
 sult of this measurement, or as the quotient of this division ; and 
 as expressing, at another time, that if we analyze the act of sex- 
 tupling, so as to decompose this act into two other acts, oivihich 
 one shall be the act of doubling^ then the other component act'i?, 
 found to be the act of tripling. But it cannot be necessary, at 
 this stage, to carry these particular illustrations any farther, as 
 regards equations between quotients. 
 
 127. There is however one other test, which, although inti- 
 mately connected with the foregoing, it may still be satisfactory 
 to consider; and which will have, besides, the advantage of 
 tending to render us familiar with the geometrical signification 
 of a certain symbol, which frequently occurs in the applications. 
 I refer to the symbol 
 
 j3 -^ a X y, 
 
 in which a, j3, 7 are, for the present, supposed to denote some 
 three coplanar rays, that is, rays in or parallel to one common 
 plane, and which may be interpreted in either of the two follow- 
 ing ways : the test above alluded to being the coincidence be- 
 tween the results of these two distinct processes of interpretation. 
 
 I. We may determine di fourth ray g, in the same plane, or 
 parallel thereto, so as to satisfy the equation 
 
 S -T- 7 = j3 -^ a, 
 
 in the way which has been already fully explained (in art. 103, 
 &c.) ; and then, on substituting for /3 -f- a, the equal (\\xo\\er\t S -^7, 
 the symbol to be interpreted becomes (compare articles 40, 99), 
 
 j3-f-aX7 = g-r-7X7 = S. 
 
 II. Or we may turn about the tays a, j3, or others equal to 
 them, by one common amount of rotation in their own plane, 
 until a comes to be perpendicular to 7 ; after which it will 
 always be possible to determine a new ray e, perpendicular to 
 both a and 7, and such as to satisfy the equation 
 
 7 X £ = a, 
 
 with that interpretation of a product of two rectangular lines 
 
140 ON QUATERNIONS. 
 
 which was assigned in art. 82. We shall then have also the con- 
 nected equation 
 
 7 = a -h £, 
 
 with that connected interpretation of a quotient of two perpendi- 
 cular lines which was given in article 122. And on substituting 
 this value for y, in the symbol lately proposed for interpretation, 
 that symbol becomes (compare article 49), 
 
 j3 -f- a X y = (j3 -4- a) X (a -7- e) = j3 -^- £. 
 
 But £ being perpendicular to both a and y, by construction, is 
 necessarily perpendicular also to the ray j3, which is supposed to 
 be coplanar with those two other given rays; or in symbols, 
 
 £ J_ jS, because £ ± a, £ ± 7, and 3 HI a, 7, 
 
 if we agree to use the mark ||1 as a sign of coplanarity. 
 Hence the quotient j3 -f- £ may itself he interpreted, on the plan 
 of art. 122, as a certain determined line ^', which will evidently 
 be in (or parallel to) the plane of the given rays, because 
 
 if S' = ^ -^ £, then S' _L j3, and g'_L £, 
 
 so that the quotient S' is perpendicular to the line £, which is 
 itself perpendicular to that given plane. And by equating the 
 two foregoing values of the quotient /3-^£, we find for the pro- 
 posed symbol this second interpretation, or value, 
 
 j3 -^ a X y = g'. 
 
 128. Now the test to which it still remains to submit the 
 whole foregoing theory, as regards the consistency of its parts 
 among themselves, is to be applied by our examining whether 
 the line §', thus determined, coincides with (or is equal to) the 
 line S which was found above, by the other method of interpre- 
 tation, as being at least one value of the symbol j3 -^ a x y. 
 Have we or have we not (in the present questioii) the equation 
 
 for if not, we shall have not merely two different processes of in- 
 terpretation for the important symbol /3 -i- a x y under examina- 
 tion (which might not be, of itself, a disadvantage), but also two 
 
 \ 
 
LECTURE IV. 141 
 
 different values for that symbol, both equally valid on our prin- 
 ciples, and scarcely to be distinguished from each other by any 
 new care in the notations : which would produce an intolerable 
 confusion, or at least a very inconvenient ambiguity, occurring, 
 as it would do, in a symbol so elementary. And happily the 
 equation ^'= S is found, in fact, under the conditions above sup- 
 posed, to be true : so that the ambiguity does not exist. For the 
 equations 
 
 give 
 
 ^' -r- 7' = j3-f-a = S-f-'y; 
 
 but it has been shewn that the quotient of two given rays is a 
 given quaternion, and conversely that any essential change in 
 either of those two rays, the other ray remaining unchanged, 
 makes a real alteration in this quotient ; consequently the quo- 
 tients S' -4- 7 and S-i- y could not be equal, as we have just now 
 found that they are, if the rays S' and S were unequal, that is if 
 they diifered from each other either in length or in direction. 
 All this may be illustrated by a reference to figure 22 of arti- 
 cle 103, in connexion with the remarks which were made in the 
 more recent article 123 ; where, with the same significations of 
 the letters, the value of the quotient (5 ~- s, that is (by art. 127), 
 an equivalent for the line g', was found in fact to be §. 
 
 129. Thus the ttio methods of interpretation of the symbol 
 
 /3 -f- a X y, where y 111 a, /3, 
 
 conduct to one common result, namely to the determined line d; 
 although one of these methods introduces only the consideration 
 of a single rotation, namely that from a to j3, or from y to ^, 
 while the oMer introduces (as in 126) the consideration oi two suc- 
 cessive rotations, performed in two different planes, namely the 
 rotations from e to a and from a to j3, compounded together into a 
 third rotation in a tJdrd plane, namely the rotation from e to Q, 
 performed round S as an axis. And with respect to this value 
 of the above written symbol, or the length and direction of the 
 line S which thus satisfies the equation 
 
 /3 -^ a X y = g, 
 
142, ON QUATERNIONS. 
 
 or the proportion 
 
 a : /3 : : 7 : S, 
 
 by which that equation may be replaced, we see, first, that this 
 fourth line S is coplanar with the three given lines a, j3, 7, which 
 were supposed to be coplanar with each other. We see also that 
 its length is (in the old geometrical sense) di fourth proportional 
 to their three lengths ; so that, by art. 110, we may write the 
 following proportion between tensors^ 
 
 Ta:Tj3::T7:Tg. 
 
 We see too that its direction also is, in a certain modern sense {not 
 however peculiar to quaternions), a. fourth proportional to their 
 three directions ; meaning hereby that the rotations from a to j3 
 and from 7 to S are equal in amount, and similar in direction : 
 which relation, at least when combined with the two relations of 
 coplanarity, namely with the following, 
 
 7llla,/3, andglll«,/3, 
 
 may conveniently be symbolized in this calculus, by the follow- 
 ing proportion between versors, 
 
 Ua:U/3::U7:US. 
 
 Indeed this interpretation of the symbol j3 -r- o x 7, for the case 
 of coplanar lines, had been familiar to a certain class of thinkers, 
 and had been well known to myself, before the quaternions were 
 perceived, although some of the foregoing notations connected 
 with it are new. But on account of my having departed from 
 many other usages^ and having found myself obliged to give up 
 (as unsuited to my purposes) many other results, of those who 
 had thus speculated before myself, even as regards combinations 
 of lines in one plane, it became necessary, for the sake of clear- 
 ness, and even for the sake of logic, that I should explain dis- 
 tinctly on what grounds I retain the previously proposed signi- 
 fication of the symbol /3 -^ a x 7, as denoting a certain definite 
 fourth line 8, at least when the three given lines a, /3, 7 are in one 
 common plane : together with the equation j3 -7- a x 7 = 8, and 
 with the proportion a : |3 : : 7 : §. 
 
 \ 
 
LECTURE IV. 143 
 
 130. As additional examples of such signification, we may 
 remark that if, in fig. 25 (art. 119), we make 
 
 a = A-o, /3 = B-d, 7 = c-o, 
 
 we shall then have 
 
 S = j3 -j-axY=D-o; 
 
 and that, generally, the fourth proportional to any three rays of 
 a logarithmic spiral is (in length and in direction) ihoX fourth 
 ray of the same spiral, which is angularly related to the third 
 ray as the second is to the first. It is evident that whenever the 
 equation 
 
 S = j3-r-axy, orS-r-7 = /3-^a, 
 
 interpreted as above, holds good, we then have also the inverse 
 equation 
 
 y _i_ S = a -f- j3, 
 
 and the alternate equation 
 
 8 -1- j3 = y -f- a ; 
 
 results which may also be expressed as inversion and alternation 
 oi Q. proportion, and from which it follows (compare art. 99) that 
 
 /3 -^ a X y = y -T- a X /3, if 7 III a, /3, 
 
 the line S, above determined, being the common value of the two 
 members of this last equation, under this condition of coplana- 
 rity. We may also write more concisely (see art. 118), 
 
 g = |3a-i.y = ya-i.j3. 
 
 What happens when the three lines a, j3, y are not in nor pa- 
 rallel to any one common plane ; or in other words, what is to 
 be regarded as being the fdu?'th proportional to three lines not 
 coplanar, is a question which must be reserved for investigation, 
 at a stage a little more advanced. But at least we may already 
 see that in this more general and reserved case of non-coplana- 
 rity, the sought fourth proportional j3 -4- a x y, cannot (con- 
 sistently with the foregoing theory) be equal to any fourth 
 LINE 8 : for the equation g _i- y = j3 -f- a requires, by the princi- 
 ples already laid down, that the four rays compared should be 
 
144 
 
 ON QUATERNIONS. 
 
 coplanar, and by still stronger reason that the three rays a, /3, 7 
 should be so. In fact it was this very difficulty, respecting the 
 interpretation of the symbql j3 -^ a x y for the general case of 
 non-coplanarity which had pressed most upon my own mind, as 
 seeming to be insoluble upon known principles, before I was 
 led to conclude (what will soon be proved) that "Me Fourth Pro- 
 portional to three Lines which are not coplanar is generally a 
 Quaternion." 
 
 131. When the three lines a, j3, 7 are coplanar, the following 
 is a simple and somewhat neat construction, for that fourth line 
 8 which is then their fourth proportional. As there is never any 
 difficulty about the length, or tensor, of this fourth line, since we 
 have always the arithmetical equation, 
 
 TS = Tj3 -- Ta X T7, 
 we need only attend to the directioti or to the versor of S; and 
 in seeking this fourth versor, US, may dispose at pleasure of the 
 lengths or tensors of a, /3, 7, provided that we leave unaltered 
 their directions, or their three versor s Ua, Uj3, U7. It is ob- 
 vious also that a reversal of any one of these three versors, or 
 directions, merely reverses the direction of the result. Conceive 
 then that the three proposed lines a, j3, 7 are made the successive 
 sides of a triangle, bca, by some suitable changes of their 
 lengths, without any change in their directions, or at most with 
 simple reversions ; so that we shall have the values, 
 
 a=C-B, |3=A-C, 7=B-A, 
 
 with the relation 
 
 7 + /3 + a= 0. 
 
 Circumscribe a circle about this trian- 
 gle, as in Fig. 26 ; take the arc ad 
 equal to the arc ac, and prolong the 
 chord BD to meet in e the tangent to 
 the circle at a; take also on the same 
 indefinite tangent the portion af equal 
 in length to the portion ae, but lying Uq 
 to the other side of the point a of con- 
 tact. Or draw the chord bg parallel to 
 the tangent at a, and prolong the chord 
 
LECTURE IV. 145 
 
 GC to meet that tangent in f. Then if we denote by S and £ the 
 lines 
 
 g = F-A=:A-E, £==E-B, 
 
 we shall have not only the relation 
 but also the values 
 
 For it results from the similarity of the two triangles bca, bae, 
 and from the equality of ea and af, that the proportions 
 
 BC : CA : : BA : AE : : AB : af, and bc : ab : : ab : be, 
 
 hold good, even when the directions as well as the lengths of the 
 lines are compared ; that is, we have here the proportions between 
 vectors, 
 
 a : j3 : : y : S, and a : 7 : : 7 : e. 
 
 The curved arrows in the figure may assist the perception of the 
 relations between the directions of these lines ; and a student 
 might find it worth while to vary this figure 26, by supposing the 
 angle abc to be obtuse instead of acute, or by placing b between 
 a and c, leaving those two points unaltered in the figure. In this 
 new case, the chord bd would require to be prolonged through b, 
 in order to meet the tangent at a in a point which might still be 
 called E, but which would now lie at the other side of the point 
 of contact a, or at the same side as the old point f ; while the 
 new point f would thus come to lie at the same side of a as the 
 old point E. But the new triangles bca and bae would still be 
 similar to each other, and the requisite relations between direc- 
 tions, as well as between lengths, would still be found to hold 
 good. We should therefore still have the proportion between 
 four vectors, 
 
 c-b:a-c::b-a:f-a; 
 
 as also the following continued proportion between three vectors, 
 
 c-b:b-a::b-a:e-b; 
 
 although the positions of the points b, e, f would (as above ex- 
 plained) have, all three, changed together. And if the angle 
 
 L 
 
146 ON QUATERNIONS. 
 
 ABC were rights the only modification of the construction would 
 be that the points c and d would coincide. We may then enun- 
 ciate generally this result, which it will be found advantageous 
 to remember : " The Fourth Proportional to the three succes- 
 sive sides of a Triangle inscribed in a Circle is equal to a fourth 
 Line, which touches the circle at the corner of the triangle oppo- 
 site to ihejirst side." Or somewhat more fully, we may say 
 that the fourth proportional to the base bc and the two successive 
 sides CA and ab, of any plane triangle bca, regarded as three 
 vectors, is equal to di fourth vector af, drawn from the vertex a, 
 so as to touch, at that vertex, the segment bca of the circle 
 which circumscribes the triangle. In the figure 26 itself, this 
 segment does not contain the point d, and the tangential vector 
 AF touches the shortest (rather than the longest) arc of the circle 
 from A to c ; but if b were placed upo7i that shortest arc ac, as 
 in a recently suggested variation of that figure, the segment bca 
 would then contain the point d, and the required tangent at a 
 would take (as was above observed) the opposite direction, so as to 
 touch the shortest arc from a to d, rather than that from a to c. 
 In each case, however, in conformity with the last enunciation of 
 the rule for constructing the direction of the fourth proportional 
 AF, or d, or j3a"^. 7, to the three directed sides c - b, a- c, and 
 b - A, that sought direction of the line af may be found by the 
 condition of touching the segment bca, or of coinciding with the 
 initial direction of motion along the circumference, //'om a ifo b, 
 through c. If we had adopted the plan of determining the point 
 F from the point g, without employing e or d (namely, by draw- 
 ing, as above suggested, the chord bg parallel to the tangent at 
 A, and by prolonging the chord gc to meet that tangent in f), 
 the similar triangles to have been compared would then have 
 been the original triangle bca and the triangle acf: and the 
 figure might have suggested the proposed proportion under the 
 form 
 
 a :-7 : : -/3: ^; 
 
 which is in fact (see 130) a legitimate transformation of it, in 
 quaternions as in ordinary algebra. 
 
 132. All the remarks which have been made in the foregoing 
 
LECTURE IV. 
 
 147 
 
 article, so far as they regard only proportions of directed lines in 
 one pla?ie, depend (as it has been already stated) on principles 
 which are not peculiar to the theory of quaternions, but are com- 
 mon to some other modern systems also. Yet it appeared useful 
 to introduce them in this place; and before wq resume the con- 
 sideration of things peculiar to quaternions, it seems worth while 
 to mention here another construction, depending on the same 
 principles, and involving only (like the former) some elementary 
 properties of the circle, which construction serves to form a geo- 
 metrical representation for the fourth proportional to any three 
 coplanar lines, when directions as well as lengths are attended to. 
 Let the three given coplanar lines a, j3, y, to which we wish 
 to construct the fourth proportional j3a"^ .7, be conceived to be 
 respectively arranged as the second, first, and third sides, bc, ab, 
 CD of a quadrilateral abcd; and let it be at first supposed that 
 this quadrilateral is inscribed in a circle, as in figs. 27, 28. 
 
 Fig. 28. 
 
 Draw the chord be parallel to the fourth side da, and prolong (if 
 necessary) the new chord ce, to meet this side da in r ; and de- 
 note the line df by S, so that 
 
 a = C - B, /3 = B-A, 7=0-0, g=F-D. 
 
 Then by the similar triangles cba, cdf, and by the curved arrows 
 in the figures, we have the required proportion, 
 
 c-b:b-a::d-c:f-d, ora:j3::7:8; 
 
 so that the line df or § is the sought fourth proportional, or is 
 
 L 2 
 
148 ON QUATERNIONS. 
 
 the result obtained when the first side ^ or ab of the inscribed qua- 
 drilateral 1% divided hy the second side a or bc, and the resulting 
 quotient or quaternion, /3a~^, is then multiplied as a factor into 
 the third side y or cd. And accordir)g as the inscribed quadrila- 
 teral ABCD is an uncrossed one (as in fig. 27), or a crossed one 
 (as in fig. 28), we see that this resulting line S is in the direction 
 opposite to the fi)urth side da, or in the direction of that fourth 
 side itself. And if for greater generality the third of the given 
 lines be now supposed longer or shorter than the third side cd of 
 the quadrilateral inscribed in the circle abc, or even opposite in 
 direction to that side, we may still conceive it placed so as to 
 begin at c, and may represent it by 
 
 7 
 
 = D - C 
 
 and then by drawing from it?, final point d' a parallel to ad or to 
 BE, so as to meet the old chord ce in a new point f', we shall 
 find a new line 
 
 S' = f'-d', 
 
 as in the same figs. 27, 28, which will be the new fourth propor- 
 tional sought, or will satisfy the equation 
 
 For example, in fig. 27, if a be the intersection of the lines cd 
 and BE, then ge is, in length and in direction, the fourth propor- 
 tional to BC, AB, and cg. 
 
 133. The same principles give easily, as has been seen, a 
 simple construction for the third proportional to any two directed 
 lines, such as a and y in fig. 26 (art. 131); and the inspection 
 of the same figure shews easily, as was to be expected, that the 
 line € so found is the third proportional also to a and-y; for in 
 that figure it is evident that 
 
 c-B :a-b : :a-b:e-b. 
 
 But it is important to observe that when we have thus a conti- 
 nued proportion between three vectors, 
 
 a : y : : y : £, or a : - y : : - y : s, 
 
LECTURE IV. 149 
 
 we must not in quaternions write generally, as in ordinary alge- 
 bra, an equation between square and product, such as 
 
 7^ = as, or y^ = ea', 
 
 for y^ is, in our system (see art. 85), a negative scalar, while as 
 and ea are in general (by arts. 89, 91) two conjugate quaternions, 
 of which neither reduces itself to a scalar, positive or negative, 
 unless the vectors n and e have coincident or opposite directions. 
 This new departure from ordinary usages (from which it may be 
 noticed that 1 aim at departing as seldom as I can), arises from 
 that fundamental pecw/m/^Y?/ of quaternions whereby they, and 
 even the vectors which they involve, are not generally commuta- 
 tive as factors (arts. 74, 82, &c.) In fact if we could infer gene- 
 rally the equation 'y^= ae, from the continued proportion between 
 three vectors a : 7 : : 7 : e, then since this proportion may be in- 
 verted (art. 130), or written thus, e : 7 : : 7 : a, we should be equally 
 well entitled to conclude the equation 7^ = 60, and therefore also 
 ia = ae ; which (as a general inference) would contradict the non- 
 commutative principle, respecting the multiplication of vectors. 
 It is therefore satisfactory to know, what is easily shewn on our 
 principles, that the continued proportion above supposed, between 
 three vectors a, 7, £, gives still, as in ordinary algebra, and as in 
 those other and more modern systems also to which allusion has 
 been made, the equations, 
 
 ea"^ = (7a"^)^, oe"^ = (76"^)^ ; 
 
 provided that we retain in quaternions, as the definition of a 
 square, or second power, the formula 
 
 which will agree with what has been already laid down respect- 
 ing the squares or second powers of vectors. In fact if we make 
 
 q = -fa^, ovqa = y, 
 we shall then have 
 
 q'^a = q>^qa = ya~^ ' y = t = ia'^ . a, 
 
 and therefore 
 
150 ON QUATERNIONS. 
 
 134. Conversely, by an introduction of the notion of the 
 power of a quaternion, with an exponent = ^, which includes what 
 has been shewn respecting such a power of a vector, I should still 
 write generally, 
 
 ya"^ = ± (£a"^)i, when a : 7 : : 7 : e ; 
 
 although I am not at liberty to write generally, under the same 
 condition of proportionality, the equation 
 
 7 = + \/(a£), 
 
 as might be done in commutative algebra. Thus the mean pro- 
 tional 7 between any two proposed vectors, a and e, is not (with 
 me) equal generally to the square root oj^ their product ; but ?*/* 
 this mea7i 7, and the third vector e, be each divided by the^r*^ 
 vector a, the former of the two quotients (or quaternions) so ob- 
 tained is still (as in algebra) a species of square-root of the 
 latter. And accordingly I write, as an expression for this mea?i, 
 the formula 
 
 7 = + (fa"^)* a ; 
 
 where, to remove generally the ambiguity of sign, I may here 
 state that I take the upper sign (+) when 7 has the direction of 
 the bisector of the angle between the directions of a and e; but 
 the lower sign (-), when, as in figure 26, 7 has the opposite of 
 that direction. And when I have occasion to speak definitely 
 of THE MEAN pvoportional between two given vectors a and e, I 
 adopt then the upper sign in preference, or take the bisector it- 
 self of the angle between the two extremes, in preference to the 
 opposite of that bisector. There is thus only owe case left, in 
 which the direction of the mean remains ambiguous, or rather in- 
 determinate, if the directions of the extremes be given, namely, 
 the case when those two given directions are opposite to each 
 other: for then the resulting symbol, suppose 
 
 -y := (_ x^a . a'^y^ a, or 7 = (- x -y a, 
 
 where x represents some positive scalar, may on the foregoing 
 principles, denote any line 7 which satisfies the two conditions, 
 
 T7=a;To, 7X0; 
 
 \ 
 
LECTURE IV. 151 
 
 SO that THIS MEAN y may have any direction in a plane per- 
 pendicular to a. Accordingly it is evident that the third pro- 
 portional to any two rectangular vectors is a third vector with a 
 direction opposite to the first, whatever the plane of the two vec- 
 tors may be. It is obvious also that the third proportional to any 
 two parallel vectors is a third vector, whose direction coincides 
 with that of the first given vector. And there can be no diffi- 
 culty in perceiving (what indeed does not depend on anything 
 peculiar to quaternions) that the mean proportional between any 
 two rays of a logarithmic spiral, at least if they be taken, for 
 simplicity, as belonging to one common semispire, is constructed, 
 in length and in direction, by that other ray of the same half- 
 spire which bisects the angle between them. 
 
 135. It is natural to interpret, on the same general plan, the 
 symbol 
 
 (/3-^a)^xa, or(/3a"^)^a, 
 
 as denoting the first of two mean proportionals (in length and in 
 direction), inserted between the two lines a and j3; the second of 
 these two mean proportionals, thus inserted, being denoted by 
 the analogous symbol, 
 
 (/3 -r- a) 3 X a, or (/3a "^)^ a. 
 
 For example, if a and j3 should be chosen so as to denote the 
 rays oa and od of the logarithmic spiral in fig. 25 (art. 119), 
 then the two means, symbolized above, would be the two inter- 
 mediate rays of the same spiral, ob and oc. In symbols, the 
 two means between i andj y/8 are hi i \/2 and 2 k^i. (Such is 
 at least the simplest pair of means between the given extremes; 
 for we shall soon see that is possible, although in a less simple 
 way, to insert other pairs.) Indeed this notation is, so far, con- 
 sistent with the principles of other systems also ; but it is impor- 
 tant to observe that in our system of notation we must not de- 
 note these two means between a and ]3 by the symbols 
 
 which would, in common or commutative algebra, be merely 
 transformations of the foregoing ; whereas they denote, on the 
 
152 ON QUATERNIONS. 
 
 principles of the present theory, no two lines whatever, unless 
 the directions of a and j3 should happen to coincide, but two 
 QUATERNIONS, of which the tensors and versors shall be assigned 
 hereafter. Meanwhile it is clear that since (by what precedes), 
 
 (/3 -^ a)i = y -^ a, (/3 -f- a)^ = y' -^ a, 
 
 if -y, j' denote the two means above considered, so that 
 
 a : 7 : : 7 : 7 : : 7 : /3, 
 
 the powers of any proposed quaternion j3 -f- a with the exponents 
 ^ and f, or in other words the cube-root of /3a"^ and the square 
 of that cube-root, are generally themselves quaternions ; whose 
 tensors are the corresponding powers of the tensor of the given 
 quaternion, 
 
 T . (/3a-i)* = (T . jSa-i)* == (T/3 -^ Ta)^, 
 T.(j3a-i)* = (T.j3a-i)t=(T/3^Ta)t; 
 
 while the axes of the new versors are the same with the axis of 
 the given versor of j3a"S and the angles of those versors are re- 
 spectively equal to one third and to two thirds of the given angle 
 between a and j3 : so that we may write, with reference to the 
 versors, in consistency with former results, 
 
 U . (/3a-i)i= (U . f3a-0* = (Uj3 ~ Ua)i, 
 
 U.(/3a-i)* = (U.j3a-i)f=(U)3^Ua)^, 
 
 and also, with reference to the angles, the equations, 
 
 Z.(i3a-i)* = ^Z(/3a-i), 
 
 136. More generally we may now interpret the symbol q*, or 
 the POWER OF A QUATERNION q, with any scalar exponent t, 
 as denoting a neiv quaternion, of which the tensor and the ver- 
 sor are respectively the same {t"') powers of the tensor and ver- 
 sor of the old or given quaternion ; in such a manner that we 
 may write, generally (compare art. 116), 
 
 T.q^=(Tqy^Tq^; 
 
 \J .q' = i\Jqy=Uq'; 
 
 the points and parentheses being omitted in these last symbols, 
 
LECTURE IV. 153 
 
 Tq* and \Jq% 
 
 as being not required for ike prevention of ambiguity . The ten- 
 sors being simply positive or (more properly) signless numbers 
 (by articles 63, 113), their powers are to be formed by the or- 
 dinary rules of algebra, or r'dther of aritlmietic. And with re- 
 spect to the formation of powers of versors, or the interpreta- 
 tion of the symbol Uq\ it is natural to consider each such power 
 as being a new versor, which has the effect of turning any line 
 a, in a plane perpendicular to the axis ofq, through an angle, or 
 an amount of rotation round that axis, which is represented by 
 the product 
 
 tx iq ; 
 
 the rotation being right-handed or left-handed, according as this 
 product is a positive or a negative number. All this is evidently 
 consistent with, and includes, what has been already laid down 
 respecting powers of vectors, or oi quadrantal versors {\x\ articles 
 86, 115, 116, &c.) ; and it enables us to write, in the calculus of 
 quaternions, as well as in ordinary algebra, the formula, 
 
 where m and n are any positive or negative whole numbers^ or 
 zero. For example, we have the identities 
 
 q -q'^ =q'^ q = q^~^ = (f = 1; 
 
 so that (compare arts. 44, 117), we may call the power §'"i, 
 with negative unity for its exponent, the reciprocal of the qua- 
 ternion q. We have also, for any such ivhole values of m and n, 
 the usual algebraic identity, 
 
 But before we can decide whether these two last formulse (with 
 m and n) are true generally iov all scalar values of the expo- 
 nents m and w, mc\\\^\ng fractions and incommensurables, we 
 must consider more closely, and define more precisely, than has 
 yet been done, what is to be understood in general by the angle, 
 or AMPLITUDE, Z g", of a quaternion, or of a versor. 
 
 137. It will be remembered that whenever we have supposed 
 that an equation of either of the two following forms, 
 
154 
 
 ON QUATERNIONS. 
 
 q = (3-~-a, or q X a = (5, 
 holds good, we have always conceived (see arts. 40, 90, &c.) 
 that the quaternion q, regarded as a metrographic operator, 
 produces the complex (metrographic) f^ec^ of changing j^z-sHhe 
 length of a to the length of j3, according to the rule expressed 
 by the formula (compare art. 110), 
 
 Tq X Ta == Tj3 ; 
 and also of changing, secondly, the direction of a to the direction 
 of /3, as is expressed by this other formula (compare art. 113), 
 
 U^x Ua=U,8: 
 
 and this change of direction, of the line a thus operated upon, 
 has been always supposed to be accomplished by a rotation in 
 THE PLANE of the two ruys a and j3, round an axis perpendicular 
 to that plane, but coincident with (or parallel to) the axis of the 
 operatijig quaternion q. Now it is evident that if we only care 
 for obtaining, in some way, the direction of ihefnal ray j3, re- 
 garded as the RESULT of such a rotation, we may add (or sub^ 
 tract) ANY WHOLE NUMBER OF COMPLETE REVOLUTIONS (per- 
 formed in the same plane) ; because each such revolution, forward 
 or backward, restores the direction of the revolving line or 
 ray. For example, a rotation through + 60° in any plane is equi- 
 valent, as far as regards only its final effect, to a rotation 
 (round the same axis) through + 420°; or through - 300° ; or 
 through + 780°, &c. Conceive then that we wish to form, on 
 the general plan of the foregoing article (136)', the power q^ with 
 exponent I of the versor q = j3a'^, where a and j3 shall be sup- 
 posed to denote, as in fig. 29, two coinitial sides oa and ob 
 of an equilateral triangle aob in a 
 horizontal plane, the side oB lying 
 towards the right hand, with refe- 
 rence to the side oa. If we select, 
 for the present pair ot rays, the sim- 
 plest value for the angle between o 
 them, and the one which agrees best 
 with ordinary geometry, and with 
 the analogy of former articles, 
 namely, the following value of the 
 
 Fig. 29. 
 
 +280' 
 
 100 
 
LECTURE IV. 155 
 
 rotation (round an upward axis) from the direction of a to that 
 ofjS, 
 
 Z.q = L((5^a) = + 60°, 
 
 the general expression in article 136 for the amount of the rota- 
 tion performed by the power g\ considered as a new operator 
 on a, will then supply us with the following value for this new 
 rotation (round the same upward axis) : 
 
 txLq = ix{+60°)= + 20°. 
 
 We shall thus be led to write the equations 
 
 where c is conceived to denote the point on the circumference 
 AB (with the origin o for centre), which is advanced by 20° be- 
 yond the point a in the order of right handed rotation; and this 
 result will agree perfectly with article 135, because it will give 
 the ray y as the first of two mean proportionals, y and j', inserted 
 between a and j3 ; so that 
 
 a : y : : 7 : 7' : : 7' : /3, where 7' = c' - o, 
 
 c' being the^wa/ point of a positive arc of 40°, measured from 
 the point a of the circumference, which latter is assumed as the 
 initial point : the Jour rays, 
 
 A-o, c - o, c'-o, B-O, 
 
 thus forming, by their directions, a continued proportion. 
 
 138. But we might also, although less simply, have supposed 
 that after turning the radius oa, as above, through an angle of 
 60°, and so bringing it to coincide with the position of ob, we 
 then continue the rotation through an additional and complete 
 revolution, passing successively through the points de', ed', acc' 
 in the figure, and thus returning to the position ob again. 
 And if we adopt this supposition, respecting the amount of rota- 
 tion performed, that is, if we suppose it to be =+ 420°, we shall 
 then have, by the general formula of art. 136, the following value 
 for the corresponding rotation effected by the required po^^-e/- ^^ : 
 
 <x^:g = ix(+420°)= + 140°. 
 
156 ON QUATERNIONS. 
 
 In this manner we shall be led to consider the point d in the 
 figure, namely, the termination of a positive are of 140° from a, 
 together with the connected point d' which is the termination of 
 the same arc doubled, as the extremities of two new rays, 
 
 g=:D-o, andS'=D'-o, 
 
 which are, although in a less simple sense than before, two mean 
 proportionals inserted between a and |3, and satisfy the conditions 
 of the formula, 
 
 a : S : t S : S' : : S' : /3 ; 
 
 while the first of these two new means satisfies also, in the same 
 sense, the equations, 
 
 g*a = g, (/3a-i)i = ga-i. 
 
 139. Or again we might conceive ourselves as passing from 
 a to ]3, or from a to B, by a rotation in the opposite order, through 
 the points d'e, e'd of the figure ; which new rotation would thus 
 be expressed by the symbol - 300° : and then the general formula 
 of art. 136 would give, for the rotation caused by the operation of 
 the sought j»OM;er q^ of the versor q, the value 
 
 txLq = ix (-300°) =-100°. 
 
 And thus we should be led to consider the two new points e and 
 e' in the figure, which are the extremities of two negative arcs 
 from A, namely, arcs of 100° and 200° respectively, measured in 
 an order opposite to that adopted in recent articles, in fact if, 
 after finding these two new points (or at least conceiving them 
 to be found), we write 
 
 £ = E - o, £' = e' - o, 
 
 we shall have the new continued proportion, 
 
 and shall be led to write, in connexion therewith, the new equa- 
 tions, 
 
 q^a = £, (j3a"^)^= ea'^ 
 
 140. And NO new variety of positions for the line q^a would 
 be obtained hy any further addition or subtraction of revolutions, 
 
LECTURE IV. 157 
 
 in estimating the amount of the rotation from a to j3; because a 
 change of three such revolutions, in the estimate of that rota- 
 tion, produces merely a change of one complete revolution when 
 we come to trisect the whole angle (or at least to conceive it as 
 trisected), or to multiply Lq by the given exponent ^. For ex- 
 ample, if, instead of treating the rotation from a to j3 as being 
 = the negative arc - 300° (as in the preceding article), we were 
 to treat it as equal to the positive arc + 780°, which is greater by 
 three circumferences, we should be led, by the supposed trisec- 
 tion, to conceive an arc of + 260°, which would still conduct us 
 from A to E (in fig, 29), although by an order or direction of ro- 
 tation^ opposite to that which was considered in the foregoing 
 article.- 
 
 141. It appears then that ifyse allow the symbol 
 
 L q, or Z (/3 -^ «), or Z (/3a"i), 
 
 to represent not merely 60° (in the example recently discussed), 
 but any one of the angles or rotations which differ from this by 
 multiples of 360°, the power qi, or the cube-root of the quater- 
 nion q, may represent, or be interpreted as being equal to, any 
 one o/"three distinct quaternions; namely (with the recent 
 significations of the letters), by arts. 137, 138, 139, any one of 
 the three following : 
 
 (j3a"')^=7a"\ ov=Sa'^, or = ea"^; 
 
 but not (by art. 140) any other quaternion, distinct from these. 
 In fact if we define the cube or the third power of a quaternion 
 by the formula 
 
 which agrees not only with common algebra but with the general 
 definition of ^* in art. 136, we shall have, in the recent example, 
 the equations. 
 
 In short, we reproduce here, by this way of viewing the subject, 
 precisely that kind and degree of multiplicity of value, 
 which is so well known to present itself in the ordinary algebra 
 of imaginaries, and indeed in some known systems of geometrical 
 
158 ON QUATERNIONS. 
 
 interpretation also, in connexion with the roots of unity : 
 although it was necessary, iox the purpose of a logical developement 
 of the present theory, that I should not assume, without a new 
 and independent investigation, so important an element of any 
 other system, with which the principles of the Calculus of Qua- 
 ternions come on some points into opposition. It would not have 
 been a legitimate process for me to have borrowed, without in- 
 quiry, the Theorem that " three distinct and unequal ex- 
 pressions (as here ya'^, Sa'^, and ea"^) may have one common 
 CUBE," from any system of calculation in which the order of two 
 factors is supposed to be generally indifferent to the result ; nor 
 from any system of interpretation^ in which the three distinct 
 cube-roots of one common expression (as here of |3a"^) are sup- 
 posed generally to represent three lines, having directions in one 
 plane, instead of representing (as with me) three quaternions. 
 142. Had the exponent t denoted any other fraction, 
 
 n 
 m 
 
 where m is supposed to be prime to n, so that the fraction t is 
 expressed in its lowest terms, there would have been no difficulty 
 in proving, in like manner, what is analogous to known results 
 in other systems, that m distinct quaternions, that is, as many as 
 there are units in the denominator of the fractional exponent t, 
 might all be considered as values of the t"'- power of any pro- 
 posed quaternion q, or as included amo7ig the different interpre- 
 tations of the symbol q^ ; provided that in calculating the rota- 
 tion denoted (see 136) by the general expression 
 
 tx Lq, 
 
 we still allow (as was lately done) the symbol Z^' to denote any 
 one of those infinitely many angles, or xdithex amounts of rota- 
 tion about a given axis, which can be formed as above, by addi- 
 tions or subtractions of circumferences, or complete revolutions. 
 For example, the' square-root qi of a given quaternion q would, 
 on this plan, be found to have in general two values, of which 
 one would however be merely the negative of the other, or might 
 be formed from that other by multiplying it by - 1 : which re- 
 
LECTURE IV. 139 
 
 suit is seen, of course, to bear the closest possible analogy to 
 algebra. And if the exponent t were incommensurable, the 
 values of the power c^ would then, on the same "plan, be found 
 to be INFINITELY MANY. But a power of a given quaternion, 
 with a given whole number for its exponent, such as the square, 
 cube, reciprocal, &c.j is still, even on this plan, itself a deter- 
 mined quaternion ; in the sense that by operating as a factor 
 on any given line, in a plane perpendicular to its axis, it produces 
 a determined line in that plane as the result. 
 
 143. If then we were to adopt the plan mentioned in the last 
 few foregoing articles (137, &c.), for estimating the angle of a 
 quaternion, whereby the symbol L q for that angle, or for that 
 rotation, should not be confined to its simplest and most geo- 
 metrical value or signification, as denoting generally some 
 acute, or right, or obtuse angle, such as are treated of in Euclid's 
 Elements, and which for the moment we may here denote by this 
 other symbol q : we might then write generally 
 
 Aq = q + 2l7r, 
 
 where / is employed as a sign for any positive or negative whole 
 number, or zero, and the angular value of tt is (as usual) 180°, 
 And then, on the same plan, we might write (see art. 136), 
 
 Z (q') = tx{q +2lTr) + 2l'7r = t.q+2{lt + l')7r, 
 
 where I' denotes any new whole number, whether positive or 
 negative or zero. In the same manner we might write 
 
 Z (§-") =u.q+2 {mu + m) tt ; 
 
 where m and m would be allowed to denote any new pair of 
 whole numbers ; the new exponent u, like t, being still supposed 
 to be scalar ; but being still allowed, like it, to become fractional 
 or incommensurable. And if we seek, on the same plan, the 
 angle of that other power of q, which shall have u + t (or t + ii) 
 for its exponent, we find this other expression, 
 
 Z . g'»+'^ = (m + ^) ^ + 2 jo (^^ + TT + 2 p' TT, 
 
 where p and p are two new arbitrary integers. 
 
160 ON QUATERNIONS. 
 
 144. This being perceived, there can be little or no difficulty 
 in seeing that because generally the multiplication ofversors cor- 
 responds in the theory of quaternions to the composition of ver- 
 sions (see art. Q5)^ and because the axes of the rotations answer- 
 ing to the powers (f and (^ may be regarded as coinciding with 
 the axis of the base, or with that of the given quaternion q, we 
 may form (on the present plan) a general expression for the 
 angle of the product of two powers, 
 
 by adding the two separate expressions (found as above) for the 
 angles of the factors, and afterwards admitting or introducing a 
 term which shall be some multiple of a circumference. In this 
 way we should be led to infer that 
 
 L {q^ X q*) = {u + t) q +2 {It + mu + n) tt, 
 
 where n denotes some new positive or negative whole number or 
 zero : provided that in interpreting the symbol for the angle of 
 the product we allow every value of the one factor power to be 
 combined with eve}-g value of the other. 
 
 145. Comparing now the results of the two foregoing articles, 
 we find that in order to justify our establishing the following 
 equation, 
 
 where the exponent of the product is represented as being equal 
 (as in arithmetic) to the sum of the exponents of the factors, 
 we must endeavour to select the five whole numbers /, m, 7i,p,p"in 
 such a way that the part independnet of g, in the difference of 
 the angles of the two equated quaternions may either vanish, or at 
 least be equal to some multiple of the whole circumference ; or that 
 the coefficient of 27r in the expression of this difference may be 
 equal to some whole number p", whether positive or negative, or 
 zero ; since otherwise the two compared quaternions would not 
 be equal, because they would give unequal vectors as the results 
 of their operating as versors on one common vector, perpendicu- 
 lar to the axis oiq. In this manner we are led to the condition, 
 
 p {t + u)- {It + mu) + p -n= p" ; 
 
LECTURE IV. 161 
 
 or more concisely, 
 
 (p- I) t+ (p-m)u = n, 
 
 n denoting some new whole number which may be chosen at 
 pleasure. 
 
 146. Now without entering here into a minute discussion of all 
 the CASES which may arise from varieties of selection of the sca- 
 lar exponents t and u, it may suffice to observe that for general 
 and INCOMMENSURABLE valucs of those two scalars, not con- 
 nected by any relation with each other, the condition recently 
 written can be satisfied only by supposing that p-l, p-m, and 
 n all separately vanish ; or by our establishing the equations, 
 
 p = l=:m, and n = 0. 
 
 For example, if we assume 
 
 we shall find that the equation 
 
 a\/2 + b-^/3 = c 
 
 cannot be satisfied by any scalar and whole values of a, b. c, dis- 
 tinct from zero. We are therefore led to conclude that the pro- 
 duct of the two powers q* and q^ will not generally (on the 
 present plan) be equal to that other power g^'^*, of which the 
 exponent is the sum of the exponents of the factors, unless the 
 three whole numbers, denoted above by I, m,p, are equal to each 
 other; that is, unless, in forming the three powers, 
 
 by the three multiplications (see art. 136), 
 
 t /.q, u Lg, {u + t) Lq, 
 
 we assign one common value, such as 
 
 Lg = q + 2/7r, 
 
 to the angle of the base, or to the amount of the rotation 
 which is conceived to be produced by the operation of the qua- 
 ternion q. But if, conversely, we do thus choose m = l and/? = Z, 
 that is, if we do thus assign one common value to L g, in form- 
 ing the three powers to be compared, we shall then have 
 
 M 
 
162 ON QUATERNIONS. 
 
 p (t + u) =lt+ tnu, 
 
 independently oft and u; and the expression for the angle of the 
 product, assigned in art. 144, can only differ from the last ex- 
 pression in art. 143 by some whole multiple of the circumference. 
 And therefore, even if the quaternion q were not a simple vei'sor, 
 but had a tensor different from unity, we should be able to infer 
 from this supposed ^^r?'^?/ of its angle L q^ for any two scalar ex- 
 ponents t and Uy the equation 
 
 q^(f = f^\ 
 
 which was proposed for investigation near the beginning of the 
 foregoing article; and also, under a slightly different form, 
 towards the end of article 136. 
 
 147. With respect to the equation 
 
 which also was proposed for investigation in the place last re- 
 ferred to, the exponents t and u being still scalar, but otherwise 
 general, if we adopt, for the angle oicf, the value assigned in art. 
 143, we shall have, on the plan of that article, the expression 
 
 L . {qj' = Ut . q+2 {lut + I'u + I") TT, 
 
 where /, l\ I" are any three whole numbers. And on the other 
 hand we have, on the same plan, 
 
 / . q^'^ = ut .q +2 (inut + m) it ; 
 
 where m and fn denote some two new whole numbers. Equating 
 then the difference of these two last angles to a multiple of the 
 circumference, we find, as the condition for the correctness of the 
 equation above proposed, 
 
 {l-m) ut + l'u-m', 
 
 where m' is a new whole number, which may be chosen at plea- 
 sure. But because the scalar exponents u and ut are supposed 
 to be generally incommensurable, and not to be connected with 
 each other by any such relation as the foregoing, we can only 
 satisfy the recent condition by supposing that we have separately, 
 
 m = /, and Z' = 0. 
 
 \ 
 
LECTURE IV. 163 
 
 We are therefore still to suppose the angle of the original base 
 q to be FIXED, as in the immediately foregoing article; or to con- 
 ceive that one common value of Z q is employed, in forming the 
 
 two powers, 
 
 (^ and g'". 
 
 But besides this supposition, which answers to the condition 
 m = L the other condition recently found, namely, the equation 
 /'=0, shews that in proceeding to form the power (9*)" from the 
 power q* as a base, we must in general retain that value of the 
 angle of q^ which is immediately given by the rule of art. 136, 
 namely, the value (compare 143), 
 
 L . q* = tv. L q= t {q ^r'iy); 
 
 and must not (generally) add to this value any multiple (different 
 from zero) of the whole circumference, such as the multiple 2 I'tt 
 which was added in art. 143, before proceeding to multiply by 
 u; at least if we desire to obtain generally a neiv power (</*)", of 
 the intermediate base q*, which shall be equal to the power q^^* of 
 the given quaternion q. 
 
 148. But on reviewing the whole investigation contained in 
 the eleven foregoing articles (137 to 147), it appears to me 
 that you are likely to admit that although it was perhaps useful 
 thus to study j^r a while son>e of the ways in which the theory 
 of Quaternions might have led to symbols with multiple 
 VALUES, analogous to the known roots of unity (compare art. 
 141) ; yet it may now be desirable, with a view to simplicity 
 and clearness in owx future researches, that we should call in the 
 aid of definition to j^rrc, as precisely as we can, which one sig- 
 nification, or value, out of all the possible values or interpreta- 
 tions recently considered, we shall hereafter choose to adopt, in 
 preference to all the others, and indeed to ihau future exclusion^ 
 in the further developement of this Calculus. And I conceive 
 that we cannot better attain this object, than hy adopting hence- 
 forth expressly what has indeed been often adopted already, at 
 least tacitly and by anticipation, in earlier articles of these Lec- 
 tures, namely, the simplest value of the angle of any proposed 
 quaternion q, or in other words the one which most conforms to 
 ordinary geometrical usage ; that is to say, an angle in the first 
 
 M 2 
 
164 ON QUATERNIONS. 
 
 positive semicircle: and by regarding this as the value of the 
 symbol Z q. This comes in the notation of art. 143, to suppos- 
 ing that / is zero, or to establishing generally the equa- 
 tion, 
 
 or (more fully), it comes to assigning the limitations^ 
 
 ^ §■ > 0, < TT, 
 
 where > and < are, as usual, signs for " greater than" and " less 
 than" (compare art. 113); which will dispense with the future use 
 of the recent symbol q ., and will allow us to consider the prefixed 
 mark Z as being (in quaternions) the characteristic of a cer- 
 tain definite operation, which maybe called the operation of 
 taking the angle of any proposed quaternion. And the sym- 
 bol Z q will thus denote, with us, henceforth^ simply an acute or 
 right or obtuse angle, such as Euclid usually treats of, to the 
 exclusion oi negative values, and of values greater than two right 
 angles : although null angles, and angles equal to two right 
 angles, shall still be admitted as limits. 
 
 149. It was thus that (in art. 77) we regarded unit-vectors, 
 such as i,j^ k, &c., as being simply quadrantal versors, and not 
 as operating to turn a perpendicular line through five nor nine 
 positive quadrants, nor through three nor seven negative quad- 
 rants, &c., round the given unit- vector as an axis : and that 
 accordingly we regarded (in art. 86) the symbol t* as denoting a 
 versor, which turns a line k, perpendicular to t, through a def- 
 nite amount of rotation, and in a definite direction, which were 
 expressed (in quantity and sign) by the product ^x 90°. It was 
 thus, again, that (in art. 116) we interpreted more generally 
 the symbol p* as denoting a quaternion, which multiplies the 
 length of a line o- perpendicular to the base-line p by the tensor 
 T/o*, and turns that perpendicular line a round p as round an 
 axis, through the same definite rotation t x 90° as before, butMo^ 
 generally through any of the following odd multiples thereof, 
 
 -3^x90°, + 5^x90°, &c.: 
 
 which came to establishing the equation 
 
LECTURE IV. 165 
 
 as holding good for evej'y vector p, to the exclusion of the less 
 simple values^ - 270°, + 450°, &e., which the angle L p of the vec- 
 tor might otherwise have been supposed to receive, when this 
 vector p is regarded as being in part a versor also. It was thus, 
 once more, that (in art. 134) we proposed to remove the ambi- 
 guity of sign in the expression for a square root of a quaternion, 
 by interpreting the symbol (ta'^)^ as equivalent generally to one 
 definite quotient, such as r\a^ ; where the symbol rj (not expressly 
 introduced in 134) denotes that definite vector which bisects the 
 (acute or right or obtuse) angle between a and e, and not the op- 
 posite of that bisector (in fig. 26 the line -y, and not the line + j). 
 And a leaning towards the same view may have been observed 
 in art. 135, and in other earlier articles. But I now propose to 
 FIX it, by DEFINITION, as what 1 shall henceforth always adopt, 
 in these Lectures, unless and until some special notice shall be 
 given, of the temporary adoption of any other and less simple mode 
 of estimating the angle of a quaternion, and of calculating its 
 powers thereby. And then the power q\ so calculated, by com- 
 bining this value of L q with the rule in art. 136, will be always 
 A DETERMINED QUATERNION, if the quatcrnion q and the scalar 
 exponent t be themselves determined: with the single exception 
 of that limiting case (to be afterwards more closely considered), 
 where the base q becomes a negative scalar, by its angle taking 
 the limiting value, 
 
 Lq = ir; 
 
 in which case the axis of the power (like the axis of the base) has 
 an entirely indeterminate direction; although the angle of the 
 power will still have a determinate value. 
 
 150. From the fixity of value which we have now assigned 
 to the symbol Z q, when q is any fixed quaternion, we may see 
 at once, by the considerations of art. 146, that the formula 
 
 q q = q J 
 
 which was lately proposed for discussion, does in fact hold good 
 generally, or as an identity, in quaternions as well as in alge- 
 
166 ON QUATERNIONS. 
 
 bra : the exponents still being scalars, and the case where the 
 base is a negative number being- still excepted or reserved. And 
 we see that (abstracting from tensoi's, respecting which there is 
 never any difficulty), this formula simply expresses, that whether 
 we cause a line perpendicular to the axis of q to turn round that 
 axis, from some given initial position, through two successive 
 amounts of rotation, denoted as to their quantities and directions 
 by the symbols 
 
 t Lq and u Lq, 
 
 or through a single resultant rotation round the same axis, de- 
 noted by the symbol 
 
 {u-\- t) L q, 
 
 the Jinal position of the revolving line will be the same, for the 
 one process as for the other. 
 
 151. It is important to observe, however, that although the 
 rotation round the axis of the base q, produced by the operation 
 of the power q\ is correctly expressed (on the plan which we 
 have adopted in recent articles) by the symbol tLq, yet the 
 angle of that power cannot now be generally expressed by the 
 same symbol: because the value oH the product, 
 
 txlq, 
 
 is ?iof generally confined between the limits and tt, between 
 which limits it has been thought convenient to confine the angle 
 of any quaternion or power (art. 148). It may (and often will) 
 be necessary, in the applications, to add or subtract some whole 
 number of circumferences, or in other words some multiple of 27r, 
 to or from the product t Lq, in order to obtain hereby a result 
 which shall be comprised within \\\e first positive or negative 
 semicircle. And if the result of such addition of the multiple 
 2«7r, where n is some positive or negative whole number, shall 
 be an arc different from zero, and contained in the first negative 
 semicircle, so that 
 
 2w7r + ^ Z. 5'< 0, >-7r, 
 
 we must then change the sign of this result, in order to get a 
 positive angle : taking care, however, at the same time, to reverse 
 
LECTURE IV. 167 
 
 the axis, in order that the rotation may still be right handed. 
 We must therefore not write, as a general formula, 
 
 although this equation will often be true : but we may write ge- 
 nerally, 
 
 L {q^) = 2mr ±t Lq, 
 
 the integer n and the sign + being determined (when the angle 
 L q and the exponent t are given) by the conditions that 
 
 Init ±t Iq >0, <7r; 
 
 and the axis of the power (f being in the same direction with the 
 axis of the base q, or in the opposite direction, according as it is 
 necessary to take the upper or the lower sign (+ or -), in con- 
 formity with the foregoing conditions. 
 
 152. For example, if the exponent t be ^, or ^, or |, or ge- 
 nerally if it have any value between and 1, whether commen- 
 surable or incommensurable, the product t Lq will then be con- 
 fined between the same given limits (0 and tt) as the angle Z. q 
 itself; and therefore this product /^^e//" expresses the angle of the 
 power (f : while the axis of this power coincides with the axis of 
 the base. The formulae at the end of art. 135 remain therefore 
 undisturbed ; and the square-root of any proposed (non-scalar) 
 quaternion has always its angle acute, as being the half of an 
 angle in the first semicircle, 
 
 L{q^^\Lq<-^; 
 
 while the direction of the axis of this square-root qi \% coincident 
 with (not opposite to) the direction of the axis of q. 
 
 153. In like manner the square of an acute-angled quaternion 
 has, as compared with that quaternion itself, a double angle and 
 a coincident axis; so that, 
 
 if /.q<-, then z {q~) = 2 Lq, and Ax . 5^ = Ax . q, 
 
 where Ax . q is used as a (temporary) symbol for the unit-vector 
 which is drawn in the direction of the positive axis ofq. And the 
 square of a right-angled quaternion is a negative scalar (compare 
 
168 ON QUATERNIONS. 
 
 arts. 75, 85, &c.), which must be regarded as having its angle = tt, 
 and its axis indeterminate; or in symbols, 
 
 if Z ^ = ^, then l {(f) = tt, g-^ < ; Ax . f^ indet. 
 
 But the square of an obtuse-angled quaternion q is another qua- 
 ternion, with an opposite axis, and with an angle which is the 
 double of the supplement of the given obtuse angle; or in symbols, 
 
 ii Lq> ^, then Z {q^) = 2 tt - 2 Z g ; Kx.q^ = - kx.q. 
 
 154. For example, in fig. 29, art. 137, 
 
 \iq = la^, then f = ^a^\ 
 
 but while the angle of Sa"^ is 140°, and the axis of the same qua- 
 ternion is upward (by 137, 138), the angle of the square, or of 
 the quaternion S'a"^, is (on the plan of recent articles) regarded 
 as being not the double (namely 280°) of the angle 140° itself 
 but the double (namely 80°) of its supplement (namely 40°) ; the 
 axis of the new or squared quaternion being at the same time 
 treated as a downward line ; because when we compare imme- 
 diately the ray S' with the ray a, without introducing the conside- 
 ration of any other ray, such as S, we find it simpler to conceive 
 a right handed rotation of 80° from a to S' round such a down- 
 ward axis, than to conceive another rotation, also right-handed, 
 although round an upward axis, but extending through a more 
 considerable amount, namely 280°, from the same initial to the 
 same final ray. In fact we do not now regard 280° as being, in 
 a sufficiently simple sense for our present purpose, an angle at 
 ALL ; and therefore we adopt, instead of it, what it ivants of four 
 right angles, taking care, however, at the same time, to reverse 
 the axis. 
 
 155. Again, we saw (in art. 141) in connexion with the same 
 fig. 29, that the three quaternions, 
 
 ya'^, Sa"^, fa'^. 
 
 had all one common cube, namely the quaternion 
 
 i3a-i; 
 
LECTURE IV. ]69 
 
 and the values of the angles of the three quaternions just men- 
 tioned may now be definitely stated as follows (see arts. 137,138, 
 139): 
 
 Z(7a-i) = 20°; Z(ga-i) = 140°; Z (^a-i) = 100° ; 
 
 their axes being respectively upward, upward, and downward; 
 while the axis of their common cube is upward, and its angle has 
 (by 137) the following value : 
 
 Z(/3a-i) = 60°. 
 
 We have then, indeed, in this example, 
 
 £.(7a-i)3-3z(7a-i); 
 but we have also, 
 
 Z.(ga-i)3 = 3z(ga-i)-2 7r; 
 and 
 
 Z.(Ea-i)3 = 27r-3z(£a-i); 
 
 all which illustrates and exemplifies what was said in art. 151. 
 
 156. If with the recent significations of a, j3, y, 2, £ (in con- 
 nexion with fig. 29), we denote as follows the four quaternions 
 considered in the foregoing article, 
 
 we shall have (by art. 141), the equations, 
 
 and, by what has just been shewn, we shall have also, 
 
 Lq=Sir=3 Lr'-2Tr=27r-3A r". 
 
 These last expressions for L q give, 
 
 , 2n- , „ 27r J 
 
 Lr=^Lq; Ar= — + iLq; Lr ^—-^Lq; 
 
 but (by 135, 152) we have, generally, 
 
 L{q^) = ^Lqi 
 
 and the only one of the three distinct quaternions r, r\ r", with 
 q for their common cube, which satisfies this last condition, is r. 
 We must, therefore, by our recent definitions, regard /• as the 
 
170 ON QUATERNIONS. 
 
 {unique) cube-root of g, in this example ; and accordingly must 
 establish the equation, 
 
 to the exclusion of the two other equations, 
 
 gi = r\ 93 = /■", 
 
 these last being inconsistent with that definite signification of a 
 ■power (or root) of a quaternion which has been recently assigned ; 
 although, in that vaguer sense which was considered by us not 
 long ago, each of these two other quaternions, / and r", might 
 also^ as well as r, have been regarded (see arts. 138, 139) as being 
 among the values of the cube-root of the quaternion ^, or as being 
 one of the interpretations of the symbol cp. 
 
 157. Continuing then to adopt that definite interpreta- 
 tion of a symbol such as 5', which was assigned in articles 148,. 
 149, we see that (with the recent significations of the symbols) 
 we MAY write, definitely, for lh.e particular quaternion lately 
 denoted by r^ the equation 
 
 (/■3)i = r ; 
 
 but MUST NOT regard this equation as being an identity, since 
 it will not be true to assert that, for the two other particular 
 quaternions r and r", either one or other of the two following 
 equations, as at present interpreted, holds good ; 
 
 (r'^)~s=:r; (r'^y^ — r'. 
 
 On the contrary it is easy to see, with the help of fig. 29, that in 
 the present example, we have (compare art. 86), 
 
 (/•'3^i = ;• = ^-f r ; (r"^)3 = r -k^ r"; 
 
 (results which will soon be generalized :) because the line y, or 
 qia, or ra, is less advanced by 120° (in the figure) than the line 
 g, or r'a; but is more advanced, by the same angular amount, 
 than the line £, or r"a. The cube-root of the cube of a quater- 
 nion is therefore not generally equal to that original quater- 
 nion itself; although it may well be suspected, from the recent 
 example, to have at least (what it has in fact) some simple rela- 
 
LECTURE IV. 171 
 
 Hon thereto : and although a quaternion is always (like a 
 number^ the cube of its own cube-root. In short, the pro- 
 perty of having a given cube q, is shared in common (see art. 
 141) by three distinct quaternions; of which one alone is, by 
 our recent definitions (see arts. 148, 149, 152), regarded as being 
 the cube-root. 
 
 158. With the same deji7iite interpretation of q\ it is still more 
 easy to see that the square-root of the square of a quaternion 
 is not necessarily equal to that quaternion ; since it may just as 
 often happen to be the negative thereof {-q instead of + §■) ; be- 
 cause the original quaternion q may be as often oi^Mse-angled as 
 «CM^e-angled. In fact, by the foregoing principles, 
 
 if iq<'^, then ((?0* = 5' • 
 but if /.q> -, then {q^y = - ^- 
 For example, in fig. 29, 
 
 but, in the same figure, 
 
 {{Sa-'y}i=(^a-^)i^-Sa-''', 
 
 because the bisector of the angle of 80° between a and S' is not 
 the line 3 itself, but the opposite line - S (terminating at the ex- 
 tremity of an arc of - 40°, instead of an arc of + 140° from a) ; or 
 because (see 153, 154) the half of 27r - 2 Z 5' is = tt - Z 5', and not 
 = JL q: while a rotation from a, round an axis opposite to that of 
 q, and through an angle supplementary to Z q, conducts to a line 
 which has a direction opposite to that which would be attained 
 by revolving towards the same hand round the axis of q itself, 
 through the angle itself of q. At that intermediate stage, where 
 q is n^A^-angled, and therefore equal to some vector p, it follows 
 from what has been shewn in several former articles that the 
 square-root of its square is a vector, with an entirely indetermi- 
 nate direction: thus we may write, 
 
 (p^y= (7 ; T(7= Tp ; Uo-, indeterminate. 
 
 159. We see then that we are by no means at liberty to 
 
172 ON QUATERNIONS. 
 
 establish generally, in quaternions, at least with the definite 
 signification lately assigned to a power, and when versors as well 
 as tensors are considered, the arithmetical equation 
 
 which was one of those proposed (art. 136) in the present Lec- 
 ture for discussion. For we have found that even the less gene- 
 ral formula, 
 
 1 1 
 
 (g«)n =z g, or (r")" = r, 
 
 which is included in that equation, and in which n may be con- 
 ceived to represent some positive whole number, is an equation 
 not generally true (see arts. 157, 158), for the values n = 3, n = 2; 
 and the same formula may be easily shewn to fail (generally 
 speaking) for all higher whole values of n. In fact, the equation 
 
 r" = q 
 
 is satisfied generally, in quaternions as in algebra (compare art. 
 142), by n distinct values of r, when the quaternion g is given : 
 but onli/ one of these n values of r, suppose the unaccented r 
 
 itself, coincides with the value (compare 156, 158), of g". If we 
 start with any other, suppose r', of these n values of r, which all 
 agree in satisfying the equation r^^ = q; if we raise it to its y^"^ 
 power; and if we afterwards extract the n^^ root oj" this power, 
 namely, of the quaternion 
 
 which shall have been so obtained : we shall not hereby be 
 brought back to the value r itself, but to that other value r, 
 which has indeed the same n^^ power, namely, q, but is, notwith- 
 standing, a quite distinct quaternion. By still stronger reason, 
 therefore, we must reject, as a general conclusion, in this 
 Calculus, the equation cited at the beginning of the present ar- 
 ticle. Indeed if we remember the conditions for the general vali- 
 dity of that equation, which were assigned in art. 147, we shall 
 see that in the very act of our since satisfying one of those con- 
 ditions, hy fixing (in what appeared the simplest way) the value 
 of the angle of a quaternion, and thereby satisfying the equation 
 
LECTURE IV. 173 
 
 which (in the article referred to) was written as m = /, we have 
 made it impossible for us also to satisfy (generally) that other 
 condition of the same article 147, vvhich was there written under 
 the form I' = 0. For it is no longer possible for us, since owr fix- 
 ation of the value of the angle of a given quaternion, through 
 the limitations of art. 148, to escape the necessity (art. 151) of 
 in general adding some multiple of 27r to the product txLq, 
 and even of often changing the sign of the result, in order to 
 obtain a duly limited value of the angle of the intermediate power 
 g-*, before proceeding to raise this power, as a new base, to the 
 new power denoted by the symbol (§'*)"• 
 
 160. A little consideration, however, will suiBce to shew, 
 that although the arithmetical equation 
 
 (q*Y = 5"* 
 
 is thus not generally true in this Calculus, yet a power of a 
 power of a quaternion bears generally a simple relation to that 
 other power of which the (scalar) exponent is the product of the 
 proposed exponents, and that we may write, as a general for- 
 mula, the following, 
 
 {(fY = ( Ax . q^'' . (f, 
 
 where t and u are still two arbitrary scalars, and q an arbitrary 
 quaternion, while n is some integer number, positive or negative 
 or null, of which the value depends upon and varies with the va- 
 lues of q, t, u, but which can always be so chosen as to make the 
 formula true, in each particular case, with our present significa- 
 tion of a power. For example, if we remember that generally 
 (compare 75, 77, 153) the square of the unit-axis Ax . q is equal 
 to negative unity, so that the equation 
 
 (Ax.^)2=-1 
 
 holds good, independently of the particular value of the quater- 
 nion q ; while, for whole values of the exponents, the simple law 
 of transformation, above discussed, holds good (compare art. 
 136), and consequently, 
 
 (Ax.<?)^" = (-l)" = ±l ; 
 
174 ON QUATERNIONS. 
 
 we shall perceive that the formula above written is true for the 
 case u = ^, and that it gives, for that case, the expression, 
 
 where the choice of the sign is to be determined, for any pro- 
 posed values of q and t, by considerations of a kind already and 
 recently explained. And it will easily be found that when u = ^ 
 the same general formula is true, becoming then, 
 
 (q^)i= (Ax .q)3 .q~3. 
 
 161. For example, with the particular significations of/*, r',r\ 
 in recent articles (156, 157), we have for the unit-axes of these 
 three quaternions the expressions : 
 
 Ax . r = k; Ax . r' = ^ ; Ax . /' = - ^ ; 
 
 k still denoting an upward vector-unit ; and if we observe (com- 
 pare arts. 116, 89) that 
 
 k^= \, and (-A)-t-Af, 
 
 we shall see that the results, obtained in art. 157, may be thus 
 written : 
 
 and that they agree with the general expression, assigned in the 
 foregoing article, for a power of a power of a quaternion. But I 
 leave you to supply the general demonstration for yourselves, 
 through fear of being tedious on this subject. I may however 
 add here that the new symbol 
 
 (Ax . qYK q\ 
 
 where I denotes an arbitrary integer, has precisely that kind and 
 degree oi multiplicity of value, with our present definite signifi- 
 cation of a power of a quaternion, which was attributed provi- 
 sionally, in article 142, to the simpler symbol 
 
 6e/o/'e the fixation (in articles 148, 149) of the value of the angle 
 
 Lq. 
 
 \ 
 
LECTURE IV. 175 
 
 162. After these general remarks on powers, let us consider 
 more particularly the important and useful case where the expo- 
 nent is negative unity, and where therefore (see arts. 44, 117, 
 136) the power to be studied is the ?'eciprocal, q'^, of the origi- 
 nal quaternion q. There is no difficulty in seeing that the tensor 
 of the reciprocal of a quaternion is equal to the reciprocal of the 
 tensor; and that in like manner the versor of the reciprocal is 
 the reciprocal of the versor ; or in symbols (compare 117), that 
 
 T{q-^) = iTq)-^=i:q-\ 
 V{q-^) = {\]q)-^=\]q-^; 
 
 because an act of refaction (44) is generally compounded of two 
 other acts, of retension (63) and reversion (89) respectively. 
 Indeed these last formulae are included in the corresponding and 
 more general ones of article 136, which still hold good, for any 
 scalar exponents, with our present definite signification of 5''. 
 We have also evidently, 
 
 z(2'"^)=Z.2; Ax . 5'"^ = - Ax . g-; 
 
 because the reciprocal, q'^, considered as a re-versor, and com- 
 pared with the original quaternion q, has simply the effect of 
 turning the line on which it operates, through the same angle, 
 but round an opposite axis. And because (by art. 89) the con- 
 jugate of a versor is exactly such a re-versor, so that generally, 
 
 LK\5q=L\Jq, Ax . KUg - - Ax . U^, 
 
 and therefore also (returning from versors to quaternions), 
 
 LKq-=Lq, kx .¥.q= - kx . q, 
 
 we see that the conjugate and the reciprocal of a quaternion can 
 differ only by their tensors, which are mutually reciprocals of 
 each other, because generally (see arts. 89, 90, 114), 
 
 TKq = i:q. 
 
 Thus we may write, as a general formula for quaternions, 
 
 U^-i = KU^; 
 and may derive from it this general expression for a reciprocal. 
 
176 ON QUATERNIONS. 
 
 q-^ = Tg-K KVq; 
 
 which includes the formula of art 117 for the reciprocal of a vec- 
 tor, namely 
 
 because, by 114, 
 
 KVp^-Up. 
 
 163. We see at the same time that the following is a general 
 expression Jbr the conjugate of any quaternion, 
 
 Kq=Tq.KUq; 
 
 which may also (by the foregoing article) be written thus : 
 
 Kg' = Tq . Uq''^. 
 
 And because the quaternion q itself may (by art. 90) be expressed 
 as follows, 
 
 q=Tq .Uq, 
 
 where the tensor Tq is still (by 63, 113) a positive or absolute 
 number, and is therefore commutative as a factor with all other 
 factors, so far as the order of their multiplication is concerned, 
 we see that this other general formula holds good, as an identity 
 in the present Calculus : 
 
 qKq^Tq^', 
 
 so that the product of two conjugate quaternions is always 
 a POSITIVE SCALAR, namely the square of the common tensor. In 
 fact, when we proceed to compound with each other the two con- 
 jugate acts of faction^ of which the agents or operators are the 
 two conjugate factors q and Kg-, we find that we have to repeat 
 a tension^ and to undo a version, producing thus, upon the whole, 
 a double act of tension, or multiplying by the square of Tg, 
 without any ultimate turning of the line on which we have thus 
 operated. We arrive then at the following general expression of 
 the tensor of any proposed quaternion : 
 
 Tq=y/{qKq) = {qKqf; 
 
 which gives (see 90, 113) this connected expression for the versor, 
 
 ^q=q-^ V{qi<^q) = qiq'Kqy^; 
 
 \ 
 
LECTURE IV. 177 
 
 where it may be observed that, for reasons assigned in recent 
 articles, I abstain from writing, as a general transformation, the 
 expression 
 
 although we are at liberty to wvite, generally, or as an identity 
 in this Calculus, the formula, 
 
 {Uqy=q^ Kq. 
 
 164. In factj when q, and therefore also Kq, is an acute- 
 angled quaternion, the quotient q -^ K^ is a quaternion with the 
 samC: axis, and with a double angle ; or in symbols, 
 
 L{q^ Kq) = lLq, Ax . (g -h Kq) = Ax . q, if Z g < ^. 
 
 But when q and K^* are obtuse-aiugled quaternions, then the quo- 
 tient q -r- Kq is a quaternion with an axis opposite to that of q, 
 and with an angle equal to the double of~the supplement of L q 
 (compare art. 153) ; that is, in symbols, 
 
 L{q-^ Kq) = 27r - 2 Z ^, Ax . (g -^ K^) = - Ax . ^, if Z ^ > ^. 
 
 We may therefore, generally, establish the formula, 
 
 {q —- KqY = ± \Jq, according as z 5- ^ tt. 
 
 For example, in fig. 29, art. 137, we have the two following re- 
 lations of conjugation, 
 
 yy-^=K.ay-^; S'S'l = K . aS'i ; 
 
 and therefore, by the general formulae for multiplication and divi- 
 sion in arts. 49, 56, and by the property of a reciprocal (118), we 
 have the two quotients, 
 
 a-y'^-=rK . ay'^= (" -^7)-r-(7 -i- 7) = a -i- y' = ay'^ . 77'"^ = {ay'^y, 
 and 
 
 aS-l-- K.ag-i = aS-» -^ g'S'i = a ^ g' = aS-l . gS'"' = (aS-i)^ ; 
 
 because here 
 
 0-^7 = 7-7-7') a -7- o = S -T- S'. 
 
 But when we come to extract the square-roots of the two squares 
 
178 ON QUATERNIONS. 
 
 ofversors^ obtained by these two divisions, we find (art. 158) that 
 because the angles of the two quaternions 07"^ and a§'^ are re- 
 spectively acute and obtuse, we have, indeed, 
 
 ((a7-i)2)* = + ay-l; 
 but also, 
 
 ((„g-i)2)4 = _ag-i: 
 
 and similarly for all other cases of acute-angled and obtuse- 
 angled quaternions, when they are divided by their respective 
 conjugates, and the square-roots of the quotients taken. 
 
 165. If the quaternion 5^ should happen to be right-2iX\^eA, 
 and therefore (art. 122, &c.) to become a vector, we should have 
 (compare 114) the equations, 
 
 ^9' = |; Y^q=-q\ q^Kq=-l; 
 
 and the square-root of the quotient of these conjugates, although 
 it might be expressed by the symbol, 
 
 (q-^Kq)i={-l)i=V(-l), 
 
 would represent, or signify, on the principles of the present Cal- 
 culus, an INDETERMINATE VECTOR-UNIT, Or an unit-vector with 
 indeterminate direction. We should, however, still be allowed 
 to write, in conformity with what was remarked at the end of art. 
 163, the equation 
 
 U^'^ = q -r- Kq ; 
 
 the common value of each member being, in this case,^negative 
 unity. 
 
 166. This seems to be a natural occasion for introducing 
 some additional remarks on that important case |or indetermi- 
 NATION, in the theory of powers of quaternions, which we have 
 already several times found to present itself, when the base is a 
 negative scalar. And as the only difficulty (if any) in the ques- 
 tion arises from the power of the versor (see art. 136), which ver~ 
 sor is here equal (by art. 113) to the sign minus, or to the num- 
 ber negative unity, we have only to consider the powers of this 
 sign, or of this number, or the interpretation of the symbol 
 
 \ 
 
LECTURE IV. 179 
 
 i-y or (- 1)^ 
 
 where i is still supposed to denote a scalar. And because when 
 this exponent t is an odd number, positive or negative, the potver 
 is evidently (compare art. 60) itself equal to - 1 ; while, when t 
 is an eve7i number, positive or negative or zero, the power be- 
 comes = + 1 (as in ordinary algebra) ; we need only attend to the 
 cases where t isjractional, or incommensuTable. Now because, 
 when the base (-) or - 1 is regarded as a versor, namely (by 60) 
 as an zw-versor, its angle is tt, and its axis is indeterminate (com- 
 pare articles 149, 153), we may write, 
 
 Z (- 1) = TT ; Ax . (- 1), indeterminate. 
 
 The power under discussion, namely 
 
 (-1)', 
 must therefore, on our general principles, be conceived to be a 
 quaternion, of which it will soon be proved that the tensor is 
 unity ; and which, as a versor, has the effect (compare the end 
 of art. 149) of producing a given rotation = tir, but in a wholly 
 arbitrary plane. 
 
 167. The symbol 
 
 a/-T, or(-l)*, 
 
 regarded as a particular case of the foregoing more general power, 
 comes thus anew to be regarded (compare art. 75) as a quadran- 
 tal versor, with an arbitrary axis, or as operating in an arbitrary 
 plane ; so that we may write, 
 
 ^•V^-l= — ; Ax . "v/- 1, indeterminate : 
 
 at least until some special circumstance, oi siny particular inves- 
 tigation, by introducing some new condition, %haW Jix or limit the 
 direction of this otherwise arbitrary line. However, the tensor 
 of this power is given, being always equal to unity, because such 
 is (more generally) the value of the tensor of the power (- 1)*. 
 In fact, such a power is simply a versor, because its base is such 
 (compare art. 136) ; and we have generally, by art. 90, the equa- 
 tion 
 
 N 2 
 
180 ON QUATERNIONS. 
 
 Thus we may write, generally, 
 
 T.(-iy=l; 
 and in particular, 
 
 We are then led to regard this symbol V - 1 as having, in the 
 theory of quaternions, a perfectly real, but also a par- 
 tially INDETERMINATE, INTERPRETATION; namely as denoting 
 an ARBITRARY VECTOR-UNIT, or directed unit-line in tridimen- 
 sional space. This conclusion indeed agrees with what has been 
 already said in several former articles ; but it appeared impor- 
 tant enough to be reproduced in a new way here : since it is in 
 
 fact ONE OF THE CHIEF PECULIARITIES OF THE PRESENT CAL- 
 CULUS, in so far as its connexion ivith Geometry is concerned. 
 And if we denote by t the particular vector unit which in any 
 particular question is the value of ^/-li and at the same time 
 the axis of - 1, we shall obviously have the transformation, 
 
 (_iy=i2f. 
 
 for we shall now have 
 
 z t = -, lt= 1, 
 
 and therefore the power denoted by i^* is (by art. 86, or by our 
 more recent and more general theory: of powers of quaternions) 
 a ■yersor, which, like the power (- 1)*, turns a line k, perpendicu- 
 lar to f, through an amount of rotation expressed by the product 
 
 2t X -, or by ^tt, round the particular unit-axis t. Indeed, be- 
 
 cause t^ = - 1, the recent equation (- 1)^= t^* may be thus written, 
 
 which last equation, although not an identity in this calculus 
 (see article 159), is, notwithstanding, true, with the T^ve^ent par- 
 ticular interpretation of the symbols. 
 
 168. To give now a notion how such powers of- 1, although 
 partly indeterminate in their signification, may come to be useful 
 
LECTURE IV. 181 
 
 in the geometrical applications of this Calculus, I shall shew 
 hovf its very indetermination renders such a symbol adapted to 
 assist in forming expressions for a few simple but important loci 
 in geometry. And first let us suppose that we meet the equa- 
 tion 
 
 p^-sf-X, where |0 = P-o; 
 
 p being thus the vector of the point p (see art. 15), Avo.'wnfrom a 
 given point o as from an origin. Had the equation proposed for 
 interpretation been of the form p = a, where a is conceived to de- 
 note some given and determined vector^ the inference would have 
 been that the sought point p had itself a determined position, de- 
 noted thus (see art. 19) : 
 
 p ^ a + o. 
 
 But precisely because the symbol V-l denotes an ar6«Yra/"2/ vec- 
 tor-unit, the equation 
 
 v-o=p=^/-l, orp=v'-l+o, 
 
 leaves the position of p partly arbitrary ; and only obliges that 
 point to be situated somewhere upon a given spherical locus, 
 namely, on the surface of the sphere described about the given 
 origin oas centre, with a radius equal to the unit of length. Call- 
 ing then this surface, for shortness, the unit-sphere, and regard- 
 ing jO as the variable vector of a point upon a locus, we see that the 
 EQUATION OF THE UNIT-SPHERE is simply, with ouF notations, 
 
 p-V-1, orp3+i = 0: 
 
 a remarkable ybry//, peculiar (so far as 1 know) to the Calcu- 
 lus OF Quaternions, and one which appears to me to be very 
 extensively useful, in connexion with spherical geometry. 
 
 169. Had we chosen to form, on the same plan, the equation 
 of any other sphere, with its centre at any other given point b 
 (and not at the given or assumed origin o), and with any other 
 radius, such as b; v\e might have denoted the vector of the cen- 
 tre by /3, or might have assumed 
 
 i3 = B-o; 
 
 and might then have written the equation, 
 
182 ' ON QUATERNIONS. 
 
 p-(5 = bV-l, or (^ - j3)2 + Z»3 = 0. 
 
 Thus the symbol, 
 
 (3+bV-h 
 
 is seen to be, in this calculus, adapted to represent the variable 
 vector p, or p - o, of a variable point p, situated anytvhere on 
 the surface of the new sphere, and referred to the old point o, as 
 being still the assumed origin of vectors. And accordingly, by 
 art. Ill, the recent equation 
 
 is seen to be equivalent to the following, * 
 
 T(|0-/3) = 6; 
 where the symbol, 
 
 T(p-j3) = T(p-B)=i7, 
 
 denotes the length of the right line from b to p, that is here, from 
 the centre to the surface : which length is thus seen, in the pre- 
 sent question, to be constant, and equal to b. 
 170. The equation, 
 
 where it may be supposed that a is the known vector of a given 
 point A, so that 
 
 a= A- o, p = P- o, 
 
 would require a different (although an analogous) interpretation, 
 and would represent a different locus. For now the unit vector, 
 denoted by the symbol -/ - 1, being equal (by 118) to the quo- 
 tient of the two other vectors p and a, must (by art. 122) heper- 
 pendicidar to each ; and they (by the same article) must be per- 
 pendicular to each other : we must also have (by same art. 122), 
 the equality 
 
 Tp -^- Ta = 1, or Tp = Ta. 
 
 The line p or op must therefore now be equal in length to the 
 line a, or oa, and peiyendicular to it in direction : that is to say 
 the locus of the point p is now a circular circumference ; 
 namely a certain great ci^'cle, or diametral section, of the surface 
 
LECTURE IV. 183 
 
 of that new sphere which is described about the origin o as its 
 centre, so as to pass through the point a ; this sectioti being made 
 by a plane through o, which is at right angles to the given ra- 
 dius OA. Such therefore is the locus represented by the equation, 
 
 |oa"'=v/-lj 
 
 when interpreted on the principles of the present theory, in con- 
 formity with the notations of this Calculus. 
 
 17 1. Another mode of arriving at the same geometrical sig- 
 nification of this last equation would have been to put it first 
 under the form 
 
 and then to multiply each number into the given vector a; for 
 thus we should have found the transformation, 
 
 pa'^ . p = — a, 
 
 which would have shewn that the third proportional to a and p 
 is - a : and consequently (compare art. 134) that the symbol p 
 must here denote a line which is equal in length to the line a, 
 but perpendicular to it in direction. 
 
 172. If we wish to remove all restriction on the length of the 
 variable vector p, or to eliminate whatever depends on its tensor 
 Tp, we need only take the versors (art. 90), or write this other 
 equation 
 
 \J.pa-^= v-i ; 
 
 which latter equation therefore represents, on the same princi- 
 ples, a new and different locus, namely, that indefinite plane 
 which is drawn through the point o, perpendicular to the line 
 OA. And if we wished to form, in like manner, the equation of 
 any other plane, which might be supposed to he parallel to the 
 former plane, but to pass through some other given point, such 
 as B, we should only have to write the analogous formula, 
 
 U.((0-i3)a-i= v/-l. 
 
 In short, the two equations of the present article may be trans- 
 lated into the two following formulae : 
 
 , p 1. a ', p - (3 ± a. 
 
184 ON QUATERNIONS. 
 
 173. It may be here remarked, as an example of the use in 
 geometry of other potvers of negative unity, that the equation 
 
 interpreted on the foregoing principles, is easily seen to be the 
 equation of another circle : namely (if p and a be still conceived 
 to denote two co-initial vectors), the circle which is the locus of 
 the summits of all the equilateral triangles which can be de- 
 scribed upon the given base a. And if, taking the versors, we 
 write this other equation, 
 
 we shall thereby express or denote one sheet of a right cone, 
 or cone of revolution, described about the hne a as its interior 
 axis (or semi-axis), and with a semi-angle of sixty degrees. In 
 fact the second equation of the present article is equivalent to 
 the following angular or graphic formula, 
 
 L.pa-^ = 
 
 while the first equation includes also the metric relation, 
 
 Tp — Ta. 
 
 174. It is with some regret that I leave, for the present, this 
 class of speculations and inquiries, to which already might be 
 annexed several remarks on equations of straight lines and cy- 
 linders, and also on conic sections, and which would tend to 
 shew that you are already in possession of an organ, or of a 
 LANGUAGE, which cujoys no inconsiderable power of geometri- 
 cal expression. But for the sake of method, 1 think it better 
 to reserve the remainder of these applications^ for a later period 
 of our Course. You see, at least, already, that the degree of In- 
 determination of the Powers of Negatives (which powers 
 alone our definitions suffer to be indeterminate), is rather a re- 
 source than an embarassment, when properly managed in this 
 Calculus. I may also just remark (see art. 150), as regards the 
 theory of these powers, that the equation 
 
 (-1)«(-1)'=(-1)"^' 
 
LECTURE IV. 185 
 
 is only then to be generally regarded as true, when i\\e generally 
 indeterminate directions of the axes of those three quaternions, 
 which are here each denoted by the common symbol - 1, are 
 considered as coinciding with each other. But with these re- 
 marks on powers I must conclude the present Lecture, being 
 obliged to reserve for the next any such remarks as 1 had hoped 
 to make in this one, respecting the general multiplication and 
 division of quaternions, and especially respecting the associative 
 property of such multiplication. 
 
\ 
 
 LECTURE V. 
 
 175. Resuming without preface, Gentlemen, those investiga- 
 tions which were proposed near the beginning of the foregoing 
 Lecture, and which have already been partly entered upon, let 
 us proceed to examine whether the Associative Principle of the 
 Multiplication of Quaternions (mentioned in arts. 108, 112, 121) 
 holds good for the case of the inultiplication of three vectors^ which 
 we shall at first suppose to be coplanar. And because (by 117) 
 the reciprocal of a vector is itself another vector, with a recipro- 
 cal length, and with an opposite direction, the question at pre- 
 sent for consideration may be stated thus : 
 
 is j3 .a'^7 = /3a"^ . 7, when a||| /3,7? 
 
 176. If we retain the significations of a |3 7 S, with which 
 those letters were used in fig. 22 (art. 103), and assign to the let- 
 ter £ the same signification as in articles 123, &c., in connexion 
 with the same figure, we shall have on the one hand (by 127, 
 &c.) the equation (compare 130), 
 
 i3a-i.7 = g; 
 
 and on the other hand (by 123, 118) we shall have 
 
 a£-i = 7, i3£-i = g: 
 
 whence it follows (see 117) that we have also, 
 
 a-i7=e-\ /3.a-i7 = i3£-i = S. 
 
 It is then proved that the associative principle of multiplication 
 holds good, at least for these three vectors, a, )3, 7 ; the common 
 value of the two symbols /3a"^.7 and j3.a"^7, being (in this 
 case) equal to the fourth coplanar vector S. 
 
 177. It is easy now to see that the same reasoning may be 
 
LECTURE V. 187 
 
 employed to establish the same result, for every other case where 
 the two following conditions, of coplanarity and perpendicularity, 
 
 alJI/3,7, and 7X0, 
 
 are satisfied : it being only necessary to introduce, on the same 
 plan, the consideration of a new vector c, perpendicular to the 
 plane of a, /3, 7, and determined by the equation (compare 127), 
 
 a = 7£, or 7"^a = e : 
 
 which will give (compare 43), 
 
 ae"^ =7, a"^7 = e"^. 
 
 For, by taking S to denote the fourth proportional to the three 
 given vectors a, j3, 7, so that the proportion and equation (129, 
 130), 
 
 a : j3 : : 7 : S, 8 = /3a"^.7, 
 
 shall still hold good, we shall also have, by inversion and alter- 
 nation (art. 130), this other proportion and equation, 
 
 7:a::g:i3, or /38-i = a7-\ 
 
 Taking then the conjugates of these two last equal quaternions, 
 we find (see 89), 
 
 g-ij3=^-ia = £; 
 whence 
 
 /3 = Se, and, as before, jSt'^ = 8. 
 
 But t"^ was seen to be equal to a"^7 ; therefore we have still, 
 
 /3 . a"^7= S = j3a"^ . 7. 
 
 178. It is still more easy to perceive that when a \% parallel 
 instead of being perpendicular to 7, so that (see 59, 64, 83), 
 
 a II 7? 7 = ca = ac, a"^7 = c, 
 
 c being some scalar coefficient, the associative property holds 
 good, and the equation of art. 175 is satisfied. For we have, in 
 this case, 
 
 |3a-i . 7 = c (j3a- 1 . a) = C^ = /3c = /3 . a-i 7. 
 
188 />S QUATERNIONS. 
 
 When we come to establish, independently^ the distributive pro- 
 perty of the multiplication of quaternions, we shall be able to 
 infer, from the results of this article and of the one immediately 
 preceding it, that even when a is neither parallel nor perpendi- 
 cular to y, the equation of art. 175 still holds good : for we shall 
 only have to decompose y into two parts, or component vectors, 
 thus separately parallel and perpendicular to a, or to write, 
 
 7 = 7'+ 7"' 7' II «j 7" -J- « ■' 
 
 and then we shall have, by the distributive principle thus here by 
 anticipation spoken of, in combination with what has been re- 
 cently proved, _/o;* any three coplanar vectors^ a /3 7, 
 
 j3a"^.7 =j3a"^. 7' + /3a'^ . 7"= j3 . a'^7' + j3 . 0-^7" = /3 . a''^y. 
 
 17 9. Without assuming any knowledge of the distributive 
 principle, if the vectors a and 7, although still supposed to be 
 coplanar with j3, had not been perpendicular nor parallel to each 
 other, we might then have proceeded as follows, in order to de- 
 termine the value, or the geometrical interpretation, of the sym- 
 bol j3 -a' ^7, and to prove that this value is equal to the already 
 known value §, of j3a~^ .7. The symbol here to be interpreted is 
 seen to be expressed as a product ; namely, as the product of the 
 quaternion a'^y, multiplied by the vector j3; which last we 
 know to admit of being considered as being itself equal to a cer- 
 tain other and quadrantal quaternion (art. 122, &c.). We have 
 therefore here to resolve a particular case of the general problem 
 considered in art. 108, namely that of multiplying one quaternion 
 by another. Now the general rule, or process, for effecting such 
 a multiplication, which was assigned in the last-mentioned arti- 
 cle, may, with a slightly altered notation, be thus re-stated here. 
 To multiply one given quaternion q, as a multiplicand, by ano- 
 ther given quaternion /•, as a multiplier, we are in general to 
 find three vectors, suppose k, X, ^i, which shall satisfy the two 
 conditions, 
 
 q = \K'^', /• = juX"'; 
 
 and then the ionght product-quaternion will be the following : 
 
 rq=^fiK'K 
 
 ] 
 
LECTURE V. 189 
 
 In other words, we are to avail ourselves of the identity (com- 
 pare 49,118), 
 
 juX"^ . Xk"^ = llK'^. 
 
 Or because jc"^ and X"^ may represent any two vectors, we may 
 present the same identity under this other form, which is some- 
 times a more convenient one : 
 
 That is, we may put the given factors, q and r, under the forms, 
 
 and shall then be able to infer, for quaternions as for ordinary 
 algebra, that the product sought is 
 
 rq = ZB. 
 
 180. Applying therefore this last form of the rule to the 
 case where a'^y is the multiplicand, and j3 the multiplier, we 
 are led to seek for some three vectors, ^, ij, 0, which shall satisfy 
 the two conditions, 
 
 after which we shall have the expression, 
 
 The conditions just written give (by the last Lecture), 
 0\\\a,y; r, Hj a, 7 ; r, ± ^ ; Z ± V, K ± (5 ; 
 they give also, 
 
 er,-i = ya-i; 6 = ya' . n l T^= Tj3 -r- T„ ; 
 
 thus ^ is a line perpendicular to /3, but coplanar with a and y, and 
 thence also with j3 and 9 ; while ^ is a line whose length is the 
 quotient of the lengths of j3 and jj, this line ^ being also perpen- 
 dicular to the common plane of these five vectors, a, /3, 7, rj, 0, 
 and being directed so that the rotation round it, from n to j3, is 
 right-handed (122) : and 6 is the fourth proportional to a, 7, rj. 
 These conditions allow us to assume an arbitrary length, and 
 either of two opposite directions, for the auxiliary vector ^ ; but 
 when once these selections have been made, they serve to fix 
 
190 
 
 ON QUATERNIONS. 
 
 the lengths and directions of the two other auxiliary vectors, r\ 
 and d. But in whatever way we assume t,i consistently with the 
 foregoing conditions, we shall have 
 
 and the product Z,B will denote a certain determined vector i, co- 
 planar with a, (3, 7) rj, ; for if we double (for example) the length 
 of ^, we shall be obliged to halve the length of rj, and therefore 
 that of also, leaving the length of ^9 unchanged ; and if we 
 reverse the direction of ^, we must at the same time reverse 
 those of 7) and of 6 also, so that we shall not alter the direction 
 of the line ^6, or i. We may then write 
 
 and it only remains to examine whether this line i is equal to the 
 vector, obtained by the other mode of associating (or grouping) 
 the factors, namely, to the line 
 
 /3a-i.7 = §. 
 
 181. To render manifest this last equality, or to prove that 
 we have (under the supposed conditions) the equation, 
 
 I = d, 
 
 we have only to construct a figure, 
 suppose the annexed (figure 30), in 
 which no essential generality is 
 lost by supposing every tensor to 
 be unity. The unit vectors, a, j3, 
 y, from the centre o of a horizon- 
 tal unit-circle, may be supposed, -io\ 
 as a sufficient exemplification of _g^ 
 the nature of the question, to ter- 
 minate (as in fig. 29, art. 137), at 
 points on the circumference which are respectively graduated as 
 the extremities of three arcs of 0°, 60°, and 20°, in the direction 
 of right-handed rotation round an upward axis, from the initial 
 point A of that circumference. It is required, with these data, to 
 construct the vector t, which is the value of the symbol |3 .a'^y. 
 By the preceding article, we might choose t, so that rj should be 
 
LECTURE V. 191 
 
 directed either towards the extremity of an arc of+ 150°, or of an 
 arc of - 30°, from a ; but there may be considered to be a slight 
 convenience in adopting the latter alternative, because then the 
 direction of Z, will be upward, instead of being downward, the 
 figure being looked at from above. Taking then for X, an upward 
 vector-uniti or assuming 
 
 Z, = -\-k, (and not ^ = - A), 
 
 with that signification which we have hitherto usually attached 
 in these Lectures to this last letter^, we find that rj is the radius 
 terminating at the point graduated as -30°; because this, but 
 no other value of tj, gives (compare art. 70), 
 
 The proportion (180), 
 
 a : y : : 7) : 0, 
 
 shews next that is the radius terminating at - 10° from a. And 
 when we come to eff"ect finally the multiplication ^0, or kB, in 
 order to obtain the vector 
 
 we find that in thus forming t from 0, we must cause the extre- 
 mity of this last-mentioned unit-vector to advance through a 
 quadrant on the circle, namely from - 10° to +80°. But this last 
 point of the circumference is also the termination of the line §, 
 or |3a"' . 7, because the vector <, which is drawn to it from the 
 centre, is evidently such as to satisfy the proportion, 
 
 a : /3 : : 7 : /, or a : 7 : : /3 : t. 
 
 In short, instead of at once going forward, in this example, 
 through an angle of 20° from j3 to S, as from a to 7, we have 
 merely gone backward through 90° from j3 to rj ; then forward 
 through 20° from 17 to ; and then again forward through 90°, 
 from to i, which line i is thus found to coincide with S. 
 182. In fact we have here 
 
 a : 7 : : rj : : : Arj : /t^ : : j3 : t ; 
 
 and it is clear that the same process of reasoning applies to all 
 
192 ON QUATERNIONS. 
 
 Other cases of the same kind : the general principle on which it 
 depends admitting of being thus expressed in symbols, 
 
 niBi'.Zn'.Ze, ■in±v. and?±0. 
 
 In the language of a former Lecture, a biradial (n, 6) is only 
 changed to an equivalent biradial (^»], ^0), when both the rays 
 are caused to turn together in their own plane through a qua- 
 drant, their lengths being at the same time either left unaltered, 
 or changed proportionally. We have then generally, for any 
 three coplanar lines, a j3 7, the equation which was proposed for 
 discussion at the beginning of the present Lecture, and may 
 write, as the answer to the question proposed in art. 175, the for- 
 mula, 
 
 jSa-^y^jS.a-iy, if a Hj jS, 7. 
 
 183. The following investigation will confirm in a new way 
 this result, and will (it is hoped) be found in other respects in- 
 structive. 
 
 It can scarcely fail to have been already collected, from what 
 has been said in former articles (142, 158, 164), that the symbol 
 - q, or the negative of a quaternion, is regarded, in this calculus, 
 as being equivalent to the product of that quaternion q itself, as 
 one factor, and oi negative unity (or the sign minus), as another; 
 or, in symbols, that the following identity holds good in quater- 
 nions as in ordinary algebra, 
 
 -^=(-l)xy; 
 or, if we choose to write it so (compare art. 60), 
 
 - ^ = (-) X 5'. 
 
 With this definition oi-q, the negative of a quaternion <2 is 
 another quaternion, such that, 
 
 if 9" = j3 -h a, then - q = -fi -i- a. 
 
 In fact we have only to treat the three symbols, 
 
 q, -1, and- (7, 
 
 as representing respectively (see Lecture II.) a factor, profactor, 
 and transfactor, while a is the faciend, j3 the factum or profaciend, 
 
 \ 
 
LECTURE V. 193 
 
 and - /3 the profactum, or transfactum, in order to arrive at the 
 conclusion just now expressed. With this signification of the sym- 
 bol -g'jit is evident (compare 158) that p. 3 J 
 
 T{-q) = Tq; A(-g) = 7r-lq; 
 Ax .(-q) = - Ax .q. 
 
 See figure 31, where 5' (or + g') and 
 
 - q are pictured as two biradials. "^— /3 +/3 
 
 184. This being perceived, as regards negatives of quater- 
 nions, and what was lately said respecting conjugates being re- 
 membered, it will be seen that because, on the one hand, the 
 angle and axis of the negative are such as they were just now 
 stated to be, while the angle and axis of the conjugate are such as 
 was set forth in art. 162, the following general relations exist be- 
 tween them : 
 
 Z (- ^) = TT - Z. K<7 ; Ax . (- g) = Ax . ILq. 
 
 In words, the axes of the negative and of the conjugate (of 
 any quaternion) coincide ; but the angle of the one is supple- 
 mentary to that of the other. 
 
 185. Hence, as respects the negative of the conjugate of a 
 quaternion, or the symbol 
 
 - Kg, 
 
 we easily perceive that its tensor, angle, and axis are as follows : 
 
 T(-Kg) = Tg; l.{-Kq) = 7r-Lq', Ax . (- Kq) = Ax . g; 
 
 so that this negative of the conjugate has the efiect of turning 
 the line on which it operates, round the satne axis as the quater- 
 nion q itself, but through a supplementary angle. In fact, as re- 
 gards the angle and axis, we have only to change q to Kq, in the 
 formulae of the foregoing article, and therefore also Kq to g, be- 
 cause the conjugate of the conjugate of a quaternion is that ori- 
 ginal quaternion itself, in order to transform those earlier into 
 these more recent equations. In symbols, 
 
 KKq = q; 
 
 or more concisely, and in still more characteristically symbolical 
 language, the formula. 
 
l94 
 
 ON QUATERNIONS. 
 
 Fig. 32. 
 
 -K^ 
 
 K2=l, 
 
 holds good, whatever may be the quaternion q which is supposed 
 to be the subject of the operations. Or we might have changed 
 q to K^, in the formulae of art. 183, and have then employed the 
 values, assigned in art. 162, for the tensor, angle, and axis of a 
 conjugate. 
 
 186. To illustrate these conclusions respecting the negative 
 of a conjugate by a diagram, conceive, in 
 figure 32, that the three lines ob, oc, od are q^ 
 equally long, and that the third is opposite 
 in direction to the second ; let also the line 
 OA be supposed to bisect the angle boc be- 
 tween the two first of the three lines just 
 mentioned ; and let us write, 
 
 A-o = a, B-o=/3, c-o=7, D-o = §, 
 
 so that, by the construction, the following 
 relation shall hold good, 
 
 S = — y. 
 Then writing, for abridgment, 
 
 ^ ^a = q, 
 we shall have the two other and connected equations, 
 -y -h a = Kg', g -f- a = - Kg' ; 
 
 which are seen at once to exemplify the results of the foregoing 
 article, so far as axes and angles are concerned. 
 
 187. It is easy to prove, on the same plan, that the conjugate 
 of the negative of any quaternion is at the same time the negative 
 of the conjugate ; or that, in symbols, 
 
 K(-g) = -Kg. 
 
 Thus if we conceive, in the recent figure 32, a point e so chosen 
 that the line be shall be bisected by o, or that 
 
 E-0 = E=0-B--/3, 
 
 we shall then have, 
 
 £ -H a = -g, and S -^ a = K(£ -^ a). 
 
LECTURE V. 195 
 
 It may also be just noted here that the negative of the conjugate 
 of a vector, regarded as a quaternion, is equal (by 114) to the 
 original vector itself; or in symbols, that 
 
 -Kp = + p. 
 
 And it follows, conversely, from art. 185, that ifdi quaternion q 
 satisfy the equation, 
 
 - % = + ^, 
 
 then that quaternion must be a vector ; or that its angle must 
 have (compare 122, 149, 158, 165) the value. 
 
 It 
 ^^ = 2' 
 
 because thus only can we satisfy the condition, 
 
 Lq = Tr- Lq. 
 
 188. It was shewn in art. 1 10, that the tensor of the product 
 or quotient of any two vectors is the product or quotient of their 
 two tensors; and hence, or from articles 87, 88, 90, 113, it is 
 easy to infer that the versor of any such product or quotient of 
 two vectors is in like manner equal to the product or quotient 
 of their versors ; or in symbols, that 
 
 U.kX=U;c.UX; U(X--K)=UX-^Ufc. 
 
 Since then (by 49, 113), 
 
 Uy ^ Ua = (U7 -^ Uj3) X (Uj3 - Ua), 
 
 while it is still more obvious, from the numerical significations of 
 the symbols, that 
 
 T7 -- Ta = (Ty -^ TjS) X (TjS -f- Ta), 
 
 we see by the last cited articles, that for any two quaternions^ q 
 and r, the following relations hold good : 
 
 T .rq=i:r.Tq', \] .rq = Vr .Uq. 
 
 And in a way quite similar it may be shewn (by 50, 5Q) that 
 
 T(r-H^)-Tr^T^; \J{r ^q) = \]r ^Vq. 
 
 189. We see then that for any two quaternions, as well as for 
 
 o 2 
 
196 ON QUATERNIONS. 
 
 any two vectors, the tensor of the product is equal to the product 
 of the tensors; the tensor of the quotient is equal to the quotient 
 of the tensors; the versor of the product is the product of the 
 versors ; and the versor of the quotient is the quotient of the ver- 
 sors. And when we come to inquire into the meaning or inter- 
 pretation of these four sj^mbolical results, we easily perceive that 
 their validity depends ultimately on the mutual independence of 
 the two acts, or operations, of tension and of version ; in virtue 
 of which independence, we may compound two successive acts of 
 faction into one, or may decompose one such act into two, by 
 compounding separately, or by separately decomposing, the cor- 
 responding and component acts of tension and of version (com- 
 pare arts. 54, 5Q, 63, Q5, 90). 
 
 As a corollary it may be remarked, that we may always 
 write, 
 
 {T .')^qY = {Tr.Tqf^TrKTq^l 
 
 a tensor being subject to all the ordinary laws of arithmetic : but 
 that we have not always, nor generally, for two quaternions q 
 and ?", the analogous formula for the square of the versor of their 
 product, 
 
 {\J.rqf = VrK\]q^; 
 
 because we have not, generally, 
 
 \Jq.\Jr= U;- . U^, 
 
 these versors being not in general commutative with each other 
 as factors. 
 
 190. The conjugate of the product of any two quaternions is 
 equal to the product of their conjugates, taken in an inverted 
 order ; or in symbols, 
 
 K . r^ = Kg. Kr. 
 
 To prove this theorem, let a /3 7 be three lines chosen so that 
 (as in arts. 40, 46, 49) we may have the relations, 
 
 5'a = /3 ; ^'jS = 7 ; and therefore, rq . a = 7. 
 
 We shall then have also (see art. 163), 
 
 K/-.7==K;-.rj3-T»''./3, 
 
LECTURE V. 197 
 
 and (compare 49, 189), 
 
 (K^ . Kr) .y=Kq.(Kr.j) = Tr^ {Kq . j3) 
 = T/-2 (K^ . qa) = Tr2 Tq^ . a- (T . rqY . a 
 = {K .rqx rq) .a=K.rqx {rq . a) = K . r^- . 7 ; 
 whence, as above, 
 
 Kq .Kr =K .7'q'. 
 
 these two quaternions being thus proved to be equal, by its being 
 shewn that when they operate separately, as factors, on one com- 
 mon line y, they conduct to one common result, namely, to the 
 line denoted by the symbol 
 
 191. The rationale of the foregoing process may be said to 
 consist in this : that it puts in evidence, through the notations of 
 the present calculus, the conception, that if by any two succes- 
 sive acts of faction, whose agents or operators are here the two 
 quaternions q and r, we pass from an initial line a to a final line 
 -y ; and if we then perform, in a contrary order, the two respec- 
 tively conjugate acts, whose operators are, in this new order, Kr 
 and Kq ; we shall hereby have repeated each factor act of ten- 
 sion, but shall have reversed (and thereby annulled, as to their 
 effects^ each of the two component acts oi version (compare art. 
 114): and shall thus, upon the ivhole, have merely multiplied the 
 original line a by the product of the squares, Tg'^ and Tr^, of 
 the tensors of the two proposed quaternions q and r, or by the 
 square of the tensor T-rg* of the product of those two quater- 
 nions. But in thus passing from y, or from rq . a, to (T . ?^qY . a, 
 after passing from a to y, we have, upon the whole, repeated the 
 act of tension denoted by T . rq, and reversed the act of version 
 denoted by U . r^- ; that is, we have multiplied y, upon the whole, 
 by the conjugate K . rq, of the product rq of the quaternions. 
 
 192. A reasoning nearly similar would shew that the recipro- 
 cal of the product of any two quaternions is equal to the product 
 of the reciprocals, taken in an inverted order: or, in symbols, 
 that 
 
 {rq)'^ ■=q'^ r'^. 
 
198 ON QUATERNIONS. 
 
 Accordingly, with the recently supposed choice of the lines a, j3, 7, 
 we have (see 44, 136), 
 
 rq = y ^a, {rq)'^ = 0-1-7, 
 
 ^-1 = a -7-/3, /'-i = /3-^7; 
 
 and the recently written relation of product to factors is seen to 
 hold good, in virtue of the general formula of multiplication in 
 art. 49. It was thus, for example, that in art. 177 we had the 
 two connected equations, 
 
 £=7"^a, £'^ = a"^7. 
 
 193. The formula of art. 190 includes the equation of the same 
 kind which was established, as a definition, for the conjugate pro- 
 ducts of any two vectors k and X, in art. 89, namely 
 
 K . kX = A/c ; , 
 because (by art. 114), 
 
 K/c = — K, KX = - X. 
 
 It enables us also to infer, for any three vectors a, /3, 7, the equa- 
 tion, 
 
 K(7a-i.j3)=-i3.a-i7; 
 because 
 
 Kj3=-^, and K.ya-'^ = a-^y. 
 
 Whenever, therefore, the three lines a, j3, 7 are copla7iar, so that 
 (by arts. 129, 130) a. fourth line S may be so chosen in the same 
 plane as to satisfy the equations, 
 
 i3a-i.7=8, 7a-i./3 = g, 
 
 we see that we shall have also 
 
 i3.a-l7 = -Kg = +g = j3a-^7; 
 
 and thus we are conducted anew to the result obtained before, in 
 art. 182 ; while, in arriving at it, by this new train of investiga- 
 tion, we have had occasion to develope some useful principles and 
 general results of this Calculus. 
 
 194. It is therefore immaterial where weplace the point (or 
 other mark) of multiplication, in combining any three coplanar 
 lines, such as here 7, a"S and j3, as factors, in one determined 
 
LECTURE V. ^ 199 
 
 order, or in the order opposite to this ; the result being still equaU 
 when interpreted on our principles, to one definite vector, or 
 fourth directed line in the same plane, whichever place we choose 
 for the multiplying point or mark, and whichever of the two op- 
 posite orders of factors we may adopt. The associative prin- 
 ciple OF multiplication (referred to by anticipation in several 
 former articles) is therefore here seen to hold good; together 
 with at least a partial validity of the commutative principle also, 
 for the same case here considered : that is to say, for the case of 
 the multiplication of any three coplanar lines. And we may now 
 proceed io profit by it (compare art. 136), by dismissing, as un- 
 necessary, the point, or other multiplying mark : and by thus 
 writing simply, under the conditions of articles 129, &c., the 
 equation, 
 
 g = /3a-i7, or g = 7a-i/3: 
 
 because, whether we multiply the quaternion j3a"^ into the vec- 
 tor y, or the vector /3 into the quaternion a'^y, or -ya"^ into jS, 
 or y into a'^ft, we obtain, by each of these four processes, one 
 common line S as the result ; namely, the fourth proportional to 
 «} jSj 7j or to a, 7, j3, determined as in those former articles. And 
 we may call this fourth proportional the continued product of 
 the three vectors y, a'^, and j3; or of /3, a~^, and 7. 
 195. If we should meet with a symbol of the form 
 
 juXfc, where ^ ||1 X, /c, 
 
 without negative unity occurring as an exponent of the middle 
 factor, we might still speak of this symbol as denoting a conti- 
 nued product of three vectors, namely k,\, fx; that is, the pro- 
 duct-line obtained by multiplying k by X, and then multiplying 
 the product Xk hy fx; or we may read the product thus : ju into 
 X into K. We might also, by the recent associative principle, in- 
 terpret the same symbol /xXk as denoting the product-line obtained 
 by multiplying first fj. into X, and then the product /.tX into k. Or 
 again we may regard the symbol /xXk as being equivalent to the 
 continued product of the same three coplanar vectors, taken in 
 the contrary order, namely the order ^, X, k; or may interpret 
 it as being equal to the product " k into X into ju ;" because it fol- 
 lows from what has been already shewn, that under the supposed 
 condition of coplanarity, the equation 
 
200 ON QUATERNIONS. 
 
 is satisfied. We may also, by the last article, speak of either of 
 these two last equated symbols as denoting the fourth propor- 
 tional to X'S ju, and k, or to X"S k, and fx; because, by a princi- 
 ple which has indeed been already tacitly employed, the recipro- 
 cal of the reciprocal of a vector, or of a quaternion, is that vector 
 or quaternion itself; so that (compare 117, 136), 
 
 X=(X-i)-M q={q-^y\ 
 
 196. Since (by 117), 
 
 2-11 
 
 while the square a^ of a vector is (by 85) a scalar, namely, a ne- 
 gative number, and the place of a scalar factor among other fac- 
 tors is (compare 83) indifferent to the value of the product, we 
 see that the following ^enQXdX relation between the two products 
 
 /3a"^7 and jSay, 
 
 which are of the forms considered in the two foregoing articles, 
 holds good in quaternions as in algebra : 
 
 jSay — a^ . /3a' ^y. 
 
 If then we wish to construct the continued product ^ay of any 
 three given coplanar lines, y, a, j3, we see that we may first con- 
 struct, on the plan of either of the two articles 131, 132, the 
 fourth proportional §, to the three lines a, /3, 7, and afterwards 
 multiply the line S, so constructed, by the negative scalar a^ ; that 
 is to say, reverse its direction, and multiply its length by Ta*^ ; be- 
 cause (by HI, 116, 136), 
 
 a2 = - Ta2. 
 In symbols, 
 
 if a : /3 : : 7 : 8, then jSay = - Ta^ . g. 
 
 197. Thus, for example, if a, j3, 7 denote, as in fig. 26, art. 
 131, the three successive sides of a triangle bca inscribed in a 
 circle, the continued product j3a7, or 7oj3, denotes a vector which 
 has the direction of the tangent ae at a to the segment abc, and 
 not the direction of the tangent af to the segment bca; because, 
 in the article just cited, it was shewn that this last is the direc- 
 tion of the fourth proportional S, to a, /3, 7. As to the length 
 
LECTURE V. 201 
 
 of the line which is denoted by the symbol (5ay, it bears to the 
 length of the line af, in the same figure 26, a ratio which is the 
 duplicate of the ratio of the length of the side bc or a to the as- 
 sumed unit of length ; or in other words, this length of the line 
 (Say bears to this unit of length the same ratio which the right 
 solid, constructed with the three sides of the triangle bca as 
 edges, bears to the unit of volume, or to the cube constructed 
 with the unit of length for its edge. In symbols (compare 110, 
 188), 
 
 T.j3a7 = T/3.Ta.T7. 
 
 198. We know then hovs^ to interpret the symbol, 
 
 (a -c) (c-b) (b - a), or (b - a) (c -b) (a- c), 
 
 for any three points of space a, b, c, supposed at first to be not 
 situated on one straight line, but to be the corners of a plane 
 triangle; namely, as denoting a certain line or vector, whose 
 length represents the product of the lengths of the sides of that 
 triangle, while its direction is that of the tangent at a lo the seg- 
 ment ABC, of the circle circumscribed about it. This remarkable 
 interpretation, or construction, for the symbol (a-c) (c-b) 
 (b-a), appears to me to be frequently useful, in the applications 
 of the present Calculus to Geometry ; and it is one of those 
 which are, so far as I have hitherto been able to learn, peculiar 
 TO QUATERNIONS, from the principles of which we have seen that 
 it is a necessary and inevitable consequence. 
 
 199. If the three points abc should happen to be situated on 
 one straight line, the interpretation of the recently assigned sym- 
 bol would in that case be still more easy. For because the pro- 
 duet of two vectors which have the same direction is in this 
 theory (by art. 84) a negative scalar; while the product of two 
 vectors which have opposite directions is on the contrary (by the 
 same article) with us a positive scalar; it follows that if the 
 point A be intermediate between b and c, as in fig. 33, 
 
 5, 
 
 Fig:. 33.^0 
 
 ( /3 A y ^«y 
 
 the continued product, 
 
202 '' ON QUATERNIONS. 
 
 jSay = (a - c) (c - b) (b - a), 
 
 is constructed in this case by a line, which has the direction of 
 either of the two extreme factors b - a or a - c. But in the case 
 represented by this other figure, 
 
 ( -^- 
 Fig. 34. J c- . — — — — A ^ 
 
 i a B y f3ay 
 
 in which the intermediate point is b, the same symbol of a con- 
 tinued product denotes a line, which has indeed the direction of 
 B - A, but not that of a- c. And on the other hand, in the case 
 where c is the intermediate point, as in the figure subjoined, 
 
 Fig. 35 
 
 I a 
 
 13 Pay 
 
 the same continued product has the direction of a - c, but not 
 that of B - a. In each of these three cases, therefore, the pro- 
 duct j3a7 is constructed by a vector, which has the sa?ne direction 
 as the segments of the finite straight line on which the three 
 points ABC are situated, some two of them being at its extremi- 
 ties, and the third being in some intermediate position ; and in 
 each case, the solid under the whole line and its two segments 
 has the same numerical expression as the length of the product- 
 line. But it must again be observed that the direction thus as- 
 signed to this product-line appears to be peculiar to the present 
 calculus, or to its modes of geometrical interpretation. 
 
 200. Again, if we suppose that abcd is, as in figures 27 and 
 28, a quadrilateral inscribed in a circle, then because, with the 
 significations of the letters in those figures, we have (see 132), 
 
 ■yajS = jSay = a^ . j3a'^Y= o?^-- Ta^ . §, 
 
 it follows that the continued product, 
 
 7aj3 = (d - c) (c - b) (b - a), 
 
 is constructed by a line which has its direction opposite to that 
 of 8, and therefore similar to that of a - d in fig. 27, but opposite 
 to the direction of a - d in figure 28. Hence the continued pro- 
 
 \ 
 
LECTURE V. 203 
 
 duct of three successive sides, ab, bc, cd, of a quadrilateral 
 inscribed in a circle, is (in tliis theory) a line, which has the di- 
 rection of the FOURTH SIDE, DA, Or clsc a direction opposite to 
 the fourth side, according as the inscribed figure abcd is an un- 
 crossed or a CROSSED quadrilateral (compare 132). In symbols, 
 for every quadrilateral in a circle, we have 
 
 U . (d - c) (c - b) (b - a) = + U(a - d) ; 
 
 the upper or the lower sign being taken, according as the figure 
 is uncrossed, as in fig. 27, or crossed, as in fig. 28. And from 
 what was shewn in art. 132, in connexion with those two figures, 
 it is easy to infer that the recently written formula of versors 
 would not hold good, if d were changed to any other point on 
 the third side cd, or on that side prolonged, such as g or d' or d", 
 within or without the circle; because the versor of the continued 
 product in the first member of the formula would then either re- 
 main unchanged, or merely change its sign, while the versor of 
 the fourth side, in the second member of this same formula, 
 would be multiplied by a non-scalar quaternion. No plane qua- 
 drilateral, therefore, can satisfy the condition expressed by the 
 recent fortnula, unless it be inscriptible in a circle: for if it cannot 
 be so inscribed, the two members of that formula will represent 
 two different vector-units. And if the quadrilateral abcd were 
 what is called a gauche (or twisted) figure, that is, one not con- 
 tained in any signle plane, we shall soon see that the formula 
 would in that case fail, from the first member becoming a non- 
 quadrantal versor, while the second member would still represent 
 a vector-unit as before. It follows then that the recent equation 
 between versors expresses, in what may be regarded a remarkable 
 way, a property which belongs to inscriptible quadrilaterals alone; 
 and consequently that it expresses, at the same time, a characte- 
 ristic property of the circle, by assigning, with the notations of 
 this calculus, a general relation which exists between four con- 
 circular points, and between four such points exclusively. 
 
 201. It is time to consider now, what a recent remark may 
 remind us of, the continued products and fourth proportionals of 
 three lines not coplanar. 
 
 Suppose then that it is required to assign the value of the 
 
204 ON QUATERNIONS. 
 
 symbol j3a"^ . 7, where the line y, although not now coplanar with 
 a and /3, shall be supposed at first to be perpendicular to a, so 
 that we shall have 
 
 7not|||a,j3, but 7 _L a. 
 
 Under this last condition, we can, as in the second section of 
 art. 127, determine a line e, such that 
 
 y = a ~r- £ = ae'^ ; 
 
 and shall then have, as in that article, 
 
 j3 ^ a X 7 =j3 -r- f, or j3a'^ .7 =/3e"^. 
 
 But whereas ^q formerly concluded (in 127, II.)> that the quo- 
 tient /3 -4- £, thus obtained, was equal to a /me, because e was 
 found, in that former investigation, to be •perpendicular to j3, on 
 account of its being perpendicular to both a and 7, with which 
 lines j3 wasjbnnerly coplanar ; we must now, on the contrary, 
 infer, from the present non-coplanarity of a, j3, 7, that the line e, 
 which is still perpendicular to both a and 7, by its construction, 
 cannot also be perpendicular to /3 ; or in symbols (contrast the 
 corresponding expressions in 127), that 
 
 £ not ± j3, because £ _L a, a _L 7, and j3 not |]| a, 7. 
 
 202. We are not therefore now to consider any li?ie, such as 
 the o' of 127, but a certain non-quadraiital quaternion, to be the 
 value of the symbol /3£'^, or /3 -r- £, and therefore of |3a"^.7. 
 And if we still agree, from the analogy of former investigations, 
 to call this last symbol, namely, 
 
 j3a~^ .7, or /3 -h a X 7, 
 
 a symbol for ihe fourth proportional to the three lines a, j3, 7, 
 we find ourselves obliged to admit the following conclusion, 
 already mentioned by anticipation in art. 130, namely, that ^'The 
 Fourth Proportional to three Lines not coplanar is not 
 A Line, but a Quaternion;" at least when the first line a is, 
 as above, perpendicular to the third line 7. But we shall soon 
 see that this last condition of perpendicularity is not essential to 
 the correctness of the conclusion. 
 
LECTURE V. 205 
 
 203. Retaining, however, a little longer, this condition of 
 perpendicularity, there is no difficulty in proving, for the three 
 lines of art. 201, or rather for the three lines 7, a"^, and j3, the 
 associative property of multiplication, or the equation, 
 
 j3.a"i7 = j3a'^.7, atleastifyXa; 
 
 each member of this last formula being here = jSs"^, because, as 
 in 176, 177, the equation 
 
 y = ai'^ gives a'^ 7 = £"^. 
 
 And if we were now again, for a moment, to suppose known the 
 distributive principle of multiplication, already more than once 
 alluded to (121, 178), and of which an independent proof W\\\ be 
 given in the ensuing Lecture, we should be able to infer, by the 
 process described in art. 178, that the same associative property ^ 
 or the equation j3 . a'^7 = j3a'^ . 7, holds good for any three vec- 
 tors : namely, by decomposing 7 into two parts, or component 
 vectors, 7' and 7", of which 7' shall still be parallel to a, and 7" 
 still perpendicular to a, although this last component 7" would 
 not now be supposed (as in 178) to be in general coplanar with 
 
 204, If instead of supposing 7 _L a, we had supposed 
 
 j3 ± a, and therefore /3 = Xa, j3a"^ = X, 
 
 where X is some new line, the same associative property might 
 easily have been inferred. For in this case we should have (com- 
 pare 179), 
 
 /3 . a"^7 = Xa . a"^7 = X7 = j3a'^ . 7. 
 
 And hence by distributing any other vector j3, into two parts 
 respectively parallel and perpendicular to a, we might again infer, 
 in a way quite analogous to that mentioned in the foregoing ar- 
 ticle, that the expressions /3 . a'^y and j^a'^ .y are equal, for any 
 three vectors, if the distributive principle, for the multiplication 
 of quaternions, had been already proved. But we shall soon 
 prove generally this associative property of the multiplication of 
 vectors, without assuming any knowledge of the distributive prin- 
 ciple, as regards the multiplication of quaternions . Meanwhile 
 we see that the common value just now found for the two equal 
 
206 ON QUATERNIONS. 
 
 expressions, jS.a'^y and /3a"'. 7, in the case where /3 ± a, 
 namely the value Ay, is (like the value jSe'S found for the case 
 7 _L a) not equal to a line,hut to a quaternion ; because X, being 
 perpendicular to a and jS, cannot be also perpendicular to y, when 
 the three lines a, (3, 7 are supposed to be not coplanar with each 
 other. 
 
 205. If it happen that the three lines a, /3, 7 compose a rect- 
 angular SYSTEM, so as to be perpendicular each to each, 
 
 [5 ± a, 7 ± a, 7 ± /3, 
 
 then the line t, determined as in 201, will have its direction co- 
 incident with, or opposite to, the direction of /3, according as the 
 rotation (compare 122) round 7, from j3 to a, h positive or nega- 
 tive ; or, in other words, according as the rotation round a from 
 j3 to 7 is negative or positive. And because the symbol jSe"^, 
 which has been found (201, 203), to be the value of j3a'^ . 7, or of 
 /3 . a"^75 denotes in the first case a positive, but in the second case 
 a negative scalar, we see that " The Fourth Proportional (fia'^y), 
 to any three mutually Rectangular Lines a, j3, 7, is a Negative or 
 a Positive Number, according as the Rotation round the first (a), 
 from the second (j3), to the third (7), is of a Right-handed or of 
 a Left-handed character.'' We might also prove this Theorem 
 otherwise, by observing that in the first of these two cases the 
 line X, of art. 204, has the same direction as 7, but in the second 
 case the opposite direction (compare 82, 84). 
 
 206. For example, with the significations assigned in the 
 Second Lecture (art. 77) to the symbols 2, J, k, those symbols 
 denote three rectangular vector-units, such that the rotation 
 round i from j to k, and therefore also round J from k to i, is posi- 
 tive or right-handed. We may therefore expect, in virtue of the 
 Theorem enunciated in the immediately preceding article, to find 
 that the fourth proportional to^, Jc, and i, is a negative number, 
 which (from the value of its tensor) can be no other than nega- 
 tive unity ; or in symbols, that 
 
 k -7- j y.i = ~ 1 . 
 
 And accordingly we saw (in 76 and 75) that 
 
 k -^j = i, and i xi=- 1 . 
 
LECTURE V. 207 
 
 On the other hand, the rotation round the same j» from i to k is 
 negative ; and we have accordingly, as another example of the 
 truth of the theorem in 205, the equation 
 
 because (compare 74 and 75), 
 
 i-^j = -k, -kxk = +l. 
 
 207. Since we have still (as in 196) 
 
 a = a.^ .a'^, and a^ = - Ta^ < 0, 
 
 we see that the continued product jSay (compare 194, 195) of the 
 three vectors y, a, j3, namely, the product obtained when y is 
 multiplied by (not into) a, and the partial or intermediate product 
 ay is again multiplied by j3, may still be formed from the fourth 
 proportional to the same three vectors taken in the order a, j3, y, 
 that is to say, from j3a'^ .y, by multiplying this last quaternion 
 by the negative scalar a^. The theorem of art. 205 may there- 
 fore be thus enunciated: " The continued product (5ay, of any 
 three rectangular vectors y, a, j3, is a positive or a negative 
 number, according as the rotation round the firstly, from the se- 
 cond, a, to the third, j3, is itself positive or negative" (that is, 
 right-handed or left-handed). For this rotation, round y from a 
 to j3, has necessarily the same direction as the rotation round a 
 from j3 to y ; while the values of jSa'^y and j3ay are scalars with 
 opposite signs (as positive or negative), when a, j3, y compose a 
 rectangular system. 
 
 208. With respect to the tensor of the continued product, it 
 is obviously equal to the continued product of the tensors ; for 
 in general it is an evident consequence of the conceptions and re- 
 sults explained in former articles, that z/'any number of qua- 
 ternions be multiplied together, in any order, and with any 
 mode of association (or of grouping) among themselves as factors, 
 the tensor of the product is always equal to the product of 
 THE tensors (compare 188, 197). We may agree to denote this 
 general principle, or theorem, by writing concisely the formula, 
 
 Tn=nT; 
 
 where the Greek capital letter 11 is used as a symbol for a pro- 
 
208 ON QUATERNIONS. 
 
 duct. And on applying it to the case of the last article, we find 
 that the number, which is the value of the continued product jSay 
 of three rectangular lines, must, if we abstract from its sign, de- 
 note the product of the lengths of those three lines. 
 
 209. Thus, 
 
 j3a7 = - 7a|3 = ± T/3 . Ta . Ty, if /3 ± a, 7 J. a, 7 _L /3 ; 
 
 and if DA, db, dc, be three co-initial edges of a right solid (or 
 rectangular parallelipipedon), the continued product 
 
 (c - d) (b - d) (a - d) = + volume of solid ; 
 
 the upper or the lower sign being taken, according as the rota- 
 tion round the edge da, from the edge db to the edge dc, is di- 
 rected towards the right hand, or towards the left. 
 
 210. For example, the lines i, j, k may be regarded (by 77) 
 as three conterminous edges of the unit-cube, if we give this 
 name to the cube of which three co-initial edges are three vector- 
 units, drawn in three rectangular and standard directions from a 
 point assumed as origin of vectors ; and the rotation round i from 
 j to k is positive, but the rotation round k from j to i is negative. 
 
 And accordingly we find, in consistency with the foregoing the- 
 orem, the two following continued products (compare 206) : 
 
 kji = J2 y. kf^ i = - kj'^ i = ^- \ ; 
 
 ■ijk -p X ij'^k-- ij'^k- -\. 
 
 This last result, in connexion with those of art. 75, gives the 
 continued equation, 
 
 i^ =fi z= k^ = ijk = - 1 ; 
 
 and I cannot forbear to notice, by anticipation, here, that all the 
 rules respecting the multipUcatiotis ofi,j, k, ivill he found to he 
 included in this simple formula. 
 
 211. When the following conditions concur, 
 
 7 not 111 a, |3, and 7 not _L a, 
 
 we may conceive, as in 127, II., that the rays a and j3 are made 
 to tur 71 together in their own plane, without any alteration of 
 their relative lengths, or of their relative directions, till a comes 
 to be, in its neiv position, perpendicular to 7 ; while /3 will, at 
 
LECTURE V. 
 
 209 
 
 the same time, come to assume a certain othernew position : and 
 then these tivo new positions (or directions) of a and (5 may be 
 substituted for the two old or given ones, in order to determine, 
 on the plan of 201, a certain line e, perpendicular to the given y 
 and to the new a, but not to the new j3, and such that this new 
 j3, divided by s, shall still give, as the quotient, a non-quadran- 
 tal quaternion j3£"^ which shall be, in the present question, the 
 value of the fourth proportional /3a'^ . 7, whether both the old or 
 both the new values of a and /3 be employed, in interpreting this 
 last symbol. 
 
 212. To avoid any possible confusion which might arise from 
 the use (in the last article) of one common pair of symbols a and 
 j3, to denote two distinct pairs of lines, although these latter 
 pairs are merely the rays of two equivalent biradials (93, 94), it 
 maybe useful to employ one of the identities of art. 179; and 
 for that purpose, retaining the given pair of lines a, j3, whereof 
 the first is not perpendicular to the third given line 7, we may 
 advantageously seek to assign three other lines k, X, jh, such that 
 
 7 = Xk"^; [3a~ ^ = fx\'^ ; 
 
 for then we shall have the following expression for the fourth 
 proportional sought, 
 
 /3a"^ • 7 =jUK"^. 
 
 It is easy to see that this last symbol, fxK'^, denotes here a non- 
 quadrantal quaternion ; as, for consistency with the result of the 
 last article, it ought to do. For if k, which is perpendicular to 
 both 7 and X, could also be perpendicular to /x, then 7 would be 
 coplanar with X and fi, and therefore also with a and j3 ; but this 
 would be contrary to the hypothesis which is at present under 
 consideration. It may be remarked that the three lines ic, X, jit, 
 of the present article, may be conceived to coincide respectively 
 with the line e, and with the neiv (or altered) lines a and j3, of 
 the article immediately preceding. 
 
 213. With respect to that other and at least apparently diffe- 
 rent expression, which is formed from the expression j3a"^ .7 for 
 the fourth proportional, by disj)lacing the point of multiplication, 
 we may still write (as in 180, only changing ^ to t), 
 
 p 
 
210 ON QUATERNIONS. 
 
 but we shall now have 
 
 t not i 0, 
 
 and therefore the value lO, of j3 . a'^y, will wo^ now represent a 
 //??e, but (as in recent articles) a non-quadrantal quaternion. In 
 fact, since i is here perpendicular to both j3 and jj, if it could be 
 also perpendicular to 9, we should have j3 coplanar with rj and 0, 
 and therefore also with a and y ; but such a coplanarity of a/Sy 
 is wo^ at present supposed to exist. Thus generally, or (more 
 precisely) with the exception of the case of coplanarity, the 
 expressions j3 . a'^y and j3a'^ . 7 denote, each, a quaternion, 
 hut not a line. (Compare 202, 130.) But it remains to prove 
 that these two quaternions are always equal to each other ; or 
 that, in the notation of the present article, and of the one im- 
 mediately preceding it, the following equation holds good : 
 
 214. It may first be proper to shew distinctly that this (\\xe?,- 
 tion IS quite free from vagueness ; or that the two quaternions, 
 here to be compared, have separately determinate values, whether 
 these be equal or unequal to each other. Now with respect to 
 the quaternion 16, it is obvious (from principles respecting ten- 
 sors, already laid down) that its tensor is, 
 
 T.t9 = Tj3 Ta-i Ty; 
 while its versor is (by 188), 
 
 U.;O=Ut.U0; 
 
 where Ut and \J9 are allowed no variety of values, except that 
 which arises from their freedom to change their signs (or to re- 
 verse their directions) together, a change which will not alter 
 their pro(/wc^. For rj (by 213) is coplanar with a, y, and is also 
 perpendicular to j3 ; and j3 is not perpendicular to the plane of a, 7, 
 because it is not now supposed to be perpendicular even to a, 
 since otherwise we might at once employ the reasoning of art. 
 204, to establish the associative property : whence Uij must be 
 equal to one or other of two determined and opposite vector- 
 
"* LECTUIIE V. 211 
 
 units, because it must be parallel to the intersection of a plane 
 perpendicular to j3, with a plane parallel to both a and 7. But 
 
 < = /3 -7- ??; 9 = 'ya'^ . jj; 
 
 and therefore (see 188, 129), 
 
 Ut=Ui3--U^; Ue = {Uy ~.Ua)x Urj; 
 
 whichever, then, of the two determined values just now men- 
 tioned, we assume for U»j, we get a corresponding' pair of deter- 
 mined values for Ut and U6 ; and these three last vector-units 
 can do no more than change all their three sigtis together. The 
 value of the quaternion i9 is therefore entirely determined, because 
 the values of its tensor and its versor are so. This reasoning may 
 be usefully compared with the corresponding process in art. 180; 
 and it may serve to illustrate and confirm a remark made in art. 
 108, respecting the determinate nature oi quaternion multipli- 
 cation generally. 
 
 215. By a process quite similar, but applied to the equations 
 of 212, or to the quaternion fiK~^, we find first that the tensor of 
 this quaternion is determinate, because its value is 
 
 T.^tK-i = T/3Ta-iT7; 
 
 and that its versor is also determinate, as being the quotient of 
 two other versors, U/x and U/c, which can only change their 
 signs together. For X is coplanar with a and j3, and is also per- 
 pendicular to y, which is not nov/ supposed to be perpendicular 
 even to a, and therefore not to the plane of a and /3 ; UX must 
 therefore (like Urj) be equal to one or other of two determined 
 and opposite vector-units ; but whichever of these two values we 
 select for UX, the equations 
 
 U7=UX-f-UK, U|3 -- Ua = U^ -- UX, 
 
 derived from 212, will assign connected and determinate values 
 for Uk and U^ ; and the three vector-units Uk, UX, Ujw, are 
 only free to change their signs together. The versor and qua- 
 ternion, 
 
 U^ 4- Uic, and ^ -f k, 
 
 are therefore entirely determined, under the conditions here sup- 
 
 p 2 
 
212 ON QUATERNIONS. 
 
 posed. And there would be no difficulty in adapting (if required) 
 the reasoning of the two last articles to the cases (recently ex- 
 eluded), where 
 
 y ± a, or /3 ± a ; 
 
 which cases admit, however, as we have seen (in 203, 204), of 
 being each treated in a simpler way, as regards the proof of the 
 associative property. 
 
 216. The quaternions juk"^ and i6 (of arts. 212, 213) having 
 thus been seen to be each separately determinate, and to have 
 their tensors equal, it remains to shew that their versors are also 
 equal, in order to establish generally this associative property of 
 multiplication, so far as any three vectors are concerned. And 
 for this purpose it is clear that we need deal only with vector- 
 units ; or that we may assume, 
 
 Ta = T/3 = Ty = Tf = Tr, = T0 = Tk = TA= T^= 1. 
 
 We may therefore regard these nine vectors, 
 
 a, /3, 7, t, rj, 0, K, A, fx^ 
 
 as being so many radii of one common unit-sphere ; because they 
 may be conceived to begin all at one common origin o, namely, 
 at the centre of the sphere (compare 168); although they must 
 then in general be supposed to terminate at nine different points, 
 upon the common spheric surface, which points we shall here 
 mark, respectively, by the nine letters, 
 
 A, B, C, I, H, G, K, L, M : 
 
 in such a way that (for example) the angles of the versors (or 
 quaternions) j3a"^ and juk"^ shall (by this construction) coincide 
 with the angles aob, kom, at the centre of the sphere ; and shall 
 be represented, as to the corresponding amounts and directions 
 of rotation, by the arcs of great circles, ab and km, upon the 
 surface.^^Let us then proceed to construct the versor juk'^, by 
 constructing its representative arc, km, with the aid of some 
 simple principles of spherical geometry. 
 
 217. In general let r, q, r, s denote any four points upon the 
 surface of the unit-sphere, o being still the centre; and let q, r 
 
 \ 
 
LECTURE V. 
 
 213 
 
 denote the two following quaternions, or versors, with pq and rs 
 for their representative ares, 
 
 g'=(Q-o)-r-(p-o), /• = (s-o)-t-(r-o). 
 
 Then in order to construct, by a new representative arc, tu, the 
 product, rq, which is obtained when the former of these two ver- 
 sors is multiplied by the latter, we may (compare 49, 108, 179) 
 proceed as follows. Prolong if necessary, as in fig. 36, the 
 two given representative 
 arcs, PQ, RS, till they 
 meet in a point l upon 
 the surface of the sphere. 
 On the great circle pql 
 take a new point k, so 
 as to satisfy the equa- 
 tion 
 
 <^KL = 
 
 PQ, 
 
 which is designed to denote that the arc from k to l has not only 
 the same length, but also the same direction, as the given arc 
 from p to Q : this sameness of direction of two arcs being con- 
 ceived always to include the condition of their being parts of one 
 great circle. Again, on the great circle rls take another new 
 point M, such that 
 
 '-sLM = ^RS, 
 
 with the same full signification of equality of arcs as before. 
 Finally join the points k, m, by a great circle, and take thereon 
 at pleasure any two new points t and u, such that 
 
 ^TU =->KM. 
 
 Then we shall have the equation, 
 
 rq={\j-o) ^ (t-o); 
 
 or in other words, the arc km, or its equal tu, may be taken as 
 the representative arc of the required product, namely, the ver- 
 sor or quaternion rq. In fact either of these two equal arcs, km 
 or TU, may represent in this question (compare Q5) the transver- 
 sor, rq, the arcs kl and lm at the same time representing re- 
 
214 
 
 ON QUATERNIONS. 
 
 spectively the versor, q, and the proveisor, r, in this multiplica- 
 tion ofversors, or composition of versions or rotations. And it 
 seems that we may not inconveniently say, that the versor, pro- 
 versor, and transversor, of the Second Lecture, are now repre- 
 sented on the unit sphere, by a vector arc, kl, a provector arc, 
 LM, and a transvector arc, km, respectively. (Compare Lec- 
 ture L) 
 
 218. It may be noticed here that the foregoing process, when 
 combined with the principle (188) respecting the tensor of a pro- 
 duct, serves to accomplish generally, by the aid of arcs upon a 
 sphere, the multiplication of any two quaternions. Lideed if we 
 compare the recent figure 36 with fig. 7 of art. 53, we find that 
 Tve have only to conceive the centre o of the sphere to coincide 
 with the vertex d of the pyramid, and the edges da, db, dc, of 
 the pyramid to meet the spheric surface in the points k, l, m. 
 And the recently suggested analogy of tnultiplication ofver- 
 sors, to what may be called addition ofarcual vectors, appears 
 to be well worthy of attention; a quaternion product being (as 
 we have seen) represented by an arcual sum, if we agree to say, 
 for arcs as for lines (see 31), that " Provector, plus Vector, 
 equals Transvector." 
 
 219. The construction in art. 217 may serve to illustrate 
 some general properties of quaternion multiplication. Thus, if, 
 as in fig. 37, we pro- 
 
 long the arcs kl and 
 
 ML to k' 
 
 and m', so 
 
 as to have the equa- 
 
 tions, 
 
 
 ^ kl 
 
 = -- lk', 
 
 ■^ m'l 
 
 = - LM, 
 
 the arcs kk' and m'm 
 
 thus bisecting each 
 
 other in the point l ; 
 
 and if we still conceive that kl and lm are representative arcs of 
 
 the versors q and r, so that lk' and m'l shall also admit of being 
 
 regarded as representative arcs of the same two quaternions : 
 
 then, while the arc km will still represent the former product r^-, 
 
LECTURE V. 215 
 
 it will on the contrary be the are m'k' which shall represent, on 
 the same plan, the product qr^ of the same two factors, r and q^ 
 taken now in the contrary order. And because the two arcs km 
 and m'k', which thus represent these two products, rq and qr^ are 
 indeed equally long, but are portions of different great circles^ we 
 must not assert that they are equal, in that full sense of ar- 
 CUAL equality, which was employed in art. 217. We have, 
 therefore, the following inequality of arcs ; 
 
 ^ m'k' not = '- KM, 
 
 under the circumstances of fig. 37, when the directions, and con- 
 sequently the PLANES, of the arcs are to be compared ; or when 
 (see 93, 94) the aspects of the two corresponding biradials, 
 m'ok' and kom, are taken into account, o being still the centre 
 of the sphere. We arrive then thus anew at the following m- 
 equality of versors, which involves, as a consequence, the cor- 
 responding inequality of the two quaternions, which are denoted 
 by the same two symbols : 
 
 qr not generally = rq. 
 
 And thus we are conducted again to the important and remark- 
 able conclusion, that the multiplication of quaternions is not ge- 
 nerally a commutative operation : which result has, at least par- 
 tially, presented itself in many former articles. (Compare 74, 81, 
 82, 89, 112, 121, 133, 134, 135, 189, 207, 209, 210.) 
 
 220. In the same figure 37, the arc lk, or k'l, will represent 
 the reciprocal, q'^, of the quaternion or versor q, this reciprocal 
 being regarded as a reversor (compare 44, 89, 136); while k'm 
 will represent the product rq~^, on the recent plan of construction 
 for multiplication of quaternions ; and the triangle k'lm shews, 
 when employed on the same general plan of art. 217, that (as in 
 algebra) the following identity holds good : 
 
 rq'^.q = r. 
 
 But also, by art. 50, we have, as an identity, 
 
 {r~q)xq = r; 
 
 equating then these two last expressions for r, we arrive at this 
 other identity (compare 118) : 
 
216 ON QUATERNIONS. 
 
 r -^ q = rq'^. 
 
 We know then how to construct the quotient of any two versors, 
 and therefore also (by the principle respecting quotients of ten- 
 sors in art, 188) the quotient of any two quaternions ; namely, by 
 constructing its representative arc upon the unit-sphere : which 
 may be done (as we see) by first representing the dividend r, and 
 the divisor q, by two co-initial arcs of great circles, such as lm 
 and lk'; and then drawing a third arc k'm, to represent the quo- 
 tient, from the end of the arc which represents the divisor, to the 
 end of that other arc which represents the dividend. In short we 
 can thus (compare 36) recover the provector arc k'm, by a spe- 
 cies of ARCUAL SUBTRACTION, from the given vector and trans- 
 vector arcs, lk' and lm; and can thereby recover the pro- 
 VERSOR, rq~^, considered as a profactoi', when the versor and 
 transversor, which are here q and r, are given as factor and trans- 
 factor. But such a RETURN TO THE MULTIPLIER (in this case a 
 proversor, rq'^, regarded as a profactor), when the multiplicand 
 (in this case, q) and the product (in this case, /*) are given, is pre- 
 cisely that OPERATION, to which, in this calculus, by an extension 
 of a received phraseology, the name of Division has been as- 
 signed : whether the proposed multiplicand and product, regarded 
 thus as divisor and dividend, be simply vectors (as in 40, 41), or 
 quaternions, considered as/actors (as in 50, 54, 56). 
 
 221. It must not be forgotten that in consequence of the (ge- 
 nerally) non-commutative property (219, &c.) of quaternion mul- 
 tiplication, the product q'^r is not to be confounded with the 
 product rq~^ ; and is therefore not to be equated generally to the 
 quotient r -^ q, to which the last mentioned product {rq'^) has 
 recently been seen to be equal. In fact, this new product, q'^r, 
 would be represented, in fig. 37, by the arc m'k; but this latter 
 arc does not generally belong to the saine great circle as the arc 
 k'm, which has been seen, in art. 220, to represent rq~^, or r -r- q. 
 (Compare 219.) What is to be understood generally, by such 
 symbols as q''^r . q, or rqr'''-, will be an important subject for 
 discussion, at a subsequent stage of our inquiries. 
 
 222. The two co-initial arcs kl and km, in the same figure 
 37, might be employed, by the recent construction (220) for di- 
 
LECTURE V. 
 
 2ir 
 
 \ vision of quaternions, to put in evidence this other general rela- 
 \ tion between multiplication and division (compare art. 50) : 
 
 rq -^ q = r. 
 
 The identity of art. 192, namely, 
 
 {rq)'^ =q~'^ r'^, 
 
 may be illustrated by considering ml, lk, and mk, as an arcual 
 system of vector, provector, and transvector. Or if we choose to 
 consider conjugates rather than reciprocals of quaternions, we can 
 easily employ the construction of art. 217, to prove anew the 
 analogous theorem of art. 190, as in the annexed figure 38, where 
 the curved arrows are de- pj gg 
 
 signed to remind us that (ab- 
 stracting from the tensors) the 
 conjugates K^' and Kr may 
 be regarded as equivalent (by 
 89) to the reversors, which 
 answer to the two given ver- 
 sors, q and r. For the figure 
 shews that K^'. Kr, or that 
 the product of the two conjugates, taken in an inverted order, 
 is represented by an arc mk, which has the same length as the 
 arc KM, and is part of the same great circle, but has an exactly 
 opposite direction, and represents therefore the conjugate of the 
 product rq, which latter product is represented by the arc km it- 
 self. We are therefore again led to write, as in 190, the gene- 
 ral equation, or identity, 
 
 K . rg' = K^^ . Kr, 
 
 which is frequently useful in this calculus. 
 
 223. After these remarks on certain modes of representing 
 generally, hy spherical constructions (compare 121), the products 
 and quotients of quaternions, and some other things connected 
 therewith, let us now resume the problem proposed at the end of 
 art. 216 ; namely, to construct the representative arc km, of that 
 particular fourth proportional, or quaternion product, jSa'^.y, 
 which was considered in 211 and 212; the three unit-vectors a, 
 
218 
 
 ON QUATERNIONS. 
 
 j3, 7, that enter into its composition, being supposed (as in 216) 
 to radiate from a known and common origin o, and to terminate 
 at three given points, a, b, c, upon the surface of the unit sphere. 
 And whereas, we have already considered specially, in connexion 
 with the associative property, the cases (203, 204) where a is 
 perpendicular to j3 or toy, or, in other words, where one of the arcs 
 AB, AC is quadrantal, we shall now begin by supposing, for the 
 sake of simplicity, and in order to fix our thoughts, that each of 
 the three sides of the spherical triangle abc is an arc less than a 
 quadrant. Let us also imagine, for the purpose of making our 
 conception of the question still more completely definite, with 
 the aid of astronomical illustrations, that a and b are points on 
 the ecliptic of an ordinary celestial globe, with longitudes respec- 
 tively equal to 100° and to 70°; while c shall be that point of 
 the equator of the same globe, which has its right ascension 
 equal to six hours, or to 90°, as in the following diagram (fig. 39). 
 It is required then, under 
 these conditions, to con- 
 struct an arc km, which 
 shall represent, as to 
 amount and direction of 
 rotation, that sought qua- 
 ternion, or versor, which is the fourth proportional to the three 
 directed radii, or unit-vectors, oa, ob, oc; o being the centre 
 of the globe, and the length of each radius being unity. 
 
 224. For this purpose, I form the annexed figure 40, which 
 is designed^to be an ortho- 
 graphic projection of one 
 quarter of the globe, on 
 the plane of the equinoc- 
 tial colure; a, b, c being 
 still placed at points cor- 
 responding to those of the 
 recent and simpler figure 
 
 39 ; but the letters, l, q, L' d C E N L 
 
 and l' being now written, for convenience, instead of the as- 
 tronomical marks ^, 25, and V in that figure; and the letter 
 K being employed to mark the place of the north pole of the 
 
LECTURE V. 219 
 
 equator, so that cl, ck, and kl are quadrants, respectively, of 
 the equator, and of the solstitial and equinoctial colures. Now 
 this latter quadrant, kl, may be taken as the representative arc 
 of the multiplicand, -y, in the proposed product j3a"^ . 7, this 
 vector 7, or oc, being regarded, by our general principles (art. 
 122, &c.), as a quadrantal quaternion ; while the arc ab repre- 
 sents, on the same general plan of art. 216, the multiplier, (5q'^, 
 or OB -f- OA, regarded as another quaternion. And although this 
 last mentioned arc, ab, does not imtnediateli/, or in its actual and 
 present situation, begin where the arc kl ends, yet it can easily 
 be MADE to begin there (compare 99), without any alteration of 
 its value, or significance, as representing one dejinite versor: 
 namely, by causing (or conceiving) it to tur7i in its own plane, 
 or on the great circle to which it belongs, till it comes to take a 
 new position, such as that denoted in the figure by lm, begin- 
 ning now, as a provector arc (217), at the point l, where the 
 vector arc kl ends, and satisfying the arcual equality, 
 
 ^ LM = --^ AB . 
 
 And then by simply drawing the transvector arc of north polar 
 distance, km, from the point k where the vector arc kl begins, 
 to that new point m where the new ov prepared provector arc lm 
 ends, we shall have accomplished the construction which it was* 
 required to eiFect. For the arc km, thus drawn, will represent, 
 on the general principles already explained, that sought quater- 
 nion, fxK'^, which is, with the here supposed directions of the vec- 
 tor-units, the value 0/ the product [5a'^ . 7, or of what we have 
 already called, by analogy, the fourth proportional to the three 
 vectors, a, /3, 7- 
 
 225. Before proceeding to compare this arc km with any other 
 arc, as respects their equality or inequality, it will be useful to 
 determine its pole, and to construct thereat an equivalent sphe- 
 BiCAL angle; because we shall thus, in a new way, have con- 
 structed or determined the quaternion, or versor, j3a'^ .7, by as- 
 signing its axis, and its angle. For this purpose we need only 
 prolong (in fig. 40) the arc of north polar distance, km, till it 
 meets the equator in n ; and then take a new point d on the 
 
220 ON QUATERNIONS. 
 
 same equator, which shall satisfy the arcual equality (compare 
 217), 
 
 ^CD = -^LN ; 
 
 for then the arc nb will be a quadrant, and d will be the sought 
 pole of KM. The arc md being thus another quadrant, if we 
 oblige MR to become a quadrant also, by taking the point r upon 
 the ecliptic so as to satisfy the equation 
 
 '- QR = '-- LM, 
 
 M will be the pole of the arc dr, and the angles mdr, mrd will 
 be right. But kdn is also a right angle, kd being a quadrant of 
 north polar distance ; wherefore 
 
 RDK = MDN, and l'dR = KDM. 
 
 We may then take the spherical angle l'dr, or its equal, kdm, 
 as the representative angle of the quaternion jSa'^.y, or of 
 its equal fiK~^ ; because not merely is each of these two spherical 
 angles equal in amount to the angle or amplitude of the quater- 
 nion, so as to satisfy the quantitative or metric equation, 
 
 Z (j3a"^ . 7) = l'dR = KDM, 
 
 but also the axis of the same quaternion is the radius od, drawn 
 towards the vertex o of the same angle on the spheric surface, 
 in such a manner that m'c may establish also the following direc- 
 tional or graphic formula, 
 
 Ax . (j3a'^ . 7) = D -o. 
 
 226. Let E be a new point on the equator, such that 
 
 -- EC='-CD, 
 
 and from this point e let there be drawn the arc of latitude, or 
 perpendicular on the ecliptic, es. The right-angled triangles, 
 LSE, l'rd, shew evidently that the arcs es and dr are equally 
 long, or that the points e and d have their two south latitudes 
 equal ; they shew also that 
 
 '^ls^'^rl'; and -^ sq = -^ qr. 
 But by 225, 224, 
 
 '- QR = '^ LM = -- AB ; 
 
LECTURE V. 221 
 
 thus 
 
 '-^ SR = 2 X -- AB, 
 
 and 
 
 -SA+^BR = ^AB = ^AT + -- TB, 
 
 whatever new point t may be chosen upon the arc ab. We can 
 therefore so choose this point, as to have, at once, 
 
 - SA = -- AT, and - BR = '^ TB. 
 
 And then by erecting at t a perpendicular tf to the ecliptic, to- 
 wards the northern side, and equal in length to either of the two 
 former perpendiculars, dr or es, so that the no?-th latitude of the 
 point F shall be equal in amount to the south latitude of d or e, 
 the two pairs of right-angled triangles, drb, ftb, and esa, fta, 
 will shew that the opposite angles at b are equal in one pair, and 
 those at a in the other pair ; and also that, in each pair, the two 
 hypotenusal arcs are equal: from which it follows that if f be 
 joined by arcs of great circles to d and e, these joining arcs shall 
 pass through the points b and a, and shall be bisected at those 
 points. The vertex of the representative angle, l'dr (225), of the 
 quaternion (Ba'^ . y, which is the fourth proportional to the three 
 unit-vectors, a, /3, 7, that are drawn from the centre o of the 
 sphere to the three given points, a, b, c, on the same unit-sphere, is 
 therefore situated at a corner b of a certain new spherical trian- 
 gle, DEF, whose SIDES, EF, FD, DE, are respectively bisected by 
 the three corners of the given (or old) spherical triangle, abc. 
 And the choice o/this particular corner, d, as distinguished 
 from the two other new corners E and F, is seen to be determined 
 by the condition, that it shall be opposite to that side, ef, of the 
 new triangle, which is bisected by the first corner, a, of the 
 given triangle, abc ; or by the frst (namely, at present, a) of 
 the three given vector-units. 
 
 227. A not less simple rule for geometrically connecting the 
 ANGLE (as well as the axis) of the quaternion, j3a"^ . 7, with the 
 new triangle def, circumscribed according to the recent law 
 about the old or given triangle abc, or for constructing the mag- 
 nitude (as well as the situation) of the representative angle, l'dr. 
 
222 
 
 ON QUATERNIONS. 
 
 may be investig'ated in the following way. Let figure 41 be con- 
 ceived to denote the southern hemisphere 
 of latitude (of a celestial globe), projected 
 orthographically upon the plane of the 
 ecliptic, of which great circle the south 
 pole is denoted in the figure by p ; a', b', 
 f', in the same figure, denoting the points l1 
 diametrically opposite to a, b, f ; and 
 the other letters, a, b, c, d, e, l, l', q, 
 R, s, retaining their recent significa- 
 tions. Then, because the three points d, -^ 
 E, f' have equal southern latitudes, they are all contained on one 
 small circle, described about p as a pole, and parallel to the 
 ecliptic, or (in the figure) concentric therewith. We wish to ob- 
 tain some simple and convenient expression for the angle l'dr, 
 or for its vertically opposite angle, cdp. Now this last is one of 
 the base-angles of an isosceles spherical triangle, namely, of the 
 triangle dpe ; and each of the adjacent triangles, dpf', epf', is evi- 
 dently also isosceles. If then, in the triangle def', we deduct the 
 angle at f' from the sum of the two angles at d and e, the half 
 of the remainder will be the angle required. But in the hme ff' 
 (only partially pictured in the figure), the opposite angles at f 
 and f' are equal ; so that the angle at r, in the triangle def, 
 is equal to the angle at f', in the triangle def'. On the 
 other hand, the angles at d and e, in one of these two tri- 
 angles, are supplementary to the angles at the same two points 
 in the other. We are then to subtract the sum of the three an- 
 gles of the triangle def from ^owr right angles, and afterwards 
 to halve the remainder. And thus we find that the angle l'dr 
 or CDP, of the quaternion ivhich is the fourth proportional to the 
 three unit-vectors, oa, ob, oc, ivhich respectively bisect the three 
 sides, EF, FD, DE, of a spherical triangle def, is equal (at least 
 under the conditions lately considered) to the supplement of 
 the semisum of the angles of the triangle tvhose sides are so 
 bisected: or in symbols that (in this recent case), 
 
 Z(j3a-l.7) = 7r-i(Z> + ^ + i^). 
 
 228. It must however be observed, that by arranging the 
 
LECTURE V. 223 
 
 three points, a, b, c, as in the recent figures, we have tacitly 
 supposed that the rotation round a from j3 towards -y, or that the 
 rotation round OAfrom ob towards oc, is negative or left-handed. 
 And thus it happened that, in fig. 40, after going by a vector arc, 
 KLj from the north pole of the equator to the autumnal equinoc- 
 tial point, we went next along the ecliptic, by a provector arc, 
 LM, through thirty degrees of longitude, but in a direction con- 
 trary (in astronomical parlance) to the order of the signs, thereby 
 RETROGRADING from Libra to Virgo, and consequently approach- 
 ing to the north pole k of the equator, from which we had at first 
 set out. This was the reason for the transvector arc, km, being 
 found to be less than a quadrant, under the conditions lately con- 
 sidered. Had the rotation in the ecliptic, corresponding to the 
 proversor, |3a"^, been supposed to Redirect, instead of being re- 
 trograde, the result would, in this respect, have been different ; 
 for we should have gone, in the arcual provection upon the 
 spheric surface, still farther from the north pole than we had 
 done, in arriving, by the first vection, at the autumnal equinoc- 
 tial point ; and the arc of transvection would have been found to 
 be, in that case, greater than a quadrant. 
 
 229. For example, if, without making any change in the sig- 
 nifications of the letters lately employed, we now propose to our- 
 selves to determine the axis and angle of the following new qua- 
 ternion, 
 
 a/3-1.7; 
 or if we seek the fourth proportional to the three former unit- 
 vectors, in the new order j3, a, 7, and not now in the order 
 a, j3, 7 : we shall be led to advance (according to the order of 
 the signs of the zodiac) from Libra to Scorpio, or (by the provec- 
 tion) from L to a new point m', not opposite on the sphere to m, 
 but such that (compare fig. 37), 
 
 ^ lm'^"^ ml ^'^ ba ; 
 and the transvector arc will now be 
 
 km' >-, although KM <—. 
 
 In fact it is clear that the two transvector arcs, km and km', 
 which are also the representative arcs of the two quaternions 
 
224 ON QUATERNIONS. 
 
 j3a"^ . 7 and a(5'^ .7, are, in amount, supplementary to each 
 other ; so that if we attend only to the magnitudes of these two 
 arcs, we may write 
 
 KM'=7r - km; 
 
 or, passing to the angles of the two quaternions which corres- 
 pond, 
 
 Z(aj3-i.7)-7r-Z(j3a-^7). 
 
 But if we attend also to the planes, or poles of the arcs, or to the 
 axes of the two quaternions, we see easily (on the plan of art. 
 225), that the pole of the arc km' is the point e, and that, there- 
 fore, we may write, 
 
 Ax . (a/3~^ . 7) = e - o. 
 
 230. Still we perceive that the rule of art. 226 holds good, 
 since the pole or point e, thus determined, is (as the rule re- 
 quires) that corner of the circumscribed triangle def, the side 
 opposite to which (namely fd) is bisected by the extremity (at 
 present b) of what is now the first (namely /3) of the three given 
 unit-vectors (j3,a,7). That rule of 226, for the direction of the axis 
 of the quaternion, is therefore seen to be independent of the order 
 of the rotation of those vectors among themselves : although, as 
 we shall presently see, this order of rotation is not in all respects 
 indifferent to the result. For it is easy to perceive, from what 
 has been already shewn, that the spherical angle ces, in fig. 40, 
 may be taken as the representative angle of the quaternion 
 aj3'^ • 7 5 ^"d hence it follows (by the reasonings in 227) that we 
 may write, 
 
 z(«/3-i.7) = i(D + i'+jF'); 
 
 the SEMisuM ITSELF of the angles of the triangle def, or the 
 SUPPLEMENT of that semisum, being thus equal to the angle of 
 THE FOURTH PROPORTIONAL to the three bisecting vectors, ac- 
 cording as the ROTATION round the first of them (in the recent 
 case j3), from the second (in this last case a)", towards the third 
 (7), is POSITIVE or NEGATIVE. It is to be remembered that the 
 arcs AB, Bc, CA, or the angles between a, j3, 7, have been sup- 
 posed (in art. 223) to be all less than quadrants, or than right 
 
 1 
 
LECTURE V. 225 
 
 angles, with a view to avoiding, at first, any complex modifix?a- 
 tions of the figures. 
 
 231. Retaining still for simplicity this restriction on the 
 sides of the given triangle abc, we may proceed to prove, as 
 follows, that the problem of circumscribing about it another tri- 
 angle DEF, whose sides shall be bisected by its corners, is not 
 merely (what has been already proved, in arts. 225, 226) a pos- 
 sible problem, but also one entirely determinate, at least if we 
 attend only to those spherical triangles which have (as is usual) 
 their sides each less than a semicircle. Conceive then, conversely, 
 that three points a, b, c, at distances from each other which are 
 each less than 90°, are given as the middle points of the sides 
 EF, FD, DE, of a triangle def ; and let us study some of the re- 
 lations which connect the two triangles abc, def together, with 
 a view to inquiring whether any other triangle, such as d'eV\ 
 would admit of being substituted for the given def, without 
 change of abc. 
 
 232. Now, for this purpose, it seems sufficient to observe, 
 that if f' be the point diametrically opposite to f, the small cir- 
 cle def' must always (as in fig. 41, art. 227) be parallel to the 
 great circle ab, having a common pole therewith, which pole we 
 may still call p ; and that, therefore, the bisecting perpendicular 
 PC, of the arc de, must always cross the great circle ab likewise 
 at right angles. For hence it follows, that if we let fall a per- 
 pendicular arc CQ on ab from c, and then through c draw a great 
 circle perpendicular to cq, this last great circle must contain not 
 merely (as in figs. 40, 41) the points d and e already considered, 
 but any others, if such there be, which can be substituted for 
 them.. In like manner the points e and f, or any substitutes for 
 them, must be situated on that great circle through a, which is 
 perpendicular to the arc let fall perpendicularly from a on bc ; 
 and F and d must be on that other great circle, which is drawn 
 through B, at right angles to the perpendicular arc let fall on CA 
 from B. Thus we have three great circles, entirely determined 
 in position, which must intersect, two by two, in the three points 
 D, E, F ; and if any other points admit of being substituted, in 
 whole or in part, for these, as corners of the triangle whose sides 
 are to be bisected, they can only be the opposite intersections of 
 
226 ON QUATERNIONS. 
 
 the three great circles found as above, or the points d', e', f', 
 which are diametrically opposite to the former points d, e, f. 
 
 233. But two successive and supplementary arcs of the same 
 great semicircle cannot both be bisected by any common point ; 
 we cannot, therefore, make any partial change of the given 
 points, D, E, F, to their opposites, consistently with the conditions 
 of the question : for example, the arcs df', ef', in fig. 41, are not, 
 like the arcs df, ef, of fig. 40, bisected by the points b, a. And 
 if we make a total change of d, e, f, to the three opposite points, 
 d', e', f', we shall indeed have altered the triangle def to another, 
 namely d'e'f', such that the three following arcual equations shall 
 hold good : 
 
 ^ e'a = - af' ; - f'b = - bd' ; -V d'c = - ce' ; 
 
 but the sides e'f', f'd', d'e', of this new triangle, if, as is usual 
 and as we lately (in 231) agreed to do, we measure these three 
 sides so as to be each less than a semicircle, will not (in the 
 strictest and simplest sense of the words, which is the sense at 
 present under consideration) be bisected by the three points a, 
 B, c, BUT by the three respectively and diametrically opposite 
 points, that is, by the three points a', b', c'. The triangle abc 
 being then given and fixed, the triangle def is also deter- 
 mined, without any ambiguity whatever, under the conditions 
 lately supposed. Under certain o^Act" conditions, it will be shewn 
 hereafter that a different result may take place. 
 
 234. If then we were to propose to ourselves to investigate 
 the value of the fourth proportional to the same three given 
 unit-vectors as before, but taken now in the new order, a, 7, /3; 
 or (in other words) if we should seek to construct the represen- 
 tative arc, or representative angle, of the following new quater- 
 nion, 
 
 it is clear that we should be led, on the plan of recent articles 
 (225, '^2^, 229, 230), to circumscribe, q}oowX.\\\q same given trian- 
 gle ABC, the SAME auxiliary triangle, def, as before. And be- 
 cause what is now the Jirst of the three given vectors, namely a, 
 or OA, bisects that side, namely ef, of the auxiliary (or circum- 
 scribed) triangle which is opposite to the point d ; while the ro- 
 
LECTURE V. 227 
 
 tation round a from -y towards j3 is positive ; it follows, from the 
 rules laid down in articles 226, 230, that the axis of the new 
 quaternion, proposed for consideration in the present article, is 
 directed towards the point d, and that the angle of the same qua- 
 ternion (7a "1 . j3) is equal to the semisum itself (and not to the 
 supplement of the semisum) of the three angles of the spherical 
 triangle def. In symbols, under the conditions supposed, the 
 two following equations, or formulae, hold good : 
 
 Ax. ('ya'^ . j3) = D - o ; 
 
 As the representative a?igle of the new quaternion ya'^ . j3, we 
 may take the spherical angle rdc in fig. 40 (art. 224) ; and there 
 would be no difficulty in hence constructing, if it were required, 
 the representative arc also. 
 
 235. Comparing now the expressions (in 225, 227, 234), for 
 the axes and the angles of the two quaternions, 
 
 /3a"^ . y, and ya'^ . j3, 
 
 we find that there exist the following relations between them, 
 
 Ax.(7a-'.i3) = Ax.(/3a-^7); 
 Z(7a-i.j3)-7r-Z(/3a-i.7); 
 
 the axes being thus coincident, and the angles being supple- 
 mentary. But these are the very relations which, as was shewn 
 in art. 185, and as was illustrated by figure 32 of art. 186, exist 
 generally between 
 
 q and - Kg, 
 
 or between a quaternion and the negative of the conjugate thereof, 
 so far as axes and angles are concerned. And the only remain- 
 ing relation, between two such quaternions, namely the equality 
 of their tensors (185), exists here also, because each tensor is 
 unity. We are then entitled to establish, at least under the con- 
 ditions above supposed, the formula, 
 
 j3a-i.7 = -K(7a-^i3). 
 Q 2 
 
228 
 
 ON QUATERNIONS. 
 
 But when we come to transform the second member of this for- 
 mula, by the principles of art. 193, we find that it becomes, 
 
 -K{ya-K(5)=[i.a-^y. 
 
 We are then led to establish anew, under circumstances more 
 general than before, that associative formula of multiplication 
 of three vectors, which has been the principal subject of investiga- 
 tion during the whole of the present Lecture : namely, 
 
 /Ba"^ .y = (5 . a'^y. 
 
 236. In this method of treating the question, we have not 
 found it necessary to construct that other quaternion, or its re- 
 presentative arc, which was mentioned in art. 213 ; namely the 
 quaternion denoted in that article by the symbol i9. There would, 
 however, have been no difficulty in constructing its arc also, if 
 required. To shew this, conceive that the annexed diagram (fig. 
 42) is an orthographic projection of a he- 
 misphere with B for its visible pole, while 
 X denotes the pole of the great circle ac ; 
 the letters a, b, c, d, e, f, still denoting 
 the same points as before, and i, i' being 
 
 the positive and negative poles of the cir-H [ { ^-/("'""""t^ """t^j^^ '^ 1 h' 
 cle FED, while h, h' are the two poles of 
 the circle i'bxi ; let us also conceive the 
 arc EX to be prolonged, till it terminates, 
 on the other hemisphere, in a point e', 
 
 diametrically opposite to e : and let the arcs xb, xd, prolonged, 
 meet the great circle hach' in two other points, y and z. Then 
 taking another new point g on the circle ac, such that 
 
 Fig 
 
 42. 
 
 r 
 
 y^K^ 
 
 y\c 7\ 
 
 fw^-- 
 
 
 [ i,J-7\' B 
 
 ■ f A"^^ / 
 
 
 %J^ 
 
 GH = -- CA 
 
 we shall be at liberty to write, on the plan of 216, 
 
 G-O 
 
 H-o = i7; I -o = £ ; 
 
 and may (by 213, &c.) regard the arcs gh and ih (or hi') as re- 
 presenting, respectively, the versor rj'^0 (or a'^y), and the pro- 
 versor iy\ (or j3) ; whence it will follow that the transversor, lO 
 (or )3 . a'^y), is represented, in the same construction, by the arc 
 
LECTURE V. - 229 
 
 Gi'. But it is easy to prove, by methods recently explained, that 
 the pole of this new are gi' is the point d, and that the amount of 
 the equivalent an^fe gdi', orzDH', orxDB, at that pole, is equal to 
 the supplement of the semisum of the three angles of the spheri- 
 cal triangle def ; which last equality may be established by the 
 help of the lune ee', and of the three isosceles triangles fxd, dxe', 
 e'xf ; the quadrant I'j through g is also useful. Hence by com- 
 parison with fig. 40, and with the results of arts. 225, 227, we 
 should find ourselves entitled to infer the arcual equation, 
 
 - Gi' = - KM ; 
 
 and on passing from these representative arcs to their versors, 
 we should thus have proved the equation proposed for inquiry at 
 the end of art. 213, namely, 
 
 or, by that article, and by the one immediately preceding it, we 
 should have thus arrived anew at the associative formula of mul- 
 tiplication of three vectors, 
 
 237. The case where ab is a quadrant, or where j3 J_ a, has 
 been considered in 204 ; yet, if we wished to examine how our 
 recent and more general investigations may adapt themselves to 
 that case as a limit, we might conceive, in fig. 40, that the equal 
 arcs AB and lm are each only a very little less than 90°. Under 
 this supposition, the point m would almost coincide with q ; n 
 with c ; p and r with l'; e and s with l ; and r with t, this new 
 point T being such as almost to satisfy the connected equations, 
 
 -v LA = -- AT, -- TB = '^ Bl'. 
 
 At the same time the triangle def would tend to coincide with 
 the lune l'l ; the angle at f would be almost = tt, and each of 
 the angles at d and e would almost coincide with an angle of 
 that lune ; and therefore the supplement of the semisum of the 
 three angles of the triangle would tend to become equal to the 
 complement of the angle of the lune. We may therefore expect, 
 from our recent results, to find that as /3 tends to become per- 
 
230 ' ON QUATERNIONS. 
 
 pendicular to a, the fourth proportional ^a'^y (in which symbol 
 we do not here think it necessary to write the point) tends to be- 
 come a quaternion, whose axis is directed towards the point l' 
 (in fig. 40), and whose an^le is the complement of the angle 
 ql'c ; or in other words that the angle kl'q, or the arc kq, re- 
 presents this limit-quaternion. And accordingly it may easily 
 be shewn that this result agrees perfectly with the conclusions of 
 art. 204 ; the line, which was there called A, being now conceived 
 (in connexion with fig. 40) to be directed towards the north pole 
 of the ecliptic; and the rotation from tins pole to the point c 
 being similar in direction, and supplementary in amount, to the 
 rotation from k to q, as by our general principles of interpreta- 
 tion of the quaternion product Ay, obtained in 204, it ought to 
 be. (Compare the general construction for a product of two vec- 
 tors in 88 ; also the value of the product lQ, in the recent article 
 236.) 
 
 238. Let us now consider (although more briefly) the case 
 where the arc ab is greater than a quadrant; this arc being still 
 conceived to form part of the semicircle i/ql, in fig. 40, and the 
 point A being still advanced beyond b, in the order of right- 
 handed rotation round c. We may conceive, for instance, that 
 the longitudes of a and b are now respectively, 160° and 40°; 
 the points c, k, l, l', q, retaining their positions in the figure. 
 The points m and n, determined on the plan of 224, 225, will 
 now fall in the Jirst quadrants (instead of the second) of the 
 ecliptic and equator; and the points d, e will fall in the fourth 
 and third quadrants of the latter circle (instead of falling in the 
 first and second), so that they are now outside the hemisphere 
 depicted in the figure, as also are the new points r and s. The 
 latitudes, dr, es, are northern now ; but the arc km, or the angle 
 KDM, or l'dr, still represents, by its new position and magnitude, 
 the new value of the quaternion j3a'^ .7 ; while the angle l'es 
 still represents this other quaternion, aj3"^ . 7. The point f takes 
 now a southern latitude, while the arcs ef and df are still bi- 
 sected by A and b ; but the new arc de is bisected rather by a 
 certain new point, c', diametrically opposite to c, than by the 
 point c itself Taking still a point f' diametrically opposite to 
 F, the small circle def' is still parallel to the ecliptic as before, 
 
LECTURE V. 231 
 
 but is now situated in the northern hemisphere of latitude. If 
 p' be the north pole of the ecliptic, the three triangles, dp'e, 
 epV, f'p'd, are each isosceles ; but the angle edp', which is a base 
 angle of the first of them, and may serve, instead of the vertically 
 opposite angle l'dr, to represent the quaternion )3a'^ . 7, is equal 
 now to half the excess of the angle at F'over the sum of the two 
 other angles in the triangle def'; whereas in fig. 41, art. 227, 
 that excess was in the contrary direction. Considering then the 
 lune ff', we see that we are now to subtract two right angles 
 from the semisum of the angles of the new triangle def, whose 
 sides ef, fd, de, are bisected by the points a, b, c', instead of 
 subtracting in the opposite way ; so that while the axis of the 
 quaternion ^a'^ . y is still given by the formula. 
 
 Ax . (j3a"i.7)=D-o, 
 
 as in 225, the angle of the same new quaternion is now to be ex- 
 pressed as follows, and not as in 227 : 
 
 A((5a-\y) = i{D + E + F)-7r. 
 The relations. 
 
 Ax. (0)3-1.7) = E-o, 
 and 
 
 Z(aj3-i.7) = 7r-Z(i3a-i.7), 
 
 Still hold good, as in 229 ; but this last angle now becomes, 
 
 z(aj3-i.7)=27r-i(Z) + ^+i^). 
 
 All this will easily become clear, after what has been said in re- 
 cent articles, at least with the aid (if it be thought necessary) of 
 a common globe. (See also figures 47, 48, 49.) 
 
 239. If then it be required to determine the axis and angle 
 of a quaternion, such as 
 
 where a, j3, y are the vectors of the three points a, b, c', considered 
 in the foregoing article, the arcs ab, bc', c'a being thus each 
 greater than a quadrant (and not now each less, as was the case 
 with AB, BC, CA, in 223, &c.)5 we may proceed in the following 
 way. Since we have here 
 
232 ON QUATERNIONS. 
 
 [ia~^ •j=- (5a'^ . J, because 7' = - 7> 
 
 and have just now determined (in 238) the quaternion /3a"^ . 7, 
 we need only take the negative of that quaternion, on the plan 
 of art. 183. Reversing then the axis, and taking the supplement 
 of the angle, we find, in the present question, 
 
 Ax . (/3o" ^ . 7') = d' - o = o - D, 
 and 
 
 L {[5a-' .y') = 2Tr-i(D+E+F), 
 
 where d' is the point diametrically opposite to d. But by a simi- 
 lar process, attending (as in 228, 229) to the changes in the cha- 
 racter of the rotation, which was right-handed round a from j3 
 towards 7', and is consequently left-handed round the same a, 
 when measured from j' towards /3, while d is still .(compare 226) 
 the corner opposite to that side ef of the triangle def which is 
 bisected by a, we find, without difficulty, that the following re- 
 lations hold good : 
 
 Ax . (7'a" ^j3)=d'-o = o-d; 
 
 L{y'a-'.(5)=^{D + E + F)-7r. 
 
 In fact this triangle def, when combined with the results of 238 
 respecting the quaternion aj3'^ . 7, gives the following values for 
 the axis and angle of the quaternion ja'^ . j3 : 
 
 Ax. (70-1 . j3) = D-o; 
 z (7a- ^ .(3) = 27T-h{D + E + F); 
 
 by taking the opposite of which axis, and the supplement of 
 which angle, the recent results respecting 7'a"^.j3 may be ob- 
 tained. And on comparing the conclusions of the present article, 
 respecting the two fourth proportionals, 
 
 j3a"^ . 7' and 7'a"i . /3, 
 
 we find, by the general results of 185, that each of these two 
 quaternions is the negative of the conjugate of the other. But 
 hence again we infer, by the reasoning of 193, 235, that 
 
 (5aKy=- K{y'a'.[5)=(5.a'y'', 
 
LECTURE V. 233 
 
 or in words, tbat the associative property holds good, for the 
 multiplication of any three vectors, a, j3, 7', which make obtuse 
 angles with each other. And we had proved (in 235) that the 
 same property holds also, when the angles between the three 
 vectors to be combined are all acute. But to these two principal 
 cases it is easy to reduce all others, by a suitable use of negatives 
 and oi limits; for example, we can at once infer, from the pre- 
 sent article, by returning from y to its opposite, that 
 
 j3a"^ . 7 =j3 . a"^y, 
 
 when y makes acute angles with a and j3, while they form an 
 obtuse angle with each other. 
 
 240. The associative proiperty of the multi-plication o/three 
 VECTORS is therefore fully proved, with the assistance of a little 
 spherical geometry ; and although it will be seen in the next 
 Lecture (compare what Has been said in arts. 178, 203, 204), that 
 the same important property admits of being independently (and 
 even more simply) established, by the aid of other principles, in- 
 volving the Addition and Subtraction of Quaternions, on which 
 we have hitherto forborne to touch, yet it was judged proper to 
 develope the method of the present Lecture also, as an exercise 
 in their Multiplication and Division, and as being connected 
 with some interesting geometrical constructions, and with what 
 will be found useful interpretations of some fundamental Sym- 
 bols of this Calculus. 
 
 241. An allusion has been made (at the end of art. 233) to a 
 particular but remarkable case of the general construction, on 
 which it may be well to say a few words, on account of a diffi- 
 culty which it might present, in the way of indetermination, and 
 also in order to illustrate by it the theory already given (in 205, 
 207), respecting the fourth proportionals and continued products 
 of systems of three rectangular vectors. Suppose then that the 
 three sides of a given spherical triangle abc are all equal to 
 quadrants (instead of being all less, or all greater) ; and let us 
 seek to circumscribe about this triangle another, such as def, 
 which shall have its sides bisected by the given points a, b, c 
 (as in arts. 226, 231, &c.) ; in order that we may thus, by some 
 suitably limiting form of a more general process already ex- 
 
234 ON QUATERNIONS. 
 
 plained, determine, if it be possible to do so, the axis and angle 
 of that (sought) quaternion which is the fourth proportional to 
 the three given rectangular unit-vectors, oa, ob, oc, by deter- 
 mining the limiting values of the expressions found in 225 and 
 227 ; namely, the following, 
 
 CD (or D - o), and tt - ^ {D -t E + F). 
 
 Now the three perpendiculars from the three given points, a, b, c, 
 which are to be let fall (by the general rule of 232) on the opposite 
 sides of the given triangle abc, become, at present, indetermi- 
 nate^ in virtue of its triquadrantal character : so therefore do the 
 three great circles also become, which are to be drawn through 
 those three given points (by the same general rule of construc- 
 tion), joerpewrfecwZar to these perpendiculars ; and consequently 
 the triangle, def, which (in the general process here referred to) 
 was to be found by suitably connecting the points of intersection 
 of those great circles, becomes, in this case, itself also indetermi- 
 nate. We cannot then assign, in the present question, by any 
 limiting form of the general vxxXq, the position of the point d, nor 
 specify the particular unit-vector od, which is to be the axis of the 
 sought quaternion. Nor is it wonderful that the rule should fail 
 to do so, since it was proved, in art. 205, that i\\e fourth propor- 
 tional to three rectangular vectors is a scalar : that is to say, a 
 positive or negative number, which is indeed conceived to admit 
 of being laid down (64) on a scale extending from - go to + oo, 
 but which has no one axis in space, to be preferred to any other 
 axis. If a scalar be positive, and if we abstract from its tensor, 
 or disregard its metric effect, as multiplying a line on which it 
 operates, we can only consider it as a non-versor (60) ; if, on the 
 contrary, the scalar be negative, it is, on the same plan, to be re- 
 garded as an inversor (see same art. 60) ; but the nonversion, in 
 the one case, and the inversion in the other, may both alike 
 be conceived to be performed round any arbitrary axis of rota- 
 lion, perpendicular to the line on which it operates, and which 
 line itself is arbitrary. (Compare the results of 167, &c., respect- 
 ing the indeterminate axis of the semi-inversor ^/ {- 1), and ge- 
 nerally oi the power (- 1)*, considered in 166.) 
 
 242. To render still more clear, by the help of a geometrical 
 
LECTURE V. 
 
 235 
 
 diagram, and of an astronomical illustration, the indetermination 
 of the circumscribed triangle def, for the case where the given 
 triangle abc is triquadrantal, and at the same time to shew how 
 the scalar nature of the quaternion, ob -^ oa x oc, may yet be 
 deduced from that very triangle def, by means of the semisum 
 of its angles employed in art. 227, let us conceive that the an- 
 nexed figure 43 represents an orthogra- 
 phic projection of the western hemisphere 
 of a globe on the plane of the meridian ; c 
 being supposed to represent the (projec- 
 tion of the) west point of the horizon, while 
 A denotes the south point itself, and b the 
 zenith ; the letter o being still conceived 
 to denote the (unseen) centre of the sphere. 
 Let D denote the (projection of) some point 
 chosen arbitrarily upon the surface of the globe, except that (to 
 fix our conceptions) we shall suppose it to be above the horizon, 
 with some north-western azimuth ; and then let e represent, on 
 the same plan of projection, another point, deduced from d, by 
 the conditions that it shall deviate as much in azimuth from the 
 south point towards the west, as d deviates from the north point, 
 and shall be as much depressed below, as d is elevated above the 
 horizon ; under which conditions it is clear that the west point 
 (represented by c) will bisect the arc de. Again conceive a new 
 point, F, to be so taken on the remote (or eastern) hemisphere, 
 that it may deviate as much to the east, from the south, as e has 
 been made to deviate from the west, and that this new point f 
 may also have the same altitude above the horizon, which was 
 arbitrarily assigned to d. The figure having been thus conceived, 
 it becomes evident that the arcs ef and fd are bisected respec- 
 tively by the points a and b, at the same time that the arc de 
 was seen to be bisected by the point c, while yet the altitude 
 and azimuth of d were chosen at pleasure. It is true that we 
 might have so selected d, as to render it necessary (compare 238) 
 to change the given points A, b, c (or some of them) to points dia- 
 metrically opposite, in order that the corners of the one triangle 
 might bisect the sides of the other; but this circumstance cannot 
 be considered as aifecting the essential indetermination of the 
 
236 ON QUATERNIONS. 
 
 circumscribed triangle def, when the given triangle abc is tri- 
 quadrantal. 
 
 243. On the other hand, if we conceive a new point g, which 
 shall have the same altitude as d, and the same azimuth as e, and 
 of which therefore the projection, as indicated in the figure, 
 would be exactly superposed on that of f, the point g belonging 
 to the near half, and the point r to the far half of the globe; 
 and if we suppose arcs of great circles to be drawn, upon the near 
 hemisphere, from this point g to the three given points a, b, c : we 
 shall see that the three new spherical angles, bgc, cga, agb, 
 which evidently, when taken together, make up Jour right angles, 
 are respectively and exactly equal (in their amounts or magni- 
 tudes, though differently posited) to the angles bdc, cea, afb ; 
 which latter are precisely the angles at the three corners, d, e, f, 
 of the triangle def. It follows then that, although the circum- 
 scribed triangle, def, is allowed (in the present question) to as- 
 sume indefinitely many positions, and although its angles may 
 separately vary, yet, in each of these different forms and posi- 
 tions, the SEMisuM of its three angles is equal to two right an- 
 gles ; or in other words, the supplement of that semisum va- 
 nishes. We have then here (by 227) the following determinate 
 value for the angle of the sought quaternion, or of the fourth 
 proportional to oa, ob, oc : 
 
 TT-^{D-vE + F) = Q. 
 
 This sought quaternion is therefore definitely found, by the 
 foregoing process (compare 205, 206), to reduce itself to a posi- 
 tive SCALAR ; its axis being of course, for that very reason, in- 
 determinate, as it was otherwise found, in recent articles, to be. 
 
 244. As to the positive character of the scalar thus deter- 
 mined, or the evanescence of the angle of the quaternion, we 
 must not forget that, in the recent figure (43, of art. 242), the 
 rotation round a from b to c, or round oa from ob to oc, that is, 
 round the first of the three given unit- vectors, /row tlie second 
 to the third, has been tacitly supposed (by the arrangement 
 chosen for the figure) to be left-handed, or negative. If, retain- 
 ing the figure, we alter only the order of the vectors, and seek 
 now the fourth proportional to ob, oa, oc (instead of oa, ob, 
 
LECTURE V. 237 
 
 oc), we shall thereby reverse the order of the rotation, as esti- 
 mated still round first from second to third. And then the con- 
 sequence will be, that instead of the rule of art. 227, we must 
 employ the rule of art. 230, to estimate the angle of the sought 
 fourth proportional ; or must take, for this angle, the semisum 
 itself, and not the supplement of the semisum, of the three angles 
 of the triangle def. When therefore the last mentioned order of 
 the vectors is chosen, or when the rotation round the first from 
 second to third is positive, the angle of the fourth proportional 
 is found, by the geometrical reasonings of the last article, instead 
 of vanishing, to become equal to two right angles ; for it acquires 
 in this case the value 
 
 For this case, then, oi positive rotation among the three vectors 
 (estimated in the way just now explained), the quaternion which 
 is their fourth proportional reduces itself not (as in the contrary 
 case) to a positive, hut to a negative scalar; because (com- 
 pare 166) its angle is now = tt. It is obvious what a satisfactory 
 confirmation is thus given to the two contrasted results of art, 
 205 ; and thereby to the two connected and similarly contrasted 
 conclusions, respecting continued products of three rectangular 
 vectors, which were obtained in 207. 
 
 245. As particular (but important) cases, of such contrasted 
 results, respecting products of three rectangular lines, the for- 
 mulse 
 
 kji = +l, ijk = -\, 
 
 were given in art. 210 ; and since the course of our investiga- 
 tions has suggested those formulae to us again, it may not be in- 
 appropriate to offer here a remark or two upon them, not as a 
 new proof of their correctness (which has been perhaps suffi- 
 ciently proved already), but rather as a new interpretation of 
 whatever may appear at first to be all strange in their symbolic 
 FORMS, especially when looked at in connexion with each other, 
 and with the continued equation. 
 
 Any such illustration of the foregoing formulae appears to be 
 
238 ON QUATERNIONS. 
 
 SO much the more natural in the present Course of Lectures, be- 
 cause the three italic letters, i,j, k, used with their own appro- 
 priate LAWS OF COMBINATION, bi/ multiplication among them- 
 selves, which laws were communicated (as was stated in art. 2) to 
 the Royal Irish Academy in the year 1843, and which (as it has 
 been already noticed in article 210) are all substantially mc/MC?ec? 
 in the formula recently written, were originally the only pe- 
 culiar Symbols of the Calculus of Quaternions. 
 
 246. With respect then to the formula, 
 
 kji==+l, 
 
 I wish you to remember that euer?/ multiplication of versors 
 (and as denoting versors it was, that the symbols z,_;, k presented 
 themselves in the Second Lecture to our notice) has hitherto been 
 conceived by us (see Q5^ to correspond to some combination 
 of versions, or co7nposition of rotations. It is natural there- 
 fore that in proceeding to study the proposed continued pro- 
 duct, hji, we should look out now for some original vertend; 
 that is (compare same art. %5) for some line on which we may 
 begin to operateby turning it, and which is to be thus operated 
 on, in succession, hy each of the three versors, i,j, k; one 
 line, at each of the three stages, being the subject, and another 
 line being the result of the operation. For when such an origi- 
 nal line, suppose X, shall have been found, and suc]i, a series, or 
 succession of three other lines, suppose fx, v, ^, shall have been 
 derived from it, by the three successive turnings here con- 
 ceived ; so that, in symbols, we shall have the following expres- 
 sions for the relations between these four lines, 
 
 luL = iXl v=jfx= ji\ ; ^=kv = Jcjfx = kjiX ; 
 
 it will then only remain to compare, as regards their directions, 
 the fourth with the first of these lines, in order to discover, or 
 to investigate anew, what effect the proposed continued product, 
 kji, PRODUCES, when it is regarded as being itself a sort of resul- 
 tant VERSOR, or an instrument of compounded rotation ; and 
 when, by opera^m^ on the initial direction (of X), as its sub- 
 ject, it gives thus, as its result, the final direction (of ^). 
 
 247. Now all this can, with the greatest ease, be done, if we 
 
 \ 
 
 \ 
 
 \ 
 
LECTURE V. 239 
 
 observe that, in the recent figure 43 (art. 242), the three rectan- 
 gular radii, oa, oc, ob, which are conceived to be drawn from 
 the (unseen) centre o of the globe, and are supposed (a*s in 
 former articles) to have their lengths each equal to unity, may 
 be regarded as constructions, or representations, in the order ]\x'&t 
 now written, of the three successive and quadrantal versors, or 
 rectangular vector-units i,j, k (compare 77) ; and that the sought 
 vertend, X, of the last article, may be assumed to coincide with 
 the radius oc of the same figure, or with the vector-unit J. 
 Writing then (with this reference to fig. 43) the equations, 
 
 A- o = i; B-o = ^; c - o =j = X ; 
 
 and remembering the nature of the rotations which the three 
 successive versors separately produce ; namely, that each (sepa- 
 rately) has the effect (77) of causing a line, in a plane perpendi- 
 cular to itself, to turn in that plane, through a right angle, right- 
 handedly round itself as an axis; we find the three following 
 lines, as the results of the three successive versions : 
 
 fjL = iX = ij = k = B-o; 
 v = jfx=jk=i = A-o; 
 ^=kv =ki=j-c-o. 
 
 248. In words, the line (X or oc), which was taken as the 
 original vertend, and was directed towards the west, is changed by 
 the Jirst version, performed round a southward axis (i or oa), to 
 a line {/x or ob), which comes thus to be directed to the zenith. 
 This upward line (;z or k), regarded as a new vet^tend (or as what 
 was called, in 65, a provertend), is operated on by a new versor 
 (j or oc), which is an axis directed to the west ; and it is thereby 
 brought into another position (denoted by v or oa), becoming 
 thus a line directed to the south. And finally this southward 
 line (v or i), as a new subject of the same sort of operation, is 
 made to turn round an upward axis (k or ob), till it takes the 
 Jinal position (^ or oc), of a line directed to the west. But by 
 this TRIPLE VERSION, ?i final line (^=oc =j) is attained, which 
 has the same westward direction as the initial line (X = oc = 
 j). And hence we find that (with the lately assumed initial direc- 
 tion) the three successive versions {i,j, k) have neutralized or 
 
240 ON QUATERNIONS. 
 
 annulled the effects of each other ; or that their final product 
 (^X"^= l)isa NONVERSOR (60); which result not merely yMS^2/?es 
 in a new way, but at the same time serves to interpret, or 
 explain, that symbolic equation or formula, namely, A;7 = +l, 
 which was proposed anew for consideration, at the commence- 
 ment of the foregoing article. 
 
 249. The only oMe/* direction which it would have been pos- 
 sible to assume for the original vertend A, consistently with the 
 conditions of 246, would have been an eastward (instead of a 
 westward) direction ; and if we had so chosen A, and had sub- 
 mitted it to the same three successive versions {i,j, k), we should 
 have obtained, as the three successive results, a dowmvard line 
 for fx, a northward line for v, and finally an eastward line for ^. 
 We should therefore still (compare 71) have been brought back, 
 by this triple version, to the direction originally chosen (whe- 
 ther that had been west or east) : and should thus have been 
 still led to establish, with this sort of interpretation, the same 
 formula of art. 210, kji= 1, as before. 
 
 250. On the other hand, if we had taken the operators in the 
 opposite order, k,j, i, with a view to find, on the same general 
 plan, the value of the product ijk, we might have begun as in 
 247, with a westward line j", as the original vertend; but we 
 should then have deduced from it, successively, by the three suc- 
 cessive versions, in their new order, a northicard\\x\e (kj=-i), an 
 upward line {-ji = k), and finally an eastward line {ik = -j) ; so 
 that the Jinal direction would have been opposite to the initial 
 direction, and we should have found anew, in this way, and with 
 this interpretation, that this other formula of the same art. 210, 
 
 ijk = -\, 
 
 holds good. Or this last formula might, on the same plan, have 
 been obtained, if we had begun by operating on an eastward 
 line, which would have been changed at last to a westward one; 
 the three successive and rectangular rotations, whose axes 
 are the three lines k,j, i, being thus found again to be, in their 
 combined effects, equivalent to an inversion. But with these 
 new interpretations of these characteristic formulae, it appears 
 that we may conveniently conclude the present Lecture. 
 
LECTURE VI. 
 
 251. Although, Gentlemen, an intention was more than 
 once announced, in the foregoing Lecture, of proceeding, in 
 the present, to the consideration of the Addition and Subtraction 
 of Quaternions, and to the proof of the Distributive Principle ; 
 yet the subject has so much grown under our eyes, and so much 
 still remains which it appears to be interesting or instructive to 
 contemplate, respecting the Operations of Multiplication and 
 Division, considered in themselves, and without any express re- 
 ference to those other operations of Addition and Subtraction, 
 that I scarcely at this moment hope, without extending this 
 Sixth Lecture to a length inconvenient and unreasonable, to 
 escape the necessity of once more postponing that promised proof 
 of the Distributive Principle of the Multiplication of Quater- 
 nions : in order that we may the more fully occupy ourselves, for 
 some time longer, with the study of the Associative Principle, in 
 connexion with some constructions of spherical geometry, and 
 some expressions for rotations of solids, or of systems of points 
 and lines in space, which will, however, be more of a geometrical 
 than a physical character. I shall proceed, then, without fur- 
 ther present preface, to complete, or at least to develope more 
 fully than before, that account of certain general processes and 
 results, connected with multiplication, but 7iot immediate!?/ with 
 addition of Quaternions, to which the foregoing Lecture related. 
 
 252, After the recent remarks on systems of three rectangular 
 lines, and on their continued products, with which we know (194, 
 207) that their fourth proportionals are connected, we might, 
 as another verification oi the general theoxy oi^xxch p7'oportionals 
 which has been given in the foregoing Lecture, proceed now to 
 apply that theory (but it would be tedious at this stage to do so 
 
242 ON QUATERNIONS. 
 
 with any fulness of detail) to the case of three coplanar vectors, 
 which case had been previously and separately examined by us, 
 and indeed by others also. In returning, for a moment, to the 
 consideration of this particular case, and treating it as a limit of 
 the more general case where the lines are not coplanar, we should 
 now be led to conceive that the three proposed vector-units, 
 a, /3, 7, the fourth proportional to which is required, are radii 
 drawn to three given points, a, b, c, of some one great circle on 
 the unit-sphere ; and we should have to seek for a system of three 
 other points, d, e, f, arranged upon the same great circle, in 
 such a way that the three arcs ef, fd, de may be respectively 
 bisected by the given points a, b, c ; or at least hy these in part, 
 and partly by the points a', b', c', which are diametrically oppo- 
 site to these. Supposing for simplicity that the distances of the 
 given points a, b, c from each other are each less than a quadrant, 
 we may denote their given (positive or negative) arcual distances 
 from some assumed initial point i of the circumference by the 
 letters a, 6, c: and may denote the sought distances of the 
 points D, E, F from the same initial point by the letters .t, ?/, Z', 
 so as to have the equations, 
 
 IA=a, IB = &, IC = C; ID = iC, IE = ?/, iF = z; 
 
 where ia, &c., are arcs, each less than a semicircle. The relations, 
 
 2a=?/ + 5r, 2h = z + x, ^c = x + y^ 
 
 will then hold good, in virtue of the supposed bisections, if i 
 have been suitably chosen, and will give the values, . 
 
 x = b -a-v c, y = c-b + a; z = a-c + b; 
 
 such then are the distances of d, e, f from i. If then we denote 
 by d, e, Z the unit- vectors drawn to these points d, e, f, regarded 
 now as limiting positions of the corners of a certain circumscribed 
 triangle (226), of which triangle the spherical excess vanishes, at 
 the limit here considered, so that the semisum of its angles, and 
 the supplement of that semisum, are now each equal to a right 
 angle ; we find now (as limiting cases of other and more general 
 results) that, for the present system 0/ coplanar lines, the follow- 
 ing expressions hold good : 
 
LECTURE VI. 243 
 
 And these expressions agree perfectly with the conclusions pre- 
 viously drawn from simpler and earlier considerations. 
 
 253. For example, if we assign to a, /3, y, S the same signifi- 
 cations as in fig. 30, art. 181, placing (as in that figure) the ini- 
 tial point of the circumference at a, and measuring the arcs by 
 degrees, we shall have, 
 
 « = 0, 6=60, c = 20; x = b-a + c = 80. 
 
 The same values of a, b, c give 
 
 ?/ = c-6 + « = -40; 2; = a- c + 6 = + 40 ; 
 
 and accordingly while the points a, b, c, d fall at the extremities 
 of the radii a, j3, y, S, the points e and f will fall at the extremi- 
 ties of £ and Z,, if these last radii be the fourth proportionals to 
 j3, 7, a and to 7, a, j3, respectively, and if we take the point e 
 at 40° behind a, but the point f at 40° beyond the same initial 
 point a, with reference to the assumed order of rotation on the 
 circumference. All this may be illustrated by figure 44, where 
 the points and lines connected with the j,.^ ^^ 
 
 present example are inserted, and others jy 
 
 are suppressed as being not now required ; /<M ""\ 
 
 and where you may observe that a, b, c /vX\ \ 
 
 bisect, respectively, as by the general /""^^C^NA \ 
 
 theory they ought to do, the arcs ef, fd, -^i ~ ~/o j 
 
 DE : while od is seen to be the fourth pro- \ £/ / 
 
 portional to oa, ob, oc ; oe to ob, oc, oa ; ^S^ y 
 
 and of to oc, oa, ob. Or we might con- ^ -^ 
 
 ceive, in fig. 40 (art. 224), that c came to coincide with q (by 
 the obliquity of the ecliptic vanishing), and we should find then 
 that the points d, e, f would come to coincide respectively with 
 R, s, T ; while the relations of art. 252, between «, 6, c and 
 a;, ?/, z, would be found to be satisfied by the values of those let- 
 ters, which values would become, in this example, 
 
 «=100, Z» = 70, c = 90; .t = 60, ?/=120, ^ = 80: 
 
 the assumed initial point being here the first point of Aries, so 
 
 R 2 
 
244 ON QUATERNIONS. 
 
 that the arcs are, in this example, expressed in degrees of longi- 
 tude. 
 
 254. To illustrate similarly, by the limiting ease of eoplana- 
 rity, the theory given in 238 and 239, for the fourth proportional 
 to three vectors whicjj make three obtuse angles with each other, 
 let us conceive that the distances id, ie, if are now assumed re- 
 spectively equal to 160°, 320°, and 80°, as in the annexed figure 
 45, being thus each positive now, but 
 not each less than a semicircle. The en 
 
 points A, B, c, bisecting respectively the /^^\ vXmo 
 
 arcs EF, FD, and de, will thus be such / \ / /\.f.ft 
 
 that I A, IB, ic shall be respectively equal p^-~-^,„^^^ \//,^-'''d\ 
 
 to 20°, 120°, and 240° ; and their mutual 0° I -^^^^ 1 
 
 distances will be, 540^' / \\ / 
 
 AB = 100°; Bc=120°; ca=140°; \^__j:l>a4o 
 
 260 
 
 each of these distances, as also each of the bisected arcs, being 
 treated as an arc less than a semicircle. Regarding then the 
 circumference as the limit of a spherical triangle, def, whose 
 sides EF, FD, DE are (as above) bisected by the points a, b, c, 
 which are themselves to be considered as the limiting posi- 
 tions of the corners of another spherical triangle, we see that 
 the sides of this last mentioned triangle, abc, are each greater 
 than a quadrant; and that the angles oi the former triangle, 
 DEF, are each (at the present limit) equal to two right angles ; 
 so that we have the values, 
 
 D + E + F= 37r, 
 and 
 
 Jit 
 
 The angle of the fourth proportional to the three coplanar vec- 
 tors OA, OB, oc, taken in any order, is therefore here again found, 
 by the rule in 239, to be a right angle ; and thus (compare 122, 
 149) we find again that, in this case of coplanarity, the quater- 
 nion, which is (compare 130, 202, 204, 211, 213) the general 
 value of the fourth proportional to three lines, degenerates into a 
 line, or becomes a vector (as in 129, &c.). 
 
LECTURE VI. 
 
 245 
 
 255. As regards the directions of these various vectors, which 
 are thus the fourth proportionals to the three coplanar lines, 
 OA, OB, oc, taken in different orders, we are, by another part of 
 the same rule of art. 239, to change now the points d, e, f, to the 
 points respectively and diametrically opposite, namely to d', e', f', 
 in the figure ; and so to form the equations, 
 
 0D'= OB -t- OA X oc = oc -^ OA X OB ; 
 0E'= oc -r- OB X OA = OA -i- OB X oc ; 
 0f'= OA -H oc X OB = OB -f- oc X OA. 
 
 And these three radii od', oe, of' have evidently, as the pre- 
 sent figure shews, the precise directions which might have been 
 otherwise and more easily found, by the simpler and earlier 
 theory (129) of p?'opoi'tionals in a single plane ; although they 
 have here been obtained as limiting results of a more gene- 
 ral CONSTRUCTION, which extends to lines in space, and in- 
 troduces spherical triangles. 
 
 256. As another illustration of the general theory of fourth 
 proportionals to vectors not coplanar, I shall here offer the fol- 
 lowing modification of figure 40 (art. 224), with some letters and 
 lines suppressed, and with some others introduced, chiefly from 
 fig. 42 (art. 236), but without any changes being made in the 
 significations of the letters which are thus retained, or transferred. 
 For instance, in this new 
 
 figure 46, the letters a, b^ 
 c, D, E, f, k, l, l', m, n, q, r, 
 are merely retained from fig. 
 40 ; and, as in fig. 42, x is 
 the positive pole of the arc 
 AC ; Y and z are the feet of 
 perpendiculars let fall from 
 B and D on the same arc ac, 
 or on the great circle, of 
 which that arc is a portion ; 
 the same arc ac prolonged 
 meets the prolongation of 
 BD in h'; i' is the positive 
 pole of DB, or the negative 
 
246 
 
 ON QUATERNIONS. 
 
 pole of BD ; G is supposed to be so chosen on the great circle 
 through c and a, that the arcs h'g and ca are similar in direc- 
 tion, and supplementary in amount ; finally i'g, prolonged, meets 
 DB prolonged in j ; and k' and x' are the points diametrically 
 opposite to K and x. Hence, as in fig. 40, the arc km, and the 
 spherical angle l'de, are representations of the quaternion 
 j3a"^ . 7 ; and, as in fig. 42, the arc gi', and angle zdh', represent, 
 in like manner, the quaternion j3 . a'^'y. But the points J, g, i' 
 are easily shewn to be on the great circle through kmn ; there- 
 fore the arcs km, gi' have the same positive pole at d ; and the 
 spherical angles l'dr and zdh', subtended by these arcs at that 
 pole, are equal to each other, as being each equal to the sup- 
 plement of the semisum of the three angles of the triangle def ; 
 we have therefore the arcual equality (compare 217, 236), 
 
 -^ Gl'=^ KM. 
 
 Hence, as before, we gather the associative principle, for the 
 multiplication of three vectors, -y, a"\ j3 (compare 194), at least 
 as at present arranged ; or the formula, 
 
 j3a"^ . 7 = /3 . a'^y. 
 
 It would have been possible to have gone through all the rea- 
 sonings of several former articles upon this single Jigure 46, at 
 least with the aid of a few additional lines and letters ; but it was 
 judged expedient, for the sake of clearness, to break up the in- 
 quiry into parts, and to employ more figures than one for that 
 purpose. 
 
 257. The reasonings of articles 238, 239, and therefore also 
 those of 254, 255, may be illustrated by the three following 
 figures, 
 
 \ 
 
 
 Fig 
 
 47. 
 
 
 
 K 
 
 
 / m/ 
 
 '^ 
 
 ___T 
 
 \ 
 
 rf 
 
 
 
 
 \a\ 
 
 ~~~1^^^ 
 
 II' n 
 
 C 
 P 
 
 
 L 
 
 K' 
 
LECTURE VI. 247 
 
 to which allusion has already been made (at the end of 238), 
 and of which it seems to be almost sufficient to observe here that 
 the two first of these new figures (47, 48) are designed to be or- 
 thographic projections of two opposite hemispheres, with c and 
 c' for their poles, namely, of those two which may be called the 
 hemispheres of summer and winter, on the plane of the equinoc- 
 tial colure ; while the third new figure (49) is the corresponding 
 projection of what may on the same plan be called the hemisphere 
 of spring, on the plane of the solstitial colure. It may be noticed, 
 however (compare art. 225), that m is now the negative pole of 
 DR ; and that the angles kdr, mdn, are now supplementary ; 
 which differences from fig. 40 arise from the circumstance that 
 the point d has now (as in 238) a northern latitude. We may 
 add (compare 227), that the angles l'dr, cdp are now not oppo- 
 site, but coincident ; and that in employing, with reference to 
 the new figures, the arcual equation 
 
 -- SR = 2 X-- AB, 
 
 of art. 226, we are now to conceive that, as in fig. 40, the arcual 
 motion from s to r is measured in the same direction as that 
 from A to B. Finally, the arc kn'm', or the angle kem' (= l'es), 
 in fig. 48, represents the quaternion ajS'^.y; the point m' an- 
 swering to the one which was so named in art. 229 ; and n' being 
 so situated as to satisfy (compare fig. 47) the arcual equality, 
 
 - NL = '-^ ln'. 
 
 258. Before dismissing figure 40, we may observe that it 
 leads to a simple and remarkable expression for the half of the 
 spherical excess of the spherical triangle def, considered as the 
 angle of a certain quaternion. In fact it is clear, from what has 
 been already shewn, that the angle mdn in that figure, being the 
 complement of the angle l'dr, which last has been seen to be the 
 supplement of the semisum of the angles of the triangle def, 
 must be itself the amount whereby that semisum exceeds a right 
 angle ; and therefore must be equal to the half of what is usually 
 called the spherical excess of that triangle. In symbols (for this 
 case of fig. 40, art. 224), 
 
 MDN = ^{D + E + F-tt). 
 
248 ON QUATERNIONS. 
 
 But the arc mn is (in degrees) equivalent to the angle mdn, and 
 has the vertex d of that angle for its pole. If then we write (as 
 has in part been done already), 
 
 A = L-o, /u = M-o, v = n-o, 
 as well as 
 
 a = A-0, /3 = B-05 7 = 0-0, 
 
 and 
 
 2 = D-o, E = E-0, ^ = F-o, 
 
 the are mn, and the angle mdn, will be the representative arc and 
 angle of the quaternion vfx'^; which quaternion may easily be 
 transformed as follows : 
 
 viu-i = vX-i.A/i-i = S7-i.aj3-i; 
 where 
 
 But by the theory of square roots of quaternions, explained in 
 the Fourth Lecture, we have, for the present figure : 
 
 If then we denote the recently considered quaternion by q, so 
 that 
 
 we shall have, for the axis and angle of q, the expressions : 
 
 Ax .q = E = 'D-o; 
 and 
 
 lq = ^{D+E+F-7r)', 
 
 this ANGLE of the quaternion, q, being thus the semi-excess of 
 the triangle. 
 
 259. If it were proposed to interpret on similar principles 
 this other equation, 
 
 the symbols d, e, Z, being supposed to retain their recent signifi- 
 cations, we might proceed as follows. By figure 40, and by the 
 theory of square-roots of quaternions, 
 
 (eg-i)^ = £7-i; (^£-i)l = ae-i; {dV^)^ = ^^-^; 
 
LECTURE VI. 249 
 
 hence 
 and 
 
 We are then to go first along the arc ca, which represents the 
 factor ay"^, or along one arcually equal thereto, as along a vec- 
 tor arc ; and then along the arc bd, or some equivalent, as a pro- 
 vector arc, to represent the profactor Sj3"^ ; after which we are to 
 determine the transvector arc, in order to obtain an arcual repre- 
 sentation of the sought transfactor, or product, q. That is, in 
 fig. 42, we are to go first from g to h, and then from h to J, 
 which will bring us, upon the whole, from g to J. The arc gj, 
 in fig. 42, or 46, is therefore the sought transvector arc, and re- 
 presents the required quaternion q. We see then that it follows 
 (from what has been already shewn respecting those figures), that 
 the point d is the negative (and not the positive) poZe of the sought 
 representative arc, or that the axis of q' is directed awat/ from d ; 
 while the angle of this new quaternion q' is seen to be still equal 
 to the semi-excess of the spherical triangle def. In symbols, 
 
 Ax.q=D'-o = -d; Aq = ^{D + E + F-7r). 
 
 And the distinction between the two cases, considered in the pre- 
 sent article and in the foregoing, is seen to arise from or to con- 
 sist in this ; that the rotation round o from ^ towards e is positive, 
 but the rotation round the same S from e towards Z is negative. 
 
 260. If, instead of the arrangement in fig. 40, we adopt that 
 described in art. 238 ; and propose, on the general plan of 258, 
 to express, still, by means of square-roots, the quaternion which 
 has MN and mdn for its representative arc and angle ; we shall 
 still have for this quaternion, as in 258 (see figs. 47, 48, 49), 
 
 = Ey-' . (a?-i . ^/3-i) = §7"^ . (.<-i)^ (^n*, 
 
 because (238) the arcs ef and fd are still bisected by the points 
 A and B. But because the arc de, when treated as an arc less 
 than a semicircle, is (by same art. 238) bisected noiv by the point 
 c' opposite to c, and not by the point c itself; or because the are 
 
250 ON QUATERNIONS. 
 
 CD is, with the present arrangement, greater than a quadrant, and 
 therefore the angle between y and g is obtuse ; we must (by 158) 
 write now, 
 
 prefixing thus a negative sign to the square root. Thus, in the 
 case here considered, the expression for the sought quaternion be- 
 comes, 
 
 instead of the expression which was found in 258, and which 
 differed from this one in sign. And if we still denote by q the 
 product of the three square roots, written (as in 258) without the 
 negative sign, we shall now have the equation, 
 
 vfx-^=-q. 
 261. But we have still, 
 
 Ax.v;i-i = S; Z(i'/x'i) = mdn; 
 
 therefore, by the general theory oi negatives of quaternions (in 
 183), we have 
 
 Ax.5' = -S; zg-^TT-MDN. 
 
 Now on considering the construction described in 238, we easily 
 perceive that the angle mdn is still (see fig. 49) the complement 
 of the angle kdm, which represents the quaternion /3a' ^ . y ; but 
 this representative angle was found in 238 to be, 
 
 KDM = Z (|3a-^ 7) -1 (Z) + £+ i^) -TT ; 
 
 its com'plement is therefore (in the present case) 
 
 ^ X ri^ T? T7\ '37r-(Z) + JS + i^) 
 
 and the supplement of this angle is evidently, 
 
 Lq=\{B^-B + F--K). 
 
 The angle of the product (q) of the square-roots of the three suc- 
 cessive quotients (^S"S f^"\ Sf"^)? of the vectors (S, ^, e) of the 
 three corners of a spherical triangle (dfe), is therefore still found 
 to he equal to the semi-excess of that triangle. And vvhereas 
 the axis of this product q is now = - S, like the axis of 9-' in 259, 
 
LECTURE VI. 251 
 
 and not = + S, as it was in 258, this difference of sign, or of direc- 
 tion, arises simply from the circumstance, that in the construc- 
 tion of art. 238 the rotation round d from f towards E is nega- 
 tive, whereas that rotation was positive in fig. 40. Accordingly 
 it is easy to prove that if we still denote by q the same product 
 of square-roots as in 259, we shall have, for the case of art. 238, 
 the values (compare that of the arc m' n' in figure 48) : 
 
 Ax.q'= + d; Lq^^{D + E + F-TT). 
 
 I leave it to yourselves, as an exercise, to apply these principles 
 to the two chief limiting cases, where the three bisecting vectors 
 compose, j^r5^ (as in articles 241, 242, &c.), a rectangular, or 
 secondly (as in 252, 253, &c.), a coplanar system ; and to shew 
 that each of the recently considered products of square roots 
 reduces itself, in the first case, to a vector, and in the second 
 case to a scalar. 
 
 262. In general, the two lately studied quaternions q and q 
 are versors, with opposite axes, but with equal angles ; so that 
 
 T^'' = T^ = 1 ; Ax .§'' = - Ax . 5 ; Lq=Lq. 
 
 They are therefore (by principles and definitions already fully 
 explained) two conjugate versors, and are each the reciprocal of 
 the other ; each, as an operator, undoing what the other does, 
 (Compare 162.) We have therefore here the formula, 
 
 q = ]Lq ^q'^" 
 Now if we write, for conciseness, 
 
 ^=(£§-1)4; /=(^£-l)i; /'3=(g^-l)*; 
 
 we shall have, by 259, 
 
 q = r" .r'r\ 
 
 and therefore, by 190 and 192, 
 
 q = Kq = Kr Kr . Kr ", 
 and also, 
 
 q = q''^ = r~^ r'^ . r"~^. 
 
 But, as in algebra, by the Fourth Lecture, the two square roots, 
 (£§-i)i and (§6-1)*, 
 
252 ON QUATERNIONS. 
 
 are always reciprocals of each other ; they are also, as quater- 
 nions, conjugate, if S and £ be both unit- vectors, or even if (as 
 lines) they be equally long, that is (by 110), if their tensors be 
 equal. Admitting then this equality of lengths of the vectors 
 S, e, ^, which will not essentially affect the generality of the final 
 conclusion, we have, 
 
 263. Thus, by the foregoing article, we have the expression, 
 
 g=(gs-0* (s^')*.(?g-0*. 
 And we had, in art. 258, 
 
 ^=(g£-o*.(£ro* (^s-^i 
 
 These two expressions, for the quaternion q, differ only by 
 THE PLACE OF the]point, which is used as the mark of multipli- 
 cation ; in this new case, therefore, the associative pinnciple 
 still holds good ; the three successive factors being now 
 not vectors, but quaternions. In exactly the same way we 
 should prove that the expression (in 259) for q does not change 
 its value, when the place of the point is changed ; or that with 
 the recent significations of r, r, r", the following equation holds 
 good : 
 
 r" r' . r = r" . r r. 
 
 Yet because these three successive factors, r, r, r'\ are connected 
 with each other by the relation, 
 
 we cannot assert that we have as yet done more, in these Lec- 
 tures, as regards that general associative principle of mul- 
 tiplication OF quaternions, which was enunciated, without 
 proof, in art, 108, under the form of the equation 
 
 q" q .q = q"-q' q, 
 
 than to raise, perhaps, a sort o^ presumption in its favour, not yet 
 converted into certainty. 
 
 264. Before entering on the general demonstration of this im- 
 portant proposition, it may be useful to describe here a new and 
 
LECTURE VI. 
 
 253 
 
 Fis. 50. 
 
 R\y 
 
 GENERAL CONSTRUCTION jfer the MULTIPLICATION OF ANY TWO 
 
 QUATERNIONS, q and r, of which the representative angles 
 are given upon a spheric surface, in position as well as in mag- 
 nitude. 
 
 Suppose then, at first, that these two angles of the factors, q 
 and r, are given as the base angles, at the corners q and r of a 
 spherical triangle, qrs, as in the annexed figure 50 ; and let it 
 be required to find the reipresenta- 
 tive angle of the product, rq. For 
 this purpose we may employ the 
 identity of art. 49, namely, 
 
 7 -^ a = (7 -^ /3) X (/3 -f- a) ; 
 
 aiming, as in the article just cited, 
 to put the proposed quaternion 
 factors, q and r, under the forms 
 j3 -r- a and 7 _;- j3, respectively. The line /3 must be situated in, 
 or parallel to, the planes of both the factors ; and these two planes 
 are constructed by the two tangent planes to the sphere, at the 
 points Q and r. Conceive a cylinder circumscribed about 
 THE sphere, so as to touch it along the great circle which passes 
 through these two points ; then every tangent plane to the 
 sphere, at any point of this circle, is also a tangent to the cylin- 
 der, and is parallel to the axis thereof; the line of intersection 
 of any two such tangent planes must therefore be itself also pa- 
 rallel to this axis, and consequently perpendicular to the plane of 
 the great circle of contact qr : we know then the direction of 
 the line |3, namely that of this last-mentioned axis, or perpen- 
 dicular ; and may proceed to deduce from it, as follows, the two 
 other sought directions, of the lines a and 7. Imagine that, at 
 each of the two given points, q and r, that is at each extremity 
 of the base, a normal arc is erected, perpendicular to that given 
 base, but contained upon the spheric surface, and situated (to 
 fix our conceptions) on that hemisphere which contains the given 
 vertex s. The common initial direction of these two perpendicular 
 arcs, or (in other words) the common direction of the two corres- 
 ponding and rectilinear tangents to the sphere, may (on the plan 
 just now mentioned) be denoted by the letter j3, regarded as sig- 
 
254 ON QUATERNIONS. 
 
 nifying a certain vector^ to which both these tangents are 'paral- 
 lel^ and which is (as has been seen) perpendicular to the plane of 
 the base. And then by suitably erecting (as suggested in fig. 50), 
 at Q and r, two other normal arcs, perpendicular to the two given 
 sides, QS and rs, we shall obtain, by thei?' initial directions, the 
 two other required vectors, a and y, as the initial tangents to 
 these new normal arcs, or at least lines parallel thereto. 
 
 265. But these two new perpendiculars have the directions 
 respectively of the axes of two new cylinders, circumscribed about 
 the sphere so as to touch it along the two sides of the triangle; 
 and the tangent plane to the sphere at the vertex s of the trian- 
 gle, being a common tangent to the sphere and to these two 
 cylinders, contains two lines tangential to the sphere, and parallel 
 respectively to the two axes of the two new cylinders, or parallel 
 to a and y. The plane of the quaternion y ~- a, which is, by the 
 general theory of quaternion multiplication, the plane of the 
 sought product, rq, is therefore parallel to, and may be assumed 
 as coincident vs'ith, this last tangential plane at the vertex s. And 
 this point s itself, as distinguished from its own opposite upon 
 the sphere, is the positive pole of the required resultant rota- 
 tion, or of the sought quaternion product, at least with the ar- 
 rangement in fig. 50 ; while the angle of this product is equal 
 (as the same figure shews) to the supplement of the vertical an- 
 gle, at s, of the given triangle qrs. We have therefore only to 
 prolong one side of that triangle, suppose qs, to some point t, 
 and to take then the exterior vertical angle, tsr, as the 
 representative angle of the sought quater7iion product, rq, if the 
 tivo quaternion factors, q and r, regarded as multiplicand and 
 multiplier, be, as dihowe, represented by the two base angles, sqr, 
 and QRS, of the same given triangle, and if the arrangement of 
 the points be such as we have lately conceived it to be ; that is, 
 more fully, if the rotation round the vertex (s) of the triangle, 
 from the base angle (r) which represents the multiplier (;•), 
 towards that other base angle (q) which represents the multipli- 
 cand (q), he positive, as in the recent figure. 
 
 - 266. Many conclusions may be drawn from the foregoing 
 general construction for a product ; but it seems to be proper 
 previously to exhibit the agreement of this method of employing 
 
LECTURE VI. 
 
 255 
 
 representative angles^ with a?20if/ier general method of multipli- 
 cation, which was explained in the foregoing Lecture, and which 
 made use of representative arcs ; namely the construction in 
 art. 217. To make this agreement evident, I have drawn the 
 annexed figure 51, where qrs is the same spherical triangle as in 
 the recent figure 50 ; p is the 
 
 middle point of the base qr, -^'S- ^^• 
 
 and the hemisphere with p for I'' 
 
 pole is supposed to be ortho- 
 graphically projected ; qs pro- 
 longed meets the bounding 
 circle in t ; and k, l, m, are 
 respectively the positive poles 
 of the arcs qs, qr, sr, while l' 
 is opposite to l. The new 
 figure shewS; reciprocally, that 
 Q, r, s are the positive poles, 
 respectively, of the arcs kl, 
 LM, KM ; and that the arcs kl, 
 LM, represent the same two gi- 
 ven quaternion factors, q and r, as the angles sqr, qrs. Hence by 
 the rule of art. 217, and by the present figure, the arc km, or the 
 angle ¥.sm, represents the sought quaternion product r^* (abstract- 
 ing still from tensors). But we have the equation between angles, 
 
 KSM=TSR, 
 
 even when planes and directions are attended to ; consequently 
 the EXTERNAL VERTICAL ANGLE, TSR, of the triangle whose base 
 angles represent ihe factors, is seen anew to represent the j^ro- 
 duct sought. It will not fail to be noticed that the triangle ml'k, 
 as compared with qsr, is merely the well-known polar, or sup- 
 plementary TRIANGLE, Considered often in spherical trigono- 
 metry ; but it may be observed that I have hitherto made no use 
 of any trigonometrical formula. It may also be remarked that 
 the quadrants kq, ks, prolonged, are touched by the two lines 
 which lately received the common designation of a ; lq, lr, by 
 the two'lines named j3 ; and mr, ms, by the lines which were 
 denoted by y. 
 
256 ON QUATERNIONS. 
 
 267. Resuming figure 50, we may notice that the operation 
 of the multiplicand q^ regarded as a versor, has the effect of caus- 
 ing the line a, and the tangent to the side qs, to turn together in 
 the plane which is tangential to the sphere at q, till they take 
 respectively the positions of the line j3, and of the tangent to the 
 base QR. We may therefore conceive the same act of version to 
 cause the side, qs, itself, together with its prolongation st, to 
 turn upon the spheric surface, round the point q as a pole, till 
 this arc qst comes to coincide, at least in part, with the original 
 position of the base, qr, and of that base prolonged. Again the 
 act of proversion, of which the multiplier, r, is the agent, turns 
 the other line marked /3, in the tangent plane at r, till it takes 
 the position of ^ ; and at the same time obliges the base rq to 
 take the position of the side rs ; or causes the prolongation of 
 the base, which had originally the direction of qr (and not the 
 opposite direction of rq), to turn upon the spheric surface, round 
 the pole R, till it takes the direction of the side rs reversed.^ or in 
 other Words the direction, sr, of that side measured /ro/w the ver- 
 tex. We may then say that, in this example, which may repre- 
 sent generally (at least with some easy modifications) every 
 case of multiplication of two quaternions, the versor (q) has 
 changed the arcual direction, st, of one side prolonged through 
 the vertex, to the direction of the base, qr, or of that base pro- 
 longed; and that the proversor (r) has afterwards cAa?i^ec? ^/i?s 
 direction of the base, qr, to the direction of the other side, sr, 
 measured now fro?n vertex toivards base. But we have seen that 
 our principles establish a general connexion between multiplica- 
 tion ofversors and composition of rotations ; so that while we 
 have generally the formula (65), 
 
 Transversor = Proversor x Versor, 
 
 the effect of a transversion is always conceived to be equivalent 
 to the two successive effects of the corresponding version and pro- 
 version combined. It is therefore natural to expect, in the re- 
 cent example, that (by a sort oi elimination of the intermediate 
 direction of the base) the transversor, rq, should be found to 
 have the effect of causing the direction, st, o/one side pro- 
 longed through the vertex, to turn upon the spheric surface 
 
LECTURE VI. 257 
 
 ROUND THAT VERTEX s as a POLE, till it ttssumes the direction^ 
 SR, of the OTHER side of the triangle unprolonged ; or at least 
 not prolonged through the vertex^ but measured towards (and 
 not away from) the base. And such accordingly has been found, 
 in fig. 50, to be precisely the effect of the transversor; 
 for the external vertical angle^ tsr, has been seen in that 
 figure to represent the sought product, rq ; although the proof 
 of this result, which was given in recent articles, did not involve 
 the consideration of any rotation of arcs, but only introduced 
 and combined rotations of straight lines. 
 
 268. It was remarked in art. 218, that there exists a remark- 
 able analogy between the tnultiplicatioti oiversors, and an opera- 
 tion which may be called the addition of their representative 
 arcs. And at this stage I do not think that it will appear to be 
 altogether fanciful, or useless, if I call your attention to another 
 analogy of the same sort, connecting multiplication and addition. 
 For we have recently seen that while the factors q and r are 
 represented by the base-angles of a spherical triangle, their pro- 
 dtcct, rq, is on the same plan represented by the exterior and ver- 
 tical angle. Now, if this spherical triangle should happen to be, 
 in all its dimensions, a small one, and therefore nearly plane, it 
 is obvious that this angle of the product would be, in the most 
 simple and elementary sense of the words, equal (at least nearly) 
 to the sum of the angles of the factors. If then we agree to 
 say, by analogy, even when the sides are not small, that " the 
 exterior vertical angle of a spherical triangle, is the sphe- 
 rical SUM of the tivo base angles" (taken in a certain order, to 
 be considered presently), and remember the law of the tensors 
 (188), we shall find ourselves able to enunciate, generally, the 
 following Rule for the Multiplication of any two Qua- 
 ternions: " The tensor of the product is equal to the pro- 
 duct of the tensors ; and the angle of the product is equal to 
 the sphet'ical sum of the angles of the factors." 
 
 269. It was observed, just now,that in taking this spherical 
 su7n, the order of the summands must be attended to. In fact 
 if this were otherwise, the spherical addition of angles would 
 be a commutative operation ; and would therefore be unfit to re- 
 present generally the multiplication of quaternions, or of versors, 
 
 s 
 
258 ON QUATERNIONS, 
 
 which we know (arts. 219, &c.) to be a non-co7nmutative one. 
 Accordingly it was observed, at the end of art. 265, that in ob- 
 taining the external vertical angle tsr as a representative of the 
 product, rq^ we had assumed the arrangement of the factors^ q 
 and r, to be such as is indicated in fig. 50 ; the rotation round s 
 from R towards q being positive. Had we wished to construct, 
 on the same plan, the product, qr, of the same pair of factors^ 
 taken now in an opposite order ; and to contrast, as to their joro- 
 sitiotis on the sphere, the representative angles of these two pro- 
 ducts ; we should have been led to form a figure such as the fol- 
 lowing. In this new figure, 52, the angles rqs, rqs' are equal 
 in amount, but lie at opposite sides of 
 the coi7imon base, qr, of the two tri- 
 angles, qsr, qs'r ; and a similar rela- 
 tion connects the angles qrs, qrs'j 
 whence the old and new sides qs, qs' 
 are equal to each other in length, and so 
 are the sides rs, rs', compared among ^'' V^ U\L— "^ 
 themselves. The vertical angles of ,; ,'--''''S^Nr ^' 
 these two triangles are therefore also ^^ \^' T' 
 
 equal to each other in amount, whether 
 
 both the interior or both the exterior be compared ; but the two 
 vertices, s, s', are situated at opposite sides of the base, although 
 with a certain symmetry of situation respecting it ; in such a 
 manner that the arc ss', connecting these two vertices, is perpen- 
 dicularly bisected by this common base, or by the great circle of 
 which it is a part. And while the one exterior vertical angle, tsr, 
 still represents, as before, the product rq lately considered, it is 
 the other exterior angle, rsV, at the other vertex, s', which re- 
 presents the new product qr. These two products, 
 
 rq and qr, 
 
 are therefore again found, by this neiv construction, to differ ge- 
 nerally among themselves; because although their tensors and 
 angles are equal (in amount), their poles, s and s', have diffe- 
 rent POSITIONS on the sphere. 
 
 270. As to the reasons for this difference of positions, and 
 the rules by which it may be remembered or recovered, it might 
 perhaps be sufficient to observe that while the rotation round s 
 
LECTURE VI. 259 
 
 from R towards q is positive^ as before, the rotation round the 
 same pole s, from q towards r, is, for that very reason, negative ; 
 while it is, on the contrary, from q towards r, that the rotation 
 is positive round s'. For thus we may perceive that the general 
 relation of positions between the three poles, of multiplier, multi- 
 plicand, and product, with respect to their arrangement on the 
 sphere, or to the character of the rotation from first towards se- 
 cond round third, which in our former construction (264, 265), 
 for the multiplication r x q, was in fact satisfied by the points r, 
 Q, s* is now, for that very reason, not satisfied also by the same 
 three points, in their new arrangement, q, r,s; whereas it is sa- 
 tisfied by the three points q, r, s'. In short we are now obliged 
 to look out for some neiv point on the sphere, distinct from s, 
 and adapted to be th^pole of the new product, qr; because that 
 old pole s does not possess, with respect to q and r, regarded 
 now as poles respectively of multiplier and multiplicand, the re- 
 quisite relation of arrangement; or (in other words) is not situa- 
 ted in what is now the proper hemisphere, with respect to the 
 great circle through q and r. And in the other hemisphere, which 
 is now the proper one, v^q find a pointy namely the one called 
 lately s', which does in fact satisfy not only this condition, but all 
 the other conditions of the problem, and is therefore of course to 
 be adopted, as the pole of the new product, qr, to the exclusion 
 of the old pole, s. 
 
 271. We might also reason on the lines a, (5', y, of fig. 52, 
 as we did on the lines a, j3, y, of fig. 50. Or we might construct 
 a new diagram, in connexion with the new order of the factors, 
 but on the same general plan as fig. 51, which would enable us, 
 by comparison and contrast with that figure, to bring into play 
 again an earlier construction (fig. 37, art. 219), whereby we ex- 
 hibited, in the foregoing Lecture, the general 7ion-commutative- 
 ness of quaternion multiplication, or the non-coincidence as to their 
 planes, and therefore also as to their poles, of the tivo arcs (in 
 that former figure, km and m'k'), which were obtained when the 
 two summand arcs (kl and lm) were combined in two opposite 
 orders. Or, in fig. 51 itself, we might construct three new points, 
 k", m", s', which should be, respectively, the reflexions of the 
 three old points, k, m, s, with respect to the base qr, as l' is 
 
 s 2 
 
260 ON QUATERNIONS, 
 
 already, in the same figure, the analogous reflexion of l ; and 
 then, while the new versor r would be represented by the new 
 areual vector m"l', and the new proversor q by the new arcual 
 provector l'k", the new and sought transversor qr would be seen' 
 to be represented (on the plan of 217) by the new arcual trans- 
 vector m"k", of which the pole would be at the new vertex s', and 
 the length would be equivalent (in degrees) to the supplement of 
 the new vertical angle qs'r, or of the old vertical angle rsq ; so 
 that by prolonging the new side qs' to t', we should again be led 
 to construct the new exterior and vertical angle rs't', as a repre- 
 sentation of the new product, qr. Or finally we might employ the 
 same general mode of illustration as in the more recent article 
 267 ; and observe that in performing the new multiplication, qy.r^ 
 after the new versor (r) has changed the direction of rs' to that of 
 RQjOr the direction ofs'R to that of qr, the new proversor q changes 
 this last direction of qr to that of qs', or of sV ; whence it is natu- 
 ral to suppose (what in fact has been otherwise proved) that the 
 effect of the new transversor (qr) must be to produce at once that 
 change which the two other versors have thus done successively, 
 and upon the whole ; namely, the change of the direction of the 
 ar'c s'r to that of the arc sV. For thus it might be seen again 
 that the angle rsV, in fig. 52, may naturally be supposed to 
 represent the new product, qr, as in fact we have found it to do. 
 272. As furnishing another general rule for re7nemhering or 
 recovering, if we should ever happen to forget, the distinction 
 between the two positions of the vertex, s and s', which thus cor- 
 responds to the distinction between the two arratigemoits of the 
 two factors, q and r, we may employ the following Theorem, 
 which is easily derived from remarks lately made, and includes 
 several earlier results: "In any Multiplication of two Qua- 
 ternions, the ROTATION round the Axis of the Multiplier, from 
 the Axis of the Multiplicand, towards the Axis of the Product, 
 is POSITIVE," With the help of this theorem, or rule, there can 
 never be any difficulty experienced, in forming at least a distinct 
 co^CEVTioia of the result of the multiplication of any two 
 QUATERNIONS, whosc representative angles are given, as two 
 determined spherical angles (their order being also given) ; even 
 when these two angles do not happen to be given, as in 264 they 
 were supposed to be, as being already the two base angles of a 
 
LECTURE VI. 
 
 261 
 
 spherical triangle, whose vertex was moreover thei^e conceived to 
 he given as having (as supposed in fig. 50) a certain relation to 
 the base, depending on the order of the factors, and on the cha- 
 racter of a certain rotation. To shew this clearly, let us imagine 
 that the two arbitrary spherical angles kql, mrn, in fig. 53, re- 
 present respectively any given multiplicand 
 §', and any given multipliers; and let us 
 seek to construct another spherical angle, 
 which shall represent the sought product, 
 rq. For this purpose we have only to sup- 
 pose the vertices q and R of the two given L 
 angles to be connected by an arc of a great (i'.>Q 
 circle qr, and then to conceive a new ver- ^ 
 tex s determined in that hemisphere towards which the rotation 
 round r from q is positive, by the conditions that it shall satisfy 
 the two following equations between angles : 
 sqr=kql; qrs = mrn. 
 
 For then by prolonging qs to t, or rs to u, we shall obtain an 
 angle tsr, or qsu, which shall be, on principles recently explained, 
 the required representative angle of rq, or at least of the versor 
 of this sought quaternion product, while the tensor is simply still 
 the arithmetical product of the tensors. 
 
 273. A few corollaries from this general construction for mul- 
 tiplication, which is for angles what the construction in art. 217 
 was for arcs, may be usefully inserted here. And first we shall 
 employ it to illustrate, and to deduce anew, the general signifi- 
 cation of the symbol a]3, where a, j3 are supposed to denote two 
 unit-vectors oa, ob, terminating at two given points a, b, of the 
 surface of the unit-sphere. For this purpose, I conceive that Q, 
 in fig. 54, is. the pole of the 
 arc AB, and of the semicircle 
 aa'; and then because baq 
 and QBA are evidently repre- 
 sentative angles of the multi- 
 plier a and the mutiplicand 
 /3, considered as quadrantal 
 versors (122, &c.), it is clear 
 (from recent results) that A' D 
 
262 ON QUATERNIONS. 
 
 BQA must represent the product aj3. The axis of the product of two 
 vectors is therefore seen anew to be peoyendicular to their plane, 
 and to be such that the rotation round it from multiplier to mul- 
 tiplicand is positive ; while the angle of the same product is seen 
 to be, in amount, the supplement of the atigle between the factors ; 
 all which agrees with the earlier conclusions of art. 88. (See also 
 122, and compare 236, 237.) If b take the position p, in the 
 same new fig. 54, the angle between the factors is right, and such 
 therefore is also its supplement, namely, the angle of the pro- 
 duct; the product of two rectangular lines is therefore seen anew 
 to degenerate from a quaternion to a line, because, as a versor, 
 it is quadrantal (compare again 122). On the other hand if b 
 approach to a, the angle bqa' tends to become equal to two right 
 angles ; and the product of two coincident lines is thus anew 
 perceived to reduce itself to a negative scalar (as in 84), because 
 its angle is=7r (compare 149, 153). And finally, when b ap- 
 proches to a', the angle bqa' tends to vanish ; from which we 
 might again infer (as in same art. 84), that the product of two 
 opposite lines is a positive scalar, its angle being =0. 
 
 274. The same figure 54 illustrates also the general signifi- 
 cation of some other useful symbols, for example, the symbol 
 j3a" ^. The right angle qa'b, at the opposite corner a' of the 
 rectangular lune aa' (or more fully, the lune aba'qa), represents 
 evidently the reciprocal a~^ of that given vector a, which was 
 itself represented by the other right angle of the lune, namely by 
 baq ; because it is obvious that two quadrantal and right-handed 
 rotations, round the two opposite poles a and a', destroy the 
 effects of each other; or because (see art. 117), if a be an unit- 
 vectorf its reciprocal is equal to its negative : in symbols, 
 
 o'^=-a, if Ta= 1. 
 
 Hence the product jSa'^ is represented, in the recent figure 54, 
 by the angle aqb. And hence again we might conclude (as in 
 118), that the following equation or identity holds good : 
 
 For we see anew that the product j3 x a"^ as well as the quotient 
 |3 -r- a, has its anffle equal to the angle between the lines a and 
 
LECTURE vr. 263 
 
 |3, and has its axis perpendicular to the plane of those two lines, 
 this axis being also such that the rotation round it from the divi- 
 sor a to the dividend /3 is positive. The vector c/iaracte?^ {122, &c.) 
 of the quotient of two recfatigular lines, and the scalar character 
 (59, &c.) of the quotient of two parallel lines, together with the 
 circumstance of this last quotient becoming positive or nega- 
 tive, according as the directions of the two lines compared are 
 similar or opposite, whereas, for a product, this rule ofsigtis 
 is, as we have lately seen again, reversed, would also offer 
 themselves anew, as obvious consequences, from the recent con- 
 struction for j3a"', regarded as being at the same time a construc- 
 tion also for /3 4- a. 
 
 275. Again we may employ the same fig. 54 to interpret in 
 a new way another symbol, which often occurs in this calculus, 
 namely the symbol j3a"' . j3. Conceive the point c so chosen on 
 the arc ab prolonged, that we may have the arcual equality, 
 
 '-^ AB =-> BC ; 
 
 then the angle bqc will be a new representation for j3a"S re- 
 garded now as a multiplier ; and the triangle bqc, considered as 
 having bq for its base, and c for its vertex, will shew, by the 
 general rule of art. 265, that its external vertical angle a'cq re- 
 presents the sought product, j3a"^ . j3. But this latter angle is 
 right; therefore the corresponding joroc?Mc^, in writing which we 
 may (by the last Lecture) omit the point, is a line: namely, the 
 unit-vector Y or oc, drawn from the centre o of the sphere to the 
 point c. We may therefore write, under the conditions lately 
 supposed, the equation, 
 
 and we see that the line -y, thus found, is simply what may be called 
 the REFLEXION of the line a, with respect to the line/3; in such 
 a manner that (3 bisects the angle between a and y. Indeed 
 this result obviously agrees with what was shewn, in arts. 133, 
 134, respecting the third proportional to two directed lines. Of 
 course you do not require to be told, that from the way in which 
 the figure has been put into perspective, by the principles of or- 
 thographic projection, the supposed equal arcs ab and bc (which 
 
264 ON QUATERNIONS. 
 
 I happened to take as each = 60°) are represented hy unequal 
 lines ; and that, in all the other orthographic projections sub- 
 mitted to you, results of the same sort occur. 
 
 276. It was remarked in the last-cited article (134), that the 
 square root of the product of two vectors is not generally equal 
 to that other vector, which thus bisects the angle between them, 
 and is in a certain sense their mean proportional. Accordingly, 
 with the help of the recent figure 54, we can easily assign a repre- 
 sentation for the value of the symbol 
 
 (ay)*, 
 
 and thereby shew distinctly, in a new way, that this symbol de- 
 notes generally a quaternion, but not a line. In fact, in fig. 54, 
 the product ay is represented by the angle cqa', and its square 
 root is therefore represented, on the principles of the Fourth 
 Lecture, by the Aa//' of that angle, namely by cqd (or dqa'), if 
 we conceive the point d to bisect the arc ca'; but this new re- 
 presentative angle, cqd, is acute^ and, therefore, is not fit to be 
 the angle of a vector, regarded as a (quadrantal) versor. It is 
 true that this process of construction and of reasoning admits of 
 some limits and modifications, connected with changes of the 
 value of the arc ab ; but these do not affect the general result, 
 nor does it seem that, at this stage of our course, they can occa- 
 sion to you any diflficulty. It may, however, be noticed here that 
 the same figure 54 may serve to illustrate, for the case where the 
 arc AB is less than a quadrant, or where the angle between the 
 two vectors a and j3 is acute, the conclusions that 
 
 (7a-i)* = i3a-J, if7 = /3a-ii3, 
 and that under the same conditions the symbol 
 
 {ya'^)^ a 
 
 denotes the line j3, namely, the mean proportional between a and 
 y ; both which conclusions agree with ordinary algebra, and with 
 what was shewn in art. 134. 
 
 277. The io\\ov^\x\g product of square roots 
 
LECTURE VI. 265 
 
 is agiain not to he confounded in this Calculus, with the lincy 
 
 (i3a-i)*a, 
 
 nor with either of the two quaternions, 
 
 (/3a)i, (a^)^; 
 
 although, in common or commutative algebra, these four symbols 
 might be treated as being only transformations of each other. 
 It is easy, however, to shew what i*, on our principles, the sig- 
 nification of the symbol recently written (jS^a^). For this pur- 
 pose we may conceive that a and j3 are unit vectors, directed to 
 A and B in the annexed figure bb ; and that on the arc ab as base, 
 a spherical isosceles triangle adb' is con- 
 structed, with its base angles at a and b 
 each equal to half a right angle, and with a /^"y^ ^^ 
 
 positive direction of rotation round b from y/^/o^^^^^^ r^ ^ 
 A towards d ; for then the external vertical p//>/!^^^i^-^^ 
 angle, at the new point d thus found, will U^ \ ^^ 
 
 A C B 
 
 represent (by 265, &c.) the product of 
 
 square roots required ; because these two square roots them- 
 selves, namely a^ and ^^, are represented, in this construction, 
 by the two angles, of 45° each^DAB and abd. 
 
 278. Again, it was remarked, in art. 135, that the following 
 other products of fractional powers of vectors, 
 
 j3^a^ and j3*a^, 
 
 denote, generally, in this calculus, not the two lines which may 
 be supposed to be inserted as two mean proportionals between the 
 lines a and j3, hut two quaternions^ of which we promised to as- 
 sign afterwards the tensors and the versors. Accordingly we 
 know now that their tensors are simply, 
 
 T/3* Tat and TjS* Ta*, 
 
 namely the two mean proportionals which are in fact inserted 
 between the two tensors Ta and Tj3. And with respect to the 
 two versors, the recent figure 55 enables us to construct them, 
 or their representative angles, by merely erecting on the base ab 
 two new spherical triangles, as indicated in the figure, with the 
 
266 ON QUATERNIONS. 
 
 base angles eab, abe of one triangle respectively equal to 60° 
 and 30°, while those of the other triangle, namely, fab and ABr, 
 are on the contrary 30° and 60°, and directions of rotations are 
 attended to. For then these four base angles will represent re- 
 spectively the four fractional powers of vectors, 
 
 a^, j3^, and a*, /3* ; 
 
 and the two products required will be represented by the exter- 
 nal vertical angles at e and f. 
 
 279. More generally, if a and j3 be two unit- vectors oa and 
 OB, and t a scalar exponent which we may conceive to vary from 
 to 1, then the quaternion 
 
 is a versor, of which the unit axis. Ax .q=OF, if drawn from a 
 fixed origin o, describes, by its extremity p, a certain curve apb 
 upon the unit sphere, from the point a to the point b ; and this 
 curve is such that in each position of the spherical triangle apb, 
 the two base angles at a and b are comple7nentary to each other, 
 while the exterior and vertical angle at p is equal to the variable 
 angle of the quaternion q. It is clear that if the given base ab 
 be a small arc^ the curve apb thus described, approaches to a 
 semicircle, and the quaternion q does not much differ from az^ec- 
 tor, because its angle is not much less than a 7-ight angle ; and 
 those persons who are familiar with the doctrine of spherical co- 
 nies may easily convince themselves that in general this curve 
 APB is what is called by geometers a spherical semi-ellipse, de- 
 scribed on the arc ab as its major axis, and projected orthogra- 
 phically into the plane semi-ellipse aedfb of the recent figure 
 55, in which figure the major axis becomes the line ab. Indeed 
 it is known (and quaternions will be found to furnish a new and 
 simple proof of the result), that if the base of a spherical triangle 
 be given, and also the sum of the base angles (this sum being 
 taken in the usual sense, by mere addition of magnitudes), then, 
 whether this sum be or be not a right angle, the locus of the ver- 
 tex is still a spherical conic. 
 
 280. Combining the same general conceptions of fractional 
 powers of vectors, and of products of versors constructed by their 
 
 
 \ 
 
LECTURE VI. 267 
 
 representative angles, but not obliging now (as in the last figure) 
 the angles of the factors to be complementary, we may easily 
 see that for any spherical triangle abc, of which the corners a, 
 B, c, conceived still to be situated on the surface of the unit- 
 sphere, have a, /3, 7 for their vector units, while the magnitudes 
 of the angles at those three corners are supposed to be expressed 
 as follows : 
 
 ^^^'^ Ti^y^ r-^"^ 
 
 ^-y ^~T' 2 ' 
 
 the three following relations exist : 
 
 78-^=j32/a-^; a2-^ = 7^/32'; j32-2'=aV; 
 
 provided that, as in fig. 56, the rotation round c from b to a ig 
 positive. And hence it follows that, under 
 this last condition, we have also, 
 
 7*.|32'a*=7^72-^ = 72 = -l; 
 7^ j32' . a-^ = a2 - -^ a^ = a2 = - 1 . 
 
 The associative principle holds, therefore, 
 
 here again ; and, omitting the point, we 
 
 may write, for every spherical triangle 
 
 ABC, whose corners are arranged in the lately mentioned order 
 
 of rotation, the simple but important formula : 
 
 7^/32/0^ = - 1. 
 
 And hence, either by permuting cyclically the symbols a, /3, 7 
 on the one hand, and x, y, z on the other, or by a direct per- 
 formance of calculations similar to the foregoing, we are con- 
 ducted to the analogous formulae : 
 
 It might not be too much to say, but I cannot expect you yet to 
 feel the full force of the remark, that the whole doctrine q/" sphe- 
 rical trigonometry is included in any one of these three 
 last formulce ; at least when they are interpreted and developed 
 according to the principles and rules of the Calculus of Quater- 
 nions. Meanwhile it may be observed that by combining the 
 results of the present article with the phraseology proposed in 
 
268 
 
 ON QUATERNIONS. 
 
 Fig. 57. 
 
 art. 268, or even from the principles of that former article alone, 
 we are naturally conducted to enunciate the following general 
 proposition : " The Spherical Sum of the three Angles of any 
 Spherical Triangle, taken in a suitable Order of succession, 
 is ahvays equal to Two Right Angles." 
 
 281. The general signification of the symbols 
 
 q~^r . q and rqr'^, 
 
 which, in virtue of the non-commutative character of quaternion 
 multiplication, cannot generally be reduced to the simpler forms 
 r and g, was proposed in 221 as a subject for our future discus- 
 sion. It is easy now to interpret either of these two reserved 
 symbols, for example, the latter of them, as follows. Construct, 
 as in figure 57, a spherical triangle abc, of which the base angles 
 at a and b represent the 
 factors q and r, while the y 
 
 rotation round b from a 
 towards the vertex c is 
 positive ; and let b' be the B 
 point diametrically oppo- 
 site to B. Then the ex- 
 ternal vertical angle, acb', 
 will represent the product rq ; and the angle cb'a will re- 
 present the reciprocal r-^. To construct next the new product 
 rq.r'^i we are to reflect the triangle cab', with respect to its 
 base cb', so as to change it to a new triangle ceb', such that 
 
 cb'a = eb'c, and acb' = b'ce ; 
 
 for then these new or reflected base angles, eb'c and b'ce, will 
 represent the new multiplicand r"^, and the new multiplier rq\ 
 and the new external .vertical angle, bec, will represent the new 
 product, rq. r~^. Again, in the same figure 57, if we determine 
 a point D on the semicircle bb' by the condition that 
 
 b'aD = CAB, 
 
 the angles b'ad and db'a may represent 5' as a multiplier andr'^ 
 as a multiplicand; and therefore the angle cda, or its equal edb, 
 will represent their product, qv'^. But dbe is a representation 
 
 \ 
 
LECTURE vr. 269 
 
 for r ; and therefore deb' represents r . qr'^. And since it is clear 
 from the construction, that 
 
 deb' = EEC, 
 
 we see that we may write 
 
 the associative principle being thus seen to hold good here again. 
 
 282. We see at the same time (omitting the point), that the 
 above proposed symbol rqr~'^ denotes a quaternion which is ^ewe- 
 rally distinct from the quaternion q, but which bears a very sim- 
 ple relation thereto. In fact, we perceive, first, that not only 
 the tensors but also the angles of these two quaternions are equal 
 (in amount) ; or in symbols, that 
 
 T .rqr'^ = Tq', A.rqr'^ = Lq. 
 
 And in the second place we see that (if o be still the centre 
 of the sphere) the axis oe of the new quaternion, rqr~^, may be 
 geometrically derived from the axis oa of the old quaternion q, 
 hy a CONICAL and positive rotation, round the axis ob of the 
 other given quaternion r, through an angle equal ^o double the 
 ANGLE of that other given quaternion. In fact we may pass, upon 
 the surface of the sphere, from the pole a of g to the pole e of 
 rqr~^, or from the vertex of the given representative angle of the 
 one quaternion, to the vertex of the sought representative angle 
 of the other, by moving along an arc of a small circle^ which is 
 projected in the figure into the dotted line ae, and which has its 
 positive pole at the pole b of r, while it subtends at that pole an 
 angle expressed as follows : 
 
 ABE=2Zr. 
 
 283. An analogous interpretation may be obtained, without 
 any new difficulty, for the symbol q'^rq\ since we have only to 
 conceive that q~^ and r are written, in fig. 57, instead of r and g, 
 and consequently that q is substituted forr"\ in the same recent 
 figure. For thus we shall see that while the tensors and angles 
 of the two quaternions q'^rq and /-are equal {oi least in amount), 
 the axis of the former may be obtained from the axis of the lat- 
 ter, by causing this axis of r to revolve conicallg, in a negative 
 
270 ON QUATEENIONS. 
 
 direction, round the axis of q, through an angle equal to double 
 the angle of q. And generally, if t be any scalar exponent, 
 it will be found, with the help of the theory oi powers which 
 was explained in the Fourth Lecture, that the symbol 
 
 denotes a quaternion formed from r, by causing the axis of this 
 operand quaternion r to revolve, conically, round the axis of 
 the operator quaternion q^ through a (positive or negative) ro- 
 tation, expressed by the product 
 
 2txLq. 
 
 Thus conical (as well as plane) rotation is easily symbolized , 
 by quaternions. 
 
 284. Another construction, in appearance different from the 
 foregoing, but in reality connected with it, for a symbol of the 
 class recently discussed, may be obtained as follows, from the 
 consideration of fig. 37, in art. 219. In that figure, let us sup- 
 pose that 
 
 q~^r = s, 
 
 so that s denotes a new quaternion, or versor, represented by the 
 arc m'k. Treating that arc as a vector, and the arc kl as a pro- 
 vector, the arc m'l is seen to be the transvector (on the plan of 
 217, 218) ; and thus, or immediately from the equation just now 
 written, we derive this other equation, 
 
 qs =r. 
 
 Hence by the arcs k'l, lm, treated as a new system of vector 
 and provector, or by the construction already assigned for rq'^, 
 in the same figure 37, we see that the arc k'm represents the pro- 
 duct, 
 
 qs.q-^; 
 in which latter symbol it is easy to prove anew, by an analogous 
 construction with arcs, that the point may be omitted. But the 
 arc k'm which thus represents the resulting quaternion qsq'^, 
 has the same length as the arc m'k which represented the original 
 quaternion s, and is inclined at the same angle as that former arc 
 to the great circle of which kl, or lk', namely, the representative 
 
LECTURE VI. 271 
 
 arc of the operating quaternion q^ is a part. And the double of 
 this latter part, namely, the arc 
 
 kk' = 2-v kl, 
 
 exhibits the distance along which the arc m'k itself, or its inter- 
 section K with the great circle klk', has to be transported along 
 that circle, as by a motion of a node, without any change of the 
 inclination of the moving arc thereto, or of the length of the 
 same moving arc, in order to take that new position on the 
 sphere, wherein the intersection or node comes to be placed at 
 the point K. The interpretation of the symbol 
 
 qsq-^, 
 
 or of any other symbol of the same general form, may therefore 
 on this plan be easily and fully accomplished. 
 
 285. We know then how to interpret, in two apparently dif- 
 ferent ways, which are, however, easily perceived to have an 
 essential connexion with each other, the following symbol of 
 
 OPERATION, 
 
 where q may be called (as before) the operator quaternion^ 
 while the symbol (suppose r) of the operand quaternion is con- 
 ceived to occupy the place marked by the parentheses. For we 
 may either consider the effect of the operation, thus symbolized, 
 to be (as in 282, 283) a conical rotation of the axis of the oper- 
 and round the axis of the operator, through double the angle 
 thereof in such a manner as to transport the vertex of the re- 
 presentative angle of the operand to a new position on the unit 
 sphere, without changing the magnitude of that angle, nor the 
 tensor of the quaternion thus operated on : or else, at pleasure, 
 may regard (by 284) the operation as causing one extremity of 
 the representative arc of the same 02^erand (r) to slide along the 
 doubled arc of the same operator (q), without any change in the 
 length of the arc so sliding, nor of its iticlination to the great 
 circle along which its extremity thus slides. But it is clear that 
 these two conceptions are merely transformations oi edLch. other; 
 since they are evidently related, as, in astronomy, the rotation 
 OF THE POLE OF THE EQUATOR round the pole of the ecliptic is 
 
272 ON QUATERNIONS. 
 
 related to the precession of the equinoxes. Still, it is satis- 
 factory to observe the complete consistency between the results 
 of the two different processes of interpretation of a symbol of the 
 form qrq'^, which have been employed in recent articles ; and it 
 may just be noticed here, that, whichever of those two processes 
 we adopt, the principles of the Fourth Lecture respecting powers 
 conduct to the following important equation, 
 
 (qrq~'^y = qr^q~^i 
 
 as holding good in the Calculus of Quaternions, as well as in 
 ordinary Algebra, if t be any scalar exponent. 
 
 286. When the operand quaternion r of the last article re- 
 duces itself to a vector p, then the result, qpq'^, of the operation 
 of q{)q'^, becomes itseU another vector; for, by 149 and 282, 
 
 1 _ 
 
 TT 
 
 Z..qpq-' = Lp=- 
 
 and this new vector qpq'^ may, by the article just cited (282), 
 be derived from the old or given vector p, by simply causing it 
 to revolve conically round the axis Ax . q, though the doubled 
 angle 2 Lq, whatever the direction of p may he. Assuming, then, 
 as in several former articles, some one fixed point o, as the com- 
 mon origin of all the vectors p, which may be conceived to ter- 
 minate at the various points of some system, or hody, B ; we 
 may regard the recent symbol of operation, q{)q~^i as signify, 
 ing that we are to cause this body to revolve, through the angle 
 2 L q, round an axis Ax . q, which is drawn from or through the 
 fixed point o : and the new symbol, 
 
 qBq-\ 
 
 may be conceived to denote the position of the body B, after 
 this finite rotation has been performed. In like manner the 
 symbol, 
 
 r . ^B^-'^ . r'^, 
 
 may consistently indicate that new position of the same body B, 
 into which it is brought by performing a new and succesive rota- 
 tion, through the angle 2 Lr, round the new axis Ax . r; while 
 
LECTURE VI. 273 
 
 the result of still a third finite rotation, through a third angle 
 Its, round a third axis Ax . s, will be denoted by the symbol, 
 
 s {r .q^q'^ . r~^) s~^\ 
 
 and similarly for any number of successive and finite rotations of 
 a body round any arbitrary axes, which are, however, here sup- 
 posed to be all drawn through or from one common point or 
 origin o. 
 
 •287. The symbol 
 
 q{a + p)q-^, 
 
 where a is supposed to be a constant, and p a variable vector, 
 may easily be interpreted as follows. Let 
 
 a = A - o = o - B, |0 = P - o ; 
 then 
 
 a+jo = jo + a=P-B = Q-o; 
 
 where a, b are fixed points, at opposite sides of o, but r and q 
 are points which vary together. Conceive that a rotation round 
 the axis Ax .q, through an angle =2Lq, causes the line oq to 
 take the position oq' ; then, by what precedes, 
 
 ^(a + p)(?-i = Q'-o: 
 
 and the point p is to be conceived as having been transferred, 
 upon the whole, through the point Q as an intermediate position, 
 to the final position q'. The axis of the last rotation, as of the for- 
 mer ones, is here conceived to pass through, or to be drawn from, 
 the given point o; but if, from the point b, we draw 2i parallel 
 axis, 
 
 c - b = Ax . q, 
 
 and denote by bp' the position into which the line bp is brought, 
 by revolving, through the same angle 2 z g as before, round this 
 new axis bc, we shall have 
 
 p'-p=q'-q, q'-p'=q-p = o-b = a-o; 
 
 so that the point q' may be obtained also from the point p', 
 namely, by adding or applying (see Lecture L) the constant 
 vector OA, or a. It follows that the symbol 
 
 q{a + B)q'^ 
 T 
 
274 ON QUATERNIONS. 
 
 is adapted to denote that final position into which the body B is 
 brought, when it is first made to revolve (as above) through a 
 finite angle round the recent axis bc, which axis does not (in 
 general) pass through the given origin of vectors o ; and when 
 the body is afterwards made to move, without revolving, through 
 a finite amount of translation, expressed both in length and 
 direction by the line bo or oa, or by the vector of transla- 
 tion a. We see, however, that the same symbol may also be in- 
 terpreted as denoting a translation represented by the line a^ fol- 
 lowed by a rotation round an axis Ax . q, which axis is here again 
 supposed to be drawn from the origin o ; this latter point being 
 regarded ?l% fixed in space, and as not participating in any motion 
 of the body. By adding any other constant vector, such as /3, we 
 form an expression for the result of the foregoing operations, suc- 
 ceeded by a new translation of the body in space ; for example, 
 if we wish to neutralize the recent translation a, and thereby to 
 express that the body has only revolved round the axis bc, 
 through the angle 2 z 5-, but has 7iot otherwise changed "place, we 
 may write the expression, 
 
 - a^q (a+ B) q'^. 
 
 288. If we wish to express that a vector or body is made to 
 turn round an axis Ax . q which is drawn from the origin o, 
 through an angle of finite rotation expressed by Z q, that is 
 through the angle itself oi the quaternion q, and 7iot through 
 the double of that angle, we need only (by 283) employ this 
 other symbol of operation, 
 
 q^ ( ) ^-*. 
 
 Hence, by conceiving q to be the quotient of two given vectors, 
 for instance, by supposing 
 
 5' = j3 -r- a = j3a"S 
 and therefore 
 
 we find that the symbol 
 
 (j3«-i)* B(ai3-i)* 
 denotes that new position into which the body B is brought. 
 
LECTURE VI. 275 
 
 when it is made to revolve round an axis drawn from o, perpen- 
 dicular to both a and j3, through that amount and in that direc- 
 tion of finite rotation, which would bring the vector a into the 
 direction of the vector j3 by a rotation in one plane ; namely, in 
 the plane through the origin o, perpendicular to the last men- 
 tioned axis. 
 
 289. On the other hand, if we omit the fractional exponents, 
 and so form this other symbol, 
 
 i3a-i.B.a/3-i, 
 
 we find, on the same general principles of interpretation, that 
 this symbol denotes the result of the rotation of the same body 
 round the same axis, through double the angle of the quaternion 
 j3a "^ or through an amount which is the double of the plane ro- 
 tation from a to j3. For example, in fig. 40, art. 224, where 
 A, B, c, D, E, F are supposed to be six points upon the unit 
 sphere, with a, j3, 7, S, e, Z, for their six unit-vectors ; while the 
 three arcs ef, fd, de have been shewn to be bisected by the 
 three points a, b, c ; and (compare fig. 41, art. 227) the conical 
 rotation from e to d, round the axis or pole of the arc of a great 
 circle from a to b, is equal to the double of that arc ab, namely, 
 to the plane rotation from s to r ; we may infer, from the result 
 just stated, respecting the interpretation of the symbol 
 
 j3a-^().«i3-i, 
 
 that the following equation holds good : 
 
 i3a-i.e.aj3-i = g. 
 
 290. If the operating quaternion q reduce itself to a vector, 
 suppose 7, then since its doubled angle is equal to two right 
 angles, or in symbols, 
 
 2 Z 7 = TT, 
 
 the operation symbolized by 
 
 7()7" 
 is seen to have the eifect of simply reflecting the vector or 
 body on which it operates, with respect to the operating vector, 
 7. That is to say, this operation causes each operand vector, 
 
 T 2 
 
276 ON QUATERNIONS. 
 
 suppose p, drawn from the common origin o, to tur7i conically 
 through two right angles round the line 7, which is here con- 
 ceived to be drawn from the same origin ; and thereby brings 
 this operand p^ without change of length, into a new position p\ 
 such that while we have the equation between tensors^ 
 
 Tp =Tp, if jo' = 7joy'S 
 
 the line ± 7 at the same time bisects the angle between p and p : 
 and consequently the following equation between versors also 
 holds good : 
 
 U.p'7-' = U.7p-^ 
 
 For example, in fig. 40, 
 
 7£7-' = S; 
 also, in same figure, 
 
 |3?j3-' = 8; and «£«-' = a-i£o = C 
 291. Another mode of interpreting the symbol 
 
 7^7- 1 
 is the following. We may observe that, by ill, 117, 
 
 p = -p-i T^^ 7-' = -7T7-2; 
 and that therefore 
 
 7P7"^ = Tp2 T7"2 . yp-i y. 
 
 Now we know (133, 194) that the symbol jp'^y denotes the 
 third proportional to the two vectors p and 7 ; and therefore that 
 (see 134) the vector + 7 bisects the angle between the directions 
 of |0 and yp'^j ', or by the recent transformation, the angle be- 
 tween jO and 7j07"^ : which was the graphic part of the result of 
 the last article. And with respect to the metric part of that re- 
 sult, we know (by 129, &c.) that the tensor of a third propor- 
 tional is the third proportional to the tensors, and therefore that 
 
 T .yp-'^y = Ty^.Tp-'^; 
 
 an expression which reduces itself to Tp, when it is multiplied 
 by TjO^j and divided by T72. Indeed it is clear from the more 
 general principle of art. 188, respecting the tensor of a product, 
 that 
 
LECTURE VI. 277 
 
 T . -yjo-y'^ = Ty T|0 T-y ■ ^ =Tp. 
 
 292. With reference to fig. 40, we have, by articles 289, 290, 
 
 /3.a-i6a.i3-i=/3a-i.e.a/3-'; 
 
 the common value of both members being here the vector B : so 
 that the removal of points is here again permitted ; and the asso- 
 ciative principle of multiplication is, at least so Jar, here seen 
 once more to hold good : while the geometrical interpretation of 
 this result shews that the equation thus obtained is by no means 
 a TRUISM in this Calculus (compare 108); but expresses that a 
 certain conical rotation is equivalent in its effect to two suc- 
 cessive and PLANE rotations. In the astronomical illustration 
 here referred to (see the last Lecture), the conical rotation was 
 performed round the axis of the ecliptic, from e to d in fig. 41, 
 through an amount represented by the double of the arc ab of that 
 great circle; while the two plane rotations were performed across 
 the ecliptic, namely, from e to f, and from f to d, in fig. 40, the 
 points A and b being employed as two successive reflectors. Now 
 it was by no means obvious that these two different geometrical 
 processes must conduct to one common result. Yet they have 
 been proved in the last Lecture to do so : and the conclusion ar- 
 rived at, by this geometrical demonstration, is now seen to be 
 symbolically expressed, by the very simple and apparently obvious 
 formula, which has been given in the present article. 
 
 293. It is now time to enter on the proof already promised 
 (in arts. 108, &c.), that the Associative principle of Multiplica- 
 tion of Quaternions is valid generally, in this Calculus : and first 
 to demonstrate generally, what indeed is the chief, and (we may 
 say) the only real difficulty in the required proof, that for any 
 THREE VERSORS the asscrted principle holds good. Conceive then 
 that any three proposed versors, q, r, s, are represented by some 
 three given arcs, qq', rr', ss', upon the surface of the unit-sphere: 
 and that it is required to construct, on the same spheric surface, 
 another arc tt', which shall be the spherical (or arcual) sum of 
 those three given arcs, or shall represent the product, s . rq, of 
 the three given and corresponding versors, when the arc rr' is 
 first arcually added (on the plan of art. 218) to the arc qq', and 
 
278 ON QUATERNIONS. 
 
 the arc ss' is afterwards areually added to the result, so as to con- 
 duct to and determine 21. fourth arc tt': or when the versor of q is 
 first multiplied by the new versor r, and then the product, rq, is 
 again multiplied by the third given versor, 5, so as to conduct to 
 a fourth versor, s . rq, or t. And let us afterwards proceed to 
 COMPARE this process, as to its result, with that other combi- 
 nation of arcs, or of versors, in which the arc ss' is first added (on 
 the same plan) to the arc rr', and the resulting arc then added 
 to qq', so as to form a neiv and Jifth arc, uu': or when the versor 
 s is multiplied into r, and the product, sr, is then multiplied into 
 q, so as to conduct to a new final and ffth versor, sr.q, which we 
 may for the present call u. In other words, let us examine whe- 
 ther it be true that, under these conditions, we have the follow- 
 ing equation between arcs (to be interpreted in the sense of art. 
 
 217), 
 
 ^ uu' = - tt' ? 
 
 Or that we have the corresponding equation between versors, 
 
 u = t? 
 In short, let us inquire (compare 108) whether the following 
 formula is, in this calculus, as well as in algebra, an identity, 
 
 sr . q - s . rq ? 
 
 294. After what has been already said, and illustrated by ex- 
 amples and by diagrams, it can scarcely need to be now formally 
 shewn, that instead of the three given 6m^ wholly arbitrary 
 arcs, qq', rr', ss', from which two others, tt' and uu', are to be 
 derived (as stated in the foregoing article), we are at perfect 
 liberty to substitute any three other arcs, to which those three 
 given arcs are equal (217). We may 
 then suppose, without any real loss of , ""' ' 
 
 i^i^ ' J M T T. 
 
 generality, that the first and second 
 are two successive arcs, such as ab 
 and BC in the annexed figure 58 ; and 
 that the third given arc is the arc ef 
 in the same figure, which has its i7ii- 
 tial point e on the great circle ac, 
 connecting the initial point a of the 
 first with the final point c of the second 
 
LECTURE VI. 279 
 
 arc. Then the arcual addition (218) of the second to the first 
 given arc produces, as their sum^ or as the representative arc of 
 the product, rq, of the two first given versors, the arc ac; for which 
 we may substitute an equal arc, such as de in the figure, which 
 shall end at the point e, where the third given arc ef, representing 
 the third given versor s, begins : so that the subsequent addition 
 of this third arc, or the multiplication by this third versor, con- 
 ducts to the fourth arc df (which here takes the place of the arc 
 tt' of the last article), as representing the product s .rq. Again, 
 in order to add the third given arc to the second, or to represent 
 the product s?% we are (by 217) to find the point h where the arcs 
 BC and EF intersect, and then to determine two new points, g 
 and I, such that gh and hi shall be arcually equal to bc and ef, 
 and shall therefore be fit, like those given arcs, to represent the 
 given versors r and s; for then the joining arc gi will repre- 
 sent, as required, the product of those versors, namely sr. 
 And, finally, in order to multiply this last product, sr^ into q, 
 we are to find the point l where the arcs ab and gi, representing 
 respectively the multiplicand q and the multiplier 67', intersect ; 
 and to determine afterwards two other new points, k and m, such 
 that the arcs kl and lm may be respectively equal to those two 
 representative arcs, of the new multiplicand and multiplier ; for 
 then, by merely joining these two last points, we shall obtain an 
 arc KM (the uu' of the foregoing article), which shall, by the 
 general construction in 217, represent that other sought product 
 of versors, of which the symbol is sr.q. 
 
 295. It was proposed in 293 to examine whether the products 
 of versors, denoted there by the two symbols u and ^, or by 
 
 sr . q and s . rq^ 
 
 were equal. And we now perceive that this question may be 
 thus expressed, in connexion with the recent figure 58 : are we 
 entitled to establish the arcual equation, 
 
 - KM = ^ DF, {srq) 
 
 in \\\Q,full sense of article 217, when, in the same full sense, we 
 are given these /ye other equations between arcs, 
 
280 ON QUATERNIONS, 
 
 - AB = -- KL, (q) 
 
 - BC = - GH, (r) 
 '^ EF = '- HI, («) 
 
 '- AC = '^ DE, (rq) 
 
 ^ Gi = '- LM. (sr) 
 
 You will observe that at the margin of each of the six last lines, 
 expressing arcual equalities, I have written, within parentheses, 
 the symbol of that particular versor, which the two equated arcs 
 are given, or are to be proved, to represent. 
 
 296. To those students who are acquainted with the theory 
 of the spherical conies, and I know that here, through the ex- 
 ertions of the late and present Professors of Mathematics in this 
 University, an acquaintance with that doctrine has come to be 
 widely diffused, the following brief process may be sufficient for 
 the establishment of the result in question. Let such a conic be 
 conceived to be described upon the surface of the sphere, passing 
 through the three points bfh, with the arc ce for part of one of 
 its two cyclic arcs ; then the two equations, between the arcsBC, 
 GH, and between ef, hi, suffice to shew that the arc gi is part of 
 the other of those two cyclic arcs ; and the equation between 
 AB, KL, where a is on the first and l is on the second of the same 
 two arcs, shews next that the same conic passes also through the 
 point k; or that (if f, k be joined) this conic is circumscribed 
 about the quadrilateral kbhf : because it is known that " every 
 arc of a great circle intersects a spherical conic in two points 
 which are equally distant from the points in which this arc re- 
 spectively cuts the two cyclic arcs," if the transversal arc inter- 
 sects the conic at all. (See Section II., article 13, of a Memoir 
 by the celebrated Chasles, on the general properties of the sphe- 
 rical conies, as given at the foot of page 46 of the translation of 
 that Memoir by our present Professor of Mathematics, the Rev. 
 Charles Graves, which translation was published in Dublin in 
 the year 1841.) Conceive, in the next place, that the arc fk is 
 prolonged to meet the cyclic arcs ; it will meet the first of them 
 in D, and the second in m, in virtue of the equations between the 
 arcs AC, DE, and between gi, lm : because it is known that "if 
 through two fixed points on a spherical conic two arcs be drawn 
 
LECTURE VI. 281 
 
 .which intersect in any third point of the curve, the segment 
 which they will intercept upon a cyclic arc will be of invariable 
 magnitude." (See Section III., art. 29, of the same memoir by 
 Chasles, page 50 of the translation by Graves.) Thus the^wr 
 points D, K, F, M, are situated on one common great circle^ or trans- 
 versal arc ; and therefore, by the principle before referred to, the 
 intercepted portions dk and fm, or df and km, are equal in length, 
 while it is evident that they are similarly directed. It is there- 
 fore proved to be a consequence of these few and known pro- 
 perties of spherical conies, that, under the conditions of the pre- 
 sent inquiry, the arcual equation, 
 
 '-KM=^DF, 
 
 which was lately proposed for investigation (in 295), does in fact 
 hold good (in the full sense of art. 217) : or that the two equa- 
 ted arcs are equally long and similarly directed portions of one 
 common great circle of the sphere. 
 
 297. Although the properties of spherical conies, which have 
 been referred to in the foregoing investigation, are well known to 
 a large number of students, yet as there may be others to whom 
 they are not familiar, it appears to be useful to offer now an in- 
 dependent and more elementary proof of the result to which they 
 have conducted us. Indeed it would be doing a grave injustice 
 to the Calculus of Quaternions, and conveying a false notion of 
 the nature of its principles, if you were to be allowed to suppose 
 that, for so important and essential an element as the associative 
 property of multiplication, this Calculus was dependent on the 
 doctrine of spherical (or even of plane) conies. On the contrary, 
 I believe that the easiest and most elegant method, in the present 
 state of science, of treating those and other spherical curves by 
 calculation, will be found to be that method which is furnished 
 by the Quaternion Calculus. In order, then, to prepare for legi- 
 timately so applying this Calculus, it seems to be necessary, in 
 point of logic, that we should seek to establish the arcual equation 
 of article 295, namely 
 
 -> KM = - DF, 
 
 on which (by 294) the equation between quaternions, or between 
 versors, 
 
282 
 
 ON QUATERNIONS. 
 
 sr .q - s . rq, 
 has been made to depend, by some process of geometry, which 
 shall be of a comparatively elementary nature; and which shall 
 therefore not introduce the conception of a spherical conic (nor 
 even that of an oblique co7ie) at all : although there is no reason 
 why, at this stage, we should scruple to use the notions of plane 
 and sphere, as freely as those of the right line and circle. The 
 persons who have already studied the theories of cones and conies 
 must of course have an advantage thereby; but the object, which 
 we at this moment propose to ourselves, is to render thoroughly 
 intelligible, to persons who have not studied those theories, so 
 much as may be necessary for perfectly understanding the force of 
 the demonstration, which was given in the foregoing article : or 
 of that apparently longer, but essentially equivalent proof, which 
 we are now about to give. 
 
 298. Conceive then that, in connexion with the recent figure 
 58 (o being still supposed to be the centre of the sphere), the three 
 radii ob, oh, of, are prolonged to meet, in three points p, q, r, 
 a plane pqr, which is drawn (as we shall suppose) outside the 
 sphere, but parallel to the plane of the great circle daec ; con- 
 ceive also that these three prolonged radii op, oq, or, are cut in 
 three other points, p', q', r', by another plane p'q'r', which shall 
 be drawn parallel to the plane of the great circle glim. Round 
 the four points o, p, q, r, circumscribe a new sphere opqr, which 
 we shall call, for the present, the diacentric sphere, because its 
 surface passes through the centre o of the original or unit sphere, 
 whereon the former figure 58 has been conceived to be traced. 
 Let these two spheres be conceived to be cut by the plane of the 
 great circle gbhc, which circle 
 thus becomes itself one of the two 
 sections hereby formed, as in the 
 annexed figure 59, the other sec- 
 tion being the circle opq. Then, 
 because the comparison of the two 
 representative arcs of the versor 
 r gave us (by 295) the equation 
 - BC=--GH, we have also the 
 equation between angles, 
 
LECTURE VI. 
 
 283 
 
 COB = HOG, or COH = POG. 
 
 But oc is parallel to pq, because these two lines are the inter- 
 sections of two parallel planes, namely, of daec (in fig. 58) and 
 PQR, made by one common secant plane, namely, by the plane 
 of the recent figure ; and (compare fig. 58) the direction of oc is 
 evidently not opposite, but similar to that of pq : we have there- 
 fore this other equation between angles, 
 
 pqo = coh; 
 
 and consequently also, in virtue of the last equation, 
 
 PQO = POG. 
 
 The radius og of the unit sphere is therefore a tangent to the 
 circle opq, and consequently it is a tangent also to that diacen- 
 tric sphere, opqr, whereof this circle is a section. And because 
 the line qV is parallel to this radius og (on account of the pa- 
 rallelism of the two planes p'q'r' and glim), and has a similar 
 (not opposite) direction, we have this other equation between 
 angles, 
 
 op'q'=pqo; 
 
 which shews that the four points p, q, q', p' are on the circum- 
 ference of one common circle, and that therefore the following 
 equation between rectangles subsists : 
 
 pop'=qoq'. 
 
 299. By a reasoning exactly similar it may be shewn, that if 
 the two foregoing spheres, and the two 
 planes pqr, p'q'r', be cut, as in figure 60, 
 by that new secant plane which is the 
 plane of the great circle ehfi in fig. 58, 
 then the equation 
 
 -- EF ='- HI, 
 
 which was obtained (in 295) as the result 
 of the comparison of the two representa- 
 tive arcs of s, when combined with the 
 
 parallelisms between rq, oe, and between q'r', oi, conducts to the 
 
 angular equalities. 
 
 Fig. 60. 
 
284 ON QUATERNIONS. 
 
 RQO = EOQ = ROI = OR'q' ; 
 
 and to the following equation between rectangles, 
 
 qoq' = ror'. 
 
 .The radius oi of the unit sphere is therefore a tangent to the cir- 
 cular section oqr of the diacentric sphere, and to that sphere 
 OPQR itself; and the four points r, q, q', r', are situated on one 
 common circular circumference. And by combining the results 
 of the present article with those of the foregoing one, it becomes 
 clear that the plane glim (see fig. 58) of the two radii og, oi, of 
 the unit sphere, touches at o the diacentric sphere opqr ; and 
 also (from the equalities of rectangles), that the six points p, q, r, 
 p', q', r', are situated on the surface of a third sphere, pqrp', 
 whereof the circles pqqV and rqq'r' (in figures 59 and 60), as also 
 the circles which may be conceived to be circumscribed about the 
 triangles pqr and p'q'r', are sections. 
 
 300. Conceive, in the next place, that the radius ok of the 
 unit sphere is prolonged to meet respectively the diacentric 
 sphere and the plane p'q'r' in two new points, s and s'; and let 
 the given and diacentric spheres be supposed to be both cut by 
 the plane of the great circle akbl (see fig. 58) ; the section of 
 the unit sphere being that great circle itself, but the section of 
 the diacentric being a new circle, ops. A new figure will thus 
 be constructed, so similar to those of the two last articles that it 
 seems to be almost unnecessary to write it here ; for all essential 
 purposes you may form it, or conceive it to be formed, by merely 
 changing, in fig. 59, the letters c, g, h, q, q', to a, l, k, s, s', 
 respectively : still for more perfect clear- 
 ness I shall give it to you as figure 61. 'S- • 
 But whereas, in each of the two figures /^T'^^ 
 of the two last articles, we inferred a / \^'\ 
 tangencyy/"om a parallelism, we have / '■^\c\^ 
 now, on the contrary, a tangency^/z7e?2, / ^-Y n) 
 and a parallelism is thence to be infer- / ^ — -^}^^^^/f^ 
 red. For we now ^noi^; that the radius ^K^^"^ ..."' \// 
 
 OL of the unit sphere touches the sec- \ / 
 
 tion OPS of the diacentric, because (by ^^. ^ ^ ^^ 
 
 fig. 58) this radius is contained in the 
 
f 
 
 LECTURE VI. 285 
 
 plane glim, which plane was seen (in art. 299) to touch the dia- 
 centric sphere at o. Hence the angle bol or pol, in fig. 61, 
 between chord and tangent of the section of the diacentric, is 
 equal to the angle pso in the alternate segment ; but it is also 
 equal to aok or ads, on account of the equality of the angles aob, 
 KOL, or of the arcs ab, kl, which last equality of arcs was deduced 
 in 295 from the comparison of two different representations of the 
 versor q : we have therefore the following equation between 
 angles, 
 
 PSO = AOS, 
 
 and may itifer from it that the chord ps of the diacentric is pa- 
 rallel to the radius oa of the unit sphere. But (see again fig. 58) 
 this latter radius is contained in the plane of the great circle 
 CEAD, to which (by 298) the plane pqr is parallel ; this latter 
 plane must therefore contain the chord ps : or in other words, the 
 four points p, q, r, s are all situated in one common plane. And 
 because by the construction they are also situated on the surface 
 oi one common sphere (the diacentric), they must he four concir- 
 cular points : they are in fact all situated on the circumference 
 of that common circle, in which the diacentric and third spheres 
 intersect each other. Again, in fig. 61, the lines sV and ol are 
 parallel, as being the traces, on the plane of the figure, of the 
 two parallel planes (see 298), p'q'r' and glim ; these lines are 
 also similarly directed : thus the four points p, s, s', p' are con- 
 circular; and we have the following equation between rectangles, 
 
 sos' = pop'. 
 
 In fact the circle pss'p' is contained on the third sphere ; and 
 another circle of the 
 
 same third sphere con- 
 tains the four points p', 
 q', r', s'. 
 
 301. Comparing next, 
 as in the annexed figure 
 62, the circle pqrs of 
 the diacentric with the 
 parallel and great circle 
 CEAD of the unit sphere. 
 
 Fig. 62. 
 
286 
 
 ON QUATERNIONS. 
 
 Fig. 63. 
 
 and attending to the arcual equation - ac = '- de, which was ob- 
 tained in 295 by the comparison of the two representative arcs of 
 the quaternion rq^ we see that because (by the three last figures) 
 the three chords pq, rq, ps have respectively the directions of the 
 three radii oc, oe, oa, therefore the fourth chord rs must have 
 the direction of the fourth radius od, on account of the equality 
 of the angles spq, srq, on the one hand, and aoc, doe, on the 
 other. The point d of the unit sphere, or the corresponding 
 radius od, is therefore contained in the plane ors, which coin- 
 cides with the plane ofk ; that is to say (see fig. 58), the three 
 points F, K, D are on one common great circle of the unit sphere. In 
 a similar way by comparing, as in fig. 63, the two parallel circles 
 p'q r's' and milg, it may 
 be shewn that, because 
 the three chords q'p', 
 q'r', s'p', of the one 
 circle, have respectively ^ 
 (see figs. 59, 60, 61) the 
 same directions as the 
 three radii og, oi, ol, of 
 the other,while (by 295) 
 the arcs gi and lm are 
 
 equal, as both representing the quaternion sr\ and the angles 
 p'q'r' and p's'r' are also equal to each other, as being in one com- 
 mon segment of a circle: therefore the fourth chord s'r' must have 
 the same direction as the fourth radius cm. This radius is there- 
 fore contained in the plane orV, or in the coincident plane ofk; 
 or, in other words, the point m, like the point d, is situated on the 
 great circle fk (fig. 58). And 
 if we finally cut the unit and 
 diacentric spheres by the plane 
 of this great circle, we obtain /' 
 a new figure 64, wherein, by / 
 the present article, the radius i 
 CD of the section dkfm has the \ 
 same direction as the chord rs "' 
 of the section ors, while this 
 latter section is touched at o by 
 
 Fig. 64. 
 
LECTURE VI. 
 
 287 
 
 the radius om of the former. The angles fom and dok are con- 
 sequently equal to each other, as being each equal to the angle 
 Rso ; and therefore an equality subsists between the angles dof 
 and KOM, or the arcs df and km. These latter arcs are there- 
 fore equal to each other, in the full sense of article 217 : which 
 was (in 295) the thing proposed to be proved. 
 
 302. After the elementary investigation contained in the four 
 foregoing articles, which has established the associative principle 
 of multiplication for any three versors (compare art. 293), with- 
 out introducing (see 297) even the conception of a cone, by em- 
 ploying certain combinations of representative arc*, together with 
 some evident or well-known properties of planes and spheres, it 
 may be considered unnecessary now to establish the same prin- 
 ciple by means of representative angles also. Yet, for the sake 
 of those students who are already familiar with the properties of 
 spherical conies, or even with a few of the best known among 
 those properties, I 'shall give rapidly a proof, by them, of the 
 same general and important result {sr .q = s .qr), in which proof 
 angles, instead of arcs, shall thus be employed to represent the 
 versors. 
 
 Let then, in figure Q5 (in which it has been thought sufficient 
 to draw straight 
 linesinstead of arcs 
 of great circles ), 
 the versor q be re- 
 presented by the 
 spherical angle 
 EAB ; r by abe, 
 and also by fbc ; 
 and s by bcf and 
 ecd: moreover, let 
 the angles DEC and 
 BEA be supposed to A. 
 be supplementary. 
 
 Then (see 264) the angle dec, and the supplement of cfb, will 
 represent respectively the two binary products, rq and sr ; and the 
 supplement of cde will represent on the same plan the ternary pro- 
 duct s . rq. But to shew that this latter is equal to the other ter- 
 
288 ON QUATERNIONS, 
 
 nary product sr . q, it is necessary and sufficient to prove that the 
 angles daf and fda are respectively equal to eab and cde ; and 
 also that the angles afd and cfb are supplementary : because 
 we have to prove that the angles daf and afd represent re- 
 spectively q and sr, and that the supplement of fda represents a 
 ternary product sr . q, which is equal to the former product s . rq. 
 For this purpose, conceive a spherical conic described, with e 
 and F for foci, so as to touch the arc ab ; this conic will also 
 touch the arcs bc and cd, on account of the equalities of the two 
 angles at b which represent r, and of the other two angles repre- 
 senting 5 at c ; while by the supplementary character of the angles 
 at the focus e, it will touch also the arc ad, and therefore will be 
 inscribed in the spherical quadrilateral abcd. (See the Memoir 
 of M. Chasles already cited, at the same pages as before of the 
 translation by Professor Graves.) But this inscribed conic gives 
 the two required equalities of angles, at the corners a and d, 
 and the supplementary character of the angles at the focus f : 
 and thus the theorem is established, or the associative property 
 of the multiplication of three versors is proved anew. 
 
 303. It is therefore demonstrated, in several diiferent ways, of 
 which some are shorter while others are more elementary, that 
 the equation already often mentioned (see 293, &c.), namely, 
 
 sr .q = s . rq, 
 
 is in fact an identity, although by no meansa^r^xsm (compare 
 108, 292), in this Calculus, when q, r, s denote any three ver- 
 sors ; from which, by the properties (188, 208) oi tensors of pro- 
 ducts, it follows at once that the same equation is identical when 
 the three factors denote any three quaternions. We may 
 therefore omit generally (compare 136, 194) the point or 
 other mark of multiplication, in the expression of any such ter- 
 nary product, and may denote that product by writing simply the 
 
 symbol 
 
 srq. 
 
 We see also that when we introduce (as in 296, 302) the con- 
 sideration oi spherical conies, which, however (by 298, 299, 300, 
 301), it is not necessary for us to do, then the two partial or bi- 
 nary products, rq and sr, are represented either by portions of 
 
LECTURE VI. 289 
 
 the two cyclic arcs of a conic circumscribed about a quadrilateral, 
 or else at pleasure by angles at the two foci oi another conic, in- 
 scribed in another quadrilateral : and that certain portions of the 
 sides of the one quadrilateral, or certain angles at the corners of 
 the other, represent the three g\v en factors, q, r, «, regarded as 
 versors, and their ternary product, srq. It may be allowed me 
 here to state that this focal representation of the geome- 
 trical relations between the six quaternions q, r, 5, rq, sr, srq, 
 was perceived by me almost immediately after the notion itself 
 occurred oi quaternions generally; and was exhibited at a gene- 
 ral meeting- of the Royal Irish Academy, in November, 1843, 
 together with various geometrical corollaries, deduced from the 
 same construction. 
 
 304. It is easy now to establish the associative principle of 
 multiplication generally, for 3iX\y four or more quaternions. For 
 if t denote a fourth given factor, we shall have 
 
 t . s (rq) ==ts.rq = (ts) r . q, 
 
 by treating alternately the binary products rq and ts as if each of 
 them were a single given quaternion, and by employing what has 
 been already proved respecting the multiplication of any three 
 factors; thus we may write, 
 
 f . srq = ts .rq = tsr . q = tsrq, 
 
 the points being again found to be needless. And on the same 
 plan we should pass, with the utmost ease, from the case oifour 
 to the case oi Jive given factors, and so to that of any greater 
 number of quaternions to be multiplied together: the order of the 
 factors being still, however, in general essential to be preserved, 
 because the multiplication of quaternions has been seen in former 
 articles to be not a commutative operation, though it has since 
 been proved that it is an associative one. We may for the same 
 reason now assert, generally, if we retain the phraseology of 
 articles 218, &c., respecting the operation of arcual addition, 
 that this operation also, like the multiplication of quaternions 
 which it represents, is associative, although not generally com- 
 mutative. A similar assertion may be also made respecting 
 the operation of angular summation, if we understand by the 
 
290 ON QUATERNIONS. 
 
 spherical sum of two angles on a spheric surface what was de- 
 fined in article 268. And it is important to observe that even 
 the commutative property holds good, whenever the quaternions 
 which are to be multiplied are coplanar, or co-axal ; that is 
 (see 93) when their representative biradials are parallel, even 
 though they may have opposite aspects, or although the axes of 
 the -factor quaternions may have their directions opposite. For 
 the same reason, the addition of vector arcs is a commutative 
 operation, when the arcs to be added are portions, whether simi- 
 larly or oppositely directed, of orie great circle ; and the summa- 
 tion of spherical angles is in like manner commutative, when their 
 vertices either coincide, or else are diametrically opposite. 
 
 305. Regarded as a theorem of spherical geometry, the asso- 
 ciative property of multiplication, for the case of three versors, 
 was seen in art. 295 to admit of being stated under the following- 
 form : that a certain arcual equation, 
 
 - KM = '- DF, 
 
 interpreted as in 217, v^^^di consequence of five other arcual equa- 
 tions of the same sort, namely (see fig. 58), of these five : 
 
 ^ AB = - KL, '^ BC = '-^ GH, - EF = -^ HI, "^ AC = '"^ DE, ^ GI = - LM. 
 
 To assist ourselves in remembering this result, we may state it as 
 follows, in connexion with the same figure 58 : \ifive out of the 
 six arcual equations, 
 
 KL^- AB, ^ GH = 
 
 ^'^ EC, --- ED = ^ CA; 
 
 LG = '-^ MI, -^ HE = 
 
 --^IF, ^ DK = -- FM, 
 
 be given, the sixth may be inferred. Here abc and mif are 
 triangles, and klghed may be considered as a hexagon, al- 
 though its sides kl and gh cross ; and if we suppose this hexa- 
 gon to be given, we can always choose the two triangles, so as 
 to satisfy the two first out of the three equations on each of the 
 two foregoing lines ; namely, by the process which would be em- 
 ployed (see 217, 218) for arcually adding gh to kl, and he to 
 lg: but if the hexagon have been arbitrarily taken, neither of 
 the two remaining equations (between ed, ca, and between dk, 
 fm) can then be expected to hold good. The theorem involved in 
 
LECTURE VI. 291 
 
 the associative principle shews, however, that if one of these two 
 remaining equations between arcs be satisfied, the other will be so 
 too. We may then state this associative theorem as follows: — 
 *' Ifthejirst, third, and fifth sides (kl, gh, ed), of a spherical 
 hexagon (klghed) be respectively and arcually equal to the 
 Jirst, second, and third sides (ab, bc, ca) o/one spherical triangle, 
 then the second, fourth, and sixth sides (lg, he, dk) of the same 
 hexagon are respectively and arcually equal to the first, second, 
 and third sides o/"another spherical triangle (mif)." 
 
 306. We might also, although less simply, conceive the six 
 points A, M, B, I, c, F, as being the six successive corners of 
 another spherical hexagon; the arc ab, drawn from the first of 
 these corners to the third, might be called the first diagonal of 
 this new hexagon ; the arc mi, from second to fourth corner, 
 might be called the second diagonal ; and in like manner the 
 arcs BC, IF, CA, fm would come to be called the third, fourth, 
 fifth, and sixth diagonals, respectively, of the same second hexa- 
 gon ambicf. And then the associative principle iox the vanlti^Yi- 
 cation of three versors might be expressed as follows: " 7/'five 
 successive sides o/one spherical hexagon be respectively and ar- 
 cually equal to five successive diagonals o/" another spherical 
 hexagon, the sixth side of the former hexagon will in like man- 
 ner be arcually equal to the sixth diagonal of the latter." I 
 once proposed to call this result the theorem of the two hexagons; 
 but perhaps the comparison which afterwards occurred to me, of 
 one hexagon with two triangles (305), is simpler and more na- 
 tural. 
 
 307. The enunciation of the same fertile principle may be 
 varied in many ways. For example, since the arcual sum of the 
 three successive sides of any spherical triangle (third plus second 
 plus first) must be considered as equal to zero, on the plan of ar- 
 cual addition adopted in former articles (218, &c.), we may state 
 the result of art. 305 as follows : — " If the arcual sum o/one set 
 of three alternate sides of a spherical hexagon vanish, when 
 taken in a suitable order (fifth plus third plus first), then the 
 arcual sum of the other set of three alternate sides of the same 
 hexagon (supposed to be suitably and similarly taken, as sixth 
 plus fourth plus second) will likewise be equal to zero." If 
 
 u 2 
 
292 ON QUATERNIONS. 
 
 then we allow the mark ^ to remind us that + signifies ai'cual ad- 
 dition, when interposed between two symbols of ares so marked, 
 we may write the following formula : 
 
 if -- ED + ^ GH + '^ KL = 0, 
 then '- DK + -- HE + - LG = 0. 
 
 The first of these two equations expresses a certain relation be- 
 tween the positions of the six points k, l, g, h, e, d, upon a spheri- 
 cal surface ; the second equation expresses another relation of posi- 
 tion between the same six points ; and the theorem is, that these 
 two relations are so con7iected, that each involves the other. It 
 seems to me that we might also employ, not inconveniently, the 
 symbol d -e to denote the same directed arc, or arcual vec- 
 tor (217), as that already denoted by ^ ed ; in such a manner 
 that we might write, generally, by a comparison of these two no- 
 tations, the identity, 
 
 B-A = '-^AB. 
 
 And then the recent formula would come to be thus expressed, 
 perhaps more clearly than before : 
 
 ifD-E + H-G+L-K = 0, 
 then K-D + E-H + G-L = 0. 
 
 We may also write, 
 
 E-H + G-L = D-K, ifH-G + L-K = E-D. 
 
 308. If we denote respectively by 
 
 a,l5,y; ^,e,Z; 0, r], t; k, \, fi, 
 
 the twelve unit vectors drawn from the centre o of the unit sphere 
 to the twelve points 
 
 A, B, C ; D, E, F ; G, H, I ; K, L, M, 
 
 upon its surface, then we may consider the three versors q, r, s, 
 with their binary products rq,'sr, and their ternary products 
 s . rq, sr . q, as equal to certain quotients of these vectors : for 
 we shall have by 294, 295, and fig. 58, the equations, 
 
LECTURE VI. 293 
 
 i3 ^ . . 7 ^. ,_^ 
 
 I 
 
 9=^.=--^ r = 15=-a 
 
 s = 
 
 UK (5 £ 1? 
 
 y s I fx 
 
 s.rq^-^; sr.q = —. 
 
 C K 
 
 To justify, therefore, the omission of the point in the symbol 
 
 srq, 
 
 or to establish the associative principle, comes to shewing (com- 
 pare art. 295), that the equation between quotients, 
 
 ^ = -, 
 is a consequence of five other equations of the same sort, namely, 
 
 x/3 i?7 t z, , ^ _ y . f^ ' 
 
 K:a'0j3 V £ S a X 
 
 And this consequence respecting quotients may now be con- 
 sidered as having been already proved, through the investiga- 
 tions respecting arcs and angles, which have been given in recent 
 articles. Indeed, we lately spoke of a, j3, &c., as being unit vectors ; 
 but on inspection of the six foregoing equations, it is evident that 
 their lengths may be arbitrarily chosen, without disturbing the 
 result : because the five equations, 
 
 1^2^ Trj^^Ty Zl-H Ii-Ii Ie^^Il 
 
 TK~Ta' Te~T^' Tn~Te' Tg~ Ta' TX~T0' 
 conduct by ordinary algebra to the sixth equation, 
 
 1^ = 1? 
 T^'Tg' 
 
 since the twelve symbols Ta, Tj3, &c., denote (by 110) twelve 
 positive or absolute numbers, which represent the lengths of the 
 twelve vectors. We may therefore dismiss any restriction upon 
 those lengths, in inferring the equation 
 
 ^-£ 
 
294 ON QUATERNIONS. 
 
 from the five other equations between quotients of vectors, which 
 have been written above. 
 
 309. The six connected equations between quotients of vec- 
 tors, which have been assigned in the foregoing article, might 
 have been suggested by our general conception (art. 108) of the 
 operation of multiplication of quaternions, without any such con- 
 struction by representative arcs upon a sphere, as was given in 
 figure 58. To see this clearly, it may be useful to refresh, as 
 follows, our recollection of that earlier and (in some respects) 
 more general conception. 
 
 To multiply any one quaternion, q^ by any other quaternion, 
 r, it was shewn, in the article just cited (108), that we are in ge- 
 neral Xo prepare for the employment of the earlier formula of art. 
 49, namely, 
 
 Transfactor = Profactor x Factor, 
 
 by making the given multiplicand quaternion, q, and the given 
 multiplier quaternion, r, assume the forms of a factor, j3 -^ a, 
 and of a successive factor, or profactor, y -t- j3, respectively ; in 
 order that the sought product quaternion, rq, may then emerge, 
 under the form of a transfactor, or as equal to the new quo- 
 tient, 7 -4- a. In this preparation of the two given factors, the 
 symbols a, /3, y are supposed to denote three lines, or vectors; 
 and the conception of equality of quotients, which was de- 
 veloped in arts. 102, &c., is employed, in order to transform (ge- 
 nerally) the given quaternions, q and r, into two others, which 
 shall be equal to those given ones, but shall be better suited for 
 combination among themselves, according to the general diwA fun- 
 damental relation, above cited, between factor, profactor, and 
 transfactor. In other words, it had beenj^a;ec^ by definition, 
 for reasons assigned in the Second Lecture (arts. 49, &c.) that 
 the two equations, 
 
 ^ = qxa, y=^rx (5, 
 
 conduct to an equation of the form 
 
 y = s X a, where s = r x q; 
 
 provided that a, /3, 7 denote three vectors, whereof a at least is 
 supposed to be 7iot a null one. This was indeed the very foun- 
 
LECTURE VI. 295 
 
 dation of our interpretation of the symbol, r x q, or r . q, or 
 rq; it was by this conception op transfaction that we gave a 
 meaning, a distinct signification, to the general expression : 
 Product of two Quaternions. Thus, not indeed without re«- 
 sons assigned, but still at last by definition, we agreed to fix, 
 generally, that 
 
 y = r^' . a, if j3 = ^a, and y = r/3 ; 
 or, eliminating the symbols j3 and 7, we so interpreted the pro- 
 duct, rq, of any two quaternions q and r, as to make true the 
 associative formula, 
 
 rq. a = r .qa, 
 under the conditions that the three symbols, 
 
 a, qa, and r. qa, 
 
 SHALL denote SOME THREE VECTORS. 
 
 310. We may also say that we have chosen so to interpret 
 the product rq, as to render (compare 87) the following formula 
 an identity, for quaternions as for ordinary algebra : 
 
 rq = rqa -^ a ; 
 
 where rqa is written for r .qa; and where it is still supposed that 
 o is a LINE (not null), and that this line is so selected, that when, 
 according to the simpler and earlier conception of the 
 MULTIPLICATION OF A LINE BY A FACTOR (arts. 40, &c.). Com- 
 bined with the notion of equalities of quotients, or of factors 
 (103, &c.), this line a is multiplied ^rs^ by q, and the product 
 again multiplied by r, the two successive results, qa, and rqa, 
 shall Ulceivise both be lines. Now such a selection of the line a 
 has been seen to be always possible : namely, by taking (see 
 again 108) for the line qa, or j3, a line situated (generally) in the 
 intersection of the planes of the two given quaternions, q and r, 
 with any arbitrary length, and with either of two opposite direc- 
 tions. If the two given planes coincide, or are parallel to each 
 other, then any line, in or parallel to either plane, may be selected 
 for /3, or for qa ; but, in every case, what we may call the Defi- 
 nitional Associative Formula of Multiplication of Qua- 
 ternions, namely,^ either of the two following, in which a, qa, 
 and r . qa (or rqa) are still supposed to be lines, 
 
296 ON QUATERNIONS. 
 
 rq. a = r . qa, or rq = rqa -^ a, 
 
 gives a definite meaning and determinate value to the symbol 
 rq^ when that symbol is interpreted hereby. And for this very 
 reason, as was remarked in art. 108, we were not at liberty, 
 fl/?er establishing these formulae of association, for the case 
 whjere a, qa, and rqa were lines, to establish also, without 
 PROOF, this OTHER and more general formula of the same 
 associative kind, 
 
 q"q' . q = q". q'q, ov sr . q^s .rq, 
 
 which has been the subject of our discussion in several recent ar- 
 ticles. For we knew already how to interpret definitely the 
 four symbols rq, sr, s . rq, and sr . q ; and (/'such definite inter- 
 pretations of the two last of these symbols were fijund (as in fact 
 they have been found) to give tivo equal values, or to conduct to 
 the general associative equation above-mentioned, this equation 
 was (as stated in 108) to be considered as a theorem, and not as a 
 definition. It seemed useful, at this stage, to bring this view dis- 
 tinctly before you, although it was partially noticed before ; lest 
 it might for a moment be thought that in all our investigations, 
 past or to come, respecting the general associative property 
 of multiplication of quaternions, we were merely proving, with 
 more or less of pains, what had been previously assumed. We did 
 indeed avail ourselves of definition, so far as we logically could^ 
 to assimilate, in this important respect, the calculations of qua- 
 ternions to the operations of ordinary algebra ; but this aid was 
 only valid up to a certain point : and beyond, that point it be- 
 came necessary to have recourse to proof, and to employ geome- 
 trical demonstration. 
 
 311. But we proposed (in 309) to shew how the six con- 
 nected equations between quotients, of art. 308, might present 
 themselves, without any consideration of arcs or angles on a 
 sphere, and simply as consequences of that general conception of 
 multiplication of quaternions which has been discussed in the 
 two foregoing (as well as in some earlier) articles. Now by the 
 nature of that general conception we are immediately conducted, 
 as we have seen, to the establishment of the three equations, 
 
 q = ^-^a, r=j -^15, rq = y -i- al 
 
LECTURE VI. ~ 297 
 
 when a, j3, 7 denote as before, three lines ; such being the very 
 TYPE of the multiplication, by which j'q is conceived to be pro- 
 duced. But when we come to multiply this product, rq, as a 
 new multiplicand, by the new given multiplier, s, we cannot, 
 without danger of confusion, continue to use the same three let- 
 ters, a, j3, 7, although the type is still to be preserved. We must 
 conceive in general, that some new line, denoted by some new 
 letter, such as c, is found as the intersection of the two new 
 planes of rq and s, in the same way as j3 was conceived to be 
 found as the intersection of the two old planes, of q and r ; and 
 must then derive, or suppose to be derived, from this new line c, 
 two other new lines, S and Z,, the former in the plane oi rq, and 
 the latter in the plane of s, just as a was taken in the plane of q, 
 and j3 in the plane of r ; these new lines being moreover such as 
 to satisfy the equations, 
 
 r^ = £ _i- §, 5 = ^ -T- £, and therefore, 5 ./"</= ^ _i- S. 
 
 For the multiplication s-xr, we must in general employ another 
 line 7], namely, the intersection of the two planes of r and s ; and 
 also two other lines, B and i, taken in those two planes respec- 
 tively, in such a way as to satisfy these other equations, 
 
 r = y] -^ Q, s = I -^7], sr= I ^ 6. 
 
 And finally, to effect the multiplication sr x q, we are to take 
 two lines k and fx, in the respective planes of q and sr, and a line 
 A in the intersection of those two planes, so as to give the equa- 
 tions, 
 
 q = \ -^ K, sr = iui-i-\, sr .q = fx -^ K. 
 
 312. This process shews then how, without arcs or angles on 
 a sphere, and even without any preliminary restriction on the 
 lengths of the lines compared, we might be led, by our general 
 conception of multiplication, to establish ticelve equations between 
 quaternions and quotients; which, by comparison of the two 
 values thus assigned for each of the five quaternions, 
 
 q, r, s, rq, sr, 
 
 would conduct (as in 308) to the^'i'e following equations between 
 quotients of vectors, which are true by the foregoing construc- 
 tion : 
 
298 ON QUATERNIONS. 
 
 £ -i- S - J -r- a; jU -i- \= I -r- S. 
 
 It shews also how we may be led, on the same plan, to inquire 
 whether these five equations involve, as a consequence, that sixth 
 equation between quotients, namely the equation 
 
 which is found by comparing the values of sr. q and s . rq. For 
 unless this sixth equation can be shewn to be a consequence of 
 the other five, we shall not have proved the general associative 
 principle of multiplication of three quaternions, at least on the 
 present plan ; and if it could be shewn that the above-mentioned 
 consequence did not exist, this associative principle would be 
 overthrown. But if, conversely, this consequence shall 6e shewn 
 to be valid, we shall thereby have proved the truth of that aS' 
 sociative principle; for the five equations give, as expressions for 
 the two members of the sixth, if we adopt for shortness the no- 
 tation effractions (118) : 
 
 fi = 
 
 ^X 
 
 C^ 
 
 ( J? 
 
 .^ 
 
 ll 
 
 ^ 
 
 K 
 
 Xk 
 
 da 
 
 ~r,Q 
 
 a 
 
 = ^/3' 
 
 a 
 
 
 
 
 ^-1 = 
 £ a 
 
 £ 
 
 7^. 
 
 /3a' 
 
 
 comparing, therefore, these values, we shall have, generally, by 
 the sixth equation, the formula, 
 
 £ j3 a £ j3 a 
 where the three quotients 
 
 ^ 7 ? 
 a p £ 
 
 may represent any three quaternions, 
 
 q, r, s, 
 
 notwithstanding that £ has been supposed to be coplanar with o 
 and y. To assert then that the sixth equation of the present 
 article is a consequence of the former five equations, is merely to 
 
LECTURE VI. 299 
 
 ENUNCIATE, as a theorem about certain quotients oftivelve vec- 
 to?'s, the principle that 
 
 sr . q = s . rq. 
 
 But having thus shewn that the enunciation (or expression) of 
 this associative principle might naturally conduct, without any 
 reference to a sphere, to form the foregoing system of six con- 
 nected equations between six quotients of twelve lines in space, I 
 shall be content to allow, for the present, the demonstration of 
 the same associative principle to rest on what has been shewn in 
 the present Lecture (296, 302), in connexion with certain curves 
 upon a spheric surface ; or on the comparatively elementary in- 
 vestigation with spheres and planes, in arts. 298 to 301 : although 
 (as has been several times said) a new and independent proof of 
 the same general and important result will offer itself to our no- 
 tice hereafter, in connexion with the distributive principle. 
 
 313. The same associative principle may be stated in other 
 ways by means of quotients of vectors, and of binary products 
 thereof, without its being necessary to employ so many as twelve 
 lines, or so many as six equations. For example, this principle 
 T\'ill be sufficiently stated, if we in any manner express that the 
 following formula is in the present calculus an identity : 
 
 £ a j3 £ a j3' 
 
 because any three given quaternions may be put under the forms 
 of the three quotients, 
 
 a 7 Z, ^ 
 
 p a i 
 
 and no essential generality will be lost, if we assume at the same 
 time the coplanarity, 
 
 ellia, 7. 
 
 But this last relation allows us to introduce another vector g, co- 
 planar with a, 7, £, and such as to satisfy the following relation 
 (which is in fact the fourth of the five given equations between 
 quotients, in 308 or in 312) : 
 
 ^ =-; or by alternation (130), ^ = 
 ha 
 
300 ON QUATERNIONS. 
 
 And since this relation conducts to the value, 
 
 ?! = ?, 
 
 e a S 
 
 we see that we may express the associative principle by stating 
 that 
 
 The PRODUCT OF TWO QUOTIENTS ofvectoJ's remains therefore 
 UNALTERED IN VALUE, when the divideiid vector (j) of the ?nul- 
 tipUcand quotient {y -i- j3), and the divisor vector (t) of the tnul- 
 tiplier quotient {Z,-^ e), are changed together, to any two new 
 vectors (a and S), to which they are proportional (in the full 
 sense of arts. 103, 129, &c.). And we see that in thh form of 
 symbolical expression of the associative principle, only six vec- 
 tors (a . . . Z,) are introduced. If we choose here to bring in 
 again the quaternions, q, r, s, it is easy to see that we have 
 merely been expressing, by the last formula, the following asso- 
 ciative identity : 
 
 (s.rq) q-^^s{rq.q-^) ; 
 
 whereof each member = sr. Or if we prefer to employ sums of 
 arcs, we may say that, in fig. 58, 
 
 ^ DP + -^ BA = --^ EF + '^ BC, if -^ DA = '-^ EC. 
 
 And it would be easy to assign a geometrical interpretation for 
 this result, by means of spherical conies. 
 
 314. In the notation oi reciprocals (117, &c.), and with the 
 aid of a few inversions and alternations (130), the six equations 
 of recent articles may be expressed and arranged in two sets of 
 three, as follows : 
 
 0^-i = /3y-i; KX-i = a/3-'; S£-i = a7-i; 
 
 the sixth being still that one which is to be a consequence of the 
 other five. Now whatever arbitrary vectors may be denoted by 
 ihejive symbols t, ij. 0, k, X, we can always find two other vec- 
 tors, j3 and t, which shall satisfy the four conditions of coplana- 
 rity. 
 
LECTURE VI. 301 
 
 (i\\\v,e; /311U,X; t\\\e,v; c\\\e,X; 
 
 and can afterwards determine ^wr other vectors, a, y, Z,, fi, so as 
 to satisfy the two first of the three equations of each of the two 
 sets lately written. In this manner we shall have the two follow- 
 ing values of two binary products of quotients : 
 
 and four of the five given equations will be satisfied, without any 
 restriction being imposed on S, or on the five vectors e, jj, 6, k, X, 
 from which the six other vectors a, j3, 7, K, i, ju, have been de' 
 rived. But if we are to satisfy also the remaining given equation, 
 namely, the third of the first set, as written in the present article, 
 the comparison of the two values of ay"^ shews that the six vec- 
 tors d, £, rj, 0, K, X, are then not wholly arbitrary, but are con- 
 nected by the following relation (restricting indeed partly even 
 the^z;e vectors €, »j, B, k, X) : 
 
 Conversely, if these six vectors be connected with each other by 
 this relation, we see that we can choose the six other vectors 
 a, j3, 7, ^, t, fi, so as to satisfy the whole system of the Jive given 
 equations between quotients ; and then, by the associative prin- 
 ciple (supposed to be now known), we can infer that the sixth 
 equation also is satisfied. Hence, by comparison of the two va- 
 lues of ^^"^, we are conducted to the following formula, involving 
 only six vectors : 
 
 ifS£-i=KX-i.^r,-\ then dK-^ = ev-'^.eX-K 
 
 315. It follows then from the associative principle that when- 
 ever one quotient of vectors (such as S -f- e) is given equal to the 
 product of two other such quotients, taken in a determined order, 
 we are at liberty to interchange the divisor line (e) of this pro- 
 duct with the dividend line (k) of the multiplier (k -^ X), provided 
 that we at the same time interchange the divisor line (X) of the 
 same multiplier with the divisor line (rj) of the multiplicand 
 (9 -7- r?), leaving unchanged the two remaining dividend lines 
 (S, B), namely, those of the product and multiplicand. Recipro- 
 cally we may perceive that the assertion of the right to make 
 
302 ON QUATERNIONS. 
 
 these interchanges, without disturbing the equality between one 
 quotient and the product of two others, is a mode of enunciating 
 the associative principle. For by a process which would simply 
 be the inverse of that adopted in the foregoing article, we might 
 shew that the final formula of that article is equivalent to the 
 assertion that one of the six equations between quotients is a con- 
 sequence of the other five; but the assertion of this consequence 
 was shewn (in 312) to involve an enunciation of the principle re- 
 ferred to. In the notation of sums of arcs, the same final for- 
 mula of 314 may be stated (compare 307) as follows : 
 
 if '- LK + '- HG = '-ED, 
 
 then -- HE + - LG = -> KD ; 
 or thus : 
 
 E-H+G-L = D-K, ifK-L + G-H = D-E. 
 
 316. The final formula of 314 may also be thus written : 
 
 if (kX-1 . Br]-') e = S, then (trj-i . QA"!) k= g. 
 
 That is to say, if the^ve vectors e, tj, 6, A. k, be so related that 
 the multiplication of the vector e by the quaternion k\~^ . dri~^ (or 
 
 K 6\ . 
 by the product of fractions, - - j gives anp one line (S) as the re- 
 sult, then the multiplication of the vector k by the quaternion 
 £»}"^ . 6X'^ will give the same line (o) as the product. Under this 
 Jbrm, with the poiiits and parentheses above written, we may' be 
 considered as still only expressing in a new way the associative 
 principle of multiplication, for any three quaternions ; but if we 
 now regard that principle as having been already proved (by any 
 of the methods given in arts. 293 to 303), and remember that in 
 304 the same principle was extended to any number of factors, 
 we see that, as an inference from the associative principle, we 
 may omit those points and parentheses, and may write simply, 
 
 Or because the five factors here considered, including the reci- 
 procals of rj and A, may denote any jive vectors^ subject only to 
 the condition which the formula eY^e//" expresses, we may take 
 any other six Greek letters as symbols of these factors and their 
 
LECTURE VI. 303 
 
 product ; and may, therefore, write, with equal generality, and 
 with somewhat greater simplicity, the formula, 
 
 In words, ^^ if the continued product o/five vectors be a 
 VECTOR, when they are taken in any one order, their continued 
 product will be equal to the same vector, when they are taken 
 in the opposite order." 
 
 317. It is obvious that this last result is analogous to the 
 equation of 195, 
 
 juXk: = kXju, if ju 111 X, K ; 
 
 or to the two connected equations of 194, 
 
 S = j3a"''y, S = 7a"^j3, 
 
 where a, j3, 7 were three coplanar lines ; under which condition 
 of coplanarity alone (by the preceding Lecture), either the con- 
 tinued product of three lines, or the fourth proportional to them, 
 can be itself a line. But we are 7iow prepared to prove, more 
 generally, that ^^ if the continued product of ai^y odd number 
 OF VECTORS be a line, it is equal to the product of the same 
 vectors^ taken in an inverted order; for example, for seven 
 such factors, we have the formula, 
 
 tj?£S7j3a = aj37S£?r], if either = B. 
 
 In fact, the equation (190, 222), 
 
 }L.rq= 'Kq . Kr, 
 gives evidently 
 
 Kis.rq) = K.rq.Ks = {Kq. Kr) Ks ; 
 or simply, by the associative principle, 
 
 K . srq = Kg Kr K* ; 
 
 the points being omitted as unnecessary between the symbols of 
 the three ^c^ors K*, Kr, Kq, in the second member of this last 
 equation ; but otie point being retained in the first member, to 
 express that the characteristic K operates on all that fol- 
 lows it in that member, namely, on the ternary product srq. 
 In like manner, if t be any fourth quaternion, we have 
 
304 ON QUATERNIONS. 
 
 K {t . srq) = K.srq.Kt; 
 that is 
 
 K . tsrq = Kg Kr Ks Kt : 
 
 and so on, for any number of factors. The result of 190 may, 
 therefore, be thus extended : — " The conjugate of the product of 
 any number of quaternions is equal to the product of the conju- 
 gates., taken in an inverted order," But also (by 114) the coiiju- 
 gate of a vector is equal to the negative of that vector; thus, 
 
 Ka = -a, K|3 = -j3, &c. 
 
 We have, therefore, not only the formula (see 89, 193), 
 
 K . j3a = + a)3, 
 
 for the ease of two vectors, but also these others : 
 
 K . yjSa = - ajSy, 
 
 K.gyj3a=+a^yg, 
 
 K . eSyjSa = - ajSySe, &c. ; 
 
 the sign + or - being used, according as the number of the vec- 
 tor factors is even or odd. Hence, 
 
 if yjSa = g, then a|3y = - Kg = S ; 
 
 if £gy/3a = ?, then aj3yg£ = - K^ = ^; 
 
 if r??£gy/3a = B, then a/Byge^r, == - K = ; 
 
 and so on, for any odd number of vectors. The theorem enun- 
 ciated in the present article, respecting any such product of vec- 
 tors, is therefore proved to be true; and we see, conversely, by a 
 principle stated in 187, that " if the product of any odd number 
 of vectors be equal to the product of the same vectors taken in an 
 INVERTED o^D'&K, this product is ITSELF a vector:" because it is 
 equal to the negative of its own conjugate. 
 
 318. On the other hand, if the number of the vectors be 
 even, the same reasoning proves that their continued product is 
 changed to its own negative, if this product be a line, and if the 
 order of the factors be inverted : thus, not only have we the for- 
 mula (compare 82) for two vector factors, 
 
 aj3 = K . ^o = - /3a, if /3a = y , 
 
 but also, in like manner, 
 
LECTURE VI. 305 
 
 ajdjB = - 8yj3a, if SyjSa = e, 
 ajSySe^ = - ^eSy^a, if Ke^yl^a - »,, &c. 
 
 And conversely, ?ythe continued product of any even number of 
 vectors be equal to the negative of the product of the same vec- 
 tors taken in an inverted order, then each of these two products 
 is equal to a line. I may just notice here, what you will have 
 no difficulty now in proving for yourselves, as an extension of 
 the result of art. 192, that whatever the number of factors may 
 be, and whether they be vectors or quaternions, the reciprocal 
 of the product is always equal to the product of the recipro- 
 cals, taken in an inverted order. 
 
 319. Again, the property of being equal to their own conju- 
 gates is one which belongs (114) to scalar s, and to no other quater- 
 nions ; for it is only when the angle of a versor vanishes, or be- 
 comes equal to two right angles, that no real change in the final 
 direction of the turned line, or versum {^^)t is produced by re- 
 versing the direction of the rotation (89), in order to pass to the 
 conjugate versor. We have then not only (compare 85) the for- 
 mula, 
 
 aj3 = K . /3a = )3a, if j3a = a, 
 but also 
 
 ajdyS = K . SyjSa = SyjSa, if SyjSa = b, 
 
 and in like manner, 
 
 ajdydeZ = ^eSyjSa, if this = c, &C. ; 
 
 a, b, c being here used to denote some scalar values. And con- 
 versely, ifa(5 = /3a, or if a(5yd = By (3a, &c., then each of these 
 two equated products of some given and even number of vectors, 
 in which the order of the factors is inverted in passing from one 
 product to the other, must be equal to S07ne scalar value, such as 
 a, or b, &c. 
 
 320. Some interesting examples of continued products of vec- 
 tors are supplied by the consideration of rectilinear polygons, in- 
 scribed in'a circle, or in a sphere. And first, for the case of a 
 plane triangle, abc, we know (by 197, 198) that the product 
 
 CA X BC X AB, or (a - c) (C - b) (b - a), 
 X 
 
306 ON QUATERNIONS. 
 
 of its three successive sides, regarded as three vectors, is another 
 vector, which has the direction of the tangent at the first corner, 
 A, to the circle circumscribed about the triangle, or more parti- 
 cularly, the direction of the tangent to the segment abc of this 
 circle; namely, the tangent at in the annexed figure QQ: so that 
 the product line thus found represents 
 
 .' . . Fig. QQ. 
 
 the initial direction of the motion along 
 
 the circumference^ from a through b toe. C^ /'>;v,^ 
 
 (Contrast with this the direction found //J\ ^~~~~~1^-^^/ 
 
 in 131, for the fourth proportional to -rv'/W \ / /1\ 
 
 BC, ca, and ab.) Let d be a fourth [ \/ '---\ ; / 1 
 
 point upon the same circumference, \ / \ \ ;""/-. / 
 
 taken (as we shall at first suppose) be- bV! \ \<;,' / ^^^ 
 
 tween c and a, on the continuation of \^^^\;/x^\^ 
 
 X ^^-o<V/^- — ^ .m' 
 
 the arc abc; so that abcd is (compare A 
 
 fig. 27, art. 132) an insci^ibed and uncrossed quadrilateral; then 
 
 the continued product, 
 
 DA X CD X AC, or (a - D) (d-C) (c - a), 
 
 by the same principle respecting an inscribed triangle, is con- 
 structed by a new line, which has the direction of the same tan- 
 gent AT to the circle as before. If, on the other hand, a point d' 
 be taken on the arc abc itself, so that (compare fig. 28, art. 132) 
 the inscribed quadrilateral abcd' is a crossed one, then the mo- 
 tion along the circumference from a through c to d' is opposite 
 to that from a through b to c ; and the continued product 
 
 d'a X cd' X AC, or (a - d') (d' - c) (c - a), 
 
 is represented, as to its direction, by the opposite tangent, at', in 
 the recent figure QQ. Multiplying, then, with the help of the 
 associative principle, the product of the sides of the first triangle, 
 ABC, by the product of the sides of the second triangle, acd, and 
 observing that the product of two opposite vectors, 
 
 AC X CA, or (c - a) (a - c), 
 
 is always (by 84) a positive scalar, we see that the continued 
 
 PRODUCT, 
 
 DA X CD X BC X AB, Or (a - d) (d - c) (c -b) (B - a). 
 
LECTURE VI. 307 
 
 of the TOVR SUCCESSIVE SIDES of an uncrossed quadrilateral 
 IN A CIRCLE, ABCD, is equttl to a negative scalar ; because it 
 can only differ by a scalar and positive coefficient, or multiplier, 
 from the product at x at, or from the square of the tangential 
 vector at, which square (by 85) is negative. On the other hand, 
 for the inscribed but crossed quadrilateral, abcd', the product 
 of the four successive sides, 
 
 d'a X cd' X bc X AB, or (a - d') (d' - c) (c - b) (b - a), 
 
 may be shewn, by the same mode of reasoning, to be a positive 
 scalar; because the product of the two opposite tangential vec- 
 tors, AT and at', is positive. We have, therefore (by 113), the 
 following values for the versors of these two quaternary pro- 
 ducts : 
 
 U . (a - d) (d - c) (c - b) (b - a) = - 1 ; 
 
 U . (a - d') (d' - c) (c - b) (b - a) == + 1 . 
 
 321. We see then that the continued product of the four suc- 
 cessive sides of a quadrilateral inscribed in a circle is always 
 equal to a scalar ; a conclusion which, geometrically considered, 
 contains a characteristic property of the circle (compare 200) ; 
 and, which as a symbolic result, appears likewise to be peculiar 
 (compare 198) to the calculus o^ quaternions. The formulae re- 
 cently written to express it may also (by 113) be thus trans- 
 formed (compare again 200) : 
 
 U.(d-c) (c-B) (b-a) = U(a-d); 
 U.(d'-c) (c-b) (b-a) = U(d'-a); 
 
 or thus : 
 
 U.(c-b) (b-a) = U.(c-d) (a-d)-U.(c-d') (d'-a); 
 
 or finally thus : 
 
 -^ c-B _ c-D c-d' 
 
 A-B D-A A-D* 
 
 And under this last form, you will easily find that the result ex- 
 presses, in the notation of this calculus, the well-known supple- 
 mentary relation between opposite angles (abc, cda) of an un- 
 crossed quadrilateral in a circle, and the equally well known 
 relation of equality between angles (abc, ad'c) which are in one 
 
 X 2 
 
308 ON QUATERNIONS. 
 
 common segment. See the curved arrows in the recent figure 66. 
 And the equality of the angle abc to the angle t'ac (between 
 the chord ac and the tangent at' to the alternate segment) may 
 be expressed by writing, as the calculus allows us to do, with the 
 help of the associative principle, 
 
 U.(c-b) (b-a)=U{(c-a).(a-c) (c-b) (b-a)} 
 = U . (c - a) (t - a) ; that is, 
 
 A - B T - a 
 
 In several recent transformations, we have employed the princi- 
 ple, that the versor of the product of any number of factors 
 (whether they be vectors or quaternions) is equal to the product 
 OF the versors ; which is an extension of the corresponding 
 result of art. 188, respecting the versor of a product oi two qua- 
 ternions, and may be expressed symbolically by the formula, 
 
 un=nU: 
 
 this latter being analogous to the formula TlT = IlT of art. 208, 
 which denoted the analogous extension of the result of 188, re- 
 specting the tensor of a product. 
 
 322. In the sanie figure QQ, let e be a new point, on the arc 
 ABCD prolonged ; and complete the inscribed and uncrossed pen- 
 tagon, abcde. The ternary product, 
 
 EA . DE . AD, or (a - e) (e - d) (d - a), 
 
 is a line in the direction of at ; multiplying this line, therefore, 
 into the quaternary product of the sides of the quadrilateral abcd, 
 which has been found to be a negative scalar, 
 
 (a - d) (d - c) (c-b) (b - a) < 0, 
 
 and remembering that the following product of two opposite 
 lines is positive, 
 
 (d - a) (a- d) > 0, 
 
 we find, by the associative principle, that the following quinary 
 product of vectors, 
 
 EA . DE . CD . BC . AB = (a - e) (e - d) (d - c) (c - b) (b - a). 
 
LECTURE VI. 309 
 
 namely, the product of the Jive successive sides of the inscribed 
 and uncrossed pentagon abode, is a line having the direction of 
 the opposite tangential vector, at'. Had we chosen to consider 
 either of the two inscribed and crossed pentagons, abode', abcd'e, 
 in the same figure QQ, we should have found by similar reason- 
 ings, that the product of the five successive sides of each penta- 
 gon was equal to a line in the direction of the original tangent 
 AT itself, and not in the opposite direction. For an inscribed 
 hexagon, the product of sides would be found to be again a sca- 
 lar. And so proceeding, we might shew with ease that " the 
 product of the successive sides of a polygon inscribed in a circle 
 is equal to a soalar, if the number of the sides be even; but to 
 a tangential vector, drawn at the first corner of the polygon, 
 if the number of sides be odd." It is worth noticing that in each 
 of these two cases the product remains unchanged (by 317, 319), 
 when the order of the factors is inverted. 
 
 323. Passing now from plane to gauche polygons, that is to 
 rectilinear and closed figures which are not contained in any sin- 
 gle plane, let us consider in the first place a gauche {ov bent) 
 quadrilateral, abcd, inscribed in a spheric surface. The 
 planes of abc and acd being now, by hypothesis, distinct, they 
 cut the sphere in two different circles, which may be conceived 
 to be projected orthographically, in fig. 67, into two ellipses, on 
 the tangent plane at a : and 
 
 Fiff fi7 
 
 the same two secant planes cut 
 also this tangent plane in two 
 different straight lines, at and 
 AU, neither coincident with nor 
 opposite to each other in direc- 
 tion, but touching respectively 
 the two circles (or the two el- 
 lipses) just now mentioned. 
 We mayalso conceive that these 
 tangents are so chosen as to touch the segments, abc, acd, 
 themselves, rather than the alternate segments of the two cir- 
 cles just now mentioned; and then (320) the two ternary pro- 
 ducts of vectors, 
 
 (a-c) (c-b) (b-a), and (a-d) (d - c) (c- a), 
 
310 ON QUATERNIONS. 
 
 will be lines, in the directions, respectively, of these two tan- 
 gents, AT and AU. Hence by a process the same in principle as 
 that of art. 320, and only slightly modified to meet the present 
 question, we find that the quaternary product, 
 
 (a - d) (d - c) (c - b) (b - a), 
 
 of the four successive sides of the gauche quadrilateral, differs 
 only by a scalar and positive coefficient from that quaternion 
 which is the product of the two tangential vectors; so that the 
 versors of these two products must be equal, and we may write 
 the following equation : 
 
 U.(a-d)(d-c) (c-b) (b-a)=U.(u-a) (t-a). 
 
 324. The radius oa (if o be the centre of the sphere) is of 
 course perpendicular to both the tangents, at and au ; it is evi- 
 dent, therefore, from our general principles respecting the multi- 
 plication of any two lines (88, 273) that the unit-axis of the 
 recent quaternary product must either coincide with, or be op- 
 posite to, the direction of this radius, according as the rotation, 
 round the radius prolonged, from au to at, is positive or nega- 
 tive; we may then write, 
 
 Ax . (a - d) (d - c) (c-b) (b - a) = + U (a - o). 
 
 With respect to the angle of the same quaternary product, con- 
 sidered as a versor or as 'a quaternion, it is equal, by the same 
 general principles, to the supplement of the angle uat at a, be- 
 tween the two tangents au, at ; or to the angle between at and 
 au' (ua prolonged through a) ; or finally, to the angle at a, upon 
 the surface of jthe sphere, lb etween the two small circle arcs, 
 ABC and ADC, as suggested in the annexed 
 figure 68. We know then perfectly how to 
 interpret the continued product of four suc- 
 cessive sides of any gauche quadrilateral : 
 namely, by circumscribing a sphere about it, 
 and then proceeding as above. For the axis 
 of the product is d^ normal to this sphere at 
 the first corner a of the quadrilateral ; the out- 
 ward or inward direction of this normal being 
 determined, as above, by the character "of a 
 certain rotation : and the angle of the same U/ 
 
LECTURE VI. 311 
 
 product is the angle of the lunule abcda, if we agree to give this 
 name lunule to thejigure bounded (generally) hy two portions of 
 small circles on a sphere (as here by abc and adc), which portions 
 may be greater than halves of those small circles. With respect 
 to the tensor of the product, it is of course still equal to the pro- 
 duct of the tensors, or to the product of the numbers which ex- 
 press the lengths of the four sides of the quadrilateral. When the 
 point D approaches indefinitely to the plane of abc, the inscribed 
 quadrilateral tends indefinitely to become a plane one; and the 
 angle of the product of its sides, being still equal to the angle of 
 the lunule, tends to vanish for the case of a crossed figure, but to 
 become equal to tivo right angles for the case of an uncrossed 
 one ; and thus the results of 320, respecting a quadrilateral in a 
 circle.) are reproduced as limits of more general conclusions, re- 
 specting quadrilaterals in a sphere. 
 
 325. If we pass from the gauche quadrilateral abcd to a 
 gauche pentagon, such as abcde, inscribed in the same sphere, 
 and draw a line av at a to touch the circle or rather the sepfment 
 ade, this new tangential vector av will have the direction of the 
 vector which is equal to the ternary product, 
 
 (A- e) (e -d) (d -a). 
 
 Again, the following product of opposite lines is positive, 
 
 (d- a) (a - d) > 0; 
 and the ternary product, 
 
 AV X AU X AT, 
 
 of three coplanar tangents to the sphere at a, is another line in 
 the same tangent plane ; hence the quinary product of the five 
 successive sides of the inscribed pentagon, 
 
 (a - e) (e - d) (d - c) (c - b) (b - a), 
 
 is a line, having this last mentioned direction in the tangent 
 plane to the sphere at a. We may, therefore, write, 
 
 U . (a - e) (e - d) (d - c) (c - b) (b - a) = 
 
 U.(v-a)(u-a)(t-a); 
 
 and may construct the direction of the line, which is the value of 
 this quinary product, by means of a tangent aw at a to a new 
 
312 ON QUATERNIONS. 
 
 circle ; namely, to one situated (see the annexed figure 69) in the 
 same tangent plane to the sphere, 
 and cutting the lines at and av in 
 two points T and v', such that the 
 Joining line, or chord tV, of this 
 new circle, may be parallel to the y' 
 line AU, or to the plane acd. And 
 so proceeding, for hexagons, hepta- 
 gons, &c., inscribed in the same 
 sphere, and having their first corners 
 at A, we should always find reductions of the same general charac- 
 ter ; namely, to products of four, five, or more tangential vectors, 
 all situated in the plane which touches the sphere at a. But in ge- 
 neral it is easy to shew that not only for three coplanar lines, but 
 for any odd number of such vectors, the product is a line, in the 
 same plane ; and that not only for two, but for any even number 
 of coplanar vectors, the product is in general a quaternion whose 
 axis h perpendicular to the common plane. If then we inscribe 
 in a sphere a rectilinear polygon with any odd number of sides, 
 for example, a gauche heptagon abcdefg, the product 
 
 (a - g) (g - f) (f - e) (e - d) (d - c) (c - b) (b - a) 
 
 of its successive sides will always be a line, constructed by a rec- 
 tilinear tangent to the sphere at the first corner a of the polygon ; 
 but if we inscribe in the same sphere a polygon with an even 
 number of sides, suppose a gauche hexagon, abcdef, then the 
 product of its successive sides, 
 
 (a - f) (f - e) (e - d) (d - c) (c - b) (b - a), 
 
 will be in general a quaternion, of which the axis will be nor- 
 mal to the given sphere at the point a, while the plane of the 
 same quaternion will be tangential to the same sphere at the 
 same point ; or at least parallel to the tangent plane at that 
 point, a distinction which, however, is unimportant in the present 
 theory. 
 
 326. The theorem respecting a pentagon in a sphere, 
 which was proved in the last article, namely, that the product of 
 its five successive sides is a Wie, or a vector, involves a property 
 
LECTURE VI. 313 
 
 which is characteristic of the sphere, and suffices to distin- 
 guish this from every other curved surface. In fact if the 
 quinary product of the sides ab, . . . ea, be equal to any line aw, 
 so that 
 
 (a - e) (e - d) (d - c) (c - b) (b - a) = w - a ; 
 
 and if, as is allowed, we conceive the same three ternary pro- 
 ducts, as before, of sides and diagonals, to be constructed, in 
 lengths as well as in directions (see 198), by three other lines, 
 AT, AU, AV, which shall touch respectively the three circles abc, 
 ACD, ADE, and shall give the three equations, 
 
 (a - c) (c - b) (b - a) = t - A, 
 (a - d) (d - c) (c - a) = u - A, 
 
 (a - e) (E - d) (d - a) = V - A, 
 
 we shall then, by the associative principle, have the expression, 
 
 (v - a) (u - a) (t - a) 
 (d - a) (a - d) . (c - a) (a - c) 
 
 in which the denominator is a positive scalar (as being the pro- 
 duct of two such scalars), and therefore the numerator, like the 
 fraction, must denote a line. The three lines at, au, av must, 
 therefore, be coplanar ; because three lines which are not con- 
 tained in any common plane have (as has been shewn) a quater- 
 nion, but not a vector, for their product. The three lately men- 
 tioned circles, namely, abc, acd, ade, have therefore their tan- 
 gents at A contained in one common plane ; which (if their own 
 three planes be distinct) is evidently the tangent plane at a to 
 the sphere abcd, circumscribed about the two first circles, or 
 about the gauche quadrilateral, abcd. Thus the third tangent 
 AV must be the intersection of this tangent plane with the plane 
 of the third circle, ade; and if thh third circle could differ from 
 the circle in which its plane ade cuts the sphere abcd, we should 
 have two distinct circles, in one common plane, intersecting each 
 other in the two points a and d, and yet having a common tan- 
 gent AV, at one of those two points of intersection ; which would 
 evidently (by Euclid) be absurd. The circle ade is therefore 
 not distitict from the intersection of its plane with the sphere 
 abcd ; or, in other words, this sphere contains that circle. That 
 
314 ON QUATERNIONS. 
 
 is to say, the gauche pentagon abcde, of which the product of 
 the five successive sides has been given (in the present article) 
 to be a line, is, for that reason, a pentagon inscriptible in 
 A SPHERE : and its corners, a, b, c, d, e, are five homosph^ric 
 
 POINTS. 
 
 327. The existence therefore of such a homospharic relation 
 between any five points a, b, c, d, e, or the condition required 
 for those five points being situated upon one common spheric 
 surface, may be expressed in this Calculus by the following 
 
 EQUATION OF HOMOSPH^RICISM : 
 
 AB . BC . CD . DE . EA = EA . DE . CD . BC . AB ; 
 
 where ab is used as a symbol for the vector b - a, &c. ; because, 
 by 317, if the product of five vectors remain thus unchanged 
 when the order of the factors is inverted, that product is itself -a. 
 vector. And that other condition which is required for four 
 points A, B, c, D, being situated upon one common circle (or 
 rather on one circular circumference), or the general equation 
 OF concircularity, may (by 319, 320, 321) be written under 
 the closely analogous form : 
 
 AB . BC . CD . DA = DA . CD . BC . AB. 
 
 328. Indeed we might deduce this latter equation for the cir- 
 cle, from the former equation for the sphere. To shew this, con- 
 ceive first that ABCD is a gauche quadrilateral, and that e is a 
 point upon the circumscribed sphere, extremely near to a. The 
 vector DE, or the fourth side of the inscribed pentagon abcde, 
 will then almost coincide with the vector da, or with the fourth 
 side of the gauche quadrilateral ; but the vector ea, or the fifth 
 side of the pentagon, will be a very short line, almost tangential 
 to the sphere at a, but otherwise arbitrary in its direction, even 
 when the quadrilateral is given. Passing then to the limit, or 
 supposing that (according to a phraseology often used) the point 
 E is infinitely near to a, we see that the plane of the quater- 
 nion, which is equal to the product 
 
 DA . CD . BC . AB, Or (a - d) (d - c) (c - b) (b - a), 
 
 must coincide with (or be parallel to) the tangent plane at a to the 
 
LECTURE VI. 315 
 
 sphere abcd ; because its conjugate quaternion, ab . bc . cd . da, 
 when operating as a multiplier on a line ea of arbitrary direc- 
 tion in that plane, produces a line. This result is indeed in- 
 cluded in what was found, at the end of art. 325, respecting in- 
 scribed gauche polygons with any even number of sides ; and, as 
 relates to the inscribed and gauche quadrilateral, it agrees with 
 what was shewn in 324, respecting the normal character of the 
 axis of the quaternion da . cd . bc . ab. Still it appeared to be 
 instructive to shew how this property of the quadrilateral could 
 be obtained as a limit from the property of the pentagon in a 
 sphere : and if we now suppose the gauche quadrilateral io flat- 
 ten gradually into di plane one, without ceasing to be inscribed 
 in a sphere, it will come at last to be inscribed in a circle, 
 through which indejinitely many spheres may be conceived to 
 pass, so as to have this circle abcd for the common intersection 
 of all of them. There would, therefore, be found, in this way, 
 indefinitely many planes, intersecting each other in the tangent 
 to the circle at the point a, any one of which planes would have 
 as good a title as any other to be regarded as the (indeterminate) 
 tangent plane at a to the (indeterminate) sphere abcd; and con- 
 sequently as the plane of the product, d a .^d . bc , ab. But the 
 only case in which ih.e plane of the product of given and deter- 
 mined factors, all different from zero, and taken in a given order, 
 can (in this calculus) be indeterminate, is the case where this 
 product degenerates (122, &c.) from a quaternion to a scalar. 
 The scalar character (321) of the product of the four successive 
 sides of a quadrilateral inscribed in a circle, is therefore found, 
 by these considerations of limits, and by the rules of the calculus 
 of quaternions, to be deducible from the vector character (325) of 
 the product of the five successive sides of a pentagon inscribed 
 in a sphere. 
 
 329. From what has thus been shewn respecting quadrila- 
 terals and pentagons in spheres, several consequences may be 
 drawn, a few of which shall be stated here. Suppose then, first, 
 that it is required to express that the point p is on the plane 
 which touches at a the sphere abcd; we may do this by express- 
 ing that the quaternion product of the four successive sides ab, 
 &t'., of the quadrilateral abcd, when multiplied by the tangent 
 
316 ON QUATERNIONS. 
 
 AP, or that this latter tangent multiplied by the conjugate of that 
 quaternion, produces another line; or (see 317) that these two 
 multiplications conduct to one common result: that is, in sym- 
 bols, by the formula, 
 
 AB . BC . CD . DA . AP = AP . DA . CD . EC . AB. 
 
 Such, therefore, relatively to the point p, is one form of the 
 
 EQUATION OF THE TANGENT PLANE tO the Sphere ABCD at A. 
 
 We see then that if the sphere he finite and determinate, or in 
 other words if the quadrilateral abcd be gauche^ so that the fol- 
 lowing EQUATION OF COPLANARITY of the fi)Ur pointS A, B, C, D, 
 AB . BC . CD = CD . BC . AB, 
 
 is not satisfied, the two following equations between the five 
 points A, B, c, D, E, 
 
 AB . BC . CD . DE . EA = EA . DE . CD . BC . AB, 
 AB . BC . CD . DA . AE = AE . DA . CD . BC . AB, 
 
 must be incompatible, except under the supposition that 
 
 E = A, or AE = a null line ; 
 
 that is (when abcd are not coplanar) the two last equations be- 
 tween the five points A . . . e can only co-exist under the suppo- 
 sition that E coincides with a. In fact the first of those two 
 equations expresses (by 327) that e is o?i the spheric surface 
 abcd; while the second equation expresses (by the present arti- 
 cle) that the same point e is on the tangent plane to the same 
 sphere at A. When we come to establish and develope, in the 
 next Lecture, the distributive principle of multiplication of qua- 
 ternions, we shall be able to confirm this result by a simple pro- 
 cess of calculation. 
 
 330. Again, let it be required to inscribe, in a given sphere, 
 a gauche quadrilateral, abcd, whose four successive sides, 
 AB, . . . da, shall be respectively parallel to four given radii, 
 oi, OK, OL, CM. In the an- 
 nexed figure 70, let g be a ^ '"* 
 point of crossing of the arcs J^^T'^ 
 
 IK, LM, and take two other P -/-- — -Xr p, 
 
 points F, H, such that j/ ,.-'-' \ 
 
 -^ FG= - IK, - GH = - LM; ^ N H 
 
LECTURE VI. 317 
 
 then either pole of the great circle fh may be taken as the sought 
 position of the first corner a of the quadrilateral to be inscribed. 
 For the quaternion da . cd . bc . ab can only differ by its tensor 
 from the product of the four parallel radii, om.ol.ok.oi, or 
 from the product of the two quotients of radii, 
 
 OM -=r OL X OK -f- OI = OH -f- OF ; 
 
 the tangent plane at the sought point a is therefore />ora//e/ (by 
 328) to the plane of this last quotient of radii, that is to the 
 plane of the two radii of, oh themselves. And as to the ambi- 
 guity of pole of the great circle fh, giving two opposite points 
 upon the surface, either of which may serve as the position of 
 the first corner A, it is evident that such an ambiguity ought, by 
 the very nature of the problem, to exist ; for if there be any in- 
 scribed polygon^ ABC . . . z, and if we pass from each corner to 
 the point diametrically opposite thereto, upon the spheric surface, 
 we shall thus form a new inscribed polygon, a'b'c' . . . z', of which 
 the sides shall be respectively parallel to the sides of the old one, 
 
 a'b' II AB, b'c' II bc, . . . zV II ZA. 
 
 331. The process of the foregoing article, for inscribing a 
 gauche quadrilateral with sides parallel to four given radii, was 
 properly an analytic process ; in the sense that it assumed the 
 possibility of the required inscription ; or that it only proved that 
 if any quadrilateral could be inscribed, according to the given 
 conditions, then the first corner must have one of those two dia- 
 metrically opposite positions, a and a', which are the poles of the 
 great circle fh. A converse and synthetic process has still to be 
 assigned, which shall shew a posteriori^ though still (if we think 
 fit) with the help of the principles of quaternions, that each of 
 the two points a, a', is in fact fit to be the first corner of an in- 
 scribed quadrilateral, abcd or a'b'c'd', which shall satisfy all the 
 conditions of the question. And for this purpose it appears to be 
 useful to consider here another problem, which is also otherwise 
 interesting, respecting rectilinear polygons in spheres : namely, 
 to assign an expression for the ?^"^ radius, op„, belonging to a 
 system of n radii, 
 
 OPi, OPo, . . . OP„, 
 
318 ON QUATERNIONS. 
 
 which are formed or derived in succession from a given initial 
 radius op, by inscribing a system of n rectilinear chords, 
 
 ppi, Pi Pa, . . • P«-i Pm, 
 
 respectively parallel to n given radii of the same sphere, which 
 may be thus denoted, 
 
 oil, OI3, . . . Gin; 
 
 or to any other n given lines in space. 
 
 332. Consider for this purpose any two radii OA, ob, of a 
 circle (a great circle of the sphere), and draw, as in the annexed 
 figure 71, the diameter coc' parallel to 
 the chord ab ; draw also the diame- 
 ter bob': and let it be required to ex- , /^ ^\ 
 
 A / XT* 
 
 press OB, or its opposite ob , by means /\. X\ 
 
 of OA and oc (or oc'). Here, because a / ^^^ y^^ \ 
 
 conical rotation through two right an- ^\ y ^q- — — y- — jC 
 
 ffles, round either oc or oc' as an axis, \ ,--'' ,.1 / 
 
 would bring: the radius oa into the V' / 
 
 position ob', it results from the pre- \,^^^ ^^^ 
 
 sent Lecture (arts. 290, 291) that this 
 radius ob' may be expressed as follows : 
 
 ob' = oc X OA -7- oc = oc' X OA -f- OC'. 
 
 But ob is opposite to ob'; wherefore 
 
 OB = - oc X OA -^ oc = - oc' X OA -=- OC'. 
 
 Or writing for conciseness, 
 
 OA = a, OB = j3, 00 = ^, 
 the expression for /3 as a function of a and 7 is found to be : 
 
 333. It is worth observing that this expression holds good, 
 whatever arbitrary length may be assigned to the radius of the 
 circle, or to the two equally long lines a and /3. The same 
 expression is valid also independently of the length of y, which 
 symbol may denote any line parallel to the chord ab, with either 
 of two opposite directions, or any portion of that chord. So that 
 
/ 
 
 LECTURE VI. 319 
 
 if AOB, in fig. 72, be any isosceles triangle on the base ab, and if 
 
 D, E, F be any points on that base, 
 
 or on its prolongations, we shall have ^'^' '^" 
 
 the expressions : -^ ^ 
 
 OB =- AD X OA -r- AD 
 = - AE X OA -r- AE = - AF X OA -r- AF. 
 
 334. It is easy now to resolve 
 the problem proposed in art. 331, re- 
 specting a polygon of any number of sides, inscribed in a sphere. 
 Writing 
 
 and 
 
 we have 
 and 
 
 OP=/), OPi=^i, OP2=j02, . . . OP„ = |0„, 
 oil = li, OI2 = <23 . . . OI„ = In, 
 
 Tp = Tpi = Tj02 = . . = Tpn, 
 
 Pi- P\\ 'i> /O2 - pi II hi • • ■ pn- pn-i II In ', 
 
 therefore, by 332, 
 
 pi = -lipii'^) p2 = -hpih^', • • • pn = - Ifipn-itn'^- 
 
 Hence, by the associative principle, and by the end of art. 318, 
 
 j02 = + hiipii'^h'^ = + hti ■ p • («2ti)'^ ; 
 pz = -lzhi\pii^h^li'^ = -lil-2i-i- p • (<3t3ti)"^; 
 
 and if we make, for abridgment, 
 
 5're = Wn-i • • tshl-ii 
 
 we shall have, finally, as the expression required in 331, the fol- 
 lowing : 
 
 OPn = pn = (-yqnpqn'' ; 
 
 where qn is generally a quaternion. 
 
 335- In this expression we may, on the plan of 333, substi- 
 tute for the radii, ti, . . <„, any lines to which they are parallel ; 
 for example, any segments of the n successive chords, ppj, 
 . . . Pji-iPn. Suppose then that Ai, A2, . . . a„ are any n new points, 
 not situated on the surface of the sphere, but taken respectively 
 
320 ON QUATERNIONS. 
 
 on the n chords ppi, P1P2, &C.5 or on those chords prolonged ; 
 and let us write, 
 
 OAi = ai, 0A2 = 03,... OA„ = aw 
 Make also, 
 
 q-2 = {a2-pi) qi, 
 qs = («3 - J02) q^i 
 
 Fig. 73. 
 
 qn = {an- pn-\) qn-\\ 
 
 we shall have the following system of expressions for the n suc- 
 cessive radii, from oPi to op„, or from p\ to p„, considered as de- 
 rived (see the annexed fig. 73) in succession from the initial ra- 
 dius OP or p, and from the n points, Ai to a„, through which the 
 n chords, ppi to Pn.i Pn, or their prolongations, 
 are to pass : 
 
 pi = -qipqi-^, 
 
 P2= + q2pq2~^, 
 
 ps^-qspqf^ 
 
 Pn = {-y qnpqn'^ ', 
 
 this last expression being thus of the same form as that found in 
 the foregoing article. 
 
 336. We see then that whether the ?i chords pPi, . . . Pn-iPn 
 be parallel to n given lines, or pass through n given points, there 
 is always a certain quaternion, q„, which can be formed by suc- 
 cessive multiplication of those n lines, or of n segments of the 
 chords parallel thereto, and which is such that the final radius |0„ 
 itself, U n be even, or the opposite radius - pn, if n be odd, shall 
 admit of being derived from the initial radius p, by a conical ro- 
 tation (286, &c.) through double the angle of this quaternion, 
 performed round the axis thereof In order, then, that the points 
 p, p,, &c., may be the corners of an inscribed and closed poly- 
 gon of ?i sides, or in order that the following coincidence of points, 
 or equality of vectors, may hold good, 
 
 P„ = P, or Pn = jO, 
 
 it is necessary and sufficient, \f n be eve?i, that the quaternion qn 
 
LECTURE VI. 321 
 
 should either degenerate into a scalar^ or else have its plane per- 
 pendicular to the initial radius p, or its axis coincident there- 
 with, so that the conical rotation may leave that initial radius un- 
 changed. And if the number n be odd, then, for the closure of 
 the polygon, it is necessary and sufficient that the quaternion q^ 
 should degenerate into a vector, perpendicular to the same initial 
 radius p ; in order that the reversal of this radius may be effected 
 by a plane rotation through two right angles : into which 
 plane rotation, or semi-revolution, the conical rotation through 
 2 L qn, round Ax . qn, will under these conditions degenerate. In 
 symbols, for an even-sided polygon, the equation of closure 
 will be, 
 
 p = qnpqn'^i or pqn = qnp', 
 
 which gives generally the parallelism, 
 
 Ax . qn II p, 
 
 with inclusion of that limiting case for which the quaternion be- 
 comes a scalar, and its axis becomes indeterminate. But for an 
 odd-sided polygon the equation oj" closure is, 
 
 p = -qnpqn-i, or pqn = - qnp ; 
 
 which can only be satisfied by supposing 
 
 qn=-'Kqn _L p- 
 
 And from the composition of qn as a product of n lines, which are 
 respectively parallel to or coincident with the n successive sides 
 of the closed figure, or at least with segments of those n sides, 
 it is evident that the general results of art. 325, respecting odd 
 and even-sided polygons inscribed in a sphere, are thus confirmed 
 and reproduced. For we see that the quaternion product qn 
 either reduces itself to a tangential vector at p, or else is repre- 
 sented by a biradial (93, &c.) in the tangent plane at that point, 
 according as n is an odd or an even number. 
 
 337. It is easy now to prove, synthetically (or d posteriori) 
 by quaternions, as was proposed in 331, that either of the two 
 poles of the great circle fh in fig. 70, which were found analyti- 
 cally (or a priori) in 330, is in fact adapted to be the first corner 
 ^ of an inscribed and gauche quadrilateral abcd, whose sides 
 
 Y 
 
322 ON QUATERNIONS. 
 
 shall be respectively parallel to the four given radii drawn to the 
 points I, K, L, M, in the same figure 70. For if we start with 
 any point p upon the same spheric surface, and draw from that 
 point four successive chords, 
 
 rPi II OI, P1P2 II OK, P2P3 II OL, P3P4 i OM, 
 
 then the radius 0P4 may be derived from the radius op by the 
 formula, 
 
 where the quaternion q^, when reduced to its own versor, admits 
 (by 330, 334) of being thus expressed, with reference to fig. 70 ; 
 
 <74= OH -f- OF. 
 
 That is to say, the point P4 may be obtained from the point p, 
 by a rotation in a small circle^ parallel to the great circle fh, 
 and through an arc PP4, which in direction is similar to, but in 
 number of degrees is double of the arc fh. Now not only will 
 such a rotation effect an actual change in the position of every 
 other point on the surface, except the poles of fh, but also it will 
 leave those two points unchanged ; so that if we set out with one 
 of them as the point a, and draw three successive chords parallel 
 to three of the given radii, 
 
 AB II 01, BC II OK, CD H OL, 
 
 we shall have also i\a?, fourth parallelism, 
 
 DA II om; 
 
 but if we start with any other point for a, the three first paral- 
 lelisms will not conduct to the fourth (P4 being then different 
 from p). We have, therefore, not merely confirmed the analysis 
 of 330, but also have supplied the synthesis which was required 
 in 331. 
 
 338. From what has just been shewn, it follows that, if we 
 start with any point a on the sphere, which is not one of the 
 poles of FH, in fig. 70, and Av&m four successive chords, parallel 
 to the four given radii, 
 
 AB II 01, BC II ok, CD II OL, DE || OM, 
 
 the point e thus obtained will not coincide with a. We may, 
 
LECTURE VI. 323 
 
 however, jom it to a by 2i fifth chord, and so close the inscribed 
 pentagon, abode ; and may then draw sl fifth radius, on, parallel 
 to the fifth side of this pentagon, or to the fifth chord just men- 
 tioned, so as to have 
 
 EA II ON. 
 
 But on account of the conical rotation by which the point e can 
 be derived from a (like P4 from p in 337), we see that this fifth 
 side or chord ea must be perpendicular to the axis of that rota- 
 tion, or parallel to the plane of the great circle fh ; and conse- 
 quently that the fifth radius on must terminate in a point n 
 situated somewhere upon that great circle. Now in fig. 70, art. 
 330, we have 
 
 /^ FH = -^ LM + ^ IK ; 
 
 and the arcs ik, lm are the first and third sides of the spheri- 
 cal or SUPERSCRIBED (uot rectilinear and inscribed) pentagon, 
 IKLMN. Conversely, we might have started with an arbitrary 
 and inscribed gauche pentagon abcde, and have derived from its 
 five successive sides the five respectively parallel radii, or the five 
 points I, K, L, M, N upon the sphere ; after which we might 
 have formed the arc fh, as in fig. 70, and have shewn, as above, 
 that the point n is situated somewhere upon that arc, or on its 
 prolongation. We arrive then at the ioWo^mg graphic property 
 of the inscribed gauche pentagon, which might however have 
 been deduced more directly from the equation of homosphcericism 
 (in 327), and may be regarded as a geometrical interpretation of 
 that equation : '•^If in a sphere, the five successive sides of an 
 inscribed gauche PENTAGON (abcde) he respectively parallel 
 to the five radii drawn to the five corners of a superscribed 
 spherical pentagon (iklmn), then the fifth corner (n) of 
 the second pentagon is situated somewhere upon that great 
 circle (fh) of which a portio?i coincides with the arcual sum 
 (rsL,M + r\ ik) of the FIRST AND THIRD SIDES of that second pen- 
 tagon ;" those sides being taken in a suitable order (third plus 
 first). And this relation between the directions of the five sides 
 of an inscribed gauche pentagon may also be regarded as a gra- 
 phic property of the sphere itself ; by which property that 
 surface (compare 326) is sufficiently characterized, and dis- 
 
 Y 2 
 
324 ON QUATERNIONS. 
 
 tinguished from all other curved surfaces. In fact this relation 
 of directions is for space and for the sphere, the analogue of 
 the well-known and elementary relation for the plane and for the 
 circle, between the directions of the sides of an inscribed quadri- 
 lateral, which is given in the third Book of Euclid. And accord- 
 ingly the last-mentioned relation may be deduced, as a limit, from 
 the former; because (as we have seen in 328) the equation of 
 concircularity may be obtained, as a limiting form, from the 
 equation of homosphcericism. 
 
 339. After what has been said respecting inscribed polygons, 
 you can have no difficulty now in proving that if a gauche hep- 
 tagony abcdefg, and a gauche hexagon, a'b'c'd'eV, be both in- 
 scribed in the same sphere ; and if the Jirst six sides of the hep- 
 tagon be pat^allel respectively to the six successive sides of the 
 hexagon, 
 
 AB II a'b', BC II b'c', CD II c'd', 
 
 de II d'e', ef II e'f', fg II f'a', 
 
 then the seventh side, ga, of the hexagon will be parallel to the 
 tangent plane to the sphere, at the first corner, a', of the hexa- 
 gon. If, then, we draw successively, from the seventh corner, g, 
 of the heptagon, six new chords of the sphere, respectively pa- 
 rallel to the same six successive sides of the hexagon, and in the 
 same order, namely, 
 
 GH II a'b', hi II b'c', IK II c'd', 
 
 KL II d'e', LM II e'f', MN II f'a', 
 
 we shall have, in like manner, the closing chord ox final side, ng, 
 of the new inscribed heptagon, ghiklmn, parallel to the same 
 tangent plane at a'. And hence it follows evidently, that the 
 plane, agn, of the extreme and middle corners (first, seventh 
 and thirteenth) of the inscribed polygon of thirteen sides, 
 
 abcdefghiklmn, 
 
 is parallel to the same tangent plane, at the first corner 
 a' of the hexagon : because it contains two lines, or chords, ga, 
 NG (and of course also the third chord na), which two lines have 
 been seen to be parallel to that plane. 
 
 340. An obvious generalization of the reasoning in the fore- 
 
LECTURE VI. 325 
 
 going article, conducts to the following Theorem: — " If any 
 even-sided polygon of 2n sides, 
 
 Ai A2 . . . Asn, 
 
 be given as inscribed in a sphere ; and if, starting from any arbi- 
 trary point p on the same sphere, we draw In successive chords, 
 parallel respectively to the 2w sides of this polygon, 
 
 PPl I A1A2, P1P2 II A2A3, . . . P2n-iP2« II AgTiAi ; 
 
 and then again start from the last point p^w thus obtained, and 
 draw In other successive chords, parallel to the same 2n succes- 
 sive sides of the given and even-sided polygon, 
 
 P3nP2ra + l || A1A2, . . . P4n-lP4n II A2JiAi ; 
 
 and finally join the new point P4n to p : the plane of the extreme 
 and middle corners vv^nPin^ of the inscribed polygon of in + I 
 sideSf 
 
 PP1P2 . . . P2K-lP2nP37i+l • • • I'4n-lP4Kj 
 
 will be parallel to the plane which touches the sphere at the first 
 corner, Ai, of the inscribed polygon of2n sides." For example, 
 we might assume w = 2 (instead of 3, which was its value in the 
 last article) ; and then we should have a parallelism between a 
 certain diagonal plane of an inscribed enneagon, and the tangent 
 plane at a corner of a gauche and inscribed quadrilateral. 
 
 341. One of the most important applications of the associa- 
 tive principle of multiplication is to the composition of coni- 
 cal ROTATIONS, whose axes are supposed (at first) to pass all 
 through one common point, which may be taken for the origin 
 of vectors. In fact, by 192, 286, and by the associative princi- 
 ple, we see that the following symbols are equivalent, 
 
 rq^ {rqy^ = r . qBq'''- . r-i ; 
 
 and that they both denote one common position, into which a 
 body B is brought, by either of the two following processes. The 
 first process, represented by the right hand member of the last 
 equation, consists in making this body B revolve successively, 
 through the angles 2 L q and 2 z r, round the two successive axes. 
 Ax . q and Ax . r, which are both supposed to be drawn through 
 
326 ON QUATERNIONS. 
 
 or from the common origin o. The second process, represented 
 by the left hand member of the same equation, consists in making 
 the same body revolve round a single resultant axis, Ax . rq 
 (drawn from the same point o), through one resultant angUf 
 namely, 2 L.rq, The operation performed in this latter process 
 is therefore equivalent, as regards its effect, to the system 
 of the two successive operations, which are accomplished in the 
 former process. And thus any two successive and finite conical 
 rotations, round two axes passing through one point, are with 
 the greatest ease compounded, by the multiplication of two 
 quaternions, into a third and single conical rotation, round an 
 axis through the same point o. And in like manner may any 
 NUMBER of such given successive and conical rotations be com- 
 pounded into one, with a (generally) determined axis and angle, 
 by first multiplying together, in the given order, the quaternions 
 q,r,s, . . . , which represent, by their axes and angles, the halves 
 of the given rotations, and then taking the axis and the doubled 
 angle of that quaternion product, 
 
 p = . . . srq, 
 
 which is obtained by the foregoing multiplication. For example, 
 by art. 286, and by the associative principle, the symbol 
 
 S7'q B {srq)~^ 
 
 denotes that position into which the body B is brought, by three 
 successive conical rotations round the three successive axes. 
 Ax .q. Ax . r, Ax . s, all drawn from the origin o, and through 
 the three successive angles denoted hy 2 Lq, 2 Lr, 2 Ls\ and the 
 composition of this symbol indicates that the same final position 
 of the body B may be obtained from the same given initial posi- 
 tion (whatev erthat may be), by a single resultant rotation round 
 the axis 
 
 Ax , p = Ax . srq, 
 through the angle 
 
 2 z p = 2 z . srq. 
 
 342. As an instance of the general correspondence, between 
 the multij)lication of two quaternions, and ihe composition of two 
 
LECTURE VI. 327 
 
 conical rotations, let us consider first the following very simple 
 formula of art. 118 : 
 
 /3 -r- a = /3 X a'^ 
 
 This formula gives, by taking the reciprocals (see 44, 1 92), 
 
 o -r- /3 = a X j3'^ ; 
 
 and therefore, by the associative principle, 
 
 O -J- a) |0 (a -h /3) = j3 . a- Va • ^'^' 
 
 Hence, on the plan of the foregoing article (341), we may infer 
 that a conical rotation through two right angles round a"^, or 
 (what comes to the same thing) round the oppositely directed 
 axis a, being followed by another such rotation through the same 
 amount round j3, produces on the whole the same effect as a co- 
 nical rotation round the axis of the quaternion quotient j3 -f- a, 
 through the double of the angle of the same quaternion, that is, 
 through twice the angle between a and j3, whatever the original 
 direction of the operand vector p may be. Or if, as in the an- 
 nexed figure 74, we first reflect any arbi- 
 trary point p upon the sphere, with respect '^" ,..-—,, 
 to a given point a, till it takes the position q/ 
 Q, and then a^ain reflect the point q with r ' ex I '^^"''^^A p' 'i 
 
 . ... I 7 Q'; -^^ 1! 
 
 respect to another given point b, till it ac- -pJ/.__ ,/.' .^\l' 
 
 quires the new position r, so that 
 
 ^ PA = ^ AQ, ^ QB = ^ BR ; 
 
 the passage on the spheric surface, from the first position p to the 
 third position r, may be made along an arc of a small circle, pr, 
 which in direction is similar to, and in number of degrees is 
 double of, the arc of a great circle ab. We have already had an 
 example of the truth of this theorem in art. 292, where the points 
 E, F, D, of fig. 40, art. 224, took the places of the recent points 
 p, Q, R. But lest it should appear that this case was in some 
 way a particular one, on account of the comparative complexity 
 of fig. 40, and the number of other considerations which that 
 figure was designed to illustrate, let us conceive that, in the 
 simpler figure 74 of the present article, the arcs pp', qq', rr', 
 are perpendicular to the great circle through a, b, and are let 
 
328 ON QUATERNIONS. 
 
 fall thereon as such from the three points p, q, r. We shall 
 then have evidently, by the construction, the two arcual equa- 
 tions (217), 
 
 r\ p'a = r\ AQ', r\ q'b = /n Br' ; 
 
 and the three perpendiculars pp', qq', rr', will at least be equally 
 long, although not arcually equal, in the same full sense of art. 
 217. Hence the points p and r are equally distant on the sphere 
 from the positive pole of the arc ab ; and, therefore, we can pass 
 from the former point p to the latter point r, by a rotation round 
 that pole, along an arc of a small circle pr (represented in the 
 figure by a dotted line), which is parallel to the arc of a great 
 circle ab, having also the same direction therewith, and the 
 same number of degrees as its own projection p'r' thereon, which 
 projection is seen to be the double of the same arc ab, 
 
 rs p'r'= 2 /^ ab. 
 
 The theorem of the present article is therefore proved, or con- 
 firmed, by this simple geometrical reasoning ; and you perceive, 
 of course, conversely, that any proposed rotation pr in a small 
 circle, of any given amount and round any given positive pole, 
 may be decomposed into two rotations, performed along two 
 SMALL SEMICIRCLES ; or Still more simply, into two successive 
 reflexions with respect to two points a, b, assumed anywhere 
 on a great circle round the given pole, at an interval ab which in 
 direction is similar to the proposed conical rotation, and in amount 
 is equal to the half oi'it. 
 
 343. Consider next the fundamental multiplicational identity 
 of art. 49, 
 
 7-« = (7-i3)x(/3--a). 
 
 On the general plan of art. 341, we can infer from this equation, 
 or may interpret it as signifying, that a conical rotation repre- 
 sented by the double of any arc of a great circle ab, being fol- 
 lowed by a second conical rotation which is represented in like 
 manner by the double of any other and successive arc, bc, of 
 another great circle, produces on the whole the same effect as 
 that third and resultant conical rotation, which is (on the 
 same general plan) represented by the double of the arc ac; 
 
LECTURE VI. 329 
 
 that is, by the double of the sum of the halves of the arcs 
 which represent the two component a7id conical rotations. 
 When a conical rotation is thus said to be represented by a given 
 arc of a great circle, we are to understand that the axis and 
 angle of the rotation in question are such, that they would cause 
 the initial point of the arc to revolve, in one plane, till it should 
 take the position of the Jinal point of the same given represen- 
 tative ARC. This being clearly understood, there is no difficulty 
 in confirming, by a simple geometrical diagram, the theorem of 
 composition just now stated (which perhaps may have long been 
 known), with the help of what was established in the preceding 
 article. For let abc, in the annexed figure 75, 
 be any spherical triangle, and p any point upon 
 the sphere. Reflect p with respect to a, to the 
 position Q ; and again reflect q to r, with re- / L 
 spect to the point b. An arc of a small circle, p ^^4 ~N;----^^^^B, 
 PR, can (by 342) be drawn, which shall be pa- i^^^ 
 rallel to the arc of a great circle ab, and simi- 
 lar to it in direction, but double of it in amount. Thus r is 
 the position to which we pass from p, in virtue of the^r*^ com- 
 ponent and conical rotation, considered in the present article. 
 To accomplish the second component conical rotation, repre- 
 sented by the double of the arc bc, we may, in like manner, first 
 reflect r, with respect to b, back again to the position q, and 
 then reflect Q, with respect to c, to the new position s. On the 
 whole, then, the point which was at p will have been brought to 
 s (through Q, R, and q again, as intermediate positions on the 
 sphere). But it is clear that this complex process has (in a cer- 
 tain sense) geometrically eliminated the point b. For we may 
 pass, without using that point b (or r) at all, from the position 
 p to the position s, by first reflecting p to q through a, and then 
 reflecting q, through c, to s. But, by the foregoing article, the 
 process of double reflexion last described is equivalent to a 
 single conical rotation, represented by the double of the arc 
 AC. This one rotation is therefore seen, by this geometrical con- 
 struction, to be the resultant of the two successive rotations, re- 
 presented by the doubles of the arcs ab and bc ; which illustrates. 
 
330 
 
 ON QUATERNIONS. 
 
 and (if it had been necessary) would confirm^ the theorem stated 
 at the commencement of the present article. 
 
 344. It is extremely easy to infer, from what has just been 
 proved, the following theorem, namely, that three successive 
 and conical rotations, represented by the doubles of the three 
 
 SUCCESSIVE SIDES OF ANY SPHERICAL TRIANGLE, produCC ON THE 
 
 WHOLE, NO EFFECT. In symbols, on the plan of art. 341, this 
 theorem is expressed by the identity, written here in a fractional 
 form, 
 
 -'^2=1. 
 7/3 a 
 
 Geometrically considered, and with reference to the recent fig. 
 75, it comes simply to observing that we can pass back from s to 
 p by reflecting s to q through c, and q to p through a. Fig. 
 40 might also be used to illustrate this, and several other con- 
 nected conclusions. 
 
 345. You can have no difficulty now, in interpreting simi- 
 larly the more general identity, for any number of successive 
 quotients multiplied, which may be thus denoted : 
 
 9 
 
 7/3 o 
 
 = 1 
 
 u 
 
 nor in proving that it expresses (on the same plan of art. 341) 
 that whatever spherical polygon may be pictured, in the annexed 
 figure 76, by abcd . . . g, the 
 double of the rotation ab, fol- ^'S- ''6. 
 
 lowed by the double of the rota- 
 tion BC, followed again by the 
 double of the rotation cd, and so 
 on, till we come at last to the 
 double of the rotation ga, re- 
 stores the revolving or rotating 
 point p to its original position 
 In fact the rotation represented 
 by 2 ^ ab would be equivalent 
 to reflecting any point p, on the 
 
 spheric surface, first through a to q, and next through b to r ; 
 the rotation 2 ^ bc would be equivalent to reflecting r back to 
 
 
 NJ_-— ^ 
 
 7 
 
 "Wl 
 
 
 / \ 
 
 \y 
 
 X j- 
 
 V 
 
 P" 
 
 R ■■ 
 
 ( 
 
LECTURE VI. 331 
 
 Q, and then reflecting q through c to s ; this last point s would be 
 brought by the rotation 2 /> cd to the position t, namely the re- 
 flexion of Q with respect to d ; and so on, till after arriving at the 
 reflexion w of q, relatively to the last corner g of the given po- 
 lygon, we should be brought back from w to the original posi- 
 tion p, by the final rotation 2 /n ga ; because p is the reflexion 
 of Q, with respect to the first given corner a. (Arcs of small 
 circles are denoted in the present figure by straight and dotted 
 lines ; arcs of great circles by lines without dots, but still, for 
 simplicity, straight.) 
 
 346. Again consider the equation of art. 280, 
 
 7^)3%^ = - 1, 
 which gives, 
 
 |3%* = -7-^ 
 
 and, therefore, by the associative principle, and by the property 
 (192) of the reciprocal of a product, 
 
 /B^' . a>a-^ . /3 -2/ = 7 -^|07^ 
 
 In interpreting this equation, in connexion with fig. 56, of art. 
 280, on the plan of art. 341, we are led to introduce, what it is 
 extremely easy to form, the conception of spherical angles as 
 REPRESENTING CONICAL ROTATIONS. In fact, if ABC be any 
 spherical angle, it is natural, when once we combine the concep- 
 tion of such an angle, with the conception of a conical rotation, 
 to regard the latter as being the operator which would change, 
 by Q. plane rotation, the tangent to ihe side ba of the given angle 
 ABC, to the tangent to the other side bc of the same spherical 
 angle. Now the last written formula of the present article is 
 easily seen to express, that if the rotation round the pole a (in 
 the lately cited fig. 56), through the angle xir, be followed by a 
 rotation round the pole b (in the same figure) through an angle 
 = 2/7r, the result will be equivalent to a rotation round the pole c, 
 through an angle ^-ztt. But the angles of the triangle abc (in 
 the same figure) were : 
 
 If then, for any spherical triangle, abc, the double of the rota- 
 
332 
 
 ON QUATERNIONS. 
 
 Fig. 77. 
 
 tion represented by the angle cab be followed by the double of 
 the rotation represented by the angle abc, the result will be the 
 double of the rotation represented by the angle acb (which latter 
 is the opposite of the rotation bca). 
 
 347. To shew this geometrically, let d and e be chosen so 
 (see the annexed figure 77) that we may 
 have the following equations between an- 
 gles, 
 
 dba = abc = cbe, cab = bad, acb = bce; 
 
 and let us take as two operand poirds, to 
 be separately and successively employed, 
 the vertex c, and the base corner a, of the 
 spherical triangle abc. Operating then 
 first on the vertex c, by the two successive rotations, 
 
 2 X CAB, and 2 X ABC, 
 or by 
 
 CAD and DBC, 
 
 we change c first to d, and then back to c again ; but such 
 would have also been the final result, so far as the operand point 
 c is concerned, of any rotation whatever round that point c itself 
 as a pole ; and, therefore, in particular, such would have been 
 the result, relatively to this operand c, of the rotation repre- 
 sented by 
 
 2 X ACB. 
 
 Again, as a new and independent process, let us begin with the 
 base-corner A as an operand point. The first component rota- 
 tion, 
 
 2 A 
 
 X CAB, 
 
 being performed round this point a as a pole, leaves its position 
 undisturbed. The second component and conical rotation, re- 
 presented by 
 
 2 X ABC, 
 
 transfers the new operand point a to e. But it is clear, from the 
 figure, that the same transference might also be effected, by a ro- 
 tation round the vertex c as a pole, represented by 
 
LECTURE VI. 333 
 
 2 A 
 
 X ACB. 
 
 The theorem of the last article is therefore seen to be true, for 
 the TWO different operand points, c and A : whence it is easily 
 seen, by the general conception of rotation, to be valid for all 
 others also. (An inspection of figs. 52, 57, of articles 269, 281, 
 may serve slightly to illustrate this result.) 
 
 348. An important although particular case, of the general 
 theorem of rotation contained in the two last articles, is illus- 
 trated by fig. 43, of art. 242 : namely, the case where the trian- 
 gle ABC is triquadrantal. In such a case, because a conical ro- 
 tation through a doubled right angle is equivalent to a reflexion 
 with respect to the axis or pole, we may expect to find from the 
 general theorem, that " two successive reflexions, relatively to 
 TWO rectangular axes, are equivalent to a single reflexion, with 
 respect to a third axis perpendicular to both the former." And 
 accordingly we see in fig. 43, that if e be first reflected with re- 
 spect to A to F, and if f be then reflected with respect to b to d, 
 the final result is the same as if e had been at once reflected with 
 respect to c (to d). It is clear also that, in this case, of tri- 
 RECTANGULARITY, three succcssive reflexions (with respect to 
 any three rectangular axes), produce, on the whole, no change : 
 a conclusion which answers geometrically to the formulae (210), 
 
 ijk = -\, kji=+i; 
 
 because these give, for any operand vector p, the identities, 
 
 ijkpk ~ ^j' 1 ^ " 1 = kjipi' ^j'^k~'^ = p. 
 
 349. More generally, from the results of the two foregoing 
 articles, or from the lately cited formula of art. 280, namely 
 
 y2j3%^ = - 1, 
 which gives the equation, 
 
 we may infer, on the same general plan of interpretation (341), 
 that three successive rotations, represented respectively by the 
 DOUBLES of three successive angles of any spherical triangle, for 
 instance (see fig. 5^), by 
 
 2cAB, 2abc, 2bca, 
 
334 ON QUATERNIONS. 
 
 produce, on the whole, no effect. And it is easy to generalize 
 still farther this result, so as to prove the following theorem : 
 *' If a body B be made to revolve through any number of succes- 
 sive and finite rotations, represented as to their axes and ampli- 
 tudes by the doubles of the angles, Ai, As, . . . a„, of any 
 spherical polygon, this body B will be brought back, hereby, to 
 its own original position." You will find, by the printed Pro- 
 ceedings of the Royal Irish Academy, that I stated this The- 
 orem (with only a slight difference in its wording), at a general 
 meeting of that Academy, in November, 1844, as a consequence 
 of those principles respecting Quaternions, which had been com- 
 municated to the Academy by me, about a year before. The 
 theorem, at that time, appeared to me to be new ; nor am I able, 
 at this moment, to specify any work in which it may have been 
 anticipated : although it seems to me likely enough that some 
 such anticipation may exist. Be that as it may, the theorem 
 was certainly suggested to me by the quaternions ; nor can I 
 easily believe that any other mathematical method shall be found 
 to furnish any simpler form of expression for the same gene- 
 ral geometrical result. For there is little difficulty in seeing 
 that the theorem coincides substantially with the conclusion of 
 art. 345 ; and may, therefore, be expressed in this calculus by 
 the same identity, 
 
 a K ^ 7 i^ _ 1 
 
 K t 7 i3 a 
 
 350. But it is worth while to inquire what will happen, if 
 instead of compounding, as in some recent articles, rotations re- 
 presented by the doubles of the sides of a spherical triangle, or 
 polygon, we compound rotations represented by the sides them- 
 selves of the figure ; and with respect to this inquiry, the Cal- 
 culus of Quaternions has conducted to results which, although 
 not very difficult otherwise to prove, appear to me less likely to 
 have been anticipated. 
 
 It has been shewn, in the present Lecture (arts. 258 to 263), 
 that the product 
 
 of the square roots of the successive quotients, 
 
LECTURE VI. 335 
 
 ^g-^ eZ-\ de-\ 
 
 of the radii od, of, oe, drawn to the three corners of a spherical 
 triangle dfe, is a quaternion of which the angle is equal to half 
 the spherical excess of that triangle, 
 
 while the axis of the same quaternion q is directed to or from the 
 corner d, 
 
 Ax . g' = ± S, 
 
 according as the rotation round od, from or towards oe, is po- 
 sitive or negative. Hence, by our general principles respecting 
 rotations, if q still denote the recently mentioned product of 
 square roots, the symbol 
 
 qpq-^, or qBq-\ 
 
 denotes the position into which the vector p or the body B is 
 brought, when it is made to revolve round ± 8 as an axis, through 
 an angle expressed by 
 
 D + E + F-ir; 
 
 that is, through the whole spherical excess of the triangle 
 dfe (and not through the halfoi that excess). 
 
 351. But also, by the associative principle of multiplication, 
 we have 
 
 if we make 
 
 qpq-^^p\ 
 
 Hence (compare 288), the recently described rotation round 
 + OD, through this whole spherical excess of the triangle dfe, is 
 equivalent to the system of three successive and conical rota- 
 tions, represented respectively by the three successive sides of 
 that triangle, 
 
 df, fe, ED : 
 
 a result which appears to me interesting. It. may also be stated 
 
336 ON QUATERNIONS. 
 
 thus, if we adopt the phraseology (218, &c.) of sums of arcs: 
 " The arcual sum, 
 
 1 /^ ED + -^ ^ FE + ^ ^ DF, 
 
 of the HALVES of the three successive sides of a spherical trian- 
 gle DFE, is an ARC, which has the first corner d of that triangle 
 for its positive or negative pole, according as the rotation round 
 D from F towards e is positive or negative ; while the length of 
 the same sum-arc represents the spherical semi-excess of the 
 triangle." 
 
 352. To illustrate this conclusion geometrically, we may ob- 
 serve first that the three successive rotations, represented by the 
 three successive arcs df, fe, ed, produce evidently no final effect 
 on the point d ; since they merely transfer that point upon the 
 spheric surface, first to f, then to e, and then back to the old posi- 
 tion D again. Whatever finite rotation of a body, or of a system 
 of vectors all drawn from the centre of the sphere, may be the joint 
 or combined result of these three successive rotations, the resul- 
 tant rotation so obtained must therefore have the point d for one 
 of its poles. Again, it is clear, from what has been shewn in re- 
 cent articles (342, 343), that if, as in fig. 40 (art. 224), the sides 
 DF and fe of the triangle dfe be bisected respectively in the 
 points B and a, then, not merely for the point d, but also for any 
 other operand point on the same spheric surface, the combined 
 effect of the two rotations, represented by the two successive 
 arcs DF and fe, is equivalent to a system of two successive re- 
 flexions of the operand point in question, first with respect to b, 
 and afterwards with respect to a. That is to say (see again art. 
 343), " the system of two successive rotations represented by the 
 two successive sides df, fe of any spherical triangle, is equiva- 
 lent to a single rotation, represented by the double (2 /^ ba) of the 
 arc which is the common bisector of those two sides." This sys- 
 tem of rotations would therefore carry, for example, the point m, 
 of the recently cited figure 40, to that other position m', which 
 was spoken of in arts. 229, &c. ; or in the astronomical illustra- 
 tion used in those articles, it would, on the whole, transport a 
 point of the celestial sphere from the position Virgo to the posi- 
 tion Scorpio. The remaining rotation represented by the arc 
 
LECTURE VI. 
 
 337 
 
 ED, would then carry the same moveable point backwards in right 
 ascension, till it came to a position m\ which should be situated 
 on the arc of north polar distance km prolonged, but should have 
 the same south declination as m', that is as Scorpio (or what is 
 called the^r^if point thereof) : this new point m' being such as to 
 satisfy the arcual equation, 
 
 ^ MN = ^ Nm\ 
 
 and therefore also such that 
 
 -^ MM^ = 2 /> MN. 
 
 But MN was seen (in art. 258) to represent half the spherical 
 excess of the triangle dfe ; therefore mm^ represents the whole 
 of that excess. And the positive pole of this new arc mm' is the 
 point D : the theorem of the last article is therefore, in all re- 
 spects, confirmed. 
 
 353. You are, no doubt, familiar with the well-known theo- 
 rem, so easily and elegantly proved by lunes, and by the value 
 of the whole surface of the sphere, that the area of a spherical 
 triangle is proportional to the spherical excess, and that it has 
 the same numet'ical measure, when units are suitably chosen : 
 the excess, when treated as an arc, bearing the same ratio to the 
 length of the radius, which the area of the triangle bears to the 
 square upon that radius. And you see that this justifies us in 
 now asserting, that three successive conical rotations, repre- 
 sented by the three successive sides of any spherical triangle (and 
 not now by the doubles of those sides), compound themselves 
 into a rotation round the first corner, which is (on the plan just 
 mentioned) numerically equal to the area of the triangle. Nor 
 is there any difficulty in extending this result, so as to meet the 
 case oi any other spherical polygon. Thus in the case of the 
 pentagon abcde, of fig. 78, the five 
 successive rotations represented by 
 the arcs or sides, ab, bc, cd, de, ea, 
 are equivalent to three sets of three 
 rotations, 
 
 Fig. 78. 
 
 AB, BC, CA ; AC, CD, DA; 
 AD, DE, EA ; 
 
338 ON QUATERNIONS. 
 
 each set being represented by three successive sides of a trian- 
 gle, with A for its first corner. Hence, by the three last articles, 
 any revolving body B, or vector op, is made hereby to revolve suc- 
 cessively round this point A as a pole, or round the radius oa as 
 an axis, through three successive amounts of conical rotation, 
 equivalent to, or measured by, the respective areas of the three 
 spherical triangles, abc, acd, ade, into which the spherical pen- 
 tagon has been divided, by the diagonals, ac, ad ; and it is clear 
 that a similar process might be applied to any spherical polygon. 
 We are then entitled to infer the following Theorem, which was 
 communicated by me to the Royal Irish Academy in January, 
 1848 : — " If a solid body" (or system of vectors) " be made to 
 revolve in succession round any number of different axes, all 
 passing through one fixed point, so as first to bring a line a into 
 coincidence with a line /3, by a rotation round an axis perpen- 
 dicular to both ; secondly, to bring the line j3 into coincidence 
 with a line y, by turning round an axis to which both j3 and y 
 are perpendicular; and so on, till, after bringing the line k to 
 the position A, the line X is brought to the position a with which 
 vpe began ; then the body will be brought, by this succession of 
 rotations, into the same final position as if it had revolved round 
 the first or last position of the line a, as an axis, through an an- 
 gle of finite rotation, which has the same numerical measure as 
 the spherical opening of the pyramid (o, j3, 7, . . k, A) whose 
 edges are the successive positions of that line." For, by the 
 ^^ spherical opening of a pyramid" is understood that portion of 
 the area of the unit sphere, described about the vertex as its 
 centre, which is bounded by the spherical polygon, whose corners 
 are the points where the spheric surface is met by the edges of 
 the pyramid. 
 
 354. In symbols, this theorem comes to the following, which 
 it may be sufficient to state for the recent case of the pentagon : 
 if q denote that quaternion which is the product of the succes- 
 sive square roots of five successive quotients of vectors, 
 
 '9' (i)' (;)' sr (? 
 
 where 
 
LECTURE VI. 339 
 
 a = A-Oj j3 = B-0, ..£ = E-0; 
 
 and if the rotations round a from j3, y, 8, respectively, towards 
 y, 8, E, be positive ; then 
 
 T^=l; Ax.q = a; Lq = i {A+ B+ C + D + E -Sir) ; 
 
 where A, B, C, Z), E denote the five internal spherical angles at 
 the corners of the pentagon abode. Any changes of the lengths 
 of the vectors, a, j3, 7, 8, £, will not affect this theorem, at least 
 if we write 
 
 Ax.q= Ua. 
 
 If instead of a pentagon, we take a polygon of n sides, it will 
 evidently be (n - 2) tt, instead of 37r, which will have to be sub- 
 tracted, before halving, from the sum of the angles. And if any 
 one of the rotations round the first corner, from any other corner 
 towards the one which succeeds it, in the order of passage along 
 the perimeter of the polygon, be negative, the corresponding 
 semi-excess or semi-area of the triangle, whose corners are those 
 three points, is also to be treated as negative, in the summation. 
 With these precautions we may assert generally, that the arcual 
 SUM (218) of the halves of the successive sides, o/any closed 
 polygon on the unit-sphere, is equal to an arc, whose pole is at 
 the FIRST CORNER of that polygon, and whose length repre- 
 sents the semi-area. 
 
 355. We may even conceive, as a limit, that the number of 
 these sides is infinitely great, while their lengths are infinitely 
 small, or that the polygon becomes an arbitrary but closed curve 
 upon the sphere ; and then the arcual sum of the halves of 
 all the successive elements of the perimeter will still, in a 
 perfectly intelligible and definite sense, represent the semi- 
 area of the figure. Hence also follows, on the symbolical 
 side of this whole theory, a mode of conceiving, in an extensive 
 - class of cases, a (generally) definite value, for the product of an 
 infinite number of square roots of quaternions, each infinitely 
 little differing from unity, and succeeding each other by a deter- 
 mined law ; namely, in such a way that, in the class of cases here 
 considered, the product of all those successive quaternions them- 
 selves is unity; just as (compare 307) the sum of all the suc- 
 
 z2 
 
340 ON QUATERNIONS. 
 
 cessive elements themselves (though not the sum oiiheu halves), 
 for the perimeter of any closed figure, vanishes. And on the 
 physical or rather the geometrical side, so far as regards the ge- 
 neral theory of compositions of rotations, we arrive (on the plan 
 of recent articles) at this remarkable theorem, that the infinitely 
 many infinitesimal and conical rotations, represented by the 
 successive elements {themselves now, and not their halves) of 
 the PERIMETER o/an Y closcd figure on a sphere, compound them^ 
 selves into a single resultant and finite rotation, represented by 
 the TOTAL area of the figure; it being still understood that ele- 
 ments of this area may become negative. It would also be easy, 
 if it were thought useful, to transform most of the results of the 
 few last articles into others, which should employ external angles, 
 and their halves, instead of sides and half sides of a polygon. 
 356. Although we know that the product and sum, 
 
 « 7 ^ J ' 
 
 -■k i and -^ CA+ '^ BC+ ^ AB, 
 
 7P « 
 
 are respectively equal to unity and to zeo'O (compare 344, 307), 
 
 yet on account of the general non-comnnitativeness (304, &c.) of 
 
 the operations of multiplybig quotients (or quaternions), and of 
 
 adding their representative arcs, we are not entitled to infer that 
 
 the same values hold good, for this other quotient, and this other 
 
 sum 
 
 i3 7 « J 
 
 — ^ -, and ^ AB + ^ BC + ^ CA. 
 
 a p y 
 
 It is, therefore, worth while to inquire, what quaternion is equal 
 to the formeT product, and what arc is equal to the latter su7n. 
 And it is easy now to answer these questions, without construct- 
 ing any new diagram, if we merely conceive the point m', de- 
 scribed in the recent art. 352, to be introduced into the often 
 cited fig. 40, of art. 224 ; and if we at the same time conceive 
 that A and b are reflected, with respect to c, to new positions 
 which we shall denote by a' and b' ; in such a manner that we 
 shall not only have the equation of 352, 
 
 r\ MN = /^ Nm\ 
 
 but also these two other equations. 
 
LECTURE VI. 341 
 
 '^A^C = '^CA, r>BC = ^CB\ 
 
 For this being understood, we see that to add the arc bc or its 
 equal cb\ as aprovector arc (217, 218), to the vecto?- arc ca or 
 A^c, answers to going, on the whole, along the transvector arc, 
 
 r\ A^B^= ^ BC + ^ CA. 
 
 (Compare fig. 37, art. 219.) But from the position assigned to the 
 point m\ we have the equation (see again fig. 40), 
 
 f\ a'b' = r\ M^L. 
 
 Adding then to this as a new vector arc, the new provector arc 
 (compare 224), 
 
 rs IlE = r\ LM, 
 
 we go on the whole from m' to m, or move (compare again 352) 
 along i\ns> final transvector arc, representing that ternary sum 
 which was inquired of in the present article : 
 
 /^ AB + ^ BC + /^ ca = ^ M^M = 2 /^ NM. 
 
 That is, we move along an arc of which the point d (in fig. 40) is 
 the negative pole, because this point d is (by 225) the positive 
 pole of the arc km, and, therefore, also of the arc mn ; and the 
 arc 2 r\ NM, along which we thus move, represents, in amount, 
 the area of that triangle efd whose sides are bisected respec- 
 tively by the corners of the triangle abc: because (by 258) the 
 arc MN, or the angle mdn, represents the semi-excess of the tri- 
 angle whose sides are so bisected. 
 
 357. Knowing thus perfectly what arc (namely, m'm, or 
 2nm) is equal to the ternary sum of arcs y which was proposed 
 for discussion in the present article, it is easy to infer (as also 
 proposed therein) what quaternion is equal to the connected 
 and ternary product of quotients; namely (see again 258), the 
 following : 
 
 g r « ^ //*' 
 
 And in fact we might have more rapidly arrived at the same re- 
 sult, with the help of the associative principle of multiplication. 
 For by treating (for simplicity) a, j3, 7, as unit vectors, so that 
 
342 ON QUATERNIONS. 
 
 we have 
 
 /3a-i. 7/3-1. a7-i = -(/3a-i7)2; 
 
 but the fourth proportional /ia'^y, to a, j3, 7, was shewn in the 
 Fifth Lecture, in connexion with the above cited fig. 40, to have 
 its axis directed (225) to the point d, and to have its angle (227) 
 equal to the supplement of the semi-sum of the angles of the tri- 
 angle DEF ; that is (compare 258), to the complement of the half 
 spherical excess ; or finally (353), to the complement of the semi- 
 area of that triangle. Hence, by the Fourth Lecture, the square, 
 namely {^a'^yY, of the same fourth proportional, is a quaternion 
 which has still its axis directed to d, but has its angle equal to 
 the supplement of the whole spherical excess, or to the supple- 
 ment of the total area of the same spherical triangle def. But 
 since we are to take the negative of this square, in order to ob- 
 tain the sought quaternion 
 
 ^70 
 
 a ^y 
 
 we must (by 183) reverse the axis of that square, and take the 
 supplement of the angle thereof And thus we are led again to 
 conclude, that (under the conditions of fig. 40) the lately written 
 ternary product is a quaternion which has its axis directed away 
 from D, or has d for its negative pole ; while its angle is simply 
 equal to the total spherical excess, or is equivalent to the total 
 area of the triangle efd, whose sides ef, &c., are bisected (as 
 above) by the corners, a, &c., of the given triangle abc. And 
 hence we may (on the plan of 341) infer the following theorem 
 of rotation, with which we shall, for the present, conclude our 
 account of the applications of quaternions to theorems of this in- 
 teresting class : — " If a vector p, or body B, be made to revolve 
 in succession, through three finite and conical rotations, repre- 
 sented respectively by the symbols, 
 
 2 ^ CA, 2 r\BC, 2 A AB, 
 
 or by the doubles of the three sides of a spherical triangle, abc, 
 taken in an inverted order, as third, second, and first ; and \i ano- 
 ther triangle def be so constructed, that the sides ef, fd, de. 
 
LECTURE VI. 343 
 
 respectively opposite to its three successive corners d, e, f, shall 
 be bisected by the three successive corners a, b, c, of the old or 
 given triangle ; then the vector or body (p or B) will, on the 
 whole, have revolved round the corner d of the new triangle, as 
 a negative pole, or round the radius od' which is drawn to the 
 diametrically opposite point upon the sphere, as round a positive 
 axis, through an angle which is numerically equivalent to the 
 DOUBLED AREA of the Same new triangle, def." Indeed this 
 theorem (like some others of recent articles) has been above de- 
 duced with a reference to figure 40, in which the sides of the 
 triangle abc were supposed to be each less than a quadrant: but 
 you will find no difficulty now in adapting the reasonings and 
 their results, to cases in which this particular condition is not sa- 
 tisfied. 
 
 358, It may have seemed remarkable, that in arts. 295 to 301 
 we treated the proof oi the associative principle, for the multipli- 
 cation of any three versors, as depending on the deduction of one 
 arcual equation from five others ; whereas, in art. 302, we made 
 the proof of the same principle depend on the deduction of three 
 equations between angles, from three other equations of the same 
 sort. However, a little consideration shews that this difference 
 is only apparent, so far as respects the numbers of the things 
 given and inferred; and ihdit for arcs, as well as for angles, we 
 may prove the associative principle, by deducing three equa- 
 tions from three others. In fact, after representing, as in art. 
 294, and fig. 58, the six versors q, r, s, rq, sr, and s . rq, by the 
 six arcs ab, bc, ef, ac, gi, and df, respectively, the theorem 
 which was to be proved, or the associative equation sr . q = s . rq, 
 may be thus expressed, in the notation of sums of arcs : 
 
 r\ Gl + r. AB = r\ DF. 
 
 Here, it may be considered that there are given lis, by construc- 
 tion, the three double co-arcualities (each involving four points 
 upon the sphere), 
 
 DAEC, CHBG, and ehfi, 
 
 together with whatever additional information is contained in the 
 three equations, 
 
 '-AC='-DE, -BC='-GH, -"^ EF = '^ HI ; 
 
344 ON QUATERNIONS. 
 
 that is to say, in the three middle equations of the five which 
 were regarded as the data in art. 295. And the theorem to be 
 proved may be thus stated : that if we determine three additional 
 points, K, L, M, so as to satisfy the three other double co-arcuali- 
 ties (see the general construction for arcual addition in 217), 
 
 AKBL, GLIM, DKFM, 
 
 and suitably distinguish each of these three new points from the 
 diametrically opposite point upon the sphere, we shall have also 
 the three arcual equations, 
 
 -^AB = '-KL, -GI=-^LM, -^DF='-KM; 
 
 namely, the two other given equations of 295, and the one sought 
 equation of that article. In other words, the six double co-ar- 
 cualities being now supposed to exist, we are to shew that the 
 three last equations between arcs are consequences of the three 
 others, which were written a little before them in the present 
 article. And this inference, of the three last arcual equations 
 from the three others of the same sort preceding them, under the 
 six conditions lately indicated of double co-arcuality, may be es- 
 tablished, not only by the doctrine of spherical conies, in a way 
 differing little from that of art. 296, but also by a more elemen- 
 tary process, with the help of the figures used in arts. 298 to 
 301, through a modification of the method of those articles which 
 may be briefly described as follows. 
 
 359. The constructions of 298, 299 being retained, we may 
 prove, as in those two articles, with the help of figs. 59, 60, that 
 the plane of the great circle glim, in fig. 58, touches at o the 
 diacentric sphere opqr, in virtue of the two given equations, be- 
 tween the arcs bc, gh, on the one hand, and ef, hi, on the other. 
 The other given equation, between the arcs ac, de, will shew, 
 by fig. 62, that the four points p, q, r, s, are concircular, on ac- 
 count of the parallelisms of pq, rq, ps, rs to oc, oe, oa, od, if s be 
 now defined to be the point virhere the radius ok prolonged meets 
 the plane pqr ; and, therefore, will prove that this point s is also, 
 with this netv definition of it, what it was before defined to be, in 
 the method of art. 300: namely, the second intersection of the 
 line OK with the diacentric sphere opqr. The three given equa- 
 tions having been thus made use of, we may infer the first of the 
 
LECTURE VI. 345 
 
 three sought equations, namely, that between the arcs ab, kl, 
 from a parallelism and a tangency, with the help of fig. 61, of art. 
 300 ; although in the process of that former article, the equa- 
 tion as well as the tangency was given, and the parallelism was 
 thence to be inferred. Again, if we retain the definitions of the 
 points p', q', r', s', which were given in 298 and 300, those points 
 may easily be proved, as before, to be on one common sphere, 
 and therefore on one common circle, because they still are, by 
 construction, upon one common plane; which proof may still be 
 made to depend on the equalities of the four rectangles, 
 
 P0P'= qoq'= ror'= sos'; 
 
 and thus the second sought equation, between the arcs gi, lm, 
 may be proved, with the assistance of fig. 63. And finally, a 
 parallelism and tangency will enable us, as in 301, with the help 
 of fig. 64, to infer the third and last sought equation between 
 arcs, namely, that between df and km. 
 
 360, Although it can give you no trouble to fill up the 
 sketch of an elementary demonstration contained in the fore- 
 going article ; nor thus to prove anew the associative formula, 
 sr . q = s . rq, with the help of art. 358, by shewing, in a new wag, 
 that these two products of versors are represented by equal arcs, 
 namely, by ^-^ km and ■^ df, as before; yet it may not be useless 
 to oflFer here the following remarks respecting the numbers of the 
 things given and sought. Every assertion, then, oi a co-arcualitg 
 existing between three points upon the surface of a sphere, may 
 be observed to involve a condition, which can always be con- 
 ceived to be expressed by a single numerical equation ; for 
 such an assertion is equivalent to stating, that the perpendicular 
 distance of one of the three points, from the great circle through 
 the two others, vanishes. A statement of a double co-arcuality, 
 or an assertion that/oMr points of the sphere are situated upon 
 one common great circle, is therefore equivalent, generally, to a 
 system oi two such numerical (or scalar) equations. Now what 
 we have called (in 217, &c.) an arcual equation, is understood to 
 involve such a double co-arcuality, and also to include another 
 numerical or scalar equality besides ; for the lengths of the two 
 equated arcs are to be equal, and their directions are not to be 
 
346 ON QUATERNIONS. 
 
 opposite. Hence an arcual equation of the foregoing sort is ge- 
 nerally equivalent to a system of three scalar equations; which 
 accordingly it ought to be, because it represents an equation be- 
 tween versorSf and a versor (see 91) depends generally on a sys- 
 tem of three numbers. We might then, in the investigation of 
 295, &c., have conceived ourselves as proving that a certain sys- 
 tem of three scalar equations could be deduced from a system of 
 fifteen such equations ; because one arcual equation was to be 
 deduced {ram five equations of that class. And "when we after- 
 wards came, in 358, 359, to treat six double co-arcualities as 
 given, or known, we tacitly used thereby (or, if I might venture 
 so to speak, we absorbed) no less than twelve out of the fifteen 
 numerical data of the question. It was therefore quite natural 
 that there should remain only three other data, to be still ex- 
 pressly marked by equations, and from which it was still required, 
 as in the two last articles, to shew that three other numerical e(\Vidi- 
 tions followed. It may also be noticed, that every proof, or (tacit 
 or expressed) assumption, of any co-arcuality of (three or 
 more) points, in fig. 58, is equivalent (on certain known princi- 
 ples of rec^proc^7^/) to some corresponding proof or assumption, 
 in fig. 65, of what may be called a co-punctuality of (three or 
 more) arcs : or, in other words, a meeting of three or more arcs in 
 one point ; or rather (of course) in one pair of diametrically op- 
 posite points. 
 
 361. The construction given in the last cited fig. Q5 (of art. 
 302), may be generalized or extended as follows. Instead of con- 
 sidering only three given factors, q, r, 5, let us now consider ^owr 
 such factors, q^ r, s, t ; let us denote their total product by u, 
 
 so that 
 
 u = tsrq ; 
 
 and in studying the derivation of this total product from its fac- 
 tors, let us denote for conciseness, the fivejoar^m? products of the 
 same four factors by the letters v, w, x, ?/, z, writing 
 
 v = rq^ w = sr, x = ts, y = srq, z = tsr. 
 
 Let also the ten representative points, upon the unit sphere, for 
 these various factors and products, g, r, s, t, u, v, w, x, y, z, be 
 called, in the corresponding order, a, b, c, d, e, f, g, h, i, k, as 
 
LECTURE VI. 
 
 347 
 
 Fig. 79. 
 
 marked in the annexed fi- 
 gure 79, which may be con- 
 ceived to be constructed as 
 follows. Regarding the 
 four original factors q, r, 5> 
 t, as entirely given and 
 known, we may suppose 
 ourselves to know their re- 
 presentative points. A, B, c, 
 D, and also the angles which 
 represent them at those 
 points. Then the two an- 
 gles, 
 
 lq = FkB, Z.r = ABF, 
 
 may be conceived to determine the point f ; and in like manner, 
 G may be found by 
 
 /.r = GBC, z« = bcg; 
 and H, by 
 
 ZS = HCD, Z < = CDH. 
 
 At the same time we shall have, by principles already explained, 
 
 Av=ir-BVA; Lw = Tr-CGB; z^ = 7r-DHc. 
 
 The three binary products v, w, x being thus determined, to find 
 next the two ternary products, y and z, we may observe that the 
 equations, 
 
 y = sv, z = tw, 
 
 enable us to construct the two points i, K and the two angles 
 Ly, AZy by two new triangles, thus ; 
 
 Zw = iFC, Z* = FCi, ly = Tr-ci¥; 
 
 ZW=KGD, Z^=GDK, Z2r = 7r-DKG. 
 
 And finally, to construct the one quaternary (or total) product, 
 u or tsrq, we may employ the equation 
 
 u = ty, 
 
 which leads us to determine the point e, and the angle Z M, by a 
 new triangle, as follows : 
 
348 ON QUATERNIONS. 
 
 Z2/ = EID, lt = lT>E, ZM = 7r-DEI. 
 
 362. In this manner, then, with the help of six triangles^ 
 answering to six binary multiplications ^ we can gradually and 
 successively construct the six points, r, g, h, i, k, and e, which 
 represent the products, partial and total, of the^wr given ^c- 
 tors, represented themselves (as to their positions or the direc- 
 tions of their axes) by the four given points, a, b, c, d ; and can 
 also determine the angles of these six products, the angles of the 
 factors being supposed known. And in this process it is impor- 
 tant to observe that we have been led to construct or represent 
 Z r by two different angles, namely, abf and gbc, at the point 
 B ; Z « by three different angles at c ; and z t, by three other an- 
 gles at D. The comparison, therefore, of these various repre- 
 sentations for the angles of these three latter factors r, s, t, con- 
 ducts to Jive equations of condition, or to Jive relations between 
 the angles qfthejigure, which are true by the foregoing con- 
 struction ; namely, to the five following equations: 
 
 ABF = GBC ; {^f) 
 
 BCG=HCD = FCi; {Ls) 
 
 CDH = GDK = IDE ; (Z t) 
 
 Z q occurring only in one of the six triangles, and therefore not 
 furnishing any equation. Again the binary product v occurs in 
 two triangles ; w in two others ; but x in only one ; we have, 
 therefore, from the comparison of the representations of the an- 
 gles of the binary products, two other equations between the 
 angles of the figure, namely : 
 
 7r-BFA = IFC; {z.v) 
 
 IT- CGB= KGD. (/^) 
 
 Finally, the ternary product y occurs in two triangles ; but the 
 other ternary product z, and the quaternary product u, occur 
 each only in one triangle; we have, therefore, one more equa- 
 tion, and only one more, between the angles of the figure 79, as 
 true by the foregoing construction, namely the equation, 
 
 ^-ciF = EiD. i^y) 
 
 And conversely the establishment of these eight equations of 
 
LECTURE VI. 349 
 
 CONDITION, between the angles of the figure 79, at least if com- 
 bined with attention to the signs or directions of rotation, is suffi- 
 cient to entitle that figure to be regarded as a correct representa- 
 tion of the process recently explained, for constructing, through 
 representative angles, and with regard had to the order of the fac- 
 tors, all the products, partial and total, of Bx\y four g\vex\ versors, 
 or quaternions (with the help of the general method of 264, 265, 
 272). 
 
 363. If then we take care to establish hy construction, or if 
 ■we simply conceive as so established, the eight equations of con- 
 dition assigned in the foregoing article, in connexion with fig. 
 79, we may regard that figure as being consistent with, or as 
 furnishing, all those other angular relations which ihe associative 
 principle of multiplication involves. Thus whereas we only 
 used, in 361, the six binary products, 
 
 rq = V, sr -w, ts = x, sv = y, tw = z, ty = u, 
 
 constructing each by a spherical triangle, on the plan of art. 264, 
 we may now employ ihe?,e four other binary products, which will 
 conduct to so many new triangles : 
 
 wq = y, xr = z, xv = u, zq = u. 
 
 The six former triangles (for binary multiplications) were, 
 
 ABF, BCG, CDH, FCI, GDK, IDE ; 
 
 the four latter triangles are, 
 
 AGI, BHK, FHE, AKE. 
 
 They give two new representative angles for q ; one for r; none 
 for s nor for t ; one for v, another for w, and two for x ; one for 
 y, and two for z ; and finally, two for u. On adding these num- 
 bers of new representations for the angles of the factors, q, r,s,t; 
 of the binary products v, w, x ; of the ternary, y, z ; and finally, 
 of the quaternary product, u; namely, the numbers, 
 
 2, 1, 0, 0; 1, 1, 2; 1, 2; and 2, 
 
 to the corresponding numbers of representations for the same ten 
 angles, which were obtained from the six old triangles, namely, 
 to the numbers. 
 
350 ON QUATERNIONS. 
 
 1, 2, 3, 3; 2, 2, 1 ; 2, 1 ; and 1 : 
 
 we find in each of the ten cases, a numerical sum = 3. 
 
 364. In fact, as an inspection of the recent figure 79 may 
 shew, although perhaps the foregoing enumeration shews it more 
 clearly, each of the ten points of the figure, from a to k, is a 
 common corner o/* three out of those ten triangles, of 
 which each has lately served to construct a process of binary 
 multiplication, by combining (as multiplier and multiplicand) 
 some two (suitably chosen as to their order') of the factors q^ r, 5, ^, 
 and of their partial products v, w, x, y, z; and each of these 
 processes gives, as its result, either some one of those partial 
 products, or else the total product, u. Thus taking always sup- 
 plements of vertical angles as representations of binary pro- 
 ducts, we have for each of the te7i angles z q, &c., three dis- 
 tinct representations, at its own point of the figure : and 
 consequently, we arrive, by comparison of values, at two equa- 
 tions between angles, for each of the ten points, making a sys- 
 tem OF twenty equations in all. But of these twenty equa- 
 tions, it was seen (in 362) that eight were true by construction, 
 if the figure 79 were rightly formed : and that, conversely, these 
 eight equations sufficed (with attention to signs) to justify the 
 construction of the figure. We must, therefore, conclude that the 
 twelve new equations, which we shall here write down, 
 
 IAG = EAK= FAB, KBH =ABF ; {tq, Lr) 
 
 EFH=IFC, AGI = KGD; {LV, Lw) 
 
 7r-DHC = BHK= FHE ; (z a;) 
 
 7r-GiA = EiD; {ty) 
 
 AKE =7r-HKB=V-DKG; {Lz) 
 
 and finally, 
 
 KEA=HEF=DEI, (tt - A u) 
 
 are consequences of the eight former equations, of art. 362 : just 
 as in art. 302, and in connexion with fig. 65, it was seen that 
 three relations between angles were consequences of three other 
 equations. In fig. 79, the line ke is prolonged, to exhibit the 
 angle tt-kea, which is one of the three representations of the 
 angle of the final or total product, u, regarded as equal to tsr.q ; 
 and the apparent co-punctuality of the three arcs, ai, bk, ef, is 
 accidental. 
 
LECTURE VI. 351 
 
 365. More generally, let there be any number, n, of versors, 
 
 9\y g'2j g'3, . . . qn, 
 
 which it is required to multiply together, in their given order of 
 succession, the first by the second, the second by the third, the 
 product of second into first by the third, and so forth. We shall 
 form hereby n-\ binary products, 
 
 'r\ = q%q\-> r2 = qzq%, . . rn.i = qnqn-\', 
 n-2 ternary products, 
 
 S\ = qzq2q\, 52 = g'45'3^2, . • . Sn.2 = qnqn.\qn-2) 
 
 w - 3 quaternary products 
 
 ^1 = 9'42'39'29'lj • • • • tn-3 = qnqn-iqn-2qn-3'> 
 
 and so on, till we come to two partial and penultimate products, 
 
 Zi = qn.i qn.2 ■ . q2qi, z% = qnqn-i . . Msj 
 
 and at last to one final and total product, which we shall here de- 
 note by q, so that 
 
 q = qnqn-iqn-2, ' . q^q-zqi- 
 
 The number of all these products, partial and total, will be, 
 
 (w - 1) + (w - 2) + (w - 3) + . . + 2 + 1 = i w (w - 1). 
 
 And the number of given factors was = n ; the entire number, 
 therefore, of factors and products taken together, or collected 
 into one system, is 
 
 ^n{n-\-\). 
 
 For each of these various versors there will be a representative 
 point on the sphere, depending on two spherical co-ordinates, or 
 determining numbers of some sort : the whole number of such 
 co-ordinates, for the present system of factors and products, is 
 therefore, 
 
 n{n+ 1). 
 
 But again, each of the n proposed versors, from qi to qn, depends 
 (by 91) on three numbers, suppose on two co-ordinates and an 
 angle; and conversely, if these 3n numbers be given, all the 
 points of the spherical figure (representing products as well as 
 
352 ON QUATERNIONS, 
 
 factors) will be (in general) determined. Thus, the n{n-\ 1) 
 numbers recently mentioned, will all be determined if ^n of them 
 be so ; and consequently there must in general exist 
 
 n {n + \) - Zn = n (n - 1) 
 
 RELATIONS, between the n{n+ 1) co-ordinates of the figure. 
 
 366. It was thus, for example, that when we were merely 
 constructing, as in art. 264, a triangle of multiplication, to exhi- 
 bit (by fig. 50) the relations which exist between two factors, 
 §', r, and their product rq, the number which we have lately 
 called n was = 2; n{n-2) and n {n+ \) were respectively and 
 6 ; and there existed no quantitative relation between the six co- 
 ordinates of the figure : or in other words, the spherical triangle 
 was allowed to be arbitrarily assumed, if we merely wished it to 
 serve as an example of the multiplication of two versors ; because 
 the angles of those two versors, and, therefore, also the base an- 
 gles (as well as the base) of the triangle itself, might then be 
 chosen at pleasure. Again, when there were three factors, q, r, s, 
 as in 302, and when it was required to exhibit the relations be- 
 tween those three factors, their two partial products, rq, sr, and 
 their total product srq ; we had a figure (65) with six points, 
 between the 3.4 = 12 co-ordinates whereof there existed 3 (3 - 2) 
 = 3 relations, or quantitative conditions ; because those co-ordi- 
 nates all depended on 3 . 3 = 9 numbei'S, answering to the three ar- 
 bitrary versors, q, r, s. Accordingly, in fig. 65, after assuming 
 (suppose) the four corners a, b, c, d of the quadrilateral, we were 
 not free to assume arbitrarily even otie of the two other points 
 E, F, between the four co-ordinates of which pair of points it is 
 manifest that there exist so77ie three relations (although with the 
 lireche fo?'ms of those relations we are not now concerned) ; at 
 least if we grant the conclusion of art. 302, that these two points 
 are foci of a conic, inscribed in the quadrilateral. Or, without 
 introducing any such doctrine of spherical conies, if we only 
 grant the associative principle of multiplication of quaternions, 
 as proved by the elementary investigation of arts. 298 to 301, or 
 by the more recent but not less elementary modification of that 
 proof, which was given or sketched in 359, we can still shew 
 easily that three relations must in fact exist between the twelve 
 
LECTURE VI. 353 
 
 spherical co-ordinates of the six points of fig. 65; because after 
 assuming the four points a, b, c, e, of that figure, the angular 
 equation, 
 
 ABE = FBC, 
 
 in which both members represent the versor r, assigns a locus 
 (namely, a great circle) for the point f ; and after we have 
 chosen the position of this point f, on this locus, the position of 
 the remaining point d becomes determined. In short, the three 
 equations between angles, which were employed in constructing 
 this figure 65, and from which three others were afterwards de- 
 rived, may be regarded as being themselves (indeed under the 
 very form most suited to our present purpose) the system of 
 three relations between co-ordinates, which was spoken of above. 
 And in like manner, when there were, as in some later articles 
 (361, he), four factors, q, r, s, t, to be multiplied together, so 
 thafr n was = 4, we found (362) that there existed w (w - 2) = 8 
 equations between the angles of the figure 79, as necessary for 
 the justness of that figure, and to be considered as true by its 
 construction. 
 
 367. In general, it is not difficult to prove directly, without 
 any reference to co-ordinates as such, and by a process analogous 
 to that of arts. 361, 362, that whatever the number n of factors 
 may be, there must, by the very construction of the figure which 
 represents those factors and their products, exist n{n -2) equa- 
 tions of condition between the angles, which suffice to determine 
 the positions of its various points, or at least to fix their relative 
 positions on the sphere. For this purpose, in 365, suppose that 
 the n factors q\, . . . qn are represented by the n points Qi, . . q„; 
 the n-\ binary products, ri, &c., by the n-\ points Ri, &c. ; 
 the ternary products, *i, &c., by the points Si, &c. ; and so on, 
 till the two penultimate products, zi, Z2, are represented by Zi, 
 Z3 ; and the one final or total product q is represented by the 
 one point q. We may then conceive that all these ^n (n - 1) 
 products, partial and total, are gradually and successively de- 
 duced, without repetition, by a certain spherical triangula- 
 TiON, from the n given factors ; or that the representative points 
 of the one set are gradually constructed, from those of the other 
 
 2 A 
 
354 ON QUATERNIONS. 
 
 (the angles of the factors being known) ; for which purpose it 
 may be convenient to adopt, as in 361, 362, the rule of employ- 
 ing no other multipliers^ except those proposed or given factors 
 5'2) • • • (^nt '^^\c\\. follow the first of them. For in this way we 
 shall form a system of\n{ii-\) triangles, each serving to 
 construct the position of one of the equally numerous sought 
 points, and also the angle of the corresponding product ; and 
 accomplishing this double object for every one of those sought 
 points ; namely, that system of triangles, which answers to and 
 constructs the following system of binary products : 
 
 ri = q2q\., . . . /"n-i =qnqn-i ; 
 Si = q3ri, . . • %-2 = ^ra»'n-2; 
 
 tl = qiSi, . . , tn-S = qnSn-S ', 
 
 z\ = qn-\yu z^^qny^', 
 and finally, " 
 
 q==qnZ\. 
 
 It is clear, in fact, that every one of the sought things will be 
 successively constructed thus, without any defect or excess. 
 Each will he found once, and only once, although it may be after- 
 wards used. 
 
 368. But if we now inquire how many and what cases occur, 
 in this construction, of a point, whether it be a given or a sought 
 one, being used as a common corner for more triangles than one, 
 although, in general, no point will offer itself as a common vertex, 
 for any two triangles, because none (as we have seen) h found 
 twice ; we perceive that each partial product, except the last in 
 its own rank, presents itself ^rs^ as such a product, and after- 
 wards again as a multiplicand, but not in any other way. 
 Hence, each of the w - 2 representative points ri, . . . R„_2, is 
 a common corner of two and only two triangles; whereas r„_i is 
 a corner (namely the vertex) of one triangle, and not a corner of 
 any other. In like manner, each of the w-3 points Si, . . . s„.3 
 is common to two triangles ; but s„.2 belongs to one triangle 
 only. And so on, till we come to Zi, which point (though not 
 Zo) is a common corner of two triangles. Finally, the point Q, 
 representing the total product, belongs only to one triangle. Now 
 
LECTURE VI. 355 
 
 every point, which thus belongs to two triangles^ gives, on the 
 same general plan as in art. 362, one equation between two angles : 
 so far then as the \n{n- 1) 'products^ whether partial or total, 
 are concerned, there arise, out of this construction, equations be- 
 tween angles, of which equations the number is the following : 
 
 (w-2) + (w-3) + . . + 2 + l=i(«-l)(re-2). 
 
 369. But the n given points, or the n original ^c^or*, must 
 also be attended to. Now although the first given factor, q\, 
 does not occur as a multiplier^ and although no one of the n given 
 factors occurs as a product at all, yet q^ occurs once as a multi- 
 plicand, namely, in q^^q-z, and once as a multiplier, namely, in 
 q^iqi ; thus the point Q2 is common to two of the triangles, and 
 furnishes owe equation of condition. The factor q^ occurs once 
 as a multiplicand, in q^q^, and twice as a multiplier, namely, in 
 q^q^ and in q^ri ; the point Q3 is therefore common to three tri- 
 angles, and gives two equations of condition. In like manner, 
 qi occurring once as a multiplicand (in q^q^), and three times as a 
 multiplier (in q^q^, q^r^, qiSi), Q4 is a common corner of /our 
 triangles, and we can derive from it three equations between an- 
 gles. And so proceeding, we find easily that each simple or 
 given factor supplies us with one more equation than the factor 
 preceding it had done, with the sole exception of the last factor 
 of all, qn, which nowhere enters as a multiplicand, and therefore 
 occurs no qftener on the whole than the penultimate factor qn.xt 
 although it is true that qn does occur once oftener than qn-\ as a 
 multiplier. Hence, Qn, like Q„-i, belongs only io n-\ triangles, 
 and supplies only w - 2 equations. Thus the n-\ given factors, 
 previous to the last, furnish 
 
 + 1 + 2 + . . + (w - 3) + (w - 2) = |(w - 1) (w - 2) 
 
 equations; and the last given factor furnishes n-2 other equa- 
 tions : the n given factors, taken together^ supply, therefore, 
 upon the whole, 
 
 i(w + l)(?z-2) 
 
 equations of condition. But their products were shewn, in the 
 last article, to supply 
 
 \{n~\){n-2) 
 2 a2 
 
356 ON QUATERNIONS. 
 
 such equations. The factors and their products, or the given 
 and sought points taken altogether^ furnish therefore, upon the 
 whole, as relations between the angles of the figure, or as condi- 
 tions for the correctness of its construction, the number 
 
 n {n - 2) 
 
 of equations. It is evident that this general result includes (as 
 before) the particular case of three equations of condition between 
 the angles, when there were (as in fig. 65) three factors ; and 
 also the case where (as in fig. 79) there were /our factors, and 
 eight equations of condition. 
 
 370. The spherical triangle, qrs, in fig. 50, or 53, was 
 called in a recent article (366) a triangle of (binary) multi- 
 plication, because it serves to construct the binary product, s 
 or rq, of two given quaternion factors, q and r. In like manner 
 the spherical quadrilateral abcd, of fig. Q5, may be called a qua- 
 drilateral of {ternary) multiplication, since it serves to 
 construct, by its fourth point d, and by an angle thereat, the 
 ternary product, srq, of three given factors, q, r, s, which were 
 themselves represented by the three points a, b, c : while the 
 two inserted and auxiliary points, e, f represent (as we have 
 seen) the two partial products, rq and sr. On the same plan, 
 the spherical pentagon, abode, of the more recent figure 79, might 
 be named a pentagon of {quaternary) multiplication, be- 
 cause it constructed, by an angle at its fifth corner e, the qua- 
 ternary product, tsrq or u, oifour given factors, q, r, s, t, which 
 were themselves represented (as we lately saw) by angles at its 
 four other corners, a, b, c, d : while the five partial products of 
 the same four factors, namely, rq, sr, ts, srq, tsr, were repre- 
 sented (as we have also seen) by the five auxiliary and inserted 
 points, F, G, H, I, K, or by certain spherical angles thereat. More 
 generally we may now form the conception of a (spherical) po- 
 lygon OF continued MULTIPLICATION, 
 
 Q1Q2Q3 • • • Qn-lQnQ, 
 
 constructed on the plan described in the recent art. 367, so as to 
 represent, by an angle at its last corner q, the continued product 
 
LECTURE VI. 357 
 
 ofn given quaternion factors, qu . . . qni which are themselves 
 represented by certain angles at its n first corners, Qi to q„. 
 
 371. It is essential, however, to the complete conception of 
 such a polygon of multiplication, to remember that the partial 
 products of the same n factors, whose number is, in general, 
 
 (w - 1) + (w - 2) + . . . + 2 = 1 (w + 1) (w - 2) ; 
 
 namely, those denoted in art. 365 by the symbols 
 
 ru ' ' rn.\', si, . . Sn.^; ' • • ^i, ^s ; 
 
 are to be represented, in the same (conceived) new and more 
 complex figure or construction, by those other points (or by an- 
 gles at them) which in art. 367 it was proposed to name, respec- 
 tively, the points 
 
 Rij . • Rji-i ; Si, . . Sn-2 ; • • • Zi, Zg ; 
 
 and of which the number is expressed (as above) by the formula 
 
 i(w+l) (?2 - 2), or, ^p(p- 3), 
 
 if the number of the sides or corners of the polygon itself he. de- 
 noted more simply by the symbol, 
 
 p = n+ 1. 
 
 For without the consideration of these inserted or auxiliary 
 points, Ri to Z2, there would be nothing peculiar to the theory of 
 quaternions, in the construction or study of the polygon QiQg . , 
 Q„Q itself; which might in that case be confounded with any 
 other spherical polygon, having the same number (/i+ 1) of cor- 
 ners. Thus the spherical triangle qrs of figures 50, 53, was 
 (as we have seen in 366) an arbitrary triangle, in the sense that 
 there existed no conditions limiting its three corners, except what 
 were involved in a certain supposed direction of rotation (265, 272), 
 which conditions, however, might be eluded, if we chose to consi- 
 der negative angles. Again, the spherical quadrilateral abcd, 
 of fig. 65, remains an arbitrary quadrilateral, unless we take ac- 
 count of at least one of the two inserted points e, f, which in- 
 troduce certain equations of condition. And in like manner the 
 spherical pentagon abode of fig. 79 would be arbitrary, if we did 
 
358 ON QUATERNIONS. 
 
 not consider it in connexion with two or more of the five inserted 
 points, F, G, H, I, K, of the same recent figure. 
 
 372. But when we do thus take account of the inserted 
 points, then every polygon of multiplication (after the triangle) 
 constructed as above, possesses several interesting geometrical 
 properties, suggested by the theory of products of quaternions, 
 as has already in part been seen. The property which it seems 
 most useful to investigate at this moment, as illustrating some 
 recent but less general results, is that which regards the depen- 
 dence of one set of equations^ between certain spherical angles of 
 the figure, on another set of equations between those angles; the 
 latter set being usually (indeed always, when we once pass the 
 quadrilateral, and proceed to pentagons, &c.) less numerous than 
 that other set, which is shewn to be dependent upon it. To 
 prove this, I observe that when the triangles of construc- 
 tion, employed in the process which was described in art. 367, 
 are co^nbined (as in the case of art. 363) with those others which 
 are suggested by the associative principle of quaternion multipli- 
 cation, and which may perhaps, for that reason, be properly 
 called ASSOCIATIVE triangles, then every point of the figure 
 is a COMMON corner ofn - 1 different triangles; or the quater- 
 nion which is represented by it enters, in n-\ different ways, 
 whether as factor or as product, into formulae of binary multipli- 
 cation, of the kind admitted in the present plan. In fact, the 
 first factor qi occurs as a multiplicand in 7i-\ such formulae, 
 namely (see 365) in the following. 
 
 Ml = ^1' ^^^'i = S\, S2q\ = ^1, . . . z^qi = q, 
 
 which are all true by the associative principle, although only the 
 first of them was used, in the construction described in 367. Thus 
 the point Qi is a common corner of « - 1 triangles, each repre- 
 senting a binary multiplication, although only one of these tri- 
 angles was constructive, and the rest of them are all associative 
 (in the sense of the present article). The angle Z qi is therefore, 
 in the completed figure, represented by w - 1 different but equal 
 angles at the point Qi ; and the comparison of these different re- 
 presentations, for the common value of the angle of the factor ji, 
 conducts to ?i - 2 angular equations, namely, 
 
LECTURE VI. 359 
 
 R1Q1Q8 = S1Q1K2 = T1Q1S3 = . . = QQ1Z2. 
 
 In like manner (see 369), q^ was used twice only, in the con- 
 struction, namely, as a factor in 5-3^1 and in q^q^ ; but by as- 
 sociation it is introduced also as a multiplicand into w-3 other 
 binary products, namely, into the following : 
 
 r^q^ = S2, S3q2 = ^s? • ■ t/3q2 = ^2- 
 
 Thus the point Q2 (like Qi) is, when all are taken into account, a 
 common corner of n-l triangles, and gives, on the whole, n-2 
 equations between angles. More generally, the m"*^ given factor, 
 qm, enters, on the whole, m~ I times as a multiplier, into binary 
 products, as follows, 
 
 qm • qm-li qm • qm-1 qm-2) &c. ; 
 
 and n-m times as a multiplicand into such products, namely, 
 into the following : 
 
 qm+\ • q-m^ qm + 2 qm+\ • qnn oCC. ^ 
 
 while it nowhere enters as a product : it enters, therefore, on the 
 whole, as before, into n-l formulse of binary multiplication, so 
 that Qm is still a common corner of n-l triangles, and supplies 
 still n-2 equations between angles. 
 
 373. It is true that we have here been considering only the 
 n given factors. But if, instead of a given factor, qm, we consider 
 a partial product, such as 
 
 qm qm-l qm-2 qm-3=tm.3, 
 
 we find that altiiough this quaternion enters still only n-m 
 times into a binary product as a multiplicand, namely into the 
 following, 
 
 qm+l ' tm-3, qm+2 qm + \ • fm-3> C*C., 
 
 and enters only m- 4 times as a multiplier, namely, into the bi- 
 nary products, 
 
 '7)1-3 • qm-ii tm-3 • qm-i qm-5i OZC., 
 
 and so only enters n- 4 times as a/actor, into binary products, 
 yet it enters three times, as a product, into formulae of binary 
 multiplication ; for by the associative principle, we may place the 
 point or other mark of multiplication, in the expression for t^.s, 
 
360 ON QUATERNIONS. 
 
 after q,ni or after ^„,.i, or after qm-i' And generally if we consi- 
 der the product, 
 
 we find with the greatest ease that this quaternion enters only 
 71 -m times as a multiplicand, and only m-l-l times as a mul- 
 tiplier, into the composition of binary products ; but that it occurs 
 also / times, under the form of such a product. It occurs then, 
 still, n- 1 times in all, and gives still n-2 angular equations. 
 
 374. It is then proved (as was asserted in 372), that each 
 point of the whole complex figure is, in general, a common corner 
 ofn-\ different triangles; and, therefore, that it conducts to 
 n-2 equations between angles, by comparisons made as above. 
 And the number of all the points has been seen (in 365) to be 
 =^^n{n+\)', the entire number of the angular equations, thus 
 obtained, is therefore expressed by the formula, 
 
 iw {n + 1) (n-2). 
 
 But the number of such equations which are true by construction, 
 has been found to be (see 369), 
 
 = n (n - 2) ; 
 
 subtracting therefore this expression from the one preceding it, 
 we find that the number of the angular equations which are true, 
 as depending on the n{n-2) equations of construction, is 
 
 ^n{n-\) (n-2). 
 
 And this is the general property of polygons of multiplication, 
 which it was lately proposed (near the beginning of 372) to in- 
 vestigate. We see that it includes the two cases lately considered, 
 oi dependencies of equations derived from the associative princi- 
 ple, on equations which were true by construction ; namely, the 
 case (302) of three factors, w = 3, where three equations were de- 
 pendent on three others ; and the case (364) oi four factors, 
 where twelve equations were dependent upon eight. For the 
 hexagon of multiplication, where there are five factors, and 
 ^5 (5 + 1) or fifteen points altogether, there are fifteen (=5.3) 
 equations true by construction, and 30 (= ^ . 5 . 4 . 3) equations 
 dependent on them. And in general we see, by the present arti- 
 
LECTURE VI. 361 
 
 cle, that, in any such polygon, the number of the equations which 
 are derived by the associative principle, is to the number of those 
 other equations from which they are derived, as w - 1 to 2. The 
 equations of association are therefore more numerous than the 
 equations of construction, whenever the number of w of factors 
 exceeds three; or when the number n + \ oi corners of the poly- 
 gon of multiplication is greater ihdin four ; a result which agrees 
 with what was stated by anticipation, in art. 372. 
 
 375. Since each of the \n{n-\-\) points of the complex figure 
 has been seen to be in general a common corner of w - 1 different 
 triangles, constructive or associative, we have only to multiply 
 these two numbers together, and then divide by three, in order 
 to find the number of all those triangles of multiplication ; 
 namely, 
 
 ■^(w+ \) n{n- 1). 
 
 There is however another process, distinct from the foregoing, 
 by which the same result may be obtained, and which it may be 
 useful briefly to consider. Let us then remember that (as in 373) 
 each product, partial or total, of /+ 1 successive factors, may (by 
 the associative principle) be presented under the form of a binary 
 product, in / different ways, according to the various positions 
 which may be assigned to the point, or other mark of multiplica- 
 tion. Hence, while each of the n-l binary products ri, . . r„.i 
 gives immediately one triangle of multiplication, each of the 
 n -2 ternary products, si, . . Sn-2 gives two such triangles, and 
 so on. We are then to take the sum of the series, 
 
 1 {n-l) + 2{n-2) + 3{n-3)+ , . +l(n-l), 
 
 if we wish to find how many triangles are given by all the pro- 
 ducts ri, &c., Si, &c., which contain /+ 1 or fewer factors. But 
 this sum is, by well known principles, equal to the following : 
 
 (w + 1) (1 + 2 + 3+ .. +0- {1 -2+2.3 + 3.4 + . . + ^/+l)} 
 
 = i(w+l)(/+l) /-i(/+2) {1+1)1 
 
 = |(3n-2/-l) (/+!)/. 
 
 And if we now make l=n-l, we find, for the total number of 
 the triangles, involved in the whole complex figure, the same 
 expression as above, namely, 
 
362 ON QUATERNIONS. 
 
 ^(n+ I) n{n- 1). 
 
 For example, when there were only two given factors (as in 264), 
 there was only one triangle (the qrs of fig. 50) ; when there 
 were th'ee given factors (as in 302), there were ybwr triangles 
 (the ABE, BCF, ECD, and afd of fig. 65) ; when there were Jbu?' 
 given factors (as in 361), there were ten triangles (those enume- 
 rated in 363) : and when we consider the case oijive given fac- 
 tors, and construct a hexagon of multiplication (see 370), there 
 are then found to be twenty triangles, answering to so many 
 auxiliary processes of formation of binary products. Accordingly 
 in this last case, the figure has been seen (374) to contain j^i^eew 
 points, of which each is a common corner oifour triangles of 
 multiplication. 
 
 376. Instead of seeking how many triangles may thus 
 be formed, from a quadrilateral, pentagon, &c., as representing 
 multiplication of quaternions, we may inquire how many auxi- 
 liary quadrilaterals may be deduced from, or are to be con- 
 sidered as involved in, the complete construction (371, &c.) of a 
 pentagon, hexagon, or other polygon of multiplication. For this 
 purpose we are to determine how many products of ternary (in- 
 stead of binary) forms, can be composed from a given set of fac- 
 tors gi, . . . ^-n, without transposition, repetition, or hiatus. Or 
 we may seek, in how many ways the various partial and total 
 products, 5i, &c., ^1, &c., and q = qn . . . qi, can be decomposed, 
 each into three factors: for there is evidently no use in seek- 
 ing so to decompose any one of the n given factors, q\, &c., or 
 any of their n-\ binary products, vi, &c. It is clear also that 
 each of the w - 2 ternary products, s\, &c., gives only one decom- 
 position, of the kind now sought ; but that each of the ?z - 3 qua- 
 ternary products, ^1, &c., gives 1 + 2 = 3 such decompositions, 
 because we may write, by art. 365, and by the associative prin- 
 ciple, 
 
 t\ = 3-4^3 . q-zqi = qi • q^q^qi ; 
 
 where q^qi may be treated as a binary product in only one way, 
 but q^q^qi in two ways. In like manner a quinary product admits 
 of ternary decompositions in 1 + 2 + 3 = 6 ways ; and generally the 
 
LECTURE VI. 363 
 
 number ofways^ in which a product of / + 2 factors may be put 
 under the form of a ternary product, is 
 
 1 + 2 + 3+ . .+Z=^/(/+l): 
 
 while the number of products of this order or dimension is 
 = n-l-\. If then we wish to know how many ternary forms 
 can be obtained, by suitably placing the points of multiplication, 
 from all the products 5i, &c., ^i, &c., which involve not fewer 
 than / + 2 given and successive factors, we are to calculate the 
 sum, 
 
 1 (»^-2) + 3(n-3) + 6(/^-4) + . .+!/(/+ I) {n-l-\) 
 
 = {n+\) {1 + 3 + 6+ . . + i/(/+l)} 
 
 -{1.3 + 3.4 + 6.5 + . .+i/(/+l) (Z+2)} 
 
 = i(«+l) /(/+1) (/+2)-|/(/+l) (/+2) (/+3) 
 
 = -^-z{An-U~S){l+2){l+\)l. 
 
 And finally, by making l = n-% we find for the whole number 
 of such ternary products, or of the quadrilaterals by which they 
 are constructed on the sphere, the expression, 
 
 i, (w+l)w(r*-l) (w-2). 
 
 Thus, the pentagon of multiplication (fig. 79), for which the 
 number n of given factors v&four^ is connected with ^ue auxiliary 
 quadrilaterals, namely, 
 
 ABCI, BCDK, FCDE, AGDE, ABHE, 
 
 answering (in the notation of art. 361) to the five products of 
 ternary form, 
 
 s .r .q, t .s .r, t .s .rq, t . sr .q, ts .r. q ; 
 
 and the complete construction of the hexagon of multiplication, 
 for which w = 5, must involve the construction oi fifteen such qua- 
 drilaterals. 
 
 377. If we seek on the same plan, how many auxiliary pen- 
 tagons are connected with the hexagon, heptagon, &c., or how 
 many products of quaternary form can be composed out of n 
 given factors (without transposition, &c.), we find that the num- 
 ber of quaternary decompositions of each product of /+3 fac- 
 tors is 
 
364 ON QUATERNIONS. 
 
 ^/(/ + l)(/+2); 
 
 and that the number of such products is 
 
 {n+l)-{l+d). 
 
 Multiplying these two numbers, and summing with respect to Ij 
 we obtain the expression, 
 
 which when we make l=n-3, reduces itself to 
 
 Such then is the required number of auxiliary pentagons in ge- 
 neral ; in the construction of the hexagon, there would therefore 
 be involved six such pentagons; and twenty-one in the construc- 
 tion of the heptagon. More generally still, the same analysis 
 shews that in the complete construction oj" any spherical foly- 
 GON of multiplication (370), with p {=n-\- 1) corners {or sides) 
 and with ^p (p- 3) inserted points (371), to represent partial 
 products, is involved the construction of a number q/AUXiLiARY 
 SPHERICAL POLYGONS of inferior degree, which number is ex- 
 pressed by the formula, 
 
 p(p-\) (p-2) . . . (p-p'+l) 
 1.2.3... p ' 
 
 ifp'be the number of sides of the auxiliary and inferior polygon. 
 378. You will not have failed to observe that I am far from 
 admitting, in the construction of these ijiserted or auxiliary poly- 
 gons, all possible arcs of great circles which could be drawn, 
 connecting two points taken arbitrarily in the figure. If that 
 were done, the results would of course be much more nuinerous: 
 but you see that 1 retain only those connecting arcs which are 
 required, or are useful, for constructing some of the products, 
 partial or total, of the given quaternion factors. It was thus 
 that in fig. %5 (as was remarked in art. 375), oxAy four auxiliary 
 triangles were employed, because we had no occasion for the 
 arcs AC, BD, EF ; which again arose from the circumstance that 
 we were not seeking to connect q with s, nor r with srq, nor rq 
 with sr, by any process of binary multiplication. It would cer- 
 
LECTURE vi. 365 
 
 tainly have been unnecessary to have had recourse to any such ana- 
 lysis as the foregoing, if our object had been to prove, what every 
 body knows, that a set of p' things can be taken out of p others, 
 in a number of ways expressed by the formula recently written. 
 But the question which we had to investigate was an entirely 
 different, and (it will perhaps be felt) a much less easy one. Even 
 for so simple a case as that of the hexagon and its quadrilaterals, 
 the distinction is sufficiently striking. Of course it is very 
 well known, from elementary principles of combination, that a 
 set of four things can be taken in fifteen ways out of a given set 
 of six things ; and in so many as 1365 ways out of a set of fifteen 
 things, the arrangement of the things among themselves being 
 supposed to be unimportant. It would, therefore, have been use- 
 less to offer any proof, that after constructing a spherical hexagon 
 of multiplication, to represent five given quaternion factors and 
 their total product, and then inserting also nine other representa- 
 tive points upon the spheric surface, for the various partial pro- 
 ducts, fifteen sets of four points could be chosen out of the six 
 corners of the hexagon, and 1365 such sets out of the whole sys- 
 tem of the fifteen points of the figure, arrangement being still 
 abstracted from. But it was not obvious that vahenfour points 
 were to be selected out of i\\e?ie fifteen, so as to be corners of some 
 auxiliary quadrilateral oi T[i\x\\\^\\cSii\on, connected with the re- 
 presentation (on the principles and plan already explained) of 
 some ternary multiplication of the five given factors or of their 
 products, the rejection of all useless quadrilaterals would reduce 
 the larger number 1365 to the smaller number fifteen, which last 
 was obtained at the end of art. 376, and may be derived also from 
 the more comprehensive formula of art. 377- Still less is it evi- 
 dent, without some such investigation as that lately instituted, 
 that so great a reduction as is expressed by the same formula takes 
 place, by rejection of useless combinations, when we seek the 
 number of all the auxiliary and p'-sided polygons of multiplica- 
 tion, which are connected with and involved in the construction 
 of a polygon of multiplication of superior degree, having p sides 
 or corners, but having also '^p (p - 3) inserted points, which 
 (under certain restrictions as to the mode oi combining them) con- 
 cur with the p points themselves, in the formation of the auxiliary 
 
366 ON QUATERNIONS. 
 
 and inferior polygons, according to the laws of the multiplication 
 of quaternions. Perhaps this may be as fitting an occasion as 
 any other to remark, that the process of building up a complete 
 polygon of multiplication, of any given degree, with all its auxi- 
 liary points, may be in many ways varied from that stated in art. 
 367, and exemplified previously in 361, without disturbing any 
 of the results above obtained, respecting the number of the equa- 
 tions of condition necessary for the correct construction of the 
 figure ; or the number of the equations which follow from these 
 by the associative principle, or the number of inferior and auxi- 
 liary polygons, &c. For instance, in constructing the figure 79, 
 for the pentagon, we might have begun by assuming as known 
 the six poitits, a, b, f, and c, d, h, in connexion with the two 
 pairs of given factors, q, r, and 5, t; and might have thence con- 
 structed the four other points c, i, k, and e; but we should 5^2'// 
 have had eight constructive equations between angles, and have 
 still been conducted to twelve associative equations, as following 
 from them. 
 
 379. The foregoing investigations (361 to 377) respecting 
 polygons of multiplication have been conducted quite indepen- 
 dently of the doctrine of spherical conies, although a passing 
 allusion was made to that doctrine (in art. 366), and in particu- 
 lar to the focal character of the two auxiliary points e and f, in 
 fig. 65. But if we now admit that focal character of those two 
 points, namely, that they are (as was proved in art. 302) the two 
 foci of a conic inscribed in the quadrilateral of multiplication, 
 namely in abcd of fig. Q5, and if we agree to denote this focal 
 relation of two points to four others, by writing, for conciseness, 
 any one of the following formulae, 
 
 EF (. .) abcd, 
 or 
 
 FE (. .) ABCD, or EF (. .) BCDA, Or EF (. .) DCBA; 
 
 but not the formula, 
 
 EF (. .) ACBD, 
 
 since this would come to substituting diagonals for sides, and 
 would require a change in the inscribed conic; we shall then be 
 able to derive and to enunciate briefly a series of theorems, re- 
 
LECTURE VI. 36T 
 
 specting inscriptions of systems of spherical conics in 
 
 CERTAIN SYSTEMS OF SPHERICAL QUADRILATERALS, and the Con- 
 sequent ENCHAINMENTS OFCERTAIN SPHERICAL POLYGONSamong 
 
 themselves; of which theorems the suggestion is due (so far as I 
 know) to the Calculus of Quaternions. For since every case 
 of a ternary product may be represented or constructed, on the 
 plan of fig. Q5, by a conic thus inscribed in a quadrilateral, we 
 see by recent articles that every j9-sided polygon of multiplica- 
 tion is connected with a system of such conics, whose number is 
 expressed by the formula 
 
 while their ^cz all belong to the system of those points, in num- 
 ber 
 
 i/>(p-3), 
 
 which represent the partial products of those /> - 1 quaternion 
 factors, the representative points of which factors themselves, 
 and of their total product, are the successive corners of the poly- 
 gon in question ; and out of this system oi focal points^ another 
 polygon or polygons may generally be conceived to be formed ; 
 which will be connected with the^rmer polygon, and with each 
 other, by a species of focal enchainment. (It will be remem- 
 bered that the insertion of the%e focal points is not an arbitrary 
 process, but is subject to certain laws derived from the nature of 
 quaternion multiplication ; in fact there exist, by art. 369, (p- 1) 
 (p -3) equations of construction, between the angles of the com- 
 plex figure; and from these, by art. 374, there follow ^(p-l) 
 (p - 2) (p - 3) other equations between angles, in virtue of the 
 associative principle.) 
 
 380. If, for instance, we adopt the notation of art. 367, and 
 take the case of the hexagon, 
 
 Q1Q2Q3Q4Q5QJ 
 
 we may conceive the six points 
 
 RiR2R3R4TiT2, 
 
 which represent the four binary and the two quaternary products, 
 
368 
 
 ON QUATERNIONS. 
 
 to be, in their order, the corners of a second hexagon ; while the 
 three points 
 
 S1S2S3, 
 
 whichjepresent the three ternary products, may be considered as 
 the corners of a triangle. And then, for this system of two 
 HEXAGONS AND A TRIANGLE upon tt Sphere (not now, as in 305, 
 one hexagon and two triangles), we shall have an example of the 
 lately mentioned enchainment of spherical polygons; which en- 
 chainment is here performed through a system of fifteen 
 spherical cqnics, inscribed in certain quadrilaterals of the 
 figure, and having their foci ranged at the corners of the auxi- 
 liary hexagon and triangle, as is expressed in the following Table. 
 
 Table of Focal Relations. 
 
 R1R2 ( 
 R2R3 ( 
 RsBi ( 
 
 RiTi ( 
 
 T1T2 ( 
 
 TaRi ( 
 
 R1S2 ( 
 R2S3 ( 
 R3S1 ( 
 R4S3 ( 
 T1S3 ( 
 T2S1 ( 
 
 S1S2 ( 
 S2S3 ( 
 S3S1 ( 
 
 ) QiQaQsSi 
 ) Q3Q3Q1S2 
 ) Q3Q1Q5S3 
 ) Q4Q5Q Si 
 ) QoQ Q1S2 
 ) Q Q1Q2S3 . 
 
 ) Q1Q2R3T1 1 
 ) Q2Q3R4T2 
 ) Q3Q4T1R1 
 ) Q4QaT2R2 
 ) QoQ RiRs 
 ) Q Q1R2R4 J 
 
 ) Q1R2Q4T1 -j 
 ) Q2R3Q0T2 \ 
 ) Q3R4Q Ri J 
 
 (I.) 
 
 (II.) 
 
 (III.) 
 
 And 1 think that any attempt to sketch, in its general state, the 
 complex figure here referred to, with its fifteen conies of inscrip- 
 tion, and its numerous connecting arcs, could only impair theclear- 
 ness and symmetry of the foregoing symbolical statement. 
 
 381. There is, however, one particular or rather //w?7mp'C«5e, 
 of the general construction described in the last article, which it 
 
LECTURE VI. 
 
 369 
 
 may be interesting here to consider, and which admits of being 
 illustrated by a diagram suiBciently simple. 
 
 Round any point s of the surface of the unit-sphere, as a pole, 
 with any arcual radius sq, conceive a small circle to be described. 
 Let this small circle be cut into six successive and equal portions, 
 in the order of left-handed rotation, by five other and successive 
 arcual radii, 
 
 SQi, SQ2, SQ3, SQi, SQ5, 
 
 making with sQ and with each other successive angles of sixty 
 degrees, at their common point s, as in the annexed figure 80. 
 Let six connecting arcs 
 of great circles be drawn, ^^" ®^' 
 
 QQi, QiQ2> Q2Q35 
 QaQij Q4Q55 QoQ; 
 
 which will thus become 
 the sides of (what might 
 perhaps be called) a re- Q, 
 gular spherical hexagon : \ 
 or at least of one which 
 will be at once equi- 
 lateral and equiangular. 
 Draw also the six suc- 
 cessive diagonals, 
 
 QQ25 Q1Q3, Q2Qi> Q3Q5J Q4Q5 Q5Q1 ; 
 
 and name, as follows, the six successive intersections of these 
 diagonals : 
 
 Ri the intersection 
 
 of Q Q2 and Q1Q3 ; 
 
 R2 5) J, 
 
 Q1Q3 and Q2Q4 ; 
 
 R3 ,, ,, 
 
 Q2Q4 and Q3Q5 ; 
 
 R4 5, 5, 
 
 Q3Q5 and QiQ ; 
 
 I'l » 5» 
 
 Q4Q and Q5Q1 ; 
 
 T2 ,, ,, 
 
 Q5Q1 and Q Q3 . 
 
 The figure being thus constructed, conceive next that some five 
 successive quaternion factors, of the versor kind, ^i, qo, q^, q^^ q^, 
 are represented by five spherical angles, at the five successive 
 
 2 B 
 
370 ON QUATERNIONS. 
 
 points Qi, Q2, Q3, Q4, Qo, of the hexagon ; each of these five angles 
 being equal in magnitude to the spherical angle RiQiQo, between 
 a diagonal and a conterminous side of the hexagon. The four 
 successive binary products of the five factors, namely, q^qi^ qz^-i-) 
 qiq^i q^qi, will then be represented by angles at the four points 
 Ri, R2, R35 Ri, of which the common magnitude is that of the angle 
 Q3R1Q25 or the supplement of the spherical angle Q2R1Q1. The con- 
 struction, so far, being seen to be er\i\ve\y rigorous, and indepen- 
 dent of everything like approximation, let us conceive next that 
 the arcual radius sq becomes a swza/Z arc, although remaining still 
 an arc of a great circle ; so that the spherical hexagon becomes, 
 in consequence, 2i nearly -plane one, and approaches to coincidence 
 in shape with the regular hexagon of Euclid. The angle of each 
 of the five quaternion y«c^or,s will then differ very little from thirty 
 degrees ; and the angle of each binary product will be nearly equal 
 to sixty degrees. The three ternary products, q^q^qx, qiq^q^, q^q^qz^ 
 which are in general (see 380) represented by three distinct points, 
 Si, So, S3, come now to have their three representative points very 
 nearly coincident with each other, and with the centre s of the 
 figure ; the angle of each becoming at the same time nearly right. 
 The two quaternary products, q^q^qiqi and q^qiq^q^, will be very 
 nearly represented by angles of 120° each, at the two remaining 
 corners, Ti and T2, of the interior hexagon, namely RiRoRsRiTiTj. 
 And finally the one quinary or total product of the five given fac- 
 tors, namely q^qiqzqzqu will be nearly represented by an angle of 
 150°, at the one remaining corner q, of the outer or original hexa- 
 gon, described in the present article. All this follows easily from 
 the most elementary properties of a plane and regular hexagon, 
 considered here as the limit to which a certain spherical hexagon 
 approaches, and combined with one of our general constructions 
 (264, &c.) for the multiplication of any two versors. 
 
 382. We may then, at the limit, where the ^ewer«/ and sphe- 
 rical hexagon of multiplication becomes the plane and regular 
 hexagon of elementary geometry, conceive that hexagon, with 
 its inserted or focal points, to be constructed as in the recent 
 figure 80 ; the various letters q, r, s, t retaining, at this limit, 
 the general significations of art. 380, except that the one letter s 
 (at the centre of the figure) now takes the place of each of the 
 
LECTURE VI. 371 
 
 three symbols, which were before written as Si, So, S3. We have 
 then only this last change to make now, or to conceive as made, 
 in the recent Table of Focal Relations ; that is to say, so far as 
 concerns the twelve first of those relations, we are simply to su2J~ 
 press the indices^ which were (in art. 380) suffixed to the letter s : 
 and as regards the three last of the same system of fifteen focal 
 relations, we are to remember that an ellipse becomes a circle, 
 when its two fijci coalesce. Thus, at the limit here considered, 
 the three conies of the third system degenerate into circles ; or ra- 
 ther (as it is very easy to see) they coalesce into one single circle^ 
 concentric with the original circle, and inscribed in the interior 
 hexagon, as indicated in figure 80; wherein also two conies of 
 each of the two former systems are pictured. And an inspection 
 of the same recent figure, combined with some simple geometri- 
 cal considerations, shews easily that each of the six ellipses of 
 the^r*^ system, as, for example, the ellipse inscribed in the equi- 
 lateral quadrilateral Q1Q2Q3S, or the one which is inscribed in the 
 other and similar quadrilateral Q4Q5QS, has its major axis equal 
 in length to a side of the original hexagon ; while each of the six 
 ellipses of the second system, such as the one inscribed in the rec- 
 tangle Q3Q4T1K1, or that in the other rectangle QQ1R2R4, has its 
 minor axis equal to a side, suppose Q3Q4, of the same original or 
 outer hexagon. And finally, the one interior circle, to which the 
 three ellipses of the third system reduce themselves, and which 
 is inscribed in the interior hexagon, has its diameter equal in 
 length to a side of the same outer hexagon ; to which side we 
 have just seen that a major or a minor axis, of each of the twelve 
 ellipses of the two former systems, is equal. The diagram may 
 also suggest, what a very simple reasoning proves to be true, 
 that the eight points of contact^ of the two ellipses of the first 
 system in it depicted, with the eight sides of the two equilateral 
 quadrilaterals in which they are inscribed, are ranged on the two 
 diagonals, R2R1 and RiTi, of the interior hexagon ; that is, upon 
 the minor axes of the two ellipses of the second system in the 
 figure: or on the parameters of the two fonner ellipses. 
 
 383. All this being sufficiently obvious for the case of the 
 plane and regular hexagon, it may be worth while to inquire 
 briefly in what manner the results are modified, when the arcual 
 
 2 B 2 
 
372 ON QUATERNIONS. 
 
 radius sq is treated as only moderately (but not as infinitely) 
 small, so that the sphericity of the figure is sensible. Conceiv- 
 ing, therefore, that figure 80 represents an equilateral and equian- 
 gular but spherical hexagon, constructed according to the direc= 
 tions of art. 381 ; and supposing that the five given versors, q^ to 
 g'g, are represented, as in that article, by the five spherical angles, 
 
 /.q\ = Q3Q1Q2, Z §'0 = Q4Q2Q3!. • • • Z 9'5 = Q1Q5Q ; 
 
 the general construction for a spherical triangle of multiplication 
 shews still that the four binary products, q^qi, &c., are represented 
 by these four other spherical angles in the figure : 
 
 /• q^qi == Q3R1Q2 ; z qzq-z = Q4R2Q3 ; 
 
 Z ^4^3 = Q5R3Q4 ; Z q&qi = QRiQs- 
 
 But the three ternary products, qzq^qx^ &c., will no longer be 
 (rigorously) represented by right angles at the centre s of the 
 figure ; nor will the two quaternary products be represented by 
 angles of 120° at the points Tj, To ; nor the quinary product by an 
 angle of 150° at the sixth corner q of the equilateral and equian- 
 gular hexagon. We may then ask, for the ternary products, in 
 what directions do their three representative points, Si, S2, S3, de- 
 viate from the centre s ? And if the two quaternary products 
 be now conceived to have their representative angles at some two 
 new points, t'i, and t'j, since tJ and T2 are (by art. 381) already 
 appropriated in the figure to denote certain intersections of dia- 
 gonals, we may inquire what are the directions of the deviations, 
 Tit'i and TjT'j ? Again, if the quinary product be supposed to be 
 represented (accurately) by a spherical angle at some other new 
 point q', while q shall still denote, as in the figure, a corner of 
 the equilateral hexagon, we may demand what is the direction of 
 the deviation or displacement, qq'? And with respect to the 
 magnitudes of the various representative angles, we may inquire 
 whether I qi is now less or greater than 30°? is Z. q^qi less or 
 greater than 60°? is Z. qsqoqi acute or obtuse ? does Z qiq^q^qi ex- 
 ceed or fall short of 120° ? And finally, for the quinary product, 
 is Z q^qiq^q^qi less or greater than its limiting value of 150°, when 
 account is taken of sphericity ? 
 
 384. By the construction which is to be conceived as being 
 
LECTURE VI. 373 
 
 employed, for determining the new spherical angles at Si, So, S3, 
 t'i, T'2 q', we have the angular equations : 
 
 R1Q3S1 = Lqi^ QaQsRi ; SiQiRs = Z ^i = RsQiQg ; 
 
 because, by the associative principle, the ternary product, q%qiq\t 
 may be put under either of the two forms, 5-3 • q^q^^ q^q^ • q^. It 
 is clear, therefore, that if we denote by Mo the point where the 
 arcual radius, SQ2, bisects perpendicularly the diagonal Q1Q3 of the 
 outer, or the side R1R2 of the inner hexagon, the sought point 
 Si will simply be the reflexion of Q3 with respect to M2; in such a 
 manner that the following arcual equation will subsist: 
 
 '-> Q2M2 = ^ M2S1. 
 
 The direction of the deviation ssi must, therefore, be either to- 
 wards or from the corner Qo of the outer hexagon, according as it 
 shall be found that the arc SM2 is greater or less than half oi the 
 arcual radius SQ3. To decide this question, let us observe, that in 
 virtue of the tendency of the radial arcs to meet again upon the 
 sphere, in the point diametrically opposite to the point s from 
 which they diverge, each side, such as Q1Q3, of the hexagon, is 
 shorter than the arcual radius SQi. Comparing, therefore, the two 
 right-angled triangles, Q2M2Q1 and Q1M2S, which have a common al- 
 titude QiMs, we see that the hypoteiiuse of the former triangle is 
 shorter than the hypotenuse of the latter, and consequently 
 that the base Q2M2 of the one triangle must also be less than the 
 base M2S of the other. We have then the inequality, 
 
 r\ Q2M2 < r\ M2S ; 
 
 and by combining this inequality with the equation written above, 
 we can at once infer this other inequality, 
 
 r\ M2S1 < rs M2S. 
 
 We know then definitely the direction of the deviatio?i ssi; and 
 are entitled to assert that this deviation is directed ^om the centre 
 s, towards the corner Qo, and not in the opposite direction. And 
 it is evident that reasonings exactly similar would prove, that the 
 two other deviations SS2, SS3, of the two other representative 
 points of ternary products from the centre, are directed, respec- 
 
374 ON QUATERNIONS. 
 
 tively, towards the two other and successive corners, Q3, Q4, of 
 the same original hexagon ; while the lengths of these three de- 
 viations are at the same time evidently equal. When the arcual 
 radius is assumed as 10°, I find that the common value of these 
 three deviations amounts to about 4' 36" ; and that when the size 
 of the figure is diminished, the deviation diminishes nearly in 
 the same ratio as the cube of the radius. It is less than three- 
 tenths of a second, when the arcual radius is a degree. 
 
 385. As regards the angles of the factors, and of their binary 
 and ternary products, we may see first that if Pi denote the mid- 
 dle point of the^side Q1Q2, the two right-angled triangles Q1Q2M2 
 and P1Q2S have a common base angle at Q2, but that the hypote- 
 nuse of the former is less than the hypotenuse of the latter. The 
 area of the former triangle is therefore also less than the area of 
 the latter ; so therefore likewise is the spherical excess; and so 
 must be the vertical angle. That is to say, the angle M2Q1Q2 is 
 less than the angle Q2SP1 ; or in symbols, 
 
 Lq^< 30°. 
 
 We have then answered another of the questions proposed in art. 
 383 ; for we have come to conclude that the angle of each of the 
 ^\VQ^K\ factors, in the construction here considered, is less than 
 30°. It is, however, only a very little less than this limit-angle, 
 if the size of the hexagon be small (the sphere being supposed to 
 be fixed). Even when the arcual radius is assumed so great as 
 10°, 1 find that this representative angle of 5-1 falls short of 30° 
 by only about ten seconds and a half; and this defect is reduced 
 to about the thousandth part of a second, when the radius is taken 
 as one degree ; for it can be proved to vary nearly as the fourth 
 power of the radius, so long as the figure is moderately small.* 
 
 386. The angle of the binary product q^Qi, being equal to 
 Q3R1Q2, is the supplement of the double of the angle PiRjQi ; but 
 this last angle is equal to its vertically opposite sHiivis, and there- 
 fore exceeds the complement of the angle MoSRi, in the right-an- 
 gled triangle so denoted, by the spherical excess of that triangle. 
 But the angle M2SR1 is exactly equal to thirty degrees ; there- 
 fore, PiRiQi is greater than 60° ; its double is, therefore, greater 
 than 120°, and the supplement of its double is less than sixty de- 
 
LECTURE VI, 375 
 
 grees. We arrive, then, for the angle of the binary product, at 
 
 the inequality, 
 
 Z q.qx < 60° ; 
 
 which contains the answer to another of the questions proposed 
 in art. 383. It must be observed that the defect, thus proved to 
 exist, of the angle of the binary product from sixty degrees, is 
 much more considerable than the defect, investigated in the im- 
 mediately preceding article (385), of the angle of a factor from 
 30°. For the defect of the angle of the binary product q.q^ is re- 
 presented by the doubled area of M2SR1, or by the ^oto^ area of 
 the triangle SR1R2 ; whereas the defect of the angle of the factor 
 q^ was seen to be constructed by the difference of the two small 
 and nearly equal areas, of the triangles QoM^Qi and sPiQ,. When 
 SQi is taken as 10°, the defect of the angle of the binary product 
 from 60° amounts to so much as about 15' 20"; and even when 
 the arcual radius in the construction is assumed so small as 1°, 
 this defect is still not less than about nine seconds ; varying 
 nearly as the square of this radius, so long as the dimensions of 
 the figure are small. 
 
 387. The angle of the ternary product, ^-s^a^i, being equal to 
 the supplement of Q3S1R1, is in amount the supplement also of 
 R1Q2Q3 ; or of Q1Q2Q4 ; or of P1Q2M3, if M3 be the bisecting point of 
 the diagonal Q2Q4, as M2 was of Q1Q3. But in the quadrilateral 
 P1Q2M3S, all the angles except that at Qo are right angles ; there- 
 fore this angle P1Q2M3 exceeds a right angle by an amount repre- 
 sented by the area of this quadrilateral ; and consequently its 
 supplement falls shot^t of a right angle by the same amount. The 
 angle of the ternary product is therefore acute, 
 
 Lqzq-iq^ < 90° ; 
 
 and thus another of the questions of art. 383 is answered. This 
 defect from 90° varies nearly as the square of the arcual radius ; 
 when that radius is 10°, the defect is about half a second more 
 than 45' 34" ; and it is reduced to about twenty-seven seconds, 
 when the radius is assumed to be a degree. 
 
 388. Proceeding to consider the quaternary products, q^qzq^qi, 
 qbq^qzq^i we may put the latter under the form q^qi . q^q^, and are 
 then led to assign the following conditions for the construction 
 
376 ON QUATERNIONS. 
 
 of its representative point T'2 (see art. 383), and for its representa- 
 tive angle at that point : 
 
 T^UzRi = Z q^Qi = Q2R2Q1 ; 
 
 RgRiT'a = Z qi,qi = Q4R4Q3 ; 
 
 Z q^q^q^qi = ir- Rit'jRs. 
 
 TJie point T'2 is therefore situated somewhere on the arc sTj 
 itself, or else on that arc prolonged. To decide which of these 
 two conclusions is to be adopted, we need only observe that each 
 angle of the equilateral and spherical triangle TsRoR^ must exceed 
 60°, while the angle of the binary product q^q^ has been seen to 
 fall short of 60° ; thus 
 
 t'2R2R4 < T2R2R4J and ST'2 < ST2 ; 
 
 the displacement TsT'j of the representative point of a quaternary 
 product, is therefore directed towards s : and another question 
 of art. 383 is answered. Another problem of the same article is 
 solved, by observing that, in consequence of what has just been 
 shewn, the angle R4t'2R2 is greater than R4T2R1, which has been 
 seen to be greater than 60°; therefore, by still stronger reason, 
 the angle R4t'2R2 exceeds 60°, and its supplement falls short of 
 120° ; so that we have the inequality, 
 
 When the radius is 1 0°, this defect of the angle of a quaternary 
 product from 120° amounts to about 1° 15' 50"; it varies nearly 
 as the square of the radius, and reduces itself to about 45" 
 when the radius becomes a degree. On the other hand the dis- 
 placement ToT'2 or TjT'i of the representative point varies nearly 
 as the cube of the radius; it is found to be about 10' 32", or only 
 about six-tenths of a second, according as- we assume 10° or 1°, 
 for the value of the arcual radius. 
 
 389. As regards the quinary product, and its representation 
 at the new point q' (art, 383), since the associative principle 
 allows us to regard this product as obtained in two diiFerent ways 
 through the multiplication of a binary product into or by a ter- 
 nary, because it gives 
 
 q-oqiq^q^qi = ^5^71 • q^q^qi = q^qiq^ • q-zqi, 
 
LECTURE VI. 377 
 
 we may employ either or both of the two following" systems of 
 equations for the construction of the point and angle sought : 
 
 TSiR^q' = L q^qi = QR4Q5 ; 
 
 <{ q'siR4 = L qzq%qx = tt - Q3S1R1 ; 
 
 \j. q^qiqzq^qi = tt - Riq'si ; 
 
 and 
 
 Tq'RiSs = / ^2^1 = QsRiQa ; 
 
 ^ R1S3Q' = Z q^qiqz = TT - Q6S3R3 ; 
 
 |_^ qeqiqiq^qi = tt - Ssq'ri. 
 
 But the angles of the binary products are equal to each other in 
 amount, and so are the angles of the ternary products, in the sys- 
 tem of factors at present under consideration. Hence the angles 
 S1R4Q' and q'RiSs are equally large ; and so are q'SiR4 and R1S3Q'. 
 But also the deviations ssi and SS3 are equal in amount ; and so 
 are the angles which they subtend, respectively, at the points R4 
 and Rj. Hence the angles sr^q' and q'RiS are equally large ; 
 and the point q' is either on the arc sq itself, or else on that arc 
 prolonged. But the former of these two alternatives is to be 
 adopted, because the angle sr^q' is less than SiR^q', or than the 
 angle of a binary product, which is itself less (by art. 386) than 
 60° ; and therefore less than sr^q, which is greater than 60°. 
 Thus the deviation qq' is directed towards s, and another of the 
 questions of art. 383 is answered. This deviation or displace- 
 ment, like those already considered, varies nearly as the cube of 
 the arcual radius sq; it is nearly equal to 17' 37", when that ra- 
 dius is 10° ; and is only about one second, when the radius is so 
 small as a degree. 
 
 390. It only now remains to inquire whether the spherical 
 - angle of the quinary product at q' is greater or less than the 
 limiting value of 120°, which it takes when the figure becomes 
 plane. The supplement of this quinary angle has been seen to 
 be equal to r^q'si or Ssq'Ri ; it is therefore greater than r^q's, or 
 than sq'Ri ; but each of these two last angles, in virtue of the 
 direction just now determined of the displacement qq', is greater 
 than the angle R4QS, or sqRi, which is itself greater than 30°. 
 Therefore, by still stronger reason, the supplement of the angle 
 

 378 ON QUATERNIONS. 
 
 of the quinary product is itself greater than 30°; and conse- 
 quently, that quinary angle is itself less than 150° ; or, in sym- 
 bols, 
 
 Lq^q^q^qiqi < 150°. 
 
 When the radius sq is ten degrees, this defect of the angle of the 
 quinary product from 150° amounts, very nearly, to 1° 31' 0"; it 
 varies nearly as the square of the radius, and is reduced to be 
 only fifty-four seconds and a fraction, when that radius is assumed 
 as a degree. 
 
 391- Although the foregoing numerical values have been 
 calculated with some care, yet they are here offered merely as 
 approximations, which may assist in forming a more clear and 
 distinct conception than might easily be otherwise obtained, of 
 the process of constructing the spherical hexagon of multiplica- 
 tion Q1Q2Q3Q4Q5Q', together with its nine inserted ox focal points ^ 
 R1R2R3R4, SjSaSa, t'iT'o, Under the conditions lately considered 
 When this construction shall have been in any manner correctly 
 completed, it may be followed by the inscription of a system of 
 fifteen new spherical conies, according to the table of focal rela- 
 tions in art. 380 ; in which Table it will however become neces- 
 sary, for conformity with the recent notations, to change q, Ti, Tg 
 to q', t'i, t's, leaving the other symbols unaltered. It has not 
 seemed proper to complicate figure 80, by inserting in it any of 
 these new conies, or even any one of the nine new points, 
 Si, So, S3, t'i, T'2, q', M2, Pi, M3, which have been employed in recent 
 articles. 
 
 392. For the pentagon of multiplication, represented by fig. 
 79, of art. 361, if we use the notation of that article, the five pro- 
 ducts of ternary form, 
 
 s . r . q, t .s .r, t s . rq, t . sr . q, ts .r . q, 
 
 which were enumerated in art. 376, conduct, as in the last cited 
 article, to a system of five auxiliary/ quadrilaterals ; and, there- 
 fore, also (by 379) to a system of five inscribed cotiics, and to a 
 corresponding system of five focal relations, which may be tabu- 
 lated as follows : 
 
LECTURE VI. 379 
 
 Focal Relations for the Pentagon. 
 
 F, G (. .) ABC I ."I 
 
 G, H (. .)bcdk; I 
 
 H, I (. .) CDEF ; [ , . 
 
 I, K (. .) deag; I 
 
 K, F (. .) EABH.J 
 
 Although I thought that it would too much complicate figure 
 79 to insert in it these five ellipses, yet I may be permitted to 
 mention that this species of focal enchainment (379) o/two 
 SPHERICAL pentagons, namely, here, abcde, and figkh (or 
 fghik), with each othei\ through a system of five spherical 
 conics, of which each has xi'&foci at two corners of the second 
 pentagon, and touches two sides of the first, was among the ear- 
 liest of those geometrical results, referred to in art. 303, which oc- 
 curred to me so long as 1843, and were in that year communicated 
 to the Royal Irish Academy, as corollaries from the associative 
 principle of multiplication of quaternions, and from the general 
 focal representation, illustrated by fig. ^5^ of the relations be- 
 tween any three quaternions and their products, partial and total. 
 393. I shall conclude this long Sixth Lecture, by devoting 
 one more of its many articles to the statement of one other geo- 
 metrical deduction from the associative character of the opera- 
 tion of multiplication of quaternions, and from its focal represen- 
 tation. The deduction alluded to is no doubt a very easy one, 
 and has been long since published by me, on the same occasions 
 with the more general theorem of the foregoing article, respect- 
 ing pentagons and conics on a sphere, of which theorem it is a 
 particular or rather a limiting case. Yet as it may serve to throw 
 some little additional light on what has been already said, and 
 as it admits of being illustrated by a sufficiently simple diagram, 
 I shall therefore state it here. Suppose then that the four given 
 versors, q, r, s, t, are represented respectively by four angles, of 
 36° each, whose vertices a, b, c, d succeed each other at inter- 
 vals of 72°, in a left-handed order of rotation, on the circum- 
 ference of a circle so small that it may be treated as plane. Com- 
 plete the plane and regular pentagon, abcde ; and draw its five 
 
380 
 
 ON QUATERNIONS. 
 
 diagonals, ac, bd, ce, da, eb, intersecting each other, as in the 
 annexed figure 81, in five 
 nevv points as follows : 
 
 EB and AC, in f; 
 AC and BD, in g ; 
 BD and CE, in h; 
 CE and da, in i ; 
 da and eb, in k. 
 
 Then the three binary pro- 
 ducts rq, sr, ts, at the limit 
 here considered, will be re- 
 presented by angles of 72° 
 each, at the points f, g, h ; 
 the two ternary products, 
 srg and trs, will be represented by angles of 108° each, at the 
 two remaining corner-s, i, k, of the inner pentagon, fghik ; and 
 the one quaternary product, tsrq, by an angle of 144°, at the 
 fifth corner e of the outer pentagon. The present figure 81 is 
 therefore a limiting form of the more general and spherical con- 
 struction, which fig. 79 was designed to illustrate; and as the 
 significations of the letters correspond, the system oi Jive focal 
 relations, which was tabulated in the preceding article (392), 
 must still hold good. Thus the two points f, g are, at this limit, 
 the two foci oi -a plane ellipse, inscribed in the plane quadrilate- 
 ral ABCi ; namely, the ellipse ll'hk in fig. 81, whose points of 
 contact with the four sides of the quadrilateral are marked with 
 these four letters. In like manner the two points g, h are foci 
 of the ellipse mm'if, inscribed in the parallelogram bcdk; h, i 
 are foci of the ellipse nn'kg, inscribed in cdef ; i, k are foci of 
 oo'fh, inscribed in deag; and k, f foci of pp'gi in eabh. Ac- 
 cordingly these five focal relations can all be established geome- 
 trically, at this limit, by very simple considerations ; and it may 
 be noted that, for the same limiting case of the general construc- 
 tion of a pentagon of multiplication, with its five focal points, two 
 of the four points of contact for each of the five quadrilaterals are 
 corners of the interior pentagon ; and that the major axis of each 
 of the five inscribed ellipses is equal to a side of the exterior 
 figure. 
 
LECTURE VII. 
 
 394. If, at the stage to which we have now arrived, we cast 
 back a rapid glance on the ground over which we have passed, 
 and call our chief steps into review, we shall find them to have 
 been nearly the following. — In the First Lecture of this Course, 
 we considered the primary significations which it appeared con- 
 venient to attach to the marks + and -, or to the operations of 
 addition and subtraction in geometry ; we interpreted, in con- 
 sistence with the views thus introduced, the identities, 
 
 B-A+A=B, a+A-A = a, 
 
 and some others connected with these ; and established the fun- 
 damental relations between vector, provector, and transvector, 
 for any imagined vection (or rectilinear transport) of a point, or 
 any composition or decomposition of such vections. After which, 
 in the Second Lecture, we proceeded to study, on similar prin- 
 ciples, the marks x and -f-, or the operations of multiplication 
 and division in geometry ; we interpreted the fundamental iden- 
 tities, 
 
 j3 -^axa = j3, qx a -7- a = qi 
 
 and others therewith connected ; we developed the notions of a fac- 
 tor as a metrographic agent, and of a quotient as a metrographic 
 relation, of which each involves generally a reference to the 
 length and also to the direction of a line ; established the funda- 
 mental formula which connects factor, profactor, and transfactor, 
 in any composition of successive acts of faction ; and illustrated 
 these general principles, by applications to the cases where the 
 factors to be combined were: 1st, tensors; 2nd, scalars ; 3rd, 
 signs; and 4th, quadrantal versors, such as i,j, k; which last 
 we saw reasons for constructing by a certain system of rectangu- 
 
382 ON QUATERNIONS. 
 
 lar unit-lines, and assigned their squares and products, by com- 
 pounding certain versions or rotations ; these compositions being 
 found to conduct to the important symbolical results, 
 
 ij = k, jk = i, ki=j, 
 
 395. In the Third Lecture, we examined the cases where the 
 multiplier was a vector, but not a vector-unit, or where it operated 
 on a line which was not perpendicular to itself; the product 
 of two perpendicular lines was shewn to be a third line perpen- 
 dicular to both, and such that its direction was reversed when 
 the order of the factors was changed ; on the other hand the 
 product " vector into scalar" was found to be the same line as 
 that given by the multiplication " scalar into vector," and the 
 product of two parallel lines was seen to be a positive or nega- 
 tive number, the square of every vector being negative ; other 
 powers of lines were studied, and the product or quotient of two 
 inclined lines was decomposed into two factors, namely, a tensor 
 and a versor, and was found to involve a dependence on a system 
 of four numbers, entitling it to be called a Quaternion; while, 
 by the help of their representative biradials, a general construc- 
 tion was given for multiplying (and therefore also for dividing) 
 any one such quaternion by any other ; conjugates and recipro- 
 cals were considered, and the signs K, T, U were introduced, as 
 characteristics of the operations of taking, respectively, the con- 
 jugate, the tensor, and the versor, of a scalar, or vector, or qua- 
 ternion. 
 
 396. The Fourth Lecture related chiefly to proportions of 
 lines in one plane, and to powers of quaternions, the exponents 
 of those powers being scalar ; it assigned constructions for 
 j3a"^ . 7, and introduced the symbols Z q and Ax .q; in it were 
 also pointed out some of the uses which might be derived in 
 geometry, for the expressions of certain loci, from the partial in- 
 
 determination of the signv-1, when interpreted according to 
 the principles of the present Calculus. In the Fifth Lecture, 
 the consideration of the line which is a fourth proportional to 
 three coplanar lines was resumed ; and the continued product of 
 
LECTURE VII. 383 
 
 three such lines was shewn to be, in this theorj', a fourth line in 
 the same plane, in the symbolical expression for which product 
 the place of the mark of multiplication is immaterial ; the direc- 
 tion of this fourth line was seen to be that of the fourth side of 
 an uncrossed quadrilateral inscribed in a circle, if the three first 
 sides of that figure have the directions of the three successive 
 factors ; while the fourth proportionals and continued products 
 of three lines which are not in any one plane, were found to be 
 not lines but quaternions. 
 
 397. In the same Fifth Lecture we proceeded to study this 
 last-mentioned quaternion product, of three lines not coplanar, 
 with a view chiefly to ascertain whether in its symbolical expres- 
 sion the point or other mark of muliplication might be omitted ; 
 or in other words, whether the associative principle still held 
 good, in the multiplication of three vectors, which were not in 
 nor parallel to any one common plane. This question was de- 
 cided in the aflSrmative ; and in deciding it, we had occasion to 
 introduce and to apply some general spherical constructions, re- 
 presenting versors by arcs upon a sphere, and the multiplication 
 of any two versors by a process which was called, by analogy, 
 the addition of their representative arcs ; which arcual addition 
 is merely the composition of arcual vections, and corresponds to 
 the composition of successive versions, or plane rotations, of a 
 moveable radius of the sphere : while division of versors, or de- 
 composition of versions, is represented on the same plan by a 
 sort of arcual subtraction. The generally non-commutative cha- 
 racter of the multiplication of versors, or the dependence of the 
 product on the order of the factors, was illustrated by the cor- 
 responding character of the addition of arcs, which belong to 
 difi'erent great circles ; and the same general spherical construc- 
 tion served to illustrate other results, as for instance, that the 
 conjugate or the reciprocal of a product of quaternions is equal 
 to the product of the conjugates or of the reciprocals, taken in 
 an inverted order. 
 
 398. On applying this general construction to the symbols 
 /Sa"^ . 7, j3 . a'^y, in the case where the three vectors a, /3, 7 are 
 not coplanar, it was found that both these symbols represent one 
 common quaternion, which may still be called (as above) the 
 
384 ON QUATERNIONS. 
 
 fourth proportional to those three lines, or the continued product 
 .of y, a'S and j3 ; and of which the axis is directed to the corner 
 D of an auxiliary spherical triangle def, whose sides, respec- 
 tively opposite to the points d, e, f, are bisected by the three 
 given vectors a, j3, 7, at least if those three lines make acute 
 angles with each other ; while the angle of the same fourth 
 proportional to them is the supplement of the semisum of the 
 angles of this auxiliary triangle, or is equal to that semisum 
 itself, according to the character of a certain rotation. The mo- 
 difications of these results were inquired into, which take place 
 when the angles between a, (5, 7, or some of them, cease to be 
 acute ; and the associative principle of multiplication was still 
 found to hold good. When the three angles just mentioned were 
 all supposed to be right, a curious case of indetermination arose 
 in the construction of the auxiliary triangle, which however was 
 shewn to be connected with, and to illustrate, the scalar charac- 
 ter of the fourth proportional to three rectangular lines, and also 
 that of their continued product. And as the values, 
 
 of the squares of i,j, k, had each been deduced from the consi- 
 deration of two successive and quadrantal versions in one plane, 
 so the value 
 
 ijk^-\, 
 
 which serves to complete the continued equation 
 
 , 2*2 =j'i = k'^ = ijk = - \ , 
 
 wherein all the rules respecting the multiplication of ijk are con- 
 tained, was shewn to admit of being interpreted as expressing 
 the result of three successive and quadrantal versions, or rota- 
 tions, in three successive and rectangular planes. 
 
 399. Such having been the chief subjects of the five first 
 Lectures of this Course, we proceeded in the Sixth, after some 
 supplementary remarks on the subjects lately considered, and 
 especially after shewing how the semi-excess of a spherical trian- 
 gle might present itself as the angle of a certain product of 
 square roots, to examine whether the associative principle of mul- 
 tiplication held good for any three or more quaternions generally, 
 
LECTURE VII. 385 
 
 and not merely for any three lines. To inquire whether it were 
 universally true, in this Calculus, that 
 
 s .rq = sr . q, 
 
 and to draw forth some of the chief consequences of the truth of 
 this simple but important formula, was indeed the guiding con- 
 ception, the leading aim, of the whole of that long Sixth Lec- 
 ture, of which, in this recapitulation, I shall speak with greater 
 relative brevity than of the ones preceding it, because it may be 
 supposed to be more fresh than they in your remembrance. You 
 know that a new spherical construction, by means of represen- 
 tative angles, was given in that last Lecture, for the multiplica- 
 tion of versors, distinct from, although intimately connected with 
 the construction by representative arcs, which had been pre- 
 viously offered to your notice ; the product of two versors being 
 now represented by the external vertical angle of a spherical tri- 
 angle, whose base angles, taken in a determined order, represent 
 those two versors themselves; and you remember that this con- 
 struction by angles was employed to illustrate anew some gene- 
 ral properties of the multiplication of quaternions. The equa- 
 tion 
 
 for any spherical triangle, was established, with the help of the 
 same construction : and the symbol 
 
 qrq~^ 
 
 was interpreted, as denoting a conical rotation of the axis of r 
 round the axis of q, through double the angle oi q; or else, at 
 pleasure, the equivalent amount of the turning of one plane upon 
 another, in a mode entirely analogous to the precession of the 
 equinoxes; and thus a preparation was made for symbolizing the 
 rotations, as well as the translations, of a body, or system of vec- 
 tors, and for expressing the composition of such rotations. 
 
 400. This having been done we proceeded to translate, with 
 the help of diagrams, very copiously employed in that Lecture 
 which we are now reviewing, the statement of the Associative 
 Principle, for the case of three versors, into the language of re- 
 presentative arcs, and also into that of representative angles: and 
 
 2 c 
 
386 ON QUATERNIONS. 
 
 proved it, for each of these two connected forms of construction, 
 by means of some simple and known properties of conies upon a 
 sphere ; giving however also a more elementary proof, although 
 a somewhat longer one, which did not assume any acquaintance 
 with the doctrine of those conies, and indeed did not introduce 
 the conception of a cone at all. The associative principle of 
 multiplication having been thus established for three versors, it 
 was extended without any difficulty to the case of three or more 
 quaternions, and so shewn to be general in this Calculus: and 
 its expression was in several ways varied, by means of spherical 
 figures, and by relations between quotients of lines. The same 
 fertile principle conducted us also to many conclusions respect- 
 ing continued products of vectors, especially when the factors 
 were supposed to be the successive sides of a rectilinear polygon, 
 plane or gauche, inscribed in a circle or in a sphere ; among 
 which it is worth while to remember, that the product of the 
 successive sides of any even-sided polygon in a circle, is a sca- 
 lar; but that the product of the successive sides of any odd-sided 
 polygon in a sphere, is a tangential vector. Cases of these last 
 theorems were made to furnish equations or conditions of con- 
 circularity for four points, and of homosphsericism for five : and 
 the latter equation, which includes the former as a limit, was 
 shewn to furnish a graphic property of a sphere, in relation to 
 an inscribed gauche pentagon, which property is, for space, the 
 analogue of the elementary relation between the directions of 
 the sides of a quadrilateral inscribed in a circle. A problem re- 
 specting the inscription of a gauche quadrilateral in a sphere 
 was also easily resolved, and might with equal ease have been 
 extended. Finally, the two other chief classes of geometrical 
 applications of the associative principle of multiplication, which 
 were considered in the foregoing Lecture, may be said to have 
 been those which related to the compositions (above alluded to) 
 of conical rotations ; and to the superscription on a spheric sur- 
 face of certain polygons of multiplication, with certain connected 
 systems of focal points, and of inscribed spherical conies; in- 
 cluding some limiting cases, where the polygons and conies be- 
 come plane. But these have been so recently treated of, that 
 we may now pass to other things. 
 
LECTURE VII. 387 
 
 401. The object which we propose to ourselves in this Se- 
 venth Lecture, being chiefly to treat of the Addition and Sub- 
 traction of Quaternions, and in connexion therewith to prove and 
 to apply the Distributive Property of their Multiplication ; as also 
 to introduce and exemplify the Notations S and V, which were 
 mentioned by anticipation in art. 121, and which serve to sepa- 
 rate a quaternion into its scalar and vector parts: we may here 
 begin by observing, that since we already know how to add sca- 
 lars among themselves (by the ordinary rules of algebra), and 
 also how to add vectors to each other (by the laws of the compo- 
 sition of vections), it is natural now to consider what interpreta- 
 tion can consistently and usefully be assigned to the analogous 
 operation, not hitherto studied by us, of adding a scalar to a 
 vector. To take what seems the simplest case of this inquiry, we 
 may ask, what are we to regard as the meaning^ and what as the 
 result, of the addition of a scalar unit to a vector unit ? Can we, 
 for instance, interpret the sum 1 +^, as bearing any clear and de- 
 finite signification, if k continue to denote, as it has hitherto 
 usually done with us, an upward unit line? 
 
 402. For this purpose 1 look out for some common operand, 
 on which I can operate separately, by each of the two proposed 
 symbols 1 and k, and afterwards add the results, in order to com- 
 pare their sum with the operand thus assumed. Such an operand 
 at once presents itself in the vector unit i ; for we know that 
 1 i-i, and that ki=j ; and although it may seem at first difficult 
 to add, in any intelligible sense, the number, 1, to the line, k, there 
 is no difficulty in adding the southward line, i, to the westward 
 line,j, by drawing, as in fig. 82, the diagonal . 
 OP of a square, constructed with os and ow, ^'S- 82. 
 or with the lines i andj, for two contermi- 
 nous sides. And then by comparing this 
 south-westward diagonal, i +j, whose length 
 is = \/2, with the original operand, or side, 
 or southward unit i, we obtain the equa- 
 tion : ^, 
 
 I + k = {i + ki) -^ i = (i +j) -7- i ; 
 
 so that the required sum, \-vk, is thus put under the form of a 
 
 2 c 2 
 
388 ON QUATERNIONS. 
 
 quotient of two lines ; and therefore (by our general principles), 
 it is hereby found to be a quaternion, of which the tensor and the 
 versor are as follows : 
 
 T(l+y^)=2i; U(l+A)=/i*. 
 
 (In the annexed sketch, fig. 82, I observe that {l^-k) i has been 
 inadvertently written, instead of (1 + ^) i.) We may also, for the 
 same reason, write more concisely this equation. 
 
 And it is clear that the same quaternion would have been ob- 
 tained, as the value for this expression 1 + ^, if we had set out, on 
 the same general plan, with any other horizontal line, a, instead 
 of i, as the original operand. We should still have been led to 
 construct a square in the horizontal plane, and to compare a 
 diagonal with a side ; or more fully, to divide (in the general 
 sense already explained) the one line by the other; and to take 
 the resulting quotient, V (2 Jc), as the value of the sum in question. 
 403. Those who are familiar with the principles of the Cal- 
 culus of Finite Differences, may find the following remarks throw 
 some light on the foregoing process. We were to add the num- 
 ber 1 to the line k; and there seemed for a moment to be a difl^i- 
 culty in so doing, on account of the heterogeneity of the two 
 summands. But in the Calculus of Differences an exactly ana- 
 logous difficulty presents itself to the learner, when he first meets 
 the symbol 
 
 1 + A, 
 
 where the number 1 appears as added to the characteristic A, 
 which is not a number at all, but the sign of the operation of 
 taking a finite difference. How is this difficulty removed ? A 
 function of a;, suppose x^, or more generally y*(a;), is taken as the 
 common operand; it is operated on by each separately, of the two 
 proposed things or signs, 1 and A ; the two results, namely, 
 
 \ ■ x^= x^, and A • a;^ = 3 a;^ + 3 a; + 1 , 
 or more generally, 
 
 l/(^) =/(^), and A/(aj) =f{x^\) -f{x), 
 
LECTURE VII. 389 
 
 are added to each other, by the previously known rules of ordi- 
 nary addition in algebra ; and their sum is then, by a definition 
 suggested by analogy, and found by experience to be useful, con- 
 sidered as being the result which wouldhdive been obtained, if the 
 same function of ic had been at once operated on, by the sought 
 symbolic sum, 1 + A. In this way it has come to be agreed on to 
 write, 
 
 {l + A)-x^=l-x^ + A-a;^ = x^+{3x-+dx+l) = {x + 1)% 
 
 and more generally, 
 
 (l + A)/(.^)=/(x'+l); 
 
 and then, by abstracting from the operand, it has been inferred 
 that 1 + A is, in the Calculus of Differences, the symbol of an 
 OPERATOR, which changes any given function of x to the same 
 function ofxA- 1. We come to learn then, in that Calculus, what 
 the proposed sum 1 + A is, by learning what it does ; the ope- 
 rator becomes known, through the knowledge which is acquired 
 of its operation. And similarly, in the foregoing article, the ope- 
 rator 1 + ^ has been considered as determined, when it has been 
 found to produce the determined effect, of changing the side to 
 the diagonal of a square in the horizontal plane, exactly as is done 
 by the quaternion \/'ik; to which quaternion the sought sum 
 1 + A has therefore been concluded (in art. 402) to be equal. 
 
 404. As it is perhaps impossible to be too clear on funda- 
 mental points, and as the addition of a scalar to a vector is thus 
 fundamental in quaternions, I shall venture here to submit to 
 you, for a moment, a far more elementary illustration. Suppose 
 then that you wished to shew to a child that two and three made 
 five, or to teach him how to interpret the symbol 2 + 3, you might 
 of course, for that purpose, put down first two dots as one group, 
 and then three dots as another, and afterwards combine these two 
 groups into a single one, as indicated in this 
 
 • • Fiff 83 
 
 little diagram; on counting the dots in which 
 
 one resultant group, the child would find them i — " — , , ^ , 
 
 to he Jive. Now in this simple and obvious 
 
 process, the dot is the original operand: the partial groups, of 
 
 two dots and three dots respectively, are the results of the two 
 
390 
 
 ON QUATERNIONS. 
 
 Fig. 84. 
 
 partial operations ; the proposed numbers, 2 and 3, correspond 
 to the two partial operators (being thus analogous to the sym- 
 bols 1 and k in article 402, or to 1 and A in art. 403) ; the total 
 group^ of five dots, is the sum of the two partial results (answer- 
 ing to l^■ + ki, or to Ifx + ^fx) ; and when at last the young 
 arithmetician comes to count the dots, in this final or total group, 
 he executes, on a small scale, that sort oi abstraction from the 
 operand^ which leads, in the Calculus of Differences to the in- 
 terpretation of the symbol 1 + A, and in the Calculus of Quater- 
 nions to the conclusion that 
 
 1 + k = {li + ki) -T- i = {i +j) -=- ^ = 2^^^. 
 
 405. More generally, let it be now required to add any pro- 
 posed scalar, w, to any proposed vector, p, or to interpret gene- 
 rally the symbol w + p. We have only (see fig. 84) to assume 
 any line a, or oa, in a 
 plane perpendicular to 
 p, as the original and 
 common operand ; to 
 operate on this sepa- 
 rately, by the scalar w 
 and by the vector jo, and 
 so to produce, as the 
 two partial results, two 
 
 mutually perpendicular lines, namely, wa or ob, and pa or 
 oc ; to form next the sum of these two lines, by completing 
 the rectangle, and drawing the diagonal ; and finally, to di- 
 vide this diagonal wa + pa or od, by the assumed operand line a, 
 and to equate the required sum, w -\- p,io the quaternion which is 
 obtained as the quotient of this division. In short we have only 
 to employ the very simple formula, 
 
 W-V p = (wa + pa) -r- a, where a JL jO : 
 or (under the same temporary condition of perpendicularity) to 
 make use of the identity, 
 
 {w + p) a = W'a + pa. 
 So FAR, then, the distributive property of multiplication holds 
 good BY DEFINITION in quatcmions, as serving to interpret 
 
 
 
 
 ^^ 
 
 A 
 
 
 p * <^^ 
 
 ^ 
 
 A 
 
 4 TV 
 
 ji0^ 
 
 
 
 /<^^' 
 
 
LECTURE VII. 
 
 391 
 
 (in the foregoing way) the symbol w + p^ by first introducing', 
 and afterwards abstracting from, an auxiliary and perpendicular 
 line a, as a subject to be operated upon : and it is clear that a 
 similar process would lead to the same construction, and to the 
 same final result, if we had sought to add p to w, instead of 
 adding w to p. We know therefore how to give, by quaternions, 
 in every case, a complete and definite interpretation to the ope- 
 ration of adding together a scalar and a vector; and we see that 
 such summation is commutative ; or in symbols, that (because 
 wa+pa = pa + Wa) we may write, 
 
 W + p = p + W. 
 
 406. Conversely, let aob be any proposed biradial, repre- 
 senting an arbitrary quaternion, 
 
 ^=:j3 _i-a=OB -I- oa; 
 
 and conceive that from the extremity b of the final ray ob, a 
 perpendicular bb' is let fall, on the initial ray oa, or on that ray 
 prolonged. The vector j3 or ob will thus be decomposed into 
 two partial vectors, j3' and /3", or ob' and b b, of which the for- 
 mer (j3') has either the same direction as a, or else the opposite 
 direction, unless it happens to vanish ; while the latter (/3") has 
 a direction perpendicular thereto : and consequently, if these 
 two components of /3 be respectively divided by a, the two 
 partial quotients will be respectively equal to some scalar, such 
 as w, and to some vector, such as p, this latter vector being per- 
 pendicular to the plane of the biradial. In symbols, see the an- 
 nexed figure 85, we may 
 write, 
 
 a = A-o, /3=b-o = 
 
 (b - b') + (b' - o) = 
 
 /3"+i3', jS'lla, i3"Xa; 
 
 and therefore shall have two 
 partial quotients of the 
 forms, 
 
 (^' -i-a = w, (5" -T-a = p, 
 where p ± a, /o ± jS. 
 
 Hence, if we seek, by the 
 
 fig. 85. 
 
392 ON QUATERNIONS. 
 
 principles of the foregoing article, to form the sum^ w + p, of 
 these two partial quotients, we find, 
 
 Wa = (5', pa =15", (w + p) a=/3' + j3" = /3, 
 and finally, 
 
 w + p = P -i- a = q. 
 
 Not only then may we always compound, by addition, any pro- 
 posed number w with any proposed line p into one quaternion 
 sum, but also reciprocally, we can decompose any proposed qua- 
 ternion, q, into two parts, of which 07ie shall be some scalar 
 such as w, while the other part shall be some vector as p : and 
 it is clear from the foregoing remarks that this decomposition is 
 perfectly definite ; any change, whether of number or of line, 
 making a real and not merely an apparent change, in the quater- 
 nion which is their sum. 
 
 407. We may therefore speak dejinitely of the scalar part, 
 and THE VECTOR part, or more concisely we may speak of the 
 scalar and the vector, of any proposed quaternion. And 
 these two parts of a quaternion (already alluded to, near the 
 commencement of the Fourth Lecture) will be found to present 
 themselves so often, in the developements and applications of 
 this Calculus, that it becomes almost necessary to agree on some 
 notations, by which they may be sejoara^e/y indicated. Accord- 
 ingly I have for a good while accustomed myself to employ, as 
 among the main elements of the T^toTATiom of quaternions (see 
 arts. 121, 401), the two letters, 
 
 S and V, 
 
 as CHARACTERISTICS of the two fundamental operations, of what 
 I call, respectively, taking the scalar, and taking the vec- 
 tor, of a quaternion. More fully, I denote separately, by the 
 symbols, 
 
 S^ and Vq, 
 
 the scalar part and the vector part of any proposed quaternion, 
 q. Thus 
 
 S (w + p) = w; \ (w + p)= p ; 
 
 and with the recent significations (406) of o, j3, j3', |3", we have, 
 
1.ECTURE VII. 393 
 
 S(j3^a) = /3'--a; V(/3^a) = i3"-T-a. 
 In general for any quaternion q, we have the identities, 
 
 q-^q + Vq =Yq+ Sq, 
 which may sometimes be abridged as follows : 
 
 l = S + Y = Y+S. 
 
 With the same significations of the letters, it is clear that we 
 have also, 
 
 Sw = w; Sp = ; Vw = ; Yp = p; 
 
 that is, identically (compare 90), 
 
 SS^ = S^, SV^ = 0, VS^ = 0, YYq^Yq; 
 
 or more concisely, 
 
 s^ = s, SV=VS = 0, V2=V. 
 
 408. Conjugate quaternions have equal scalars, but opposite 
 vectors ; as will at once appear, if we compare the general de- 
 composition into scalar and vector parts, constructed by the re- 
 cent figure 85, with the equally general representation of two 
 conjugate quaternions, which was illustrated by the earlier fig. 
 32, of art. 186. In the figure last cited, we had 
 
 5' = /3-f-a = OB-T-OA; Kq-y -^ a = oc -=r OA.; 
 
 and it is evident that if the right line bc were drawn, connecting 
 the extremities of the two dividend vectors j3 and 7, it would be 
 perpendicularly bisected by the divisor line a, or by that line pro- 
 longed, in a point which might be called b'. In this way we 
 should not only have, as in 406, 
 
 i3 = i3"+/3', jS'lla, i3"JLa, 
 but also, 
 
 7 = 7 + 7 » 7 II «5 7 -L «' 
 where 
 
 7' = ob'= + j3', but 7" = b'c = - b'b = - /3"; 
 
 thus the scalar and vector of the conjugate are, respectively, 
 
 S (y -J. a) = 7' H- a = jS' ^ a = + S (j3 .^ a), 
 V (7 -^ «) = 7" -r- a = - i3" -^ a = - V (/3 4- a) ; 
 
394 ON QUATERNIONS. 
 
 or more concisely, 
 
 SK^ = +S^, VK^ = -V9; or, SK=S, VK = -V. . 
 If then, as in 406, we adopt the expression, 
 
 q-w +pi 
 
 for the proposed quaternion, we shall have also, as was stated by 
 anticipation in art. 114, this connected expression for the conju- 
 gate : ^ 
 
 Kq=w- p ; 
 
 which includes the two particular expressions there given, 
 
 Kw = + w; Kp = - p- 
 
 We may also write, as an identity in this calculus, the formula, 
 
 Kg- = S^- - \q ; 
 
 which may be abridged to the following : 
 
 Kg = (S-V)^; or K=S-V. 
 
 409. It has been seen (114, 162) that conjugate quaternions 
 have always one common tensor, or that 
 
 TK2=Tg; 
 
 we have therefore the equation, 
 
 T {w - p) = T {w + p). 
 
 Again, it was shewn in 163 that the product of two conjugate 
 quaternions is equal to the square of their common tensor, 
 
 qKq=Tq^; 
 
 we have therefore the following expression for this square, 
 
 T (w + pY = {w + p)(w-p); 
 
 whence, if we had already established generally the truth of the 
 distributive principle of multiplication, we might at once con- 
 clude, what was stated by anticipation at the end of art. Ill, 
 that 
 
 Tq=T(w + p)= Vi^'-p')' 
 
 But since that principle has not yet been generally established,! 
 
LECTURE VII. 395 
 
 must take at this stage another mode of proving the correctness 
 of this last expression, for the tensor of any quaternion. And 
 this is easily done with the help of the recent figure 85. In fact 
 since the square on the hypotenuse ob is equal to the sum of the 
 squares on the two sides about the right angle, we have evidently 
 the equation, 
 
 T/3^=Tj3'^ + T/3"^; 
 
 therefore also, by general properties of tensors already esta- 
 blished, we have 
 
 |3V fTi^yj-rP 
 
 that is 
 
 but it was proved in 11 1 that 
 
 Tw"" = + w\ and that Tp" = - p^ ; 
 
 we arrive then thus at the formula which includes these two last 
 results, namely, 
 
 Tq^ = vf - p^. 
 
 410. It is evident (see fig. 85, art. 406), that if the quaternion 
 §', or j3 -^ a, be multiplied by any scalar x, by changing ]3 to a;j3, 
 the projections, j3' and j3", of the vector j3, are at the same time 
 multiplied by the same scalar; or are changed, respectively, to 
 i»j3', and to a;/3". Hence the two partial quotients, j3' -f- a and 
 j3" -r- a, OY w and p, are changed, by this multiplication, to xw 
 and xp respectively. Such then are the scalar and vector parts 
 of the product rr^ ; or more concisely, 
 
 ^.xq = x^q, diXiA Y . xq = xY q, ifVa; = 0: 
 
 this last formula expressing, evidently, in virtue of the principles 
 and notations explained in art. 407, that x is here supposed to be 
 a scalar. In particular, by making x = -\, we have the identi- 
 ties, 
 
 S(-^) = -S^; V(-^) = -V^. 
 
 And, passing from the quaternion q to its conjugate, and attend- 
 ing to the results of art. 408, we find that 
 
396 ON QUATERNIONS. 
 
 S(-K^) = -S^; V(-K^) = + V^; 
 
 or that 
 
 -Kq = -Sq+Vq, 
 -K = V-S. 
 
 In general we have, in this calculus, as in algebra, with the fore- 
 going significations of the symbols, 
 
 X {w-\- p) = xw + xp; 
 ~ {w + p) = -w - p ; 
 - {w - p) =-w + p; 
 
 the two latter identities being included in the former. 
 
 411. It was seen (in 113) that a tensor such asT§', although 
 first conceived (see 63) as a signless number, might be equated 
 to a positive scalar; whence it follows that we may now write, 
 
 ST^ = + T^ = Tq, and VT^ = 0. 
 
 But also we have generally the decomposition (90) of a quater- 
 nion into factors, 
 
 q=Tq.Uq; 
 
 where the point or other mark of multiplication may be omitted. 
 Hence (by 410) we have the two identities, 
 
 S^=T^.SU^, Vq=Tq.yVq; 
 
 when the points may again be omitted without confusion. It is 
 also allowed (see 113), and is indeed only a particular case of 
 the more general decomposition just now mentioned, to decom- 
 pose any vector into its own tensor and its own versor, as fac- 
 tors ; thus we may write, 
 
 YVq^TVUq.UYVq; 
 where, by the present article, and by 113, 153, 
 
 UVU^ = UV^=Ax.^. 
 
 The temporary symbol Ax . q, employed in the three preceding 
 Lectures, may therefore now be replaced by this other symbol 
 UYq, which is perhaps only about as easy to be written or 
 printed as the former, but which has the advantage of connect- 
 ing itself better with the system of symbols employed in the pre- 
 
LECTURE VII. 397 
 
 sent Calculus ; and we may establish the following symbolical 
 equation, between two different characteristics of two equi- 
 valent operations : 
 
 Ax. = UV. 
 
 We have also these general transformations of any proposed 
 quaternion q : 
 
 = T^(SU^+UVg.TVUg): 
 
 in which there is no difficulty in seeing now that 
 
 SUg'^cos Lq, TVUg'=sin L q, 
 
 if we merely admit the well-known meanings of the words " co- 
 sine" and " sine," and their abridged notations, " cos" and " sin," 
 without assuming here the knowledge of any formula of trigo- 
 nometry. At the same time it results from art. 1 13, that 
 
 {lJYqy = ~l; 
 
 and thus a celebrated expression is reproduced, as a general form 
 for the versorofa quaternion, namely the following : 
 
 Uq = cos /.q+ y/-l sin Lq; 
 
 in which, however, on the plan of interpretation adopted in these 
 Lectures, the square root of negative unity that occurs is not to 
 be regarded as having any imaginary character in geometry; but 
 simply as denoting a certain vector unit : namely, that particular 
 unit-line which is more fully denoted by Ax . q, or by VYq, and 
 of which the direction is perpendicular to the plane of the pro- 
 posed quaternion q. 
 
 412. Without inquiring farther, at present, into this con- 
 nexion of quaternions with trigonometry, it may be instructive 
 to exhibit, at this stage, a few of those expressions for geo- 
 metrical LOCI, which the recent symbols S and V supply, or 
 assist in supplying, when used in consistency with the principles 
 of the present Calculus. 
 
 It is evident, from recent articles, that the scalar part of a 
 quaternion is positive, or null, or negative, according as the angle 
 of that quaternion is acute^ or right, or obtuse : in symbols, 
 
398 
 
 ON QUATERNIONS. 
 
 bq = 0, according as z 5' = -. 
 
 In fact, without assuming any thing as previously known re- 
 specting the trigonometrical character of the function " cosine," 
 or even requiring, at present, the admission of the recent formula 
 SUq = cos /.q, the equations, 
 
 S (oB -f- oa) = ob' -t- oa, S (oc -f- oa) = oc' -T- OA, 
 
 taken in connexion with fig. 85, establish at once the positive 
 character of the scalar of an acw^e-angled quaternion, and the 
 negative character of the corresponding part of a quaternion 
 which has its angle obtuse; while the evanescent (or null) cha- 
 racter of the scalar part of a n^A^-angled quaternion, may be 
 made obvious to the eye by this other and very simple figure, 
 where the projection d' of d on ao coin- 
 cides with o, and the line od' or S' va- 
 nishes, making at the same time null the 
 quotient, 
 
 S'^a = S(g^a) = S(oD-r-OA)= ^=y' 
 
 od' -^ oa = 0, if 8 _L a. 
 
 ^=0 
 = D' 
 
 Fig. 86. 
 
 <- 
 
 And conversely, if a and p be any two ac- 
 tual (or non-evanescent) straight lines, r~ 
 which do not make a right angle with each 
 other, the scalar part of their quotient cannot be equal to zero; for 
 it will be (as above) either a positive or negative number, accord- 
 ing as the angle between the two lines is acute or obtuse. To 
 write therefore the equation 
 
 S(p-a)-0, 
 
 under this supposition of the actuality oi the two lines compared, 
 is equivalent to writing theforinula of perpendicularity, 
 
 p J- a. 
 
 And it is clear that, on the other hand, with the same condition 
 of the non-evanescence of the lines, to write this other equation, 
 
 V(p-«)==0, 
 
LECTURE VII. 399 
 
 is to assert that the directions of a and p are either similar or op- 
 posite; and is therefore equivalent to the establishment of the 
 formula of parallelism, 
 
 p i «• 
 
 In short, the quotient of two "parallel lines, being a scalar, has 
 no vector part; and in like manner, the quotient oi i^o perpen- 
 dicular lines, as being (in this whole theory) equal to a vector, 
 has no scalar part diflferent from zero. 
 
 413. This being clearly seen, suppose that a, j3, p denote 
 some three vectors, oa, ob, op, which have a fixed and common 
 origin o, and of which the two former terminate at two fixed and 
 known points a, b, but the latter at an unknown or variable point, 
 p. Then, using the notation of fractions (118), the equation 
 
 s^o, 
 
 expresses that p ± a, and therefore that the locus of the point p 
 is the PLANE THROUGH THE ORIGIN o, which \% perpendicular to 
 the given line oa. In like manner, the slightly more complex 
 equation, 
 
 expresses the perpendicularity, 
 
 |0-j3 J. a, or BP ± oa; 
 
 and gives therefore, as the locus of p, the plane which is drawn 
 through the given point b, perpendicular to the same given line 
 OA, and consequently /xafrct/Ze/ to the former plane. Another ex- 
 pression for a plane parallel to the first plane is the following : 
 
 where a is supposed to denote some constant and given scalar; 
 for this equation expresses (by 406, 407) that the projection |o'of 
 the vector |0 on a is the constant line «a, or that the projection 
 p' of the point p on oa is constant, 
 
 o' = OP = aa. 
 
400 
 
 ON QUATERNIONS. 
 
 And I may just mention by anticipation here, that when the de- 
 finition of the difference of two quaternions shall have been as- 
 signed, and the distributive property of the operation of taking 
 the scalar proved, the third equation of the present article will 
 be seen to result from the second, under the form 
 
 a a 
 
 414. If, inverting the fraction, we were to write the equa- 
 tion 
 
 it would still express merely that p was perpendicular to a, and 
 would still give the first plane of the foregoing article, as the 
 locus of the extremity of p ; and in like manner, the equation, 
 
 = 0, 
 
 Fig. 87. 
 
 would give still that second or parallel plane which was drawn 
 through the end of j3, at right angles to a. But if we write 
 
 we express (see the annexed figure 87) that the projection of o on 
 p is the line p itself, or that the angle opa 
 is right ; and therefore that the locus of p is 
 now the surface of the sphere, described 
 with the given line oa as diameter. With- 
 out assuming as known those general prin- l^ V/ " ^' 
 ciples respecting difi'erence and distribu- ^ 
 tion which were recently by anticipation 
 
 alluded to, we may easily see that this last spheric locus may 
 also be represented by the equation 
 
 s— -0, 
 
 for this evidently expresses the perpendicularity, 
 a -/o ± jO, or PA J_ op. 
 
LECTURE VH. 
 
 401 
 
 We may therefore already perceive, by this simple geometrical 
 construction, although the mode of proving it as a transforma- 
 tion in this calculus is for a while reserved, that either of the two 
 last equatidns must be equivalent in its import or signification to 
 the following: 
 
 because if we bisect oa in c we shall have, 
 
 oc = -, CP^ 
 
 a 
 
 2' 
 
 and these two last lines are obviously equal to each other in 
 length, the point c being the centre of the sphere. 
 
 415. More generally, there is no difficulty in seeing, what 
 indeed is not ■peculiar to the theory of quaternions, that the 
 semiswn, i(a + j3), of any two co-initial sides oa and ob, of any 
 plane triangle aob, represents in length and in direction, the co- 
 initial bisector oc of the third side ab ; for it is (see fig. 88) half 
 of the co-i?iitial diagonal 
 OD, of the completed pa- 
 rallelogram (compare art. 
 100); and in like manner 
 the line ca, which is the 
 half of the other diagonal, 
 is represented by the semi- 
 difference ^ (a- (5). If then 
 we meet the equation, 
 
 which expresses (see fig. 89) that cp is equal 
 in length to ca, or that the locus of p is the 
 sphere with ab for diameter, the right angle 
 in the semicircle apb will enable us to infer 
 that pa _L bp, or that a-p ± p - [5, and so 
 will give this other equation, 
 
 
 2d 
 
402 ON QUATERNIONS. 
 
 which we thus see, must be a valid transformation of the former, 
 although the rules for passing-, by calculation, from either of 
 these two last equations to the other, have not as yet been 
 given. Meanwhile it is evident that if we make )3 ='0, we shall 
 thereby place the point b at the origin o, and so change the last 
 figure 89 to the figure 87 of the preceding article, returning 
 thus to the particular spheric locus there constructed, from that 
 more generally situated sphere which has been since expressed. 
 
 416. From planes and spheres we can of course pass to cir- 
 cles, as their intersections ; thence to the cone, which has a 
 circle for its base : and from this again to the well-known curves 
 of intersection of such a cone with a plane, or to the conic sec- 
 tions commonly so called, which form so important a link be- 
 tween the ancient and the modern mathematics. It is also almost 
 or altogether equally easy, so far as mere expression is con- 
 cerned, to deduce, from the same principles, equations which shall 
 represent those spherical curves, which, under the name of sphe- 
 rical coNics, have attracted so much notice from geometers of our 
 own times ; and of which some mention has already been made, by 
 anticipation, in these Lectures : namely, the curves of intersec- 
 tion of a co?ie which has a circular base, with a sphere which has 
 its centre at the vertex of the cone. 
 
 417. Thus if we conceive that p, q, r, s are four points on 
 the circumference of a circle, the point p being variable, but the 
 other three points being fixed ; while o is any other given point of 
 space, which we shall suppose to be outside the given plane qrs, 
 and A the foot of the perpendicular upon that plane, let fall from 
 o, so that OAP, OAQ, OAR, OAS, are right angles ; if also we denote 
 OA by a, and op by p ; we shall then (by 413) have the follow- 
 ing equation, 
 
 SP-=1, 
 a 
 
 to represent the plane of the circle ; and in order to complete the 
 expression of the circumference, it only remains to assign the 
 equation of some sphere, on which the same circle shall be con- 
 tained. Now we can always conceive such a sphere, oqrs, de- 
 termined so as to contain the given origin o, which has been 
 
LECTURE VII. 403 
 
 supposed external fco the plane of the circle qrs ; and can then, 
 at least in thought, draw the diameter ob of this sphere, and de- 
 note the diameter so drawn by j3. Thus opb will be a right 
 angle, and (compare 414) the sphere oqrs will consequently be 
 expressed by the equation, 
 
 P 
 
 The SYSTEM OF THESE TWO EQUATIONS, 
 
 S^=l, 8^=1, 
 a p 
 
 will therefore represent the circle qbs ; which may, by a suitable 
 choice of the two vectors a and j3, be made to coincide with any 
 proposed circle in space, under the condition that its plane shall 
 not pass through the origin o. This mode of representing a cir- 
 cle is indeed far from being the only one which the principles of 
 quaternions supply ; but it is one of those which seem to suit best 
 our present stage of the developement of this Calculus. 
 
 418. If now we multiply together the two equations just 
 found for the circle (supposing o external, as before), their pro- 
 duct, namely, the new equation 
 
 a p 
 
 may easily be proved to represent the cone, which has the point 
 o for its vertex, and the circle qrs for its base. For first, that 
 the locus represented by this equation is a cone of some sort, 
 with the origin of vectors for its vertex, appears from the circum- 
 stance that if the equation be satisfied by any one value of the 
 variable vector p, it is satisfied also by every other value xp of 
 that vector, which can be derived from the former value p by 
 multiplying it by any scalar x ; since the recent equation may 
 be written thus, 
 
 a Xp 
 
 we may therefore at pleasure shorten, lengthen, or reverse the 
 vector OP of any point p of the locus, and the new point p' thus 
 
 2 D 2 
 
404 ON QUATERNIONS. 
 
 obtained, on the indefinite right line op, will still be situated upon 
 the locus. And in order to determine, next, what particular 
 cone, with o for vertex, is represented by the equation of this ar- 
 ticle, we need only determine the form and position of some one 
 plane section, such as that made by the plane whose equation is 
 
 a 
 
 Now it is clear, from comparison of the equations, that this sec- 
 tion must be entirely contained upon that other locus, of which 
 the equation is 
 
 8^ = 1; 
 
 P 
 
 that is (see 4 14, 417), the sphere through the origin, of which one 
 diameter is the vector |3 : but the intersection of this sphere with 
 the last-mentioned plane is precisely that circle which was con- 
 structed in the article immediately preceding. We see therefore 
 that this circle is one section, and consequently that it may be 
 regarded as the base, of the cone whose equation has been as- 
 signed in the present article. 
 
 419. If then with that equation, namely, with 
 
 a p 
 we combine this other equation, 
 
 7 
 
 which represents generally a new plane, if y be a new constant 
 vector, we shall hereby express that the cone with circular base 
 is cut by a plane not passing through its vertex ; and the system 
 of these two equations will represent (416) a conic section: 
 which may be a circle, ellipse, parabola, or hyperbola, according 
 to the values assigned to the three constant vectors, a, j3, y. 
 Conversely, if there be any conic section, whose form and posi- 
 tion are given in space, and if any origin o of vectors be assumed 
 outside its plane, the expression of the curve may be reduced to 
 the form of this system of equations, 
 
LECTURE vn. 405 
 
 where y may be regarded as an entirely known and Jixed vector, 
 namely, the perpendicular from the assumed origin on the given 
 plane of the section ; but in which the two other constant vec- 
 tors, a and (5, may be chosen with some degree of arbitrariness; 
 since it is clear, for instance, that they may both be multiplied by 
 any common scalar, such as t, because the equation of the cone 
 may evidently be written as follows (compare 418): 
 
 ta p 
 
 And it is not difficult to see that the cone remains in all respects 
 unaltered, when a and /3 are changed to )3"^ and a'^ respectively. 
 420. This last transformation of the equation of the cone de- 
 serves however to be more closely considered, both as an exercise 
 in calculation, and for the sake of its geometrical signification.. 
 For this purpose I observe that, by principles already explained, 
 we have the transformations (see 118, 89, 408, 410, 85), 
 
 and 
 
 ft - = S . pa'^ = SK . a" V = S . a'^p = p-S — , 
 
 whence it follows that we have, identically, for any three vec- 
 tors a, j3, pi 
 
 P (^ o P "'^ 
 
 s-s- = s-^^s — ; ^ 
 
 a p p ^ p 
 
 and consequently that the equation of the cone, employed in the 
 two preceding articles, may be put under the form, 
 
 P P 
 thus justifying the remark which was made at the end of 419. 
 The same new form of the equation shews that the same cone is 
 cut by the plane 
 
406 ON QUATERNIONS. 
 
 in a NEW CIRCLE, contained upon the sphere 
 
 the plane of this new circle being noi generally parallel to the 
 plane of that other circle (417), which was made (in 418) the base 
 of the cone here considered. In short we find ourselves con- 
 ducted anew, by this easy process of calculation with quaternions, 
 to the recognition of that antiparallel x)x subcontrary sec- 
 tion of an oblique cone with circular base, of which the existence 
 was geometrically demonstrated by Apollonius of Perga, more 
 than two thousand years ago (in the Fifth Proposition of his First 
 Book upon Conies). And the equation found in the present ar- 
 ticle, for the plane of such a subcontrary section, expresses ano- 
 ther known and remarkable property of that section, or of the 
 cone to which it belongs ; namely, that this subcontrary plane is 
 parallel to the plane 
 
 which touches at the vertex o, the sphere oqrs, circumscribed 
 about that vertex o, and about the given circular base qrs (see 
 arts. 417, 418). 
 
 421. Again, let the same cone be supposed to be cut by a 
 concentric sphere ; that is (416), by a sphere whose centre is at 
 the vertex of the cone, and therefore (here) at the origin o of 
 vectors; while the length of its radius shall be represented by 
 some given and constant number, c. One form of the equation 
 of this sphere is (see 110), 
 
 Tp = c; 
 another form (by 111) is, 
 
 ^2 + c- = ; 
 and another is, 
 
 8^-^ = 0, 
 
 fi + y 
 
LECTURE VII. 407 
 
 if 7 be the given vector of some one point upon the spheric sur- 
 face, as appears by changing a to y, and /3 to - -y, in the last 
 equation of 4 15. If then we combine any one of these three 
 forms for the equation of the sjohere, with any one of the forms 
 lately given for the equation of the concentric cone, or any legi- 
 timate transformation of the former with any such transformation 
 of the latter, we shall obtain a system of two (scalar) equations, 
 which will represent a spherical conic (see again 416). The 
 two planes through the vertex, or centre, o, which are pa- 
 rallel respectively, to the two sets of circular sections of the 
 oblique cone, have been named by M. Chasles the two cyclic 
 PLANES of that cone; thus, for the cone whose equation is 
 
 ses2 = i, 
 
 a p 
 
 the two cyclic planes have for equations 
 
 •which may also be thus written (compare 420), 
 
 S . ajo = 0, S . j3jO = 0, 
 or thus, 
 
 S . (oa = 0, S . pj3 = 0. 
 
 The same eminent geometer has given the name of cyclic arcs 
 (compare 296), to the two great circles, wherein the sphere 
 round the vertex is cut by the two cyclic planes ; the equations 
 of one cyclic arc may therefore here be written thus, 
 
 S . ap = 0, Tp =c ; 
 
 and those of the other cyclic arc as follows, 
 
 S . /3jO = 0, T^ == c ; 
 
 but these equations admit of various transformations, which have 
 in part been indicated already. The results of this article and of 
 the one preceding it may be illustrated by reference to the figures 
 58, . . . 64, of arts. 294, ... 301. 
 
 422. As another geometrical example of the utility of consi- 
 dering the scalar parts, of the quotients or products of any two 
 
408 
 
 ON QUATERNIONS. 
 
 directed lines, and of employing the notation S^', let us propose 
 to draw from a given external point s, a rectilinear tangent st, 
 to a given sphere round o, 
 as in the annexed figure 
 90. Let o be origin of 
 vectors, and let 
 
 Fig. 90. 
 
 R^ 
 
 OS= 0-, 
 OA = a. 
 
 Ta 
 
 7-, 
 
 -a. 
 
 p 
 
 
 
 
 .--' 
 
 \^ 
 
 ,.. ,_ 
 
 
 1__T 
 
 "'■'55>.^'' 
 
 
 "% 
 
 ,// 
 
 95 
 
 T^ 
 
 ^ 
 
 \ 
 
 
 A C 
 
 "/ 
 
 ^S 
 
 A being the point where 
 the line os crosses the 
 given spheric surface ; 
 then, either because the 
 sought point of contact t 
 must be situated at once 
 on the given sphere round o, and also on that other known sphere 
 through o, which has the bisecting point c of the given line os for 
 centre, or has that line os for a diameter; or because the length 
 of OT is =a, and the angle ots is right; we have the two equa- 
 tions of condition (compare 421, 414), 
 
 -a^, S . 
 
 1 
 
 and therefore, by multiplying them together, we obtain this third 
 equation, 
 
 S . (TT = - a^ ; 
 which gives. 
 
 r a" 
 
 S~ - ;» 
 
 and expresses therefore (see 413) that the sought point t is 
 situated on a certain known jo/ane, perpendicular to a or to os, 
 and crossing that known line in a point m, of which the vector is 
 
 /x = OM = -a^cr"^ 
 
 Conversely, if the point t be taken anywhere on the circumfe- 
 rence of that circle, in which this plane intersects the given 
 spheric surface, and of which intersection the equations are 
 
 T- = - a-, S . (7r = - a"-. 
 
LECTURE VII. 409 
 
 then that point t will also satisfy the condition, 
 S . err = tS or S - = 1 ; 
 
 T 
 
 but this last equation gives, by 414, the perpendicularity, 
 er-T _L T ; and thus, the angle ots being right, the line st will 
 be, as was required, a tangent to the sphere round o. We are 
 therefore led, by this easy process of calculation, to recognise 
 the well-known cone of tangents, drawn from the external point 
 s, and the circle of contact (with m for centre), along which that 
 cone envelopes the given sphere. And as regards the plane of 
 this circle, the equation of that plane may be thus written (with 
 the recent signification of /i), 
 
 s- = i; 
 
 where, because fx = -a^a''^, we have (by principles already ex- 
 plained, respecting tensors, versors, and reciprocals), 
 
 That is to say, om has the same direction as os ; and the rectan- 
 gle under om and os is equal to the square of the given radius 
 DA : in fact we may write, 
 
 fKT = (- a^ =) a^. 
 
 423. Whether the given point s be (as above) an external, 
 or a superficial, or even an internal point, with respect to the 
 given sphere, provided that it be not actually at the centre o, 
 we can always deduce from its vector a a finite and connected 
 vector, ju = -a^(7"^, or, in other words, we can determine a con- 
 nected point M, which shall satisfy the conditions recently as- 
 signed, respecting distance and direction ; and then the plane 
 which is drawn through this point M, perpendicularly to om or to 
 OS, is said to be the polar plane of the point s, with reference 
 to the given sphere; while this point s is said, conversely, to be 
 the POLE of that plane : and any point p, upon the polar plane, 
 is said to be conjugate to s. To express these conceptions with 
 the notations of the present calculus, we may denote op by p, 
 and then shall have the following equation of the polar i^lane : 
 
4l0 ON QUATERNIONS. 
 
 g-= 1 ; or S . p(T = - tt^ ; 
 
 such then is the condition for the variable vector p (from the 
 centre o) terminating in a point p, which is conjugate to the 
 given point s, wherein the given vector a terminates. And be- 
 cause we may also write the last equation as follows : 
 
 S . crp = - a^, 
 
 we see that the relation oftivo conjugate points is one of reci- 
 procity, or that the polar plane of p passes in turn through s, 
 as is exhibited in figure 90. It is true that this reciprocal rela- 
 tion between two conjugate points is perfectly well known to all 
 who are even moderately acquainted with geometry ; but it 
 seemed to be useful to reproduce it here, as being a consequence, 
 or an interpretation, in this calculus, of the identical equation, 
 
 S . pa = S . (7jo, 
 
 which expresses that any two conjugate products, such as pa and 
 ap, have a common scalar part (compare 89, 408). And this 
 seems to be a convenient opportunity for remarking, that each of 
 these two equivalent symbols, S.pcr and S .ap, may be inter- 
 preted as denoting the rectangle under the two lilies, p and a, mul- 
 tiplied by the cosine of the supplement of the angle between 
 them ; or that, in symbols, 
 
 S . po- = Tp To- cos (tt - p^), 
 
 if per denote the angle between the directions ot p and a. In fact 
 this last formula may also be thus written, 
 
 S U . |0(7 = cos (tt - pa) ; 
 
 and accordingly, we have seen (in 411) that in general, for any 
 quaternion q, 
 
 ^\]q= cos Z q, 
 
 and also (in 88, 118) that 
 
 L . pa = TT — L . pa'^ = tt - pa. 
 
 (In the Fourth Lecture the symbol q was used in a somewhat 
 different sense, but only as a temporary notation.) 
 
LECTURE VII. 411 
 
 424. The geometrical signification of the scalar part, S . j3a, 
 of the product of any two inclined vectors, a and j3, may also be 
 deduced as follows, from principles already laid down, without 
 any reference to cosines, or polars, or circles : and may afterwards 
 be applied to form expressions for certain other geometrical loci. 
 
 Since a^ is a (negative) scalar, we have by 407, 410, and by 
 the properties (118) of reciprocals of vectors, the transforma- 
 tions (compare 420) : 
 
 S . /3a = a^S . j3a"^ = a^ . |3'a"^ =/3'a ; 
 
 if j3' denote, as in fig. 85, art. 406, the projection of j3 on a, or 
 the part or component of the given vector j3, which has either 
 the sa7n€ direction as the other given vector a, or else the oppo- 
 site direction, according as the angle j3a, between a and (5, is 
 acute or obtuse; while this projection vanishes, like the S' of fig. 
 86, art. 412, when the angle between the two given vectors is 
 right. But, by art. 84, the product of any two similarly directed 
 lines in space is (in this whole calculus) a negative number, while 
 the product of two oppositely directed lines is equal, on the con- 
 trary, to a positive number ; and when one of the lines vanishes, 
 their product vanishes also. With respect then to the sign of the 
 scalar part of j3a, since this part has been just now shewn to 
 be equal to the product j3'a, we may establish the formula : 
 
 <1 ^ ^ IT 
 
 S . /3a = 0, according as /3a = - ; 
 
 the contrast of which to the first formula of art. 412, or to the 
 following, 
 
 S . /3a'"^ = 0, according as /3a = ^, 
 
 is remarkable, but is a necessary consequence of our principles. 
 In fact, as we have seen, the product /3a may be formed from 
 the quotient jSa'^, by multiplying the latter by the square of the 
 vector a, which square (by 85) is always a negative scalar; the 
 versor of the product /3a is therefore simply the negative of the 
 versor of the quotient j3a'^ (see 188, 113); and consequently we 
 may write, 
 
 U.^a = -U./3a-S 
 
412 ON QUATERNIONS. 
 
 which gives immediately this other relation, 
 
 - SU.j3a = -SU./3a-'. 
 
 The supplementary character (referred to at the end of the last 
 article), of the angle of the product, 15a, as contrasted with the 
 angle of the quotient, (5a~^, which it is of great importance to 
 remember, in the geometrical applications of this calculus, may 
 also be deduced anew, or if it had been forgotten it might be re- 
 covered, from the consideration that since (by 111) a^ = - Ta^, we 
 have the transformation, 
 
 which shews that the two quaternions j3a and -j3a"S or the pro- 
 duct and the negative of the quotient of any two vectors, since 
 they diifer only by the scalar and positive factor Ta^, must have 
 one common angle ; while the angle of the negative of any quater- 
 nion q, is (by 183) the supplement of the angle of that quater- 
 nion itself. Thus the last formula of the foregoing article is re- 
 produced, under the form, 
 
 L ./3a = Z(-j3a"^) = 7r-Z.j3a-i = 7r-j3a. 
 
 And with respect to the magnitude, or numerical amount (ab- 
 stracting from the sign), of the scalar part of the product /3a, we 
 have, by the present article (compare 109, 110) : 
 
 TS.j3a-T./3a-Tj3'.Ta; 
 
 this sought numerical amount is therefore simply the numerical 
 value or expression for the rectangle under the one given line (a) 
 and the projection (j3') of the other line (j3) thereon. It is clear 
 that since the two conjugate products, j3ct and aj3, have always 
 (89, 408, 423) the same scalar part, so that 
 
 S . aj3 = S . j3a, 
 
 we must, by the present article, have the equation (see also 85), 
 
 a'/3 = )3'a, or j3a' = aj3', 
 
 if a' denote the projection of a on j3. And in order to ea'press 
 the projection j3', of any one line j3 on any other line a, we see 
 that we may write (compare 407), 
 
LECTURE VII. 413 
 
 j3'= S . j3a -i- a; or, j3'= S . j3a~^ X a; 
 
 or any legitimate transformation of either of these two expres- 
 sions, such as the following : 
 
 /3' = a-^S.j3a; or, /3'-aS. /3a- K 
 
 425. As a new application of these principles respecting the 
 scalar part of a product of two vectors, let us resume fig. 90, of 
 art, 422. In that figure, by the rudiments of geometry, the 
 square on the line ST is equal to the rectangle under so and sm ; 
 which last line, sm, is the projection of st on so. Now, when 
 directions are attended to, we have (by 422) the expressions, 
 
 so = -(7; ST = r-a-; SM = ju-o-; 
 
 and therefore (by recent results), 
 
 S . (c7 - r) (7 = S (ST X so) = SM X so = ((7 - /i) (7 ; 
 
 in which last product of lines the directions of the two factors are 
 similar, and therefore (by 84) the product itself is negative ; as is 
 also, for the same reason (85, 111, &c.) the square of r- tr. This 
 product and this square agree therefore in {\it''w signs, being, both 
 of them, negative scalars ; and their r\\xmenc-A\ magnitudes diX'&o 
 agree, because one expresses the area of the rectangle osm, and 
 the other the equivalent area of the square on the tangent st; 
 we may therefore equate them to each other, or may write, 
 
 ((7-/x)a = (o--r)2: 
 
 or, by the formula immediately preceding, 
 
 S . ((7 - r) cr = (cr - r)^. 
 
 In fact this is equivalent to the following, 
 
 S = 1, or S =1; 
 
 a - r r — (7 
 
 and when put under this last form, it expresses (compare 414) 
 that the projection of so on ST coincides with st itself, or that 
 the angle sto is right. But also, in the right-angled triangle 
 sto, the square of the hypotenuse is equal to the sum of the 
 squares on the two other sides, or, in symbols, 
 
 T(7^=T(<7-T)^ + Tr-; 
 
414 ON QUATERNIONS. 
 
 that is, by art. 422, and by principles with which we have now 
 become familiar, 
 
 - ff^ = - ((7 - r)^ + a~, or (r - a)^ = ct^ + a^. 
 Again, by what has been shewn in the present article, we have 
 
 {S.(T(r-a))^=(r-ay; 
 we may therefore write the equation, 
 
 {S.(r(r-o-)}2-(o-^ + a-) {t - af : 
 
 which must hold good, not merely for the particular point of 
 contact T in fig. 90, whose vector from o has been above denoted 
 by r, but for every other -pointy such as u in the same figure, 
 which is contained upon the circle of contact (perpendicular to 
 the plane of the figure). And because the formula last written 
 remains essentially unchanged, when r-o- is multiplied by any 
 positive or negative scalar, we see farther (compare the reason- 
 ing in art. 418), that if, to mark more clearly that t is now 
 treated as a variable vector^ we change that symbol to p, as in 
 some former expressions for geometrical loci^ the resulting equa- 
 tion, namely, 
 
 { S . C7 (p - cr) }-= ((T- + a-) (p - cr)-, 
 
 is the EQUATION OF THE ENVELOPING CONE, which has the ex- 
 tremity s of the vector cr for vertex, and touches the sphere, with 
 radius a, described round the origin o, along that circle of con- 
 tact of which one diameter is the chord tu. It is still more easy 
 to see, by analogous but shorter calculations, that if we conceive 
 a new cone, which shall have its vertex at the centre o of the 
 same enveloped sphere, and shall pass through the same circle of 
 contact (cutting the former cone perpendicularly along that cir- 
 cle), this new cone will have for its equation, if p be its variable 
 vector, 
 
 (S . apY + a- p"- = 0. 
 
 426. The symbol S enables us also to form with ease expres- 
 sions for right lines in space, considered as being each the 171- 
 tersection of two planes. Thus the intersection of the two cyclic 
 
LECTURE VII. 415 
 
 planes of the oblique cone (418) with circular base, of which cone 
 the equation may be thus written, 
 
 S . |oa"^ . S . /3p~^ = 1, 
 
 or the right line through the vertex of this cone, which is called 
 by Chasles the major axis, has its direction and position repre- 
 sented (see 421) by the system of the two equations, 
 
 S . OjO = 0, S . /3jO = 0. 
 
 Or to take a more elementary example, let it be required to re- 
 present by equations, on a similar plan, the polar of a given 
 RIGHT LINE, taken with respect to a given sphere, such as that 
 of which the equation is 
 
 |0^ + a- = ; 
 
 namely the sphere which has its centre at the origin o, and has 
 its radius =«. Supposing the given line to be determined by two 
 given points s, s' through which it passes, and writing 
 
 OP=jO, 08 = 0-, os'=(t', » 
 
 we may suppose that p is a variable point on the sought polar of 
 ss', and are to express that this point p is conjugate to both s and 
 s', or that it is situated in the intersection of their polar planes 
 (423) ; we have therefore, as the required equations of the polar 
 of the line ss', the following (see again 423) : 
 
 S.joar = — a"; S.p(T' = — tt^- » 
 
 Let p' be another point on this polar line, and let op'=jo'; then 
 in like manner, 
 
 S . pa = — a~, S . pa = - «" ; 
 
 we have therefore, 
 
 S . pa = - a- = S . pa, and S . pa =-a^ = S . pa ; 
 
 and consequently we see that the two given points s and s are (as 
 is well known) each situated on the polar of the new line pp' ; or 
 in other words, the continued equation, 
 
 S . pa=S . pa = S . pa = S . pa' = - a"^, 
 
 expresses that the two lines, pp' and ss', are reciprocal po- 
 lars of each other. (In fig. 90, the polar of ps would be a right 
 
416 
 
 ON QUATERNIONS. 
 
 line nn', drawn through the point n, at right angles to the plane 
 of the figure; and if n' be conceived to be on the surface of the 
 given sphere round o, the tangent plane to that sphere at that 
 point will pass through the right line PS.) 
 
 42Y. But however useful the symbol S may be, in thus form- 
 ing equations of loci, and otherwise applying the calculus of qua- 
 ternions, it is important to be familiar also with the signification 
 and employment of the connected symbol V : and indeed the 
 treatment of vectors is even more peculiarly the business of this 
 calculus, than operations upon scalars, although both must often 
 be combined. The signification of the vector part of the quo- 
 tient of two lines having been sufficiently explained in art. 407, 
 we can have no difficulty in interpreting now the vector part of 
 theiv product, on the same general plan as that by which we have 
 passed from the scalar of a quotient to the scalar of a product of 
 two lines. If |3" be, as in fig. 85, that part or component of the 
 vector j3 which is ■perpendicular to another given vector a, then 
 since, by 407, 
 
 we need only multiply both numbers by the scalar dr, and we find 
 the expression : 
 
 V./3a = i3"a; 
 
 where the symbol j3"c£ can at once be interpreted, by principles 
 laid down in former Lectures, respecting a product of two rect- 
 angular vectors. To make more clear the application of those 
 earlier principles to the present question, conceive that after 
 letting fall from B the perpendicular Bs'on OA, as in the recently 
 cited figure 85, we then, as in the 
 annexed figure 91, erect at o another 
 perpendicular ob" to the same line 
 OA, which new line ob" shall be pa- 
 rallel and equal to b'b, and shall have 
 the same (not the opposite) direction, 
 and may therefore (97, 98) be de- 
 noted by j3", as well as the former 
 line b'b itself; just as j3 may denote 
 AD as well as ob, if d be the point on e 
 
 Fig. 91. 
 
LECTURE VII. 417 
 
 b"b which completes the parallelogram aobd : although it ap- 
 pears more convenient here to make |3 still denote the final ray 
 OB of the biradial aob, which represents the quotient j3a'^, or q. 
 If now we conceive this figure 91 to be laid horizontally on a 
 table, with its face upward, it is clear that a right-handed and 
 quadrantal rotation, round the new multiplier line j3", would 
 cause the co-initial multiplicand line a to assume a downward di- 
 rection ; such therefore, by the rule of art. 82, must here be the 
 direction of the product line, /3"a, or V . /3a ; while the length of 
 that product line is, by another part of the same rule of 82, the 
 product of the lengths of the two factor lines, or is numerically 
 equivalent to the rectangle under oa and ob", or to the area of 
 the lately-mentioned parallelogram, aobd. On the other 
 hand, the axis of the quotient, namely Ax • jSa'S or \]Vq (411), 
 is, for the same supposed position or aspect (93) of the figure, a 
 line directed upward; and generally we see that the vector 
 PARTS of the PRODUCT j3a and quotient j3a"^ of any two 
 LINES, a and j3, have their directions opposite. In symbols, 
 if §' = j3a"^ = OB -^ OA, then 
 
 UV.j3a = -UV^; TV.j3a=/~/AOB; 
 
 this last symbol being employed to denote the area of the com- 
 pleted parallelogram, aobd, or the doubled area of the trian- 
 gle, AOB. 
 
 428. We know then perfectly how to interpret the symbol 
 V.j3a, or the vector of the product of any two lines proposed j 
 and with respect to the recently noticed relation of opposition^ 
 between the versors of the vectors of product and quotient, 
 
 UV./3a = -UV.j3a-', 
 
 we may regard this as connected with the analogous opposition of 
 signs (in art. 424) between the versors of the product and quo- 
 tient themselves, namely, 
 
 U.j3a = -U.i3a-i: 
 
 or with the circumstance (see again 424) that /3a only differs by 
 the positive factor IV from the negative of j3a*^; at least if we 
 combine this circumstance with the formula of art. 183, for the 
 axis of the negative of a quaternion, namely, 
 
 2 e 
 
418 ON QUATERNIONS. 
 
 Ax - (-q) = - Ax . g. 
 
 Or we may consider the opposition of the axes (or of the versors 
 of the vector parts), of the product and quotient of two lines, as 
 being a consequence of the opposite characters of the two corres- 
 ponding rotations^ from the multiplier j3 to the multiplicand a, in 
 the product (5 x a (arts. 87, 88, &c.), and from the divisor line a 
 to the dividend line j3, in the quotient /3 -f- a (40, 118, &c.) ; or 
 in the two quaternions, which are equal to this product and this 
 quotient respectively, when those quaternions are regarded as 
 operating in the way of version. And in the geometrical appli- 
 cations of this calculus, it will be found important to remember 
 that the rotation round the line V . j3a from j3 to a is positive; 
 whereas the positive rotation round V.j3a"^ conducts on the 
 contrary from a towards j3. Observe the contrasted directions 
 oit\\o%e tvio curved arrows in the recent figure 91, which are 
 marked respectively, q and |3"a ; also the similarity of the direc- 
 tion of this last arrow to that which corresponds to K^'. It may 
 also be noticed here, as one of the connexions of quaternions 
 with trigonometry^ that whereas, by 423, 
 
 A 
 
 S . j3a = - T/3 Ta COS j3a, 
 we have now, 
 
 TV.^a = + Tj3Tasinj3X 
 
 j3a Still denoting the acute or right or obtuse angle between the 
 two lines a and j3. Or we may write more simply the two trigo- 
 nometrical transformations, 
 
 SU.j3a = -cosj3a; TVU./3a = + sinj3a; 
 
 and may regard these expressions as being connected with the 
 corresponding ones of art. 411, through the supplementary cha- 
 racter (118, 423) of the angle of the product of two lines, as com- 
 pared with the angle of the factors. 
 
 429. It is evident from the two last articles, and especially 
 from the formulse, 
 
 V.j3o = j3"a; 13" ± a; jS" |I! /S, a, 
 when combined with our general principles respecting products of 
 
LECTURE VII. 419 
 
 rectangular lines, that the vector of the product^ as well as the 
 vector of the quotient, of any two inclined lines a, j3, is perpetidi- 
 cular to both those lines, and therefore to their plane : thus ge- 
 nerally, 
 
 V.j3aJ.a; V./3a±j3. 
 
 Hence, although we may write (compare the two first expressions 
 for /3', towards the end of art. 424), the two following general ex- 
 pressions for the part /3 " of any vector /3, which is perpendicular 
 to a given vector a, 
 
 /3"=V./3a-a = V.i3a-»xa, 
 yet we must not transform these expressions into the following, 
 
 /3"=a-iV./3a, /3"=aV.i3a-i: 
 
 because the two products of rectangular vectors, 
 
 o"^ X V . j3a, and a X V . jSa'S 
 
 undergo each a change of sign (by 82), when the order of their 
 factors is changed. For the same reason, however, we 7nay write 
 the two following general expressions for the component /3" of /3 
 (contrast with these the analogous expressions for the other com- 
 ponent /3', given at the end of 424) : 
 
 Again, the vector part of the product of any two lines 
 
 a, /3, CHANGES SIGN WHEN THE TWO FACTORS ARE INTER- 
 CHANGED ; or in symbols, 
 
 V.ai3 = -V.i3a, 
 
 whatever may be the angle which a and j3 make with each other t 
 in fact, by 89 and 408, 
 
 aj3 = K.)3a, andVK = -V. 
 
 This conclusion may be illustrated by the recent figure 91, in 
 which the three points c, e, c", and the two vectors 7, 7", may 
 be said to be the reflexions of the three other points b, d, b", and 
 of the two other vectors j3, /3", with respect to the line oa, or a. 
 For, in this figure 91, without «^ p-esew^ assuming any know- 
 ledge of the formula 
 
 2 E 2 
 
420 ON QUATERNIONS. 
 
 which would be given by the principles of the Sixth Lecture 
 (see arts. 290, 291), we may see that we must have the equation, 
 
 for these two last products are quaternions with equal tensors, 
 and with equal versors ; because the two parallelograms, ecoa 
 and AOBD, have equal areas and angles, and have also one com- 
 mon aspect ; or because the rotation from y to a is equal in all 
 respects to that from a to j3, while the lengths of the lines j3, y 
 are equal, so that 
 
 U.7a = U.a/3, T.7a=T.a/3. 
 Hence, 
 
 Y.a(5 = Y.ja = y"a = -(i"a = -Y.(5a, 
 
 because y'^-jS'j in the same fig. 91. We have therefore also, 
 
 V.a-ij3 = -V.j3a-S 
 
 because (by 117) the reciprocal of a vector is itself another vec- 
 tor ; and therefore are at liberty to establish the two following 
 formulae, as genet'al expressions for the component /3" of /3, which 
 is perpendicular to a : 
 
 /3"=a-^V.aj3 = aV.a-ij3; 
 
 in addition to the two other expressions for the same component 
 
 j3"=V.j3a.a-^= V.j3a-^a, 
 
 which agree with the two first of those considered in the present 
 article. 
 
 430. Let p, in fig. 91, be any arbitrary point on the indefi- 
 nite right line, which is drawn parallel to a or to oa, through the 
 point B ; and let its vector op be denoted by p. Then the com- 
 ponent of this vector p, which is perpendicular to a, is still ob", 
 or j3"; and consequently we have the equation, 
 
 V.|Oa = i3"a = V.j3a. 
 
 Conversely if we meet the equation, 
 
 V.pa=V.j3a, 
 
LECTURE VII. 421 
 
 where a is still supposed to denote some given and actual ('or 
 non-evanescent) line, we can infer from it, by the foregoing arti- 
 cle, that the components of j3 and p which are perpendicular to 
 a are equal ; and therefore that these two vectors, j3 and p, can 
 only differ in their components parallel to a ; or more concisely, 
 we can, from the last written equation, infer the parallelism , 
 
 p-/3 II o; 
 
 which may also be thus denoted, under the form of another equa- 
 tion, freed from the symbol of operation V, but introducing in 
 its stead another letter x, to denote an arbitrary scalar co-effi- 
 cient, 
 
 p = (5 + a!a. 
 
 Any one of the formulae involving p, in the present article, will 
 therefore express that this variable vector p terminates in a point 
 p, of which the locus is the right line, drawn through the ex- 
 tremity of the vector j3, and parallel to the other given vector a : 
 or in connexion with figure 91, it will express that the locus of p 
 is the indefinite right line which is drawn through b and b". And 
 because the product of two parallel lines is (by 84) a scalar, 
 which has (407, 412) no vector part, we may substitute for the 
 recent formula of parallelism, this other equation : 
 
 V.(p-j3)a = 0; 
 
 which will therefore serve to express the same rectilinear locus 
 as that expressed by the former equation, 
 
 Y.pa = Y.(da, 
 
 whereof indeed it will soon be found to be, by the distributive 
 principle, a transformoMon. It may here be noted that, by 
 making j3 = 0, we obtain the following equation for the indefinite 
 right line, whereof oa or a is a given part, 
 
 V.|oa = 0. 
 The equation 
 
 V(|oV./3a) = 0, orV.pV.^a = 0, 
 
 would express that p had the direction of + V . /3a, or (by 429) 
 
422 ON QUATERNIONS. 
 
 that it was perpendicular to the plane of a and j3 ; whereas this 
 other equation, 
 
 would express that p was perpendicular to that perpendicular, 
 or that the three lines a, j3, p, were coplanar. In general, the 
 two symbols, 
 
 V.pV.jSa-V.jSa, and'S.pV.jSaH-V.jSa, 
 
 denote those two parts or components of any proposed vector p, 
 which are respectively coplanar with a, j3, and perpendicular to 
 the plane of those two lines. 
 
 431. If with the recent significations of a, j3, j3", 7, 7", we 
 oblige the variable vector p to satisfy this other equation, 
 
 V.pa = -V./3a, 
 
 we shall then have (by 429), 
 
 V . pa = V . a/3 = V . 70 = 7"a, 
 
 and the component of p, perpendicular to a, will coincide with 
 the corresponding component 7" of 7 ; we shall therefore have 
 (by the principles of the last article) the formulae, 
 
 p - 7 II a, |0 = 7 + A'a, V . (p - 7) a = 0, 
 
 where x is still an arbitrary scalar. The locus of p will, therefore, 
 in this case, be the indefinite straight line through c, in fig. 91, 
 which is parallel to the given line oa. And if, instead of equat- 
 ing Y . pa to ± V,j3a, we should equate only their squares or 
 their tensors, writing, 
 
 {Y.pay-{V.^a)\ 
 
 or, 
 
 TV.pa = TV.j3a; 
 
 we should then express merely that the length of the component 
 of p, perpendicular to a, was equal to Tj3" ; or that such was the 
 length of the perpendicular from the point p on the indefinite 
 right line through oa: or finally, that the locus of p was a cy- 
 linder OF REVOLUTION, with that line oa for its axis, and with 
 B for one of the points upon its surface. Another mode of ar- 
 
LECTURE VII. 423 
 
 riving at this cylindrical locus for p, as the geometrical interpre- 
 tation of the last written equation in p, is to observe that this 
 equation shews (by 427) that the two triangles, aob, aop, with 
 the common base Oa, have their areas (or more immediately 
 their doubled areas) equal in amount ; from which it follows that 
 their altitudes must be equal, at least in length : or that their 
 two vertices, b and p, are at equal "perpendicular distances from 
 the common base, oa. In fig. 91, the cylinder in question would 
 be generated by the revolution of the indefinite right line be", 
 round the line oa as an axis. And if we choose to leave the dia- 
 meter, or the thickness, of the cylinder round this axis undeter- 
 mined, we have only to assume that 2aTa'^ is equal to some po- 
 sitive and constant although arbitrary scalar, denoting the length 
 of the diameter, and to write the equation, 
 
 TV.pa = a; or, (V.pa)2 + «^=0. 
 
 For the same reason the equation, 
 
 TV.pj3-^=6, or(V.joj3-0' + 62 = 0, 
 
 will represent another cylinder of revolution, whose radius is 
 = 6Tj3, and whose axis, passing through the origin, coincides in 
 position with the given vector j3, while p denotes the variable 
 vector of an arbitrary point upon this new cylindrical surface. 
 432. If this last cylinder be cut by the plane 
 
 which is perpendicular to its axis of revolution, the section must 
 evidently be a circle; and accordingly the present calculus re- 
 cognises this result, by giving, as a consequence of the two equa- 
 tions last written, another equation representing a sphere, on the 
 surface whereof this intersection of the plane and cylinder must 
 be contained, namely, 
 
 because we have, in general, by 409, for the tensor of any qua- 
 ternion q, the expression, 
 
 Tg={(S#-(V^)^}^={(S^)^ + (TV^)^}i 
 Conversely, if we cut the sphere 
 
424 ON QUATERNIONS. 
 
 T.p(5-'=l, orTp=Tj3, 
 by the pJane 
 
 S./o/3"^ = a;, where aj> - 1, a;< 1, 
 
 the circle of intersection will be contained upon that cylinder of 
 revolution which has for its equation, 
 
 Or if (under the same supposition as to the limiting values of the 
 scalar x) we conceive the last-mentioned sphere, whose equation 
 may be thus written, 
 
 to be cut by the last-mentioned cylinder, their intersection will 
 be a system of two circles, at equal distances from the centre, 
 which are situated in two parallel planes, represented by the 
 equation, 
 
 (S .|oj3"^)- = a;^ or S ./>j3"^ = + a?. 
 
 And the surface of the sphere itself may be regarded as the locus 
 of the variable circle, which has for its equations, 
 
 S.pjS-i-a;, TV.pp-i = (l-rr2)i; 
 
 and which is (by what has just been seen) a perpendicular sec- 
 tion of a certain varying cylinder made by a certain connected 
 and varying plane. 
 
 433. This being distinctly seen, let us next conceive that the 
 last cylinder in art. 431 is cut obliquely, by a plane perpendicular 
 to some new given vector a, which is inclined at some acute or 
 obtuse angle to the axis j3 of the cylinder ; we shall then have a 
 system of two equations, of the forms, 
 
 and the curve of intersection, which those equations represent, 
 will evidently be an ellipse. Now that important surface which 
 is called by geometers an ellipsoid may be generated by the 
 motion of such an ellipse, if this curve be regarded as variable in 
 magnitude, as well as in position : and the following is one mode 
 of accomplishing such a generation, or of obtaining a system of 
 
LECTURE VII. 
 
 425 
 
 ellipses, whereof the ellipsoid shall be the locus : just as the 
 sphere has recently been regarded as the locus of a system of 
 circles. 
 
 434, In figure 92, let oa, ob be two given lines drawn from 
 
 Fig. 92. 
 
 a given point o, and making a given acute or obtuse ancle with 
 each other. In the plane of these two lines, and at their re- 
 spective terminations a and b, let two perpendiculars ac, bc be 
 drawn, meeting in a known point c, and join oc ; also let ob and 
 CA (prolonged if necessary) meet in another fixed point b' : and 
 let F, f' be such that o shall bisect bf, b V. ■ In the same given 
 plane describe the circle dbef, with o for centre, and with the 
 diameter de parallel to the tangent cb ; draw also two other tan- 
 gents at D and e, and let them meet, in the points d' and e', a 
 right line drawn through o, perpendicular to oa, or parallel to 
 the line cab'. From any point g on the finite line oc, let a pa- 
 rallel to de or cb be drawn, cutting the semicircle in l and n, 
 
426 ON QUATERNIONS. 
 
 and the radius ob in m ; take also any other point q upon the 
 chord LN ; through the three points l, q, n draw three lines 
 parallel to ob, and let these three parallel lines be cut respec- 
 tively in the three points l', q', n', by a new line from g, which 
 new secant shall be drawn parallel to d'e', or to cb', and shall 
 also cut the line ob or om in a new point m'. The figure 
 being thus constructed in the plane, conceive next that the 
 indefinite right line through d and d' turns round ob as an 
 axis, till it takes the position of the indefinite line through e 
 and e', describing thus a semi-cylinder of revolution ; and con- 
 ceive, in like manner, that the indefinite line ll' turns round the 
 same axis ob, till it assumes the position of nn', describing thus 
 another semi-cylinder of revolution, co-axal with the former, but 
 having a smaller radius (namely ml, instead ofoo). Imagine 
 that the first semi-cylinder is cut by a pair of planes, perpendicu- 
 lar to the plane of the figure, and passing through the lines de, 
 d'e'; and that the second semi-cylinder is cut by another pair of 
 planes, which shall be parallel to the former pair, and shall pass 
 through the lines ln, l'n'. And finally, let the second semi-cy- 
 linder be also conceived to be cut in two points p, p', by two 
 right lines qp, q'p', which are erected at q and q', perpendicu- 
 larly to the plane of the figure: and let us consider what the 
 LOCI of these two new points, p and p', not expressly marked in 
 the diagram, or what the loci of the two sections of the second 
 and varying semi-cylinder must by this construction be. 
 
 435. I say then that while the locus of the point p, con- 
 structed as above, is very easily found to be the quarter of the 
 surface of a sphere, resting upon the semicircle dlbne (if we 
 still oblige the auxiliary and variable point q to be inside that 
 semicircle, and employ still only sem-cylinders), the locus of the 
 connected point p' is (under the same restrictions) the quarter of 
 the surface of an ellipsoid, resting on the semi-ellipse d'l'b'n'e', 
 and having the same point o for its centre. In other words, I re- 
 mark that as the above-mentioned portion of the sphere is (com- 
 pare 432) the locus of the varying 5e»^^c^rc/e which has ln for its 
 varying diameter, while the centre m of that semicircle moves from 
 o to B, so the corresponding portion of a certaiji derived ellipsoid 
 is (compare 433) the locus of the vanjing semi-ellipse, which rests 
 
LECTURE, VII. 427 
 
 on l'n' as its variable major-axis ^ while its centre m' changes its 
 position, from o to b' : each of the two last-mentioned curves being 
 a section of the inner and varying semi-cylinder made by a vary- 
 ing plane, which moves so as to be always parallel to itself, or to 
 a fixed plane, and perpendicular to the plane of the figure. In 
 fact, for the point p we have evidently, by the circular section of 
 the inner cylinder, 
 
 MQ2 + Qp2 = MP2 = ML^ = OL^ - OM% 
 
 and therefore 
 
 OP^ = OM^ + MQ2 + QP"^ = OL^ = 0B% 
 
 so that the locus of p is (as above stated) a portion of the sphere 
 round o, with ob for its radius ; or is simply the whole surface 
 of that sphere, if we now allow it to belong at pleasure to the 
 other variable semi-cylinder, at the other side of the plane of the 
 figure, and to have its projection q, on that plane, situated within 
 the other semicircle, dfe, which is described on de as diameter. 
 And (with the analogous removal of restrictions) the locus of the 
 connected and variable point p'is almost as easily shewn to become 
 (as above asserted), after the foregoing process of deformation of 
 this spheric surface, what is called by geometers an ellipsoid. 
 For we have, by similar triangles in the plane of the figure, the 
 relations. 
 
 OM 
 
 OG OM 
 
 MQ 
 
 MQ 
 
 
 
 
 ^ — 
 
 OB 
 
 OC OB 
 
 od' 
 
 OD 
 
 and, by the rectangle qpp'q' perpendicular to that plane, we have 
 an equality between the two ordinates qp and q'p', which termi- 
 nate on one common side, or rectilinear generatrix, pp', of the 
 inner cylinder; hence 
 
 qV -4- 0C'= QP -r- OC', 
 
 where oc' may be supposed to be an ordinate or perpendicular to 
 the plane of the figure, erected at the centre o, and terminating 
 on the sphere, or on the outer cylinder, at a new point c'. Hence 
 p' must satisfy the equation, 
 
 OB y V OD / V OC 
 
428 ON QUATERNIONS. 
 
 because the point p, on which it depends, is subject to the analo- 
 gous equation, 
 
 omV /mqV /qpY_, 
 
 OB/ yOTiJ \OC'J 
 
 I suppose that many of you may have already perceived 'that 
 b', c, d' are three conjugate summits of the ellipsoid, or that ob'^ 
 oc', o'D ^xe three conjugate semi-diameters thereof: oc' being the 
 mean semi-axis, and ob', od' being contained in the principal 
 plane, or in the plane of the focal hyperbola, whereof one asymp- 
 tote coincides in position with ob'; because this last line is the 
 axis of a cylinder of revolution, circumscribed about the ellipsoid, 
 namely, the outer cylinder in our construction : but it is by no 
 means necessary to be acquainted with these latter properties of 
 the ellipsoid, in order to understand that translation of the con- 
 struction of the foregoing article into the language of quater- 
 nions, which we are now about to give. 
 
 436. The two lines oa, ob, in fig. 92, from which, as data, 
 everything else in the figure has been constructed, being treated 
 as two given vectors a, /3, it is clear from the principles of 
 this calculus (see art. 413, and other recent articles), that the two 
 planes through o which are respectively perpendicular to these 
 two lines, and which cut the plane of the figure along d'e' and 
 DE, have for their respective equations : 
 
 S.^a-i-0; S.p[5''=0; 
 
 while the two planes parallel to these, which have CB'and cb for 
 their traces on the same plane of the figure, have for their equa- 
 tions the following : 
 
 In like manner, if we make]^for abridgment, in reference to the 
 same fig. 92 (compare 435), 
 
 aj = OG 4- oc = OM -^ oB = om' -t- ob', 
 the equations 
 
 S.pa'^ = X, S.p(5'^ = x, 
 
 will denote those two other planes, which cut the plane of the 
 figure perpendicularly along the lines gm', gm ; or which cut oa. 
 
LECTURE VII. 429 
 
 OB perpendicularly at points whose vectors are xa, x[5 (the latter 
 of these two points being m). Again the equations of the outer 
 and inner cylinders (through DD'and ll'), which have the hne ob 
 or j3 for their common axis, are respectively, by the principles of 
 431, 432, 
 
 TV.pi3-^=l; TV.pj3-i = (l-^0^; 
 or 
 
 because the radius od of the former has the same length as ob 
 or as j3 ; while the radius ml of the latter, when divided by od, 
 gives {\ - x'^)^ for the quotient. Thus whereas the Jixed cir- 
 cle on DE, perpendicular to the plane of the figure, in the con- 
 struction of art. 434, is represented by the two equations, 
 
 the corresponding^o^e^ ellipse on d'e', in the same construction, 
 is represented by this other pair of equations, 
 
 which are included in the general equations of art. 433. And 
 while the varying circle on ln is represented by the two last 
 equations of art. 432, or by the following, 
 
 S.p{i-' = x, {Y.p(5-^y = x^-l, 
 
 the equations of the varying ellipse on l'n' may be thus written : 
 
 S.pa-^ = X\ (V.pj3-0' = ^'-l. 
 
 Finally, as one form for the equation of the sphere, which is 
 the locus of the system of circles, may be obtained by elimina- 
 tion of X between the two equations of a variable circle of that 
 system, and may (as in 432) be written thus, 
 
 (S.p/3-y-(V.^/3-0^=l; 
 
 so may the corresponding form of the equation of the ellipsoid, 
 which is the locus of the system of ellipses (in the recent con- 
 struction), be obtained by an analogous and equally easy elimi- 
 nation of the same variable x, between the two equations of a 
 
430 
 
 ON QUATERNIONS. 
 
 variable ellipse : and this equation of the ellipsoid is in this 
 way found to be, 
 
 or, 
 
 And we may here remark that another form of this important 
 equation is the following : 
 
 T(S.^a-i + V.pj3-0 = l; 
 
 because (by 409, or 432) the square of the tensor of the quater- 
 nion, whose scalar and vector parts are, respectively, 
 
 S.pa-i and Y . p^-\ 
 
 is equal to the square of the scalar, minus the square of the vec- 
 tor part. When the distributive principle of multiplication of 
 quaternions shall have been established generally, it will be 
 found that this last form of the equation admits of a new and in- 
 dependent geometrical interpretation ; and that it conducts 
 thereby to an entirely new mode of constructing (or generating) 
 the ellipsoid. 
 
 437. After the foregoing Fig. 93. 
 
 details respecting one mode 
 of constructing the ellipsoid, 
 and of expressing that con- 
 struction by quaternions, it 
 may suffice to state more 
 briefly the analogous methods 
 of constructing and expressing 
 certain other surfaces of the 
 second order, especially the 
 hyperboloids and the cone, 
 and of connecting each of 
 these surfaces with the sim- 
 plest surface of its own spe- 
 cies. In the annexed figure 93, 
 although for the sake of con- 
 venience reduced in size, the 
 letters 0, a, b, c, d, e, f, b', d', 
 
LECTURE VII. 431 
 
 e', f', may be conceived to denote the same points which were so 
 marked in the recent diagram 92 ; the point g is now taken on 
 DC prolonged, and h is such that o bisects gh ; lbn is an arc of 
 an equilateral or rectangular hyperbola, with bf for its transverse 
 axis, and zox, woy for asymptotes; the two secants from g, 
 which are now the lines gxlmqny and gx'l'm'q'n'y', are still pa- 
 rallel to the two fixed lines cb, cb', to which the lines hzw, 
 HzV are also parallel ; q is still an arbitrary point on the chord 
 LN, and the lines ll', qq', nn' are still perpendicular to de, or 
 parallel to f'fobmb'm', as also are the new lines ww', xx', yy', 
 zz' ; ll' is still imagined to generate a cylinder of revolution, by 
 turning round ob as an axis, and qp, qV are still supposed to be 
 ordinates, perpendicular to the plane of the figure, and terminat- 
 ing on one of the generating sides pp'of this cylinder ; oc'is still 
 conceived to be a parallel ordinate, which terminates on the co- 
 axal cylinder described by the revolution of dd', or on the sphere 
 with DE for diameter; finally we are to conceive that qr, q'r' 
 are two other ordinates to the same plane of the figure, termi- 
 nating on a side rr' of the cylinder formed by the revolution of 
 xx' round the same axis ; and the two infinite branches of the 
 hyperbola lbn, together with its asymptotes zox, woy, are sup- 
 posed to turn through 180° round the same line ob, and so to 
 generate the two sheets of an equilateral hyperboloid of 
 REVOLUTION, together with the two corresponding sheets of its 
 ASYMPTOTIC CONE. This process (which closely resembles that 
 of art. 434) being once distinctly conceived, and combined with 
 elementary properties of the hyperbola, it becomes clear that the 
 hyperboloid and cone, thus formed, are respectively the loci of 
 the points p and r, and that these two points satisfy respec- 
 tively the two equations, 
 
 MQ^ + Qp2 = OM^ - OB^ ; 
 
 mq'^ + qr2 = OM^ : 
 
 whence the two connected or derived points, p' and r', must sa- 
 tisfy the two connected equations, 
 
 m'q'V /q'p'V /om'\2 , 
 
432 on quaternions. 
 
 m'q'Y /q'r'Y /om' 
 on' J \oc J \ob' 
 
 And hence again it follows, if we here admit as known some ge- 
 neral and simple results respecting surfaces of the second order, 
 that the locus of p' is another hyperboloid of two sheets, 
 and that the locus of r' is another cone of the second de- 
 gree, namely the asymptotic cone of the new hyperboloid ; 
 although neither of these two new surfaces, produced by this sort 
 of deformation, will be (with the construction here employed) a 
 surface oi revolution. A section of one sheet of the new hyper- 
 boloid is the hyperbolic curve l'b'n'; and two sides of the new 
 cone are the two asymptotes to this curve, namely the lines z'ox' 
 and w'oy'. The hyperboloid, which is in this article the locus of 
 p', touches the ellipsoid of art. 435, at the two points b' and f'; 
 as the other hyperboloid of two sheets touches the concentric 
 sphere, described on de as diameter, at the points b and f. 
 
 438. To translate now the foregoing construction into the 
 language of quaternions, we may adopt nearly the same plan as 
 in art. 436. The varying circle in which the hyperboloid of re- 
 volution lbnp, or the cylinder ll'nn', is cut by the plane lpn, 
 has for its equations, 
 
 S.|o/3"^ = a;, TV. pj3'^ = (a;^ - 1)^5 where a; = OG -i- oc ; 
 
 and the varying ellipse in which the same cylinder of revolution 
 through ll' is cut obliquely by the plane l'p'n', has for equations, 
 
 ^.pa-^ = x; TV.joj3-i = (A'^--l)^. 
 
 Eliminating therefore the variable scalar x, between the two 
 equations of the circle, we find for the hyperboloid of revolu- 
 tion, or for the locus of that circle, the equation, 
 
 or 
 
 (S.p/3-T- + (V.pj3-)^ = l. 
 
 And in like manner, if we eliminate x between the two equations 
 of the oblique section, we find for the derived hyperboloid of two 
 sheets, considered as the locus of the varying ellipse, the ana- 
 logous equation. 
 
LECTURE VII. 433 
 
 In a similar way, the equations of the right and oblique cones, 
 which enter into the construction of the foregoing article, are 
 found to be, respectively, in quaternions, 
 
 (S.p/3-0^ + (V.pi3-0-O, 
 and 
 
 439. By a quite analogous deformation of the equilateral 
 HYPERBOLOiD OF ONE SHEET, which has for its equation, 
 
 (S.p/3-)^+(V.pi3-T- = -l, 
 
 and is generated by the revolution round ob of that other equila- 
 teral hyperbola (not traced in fig. 93) whose transverse axis is 
 DE, we should obtain another hyperboloid of one sheet, 
 which would not be a surface oi revolution, and whose equation 
 would be, 
 
 In fact, each circle on the former of these two last hyperboloids 
 will (as in the recent constructions) correspond to an ellipse on 
 the latter; these two curves being still sections of one common 
 cylinder of revolution ; and iheiv planes being still joa/'a//e/ to two 
 given planes, and intersecting each other on a third fixed plane 
 (these three planes being those which are drawn through the 
 three lines gl, gl', gc, and are perpendicular to the plane of the 
 figure). Hence with the recent (or analogous) significations of 
 the letters, the variable points p and p' of the two hyperboloids 
 of the present article must respectively satisfy the two conditions : 
 
 mq2 + QP- - OM^ = OB- ; 
 
 m'q'Y /q'p'Y /om'Y_ , 
 od'/ Voc'/ \o-r' ) 
 
 which are forms familiar to geometers, but are (I think) in some 
 small degree less simple than those equations in quaternions.^ to 
 which the present calculus conducts as above. It may be noticed 
 that this new oblique hyperboloid (if we may venture so to call it) 
 would still have, as asymptotic to itself, the last-mentioned ob~ 
 
 2 F 
 
434 ■ ON QUATERNIONS. 
 
 lique cone : and that it would touch the ellipsoid (of arts. 434, 
 &c.), and the circumscribed cylinder dd', along the ellipse de- 
 scribed on d'e' as major axis, in a plane perpendicular to the 
 plane of the figure ; that is to say, along the oblique section of 
 this cylinder dd', for which section the following equations were 
 assigned in art. 436 : 
 
 The equations of the varying circle of the present article would 
 be, 
 
 and the corresponding equations of the varying ellipse would be- 
 come, 
 
 440. These results, so far as they are geometrical, require 
 for their proofs only a moderate acquaintance with the theory 
 of surfaces of the second order ; they have here been brought 
 forward, chiefly for the purpose of exemplifying some of those 
 MODES OF EXPRESSION, for geometrical loci, &c., which the cal- 
 culus of quaternions suggests ; and it would be easy to extend 
 them, so as to obtain analogous expressions for non-central sur- 
 faces, whether those be or be not of revolution. For example, 
 two ELLIPTIC PARABOLOIDS, connected with each other on the 
 same general plan, whereof the former is, and the latter is not 
 a surface of revolution, may be represented by the two equa- 
 tions, 
 
 S.p/3-^ + (V.pj3-iy-=0; 
 
 S.pa-i+(V.|Oj3-0' = 0: 
 
 their tangent planes, at the origin of vectors, which is a point 
 common to both of these two paraboloids, being represented by 
 these other equations, 
 
 S.pj3-^ = 0; S.pa-^=0: 
 
 while the following equation, which does not involve the sym- 
 bol V, 
 
 S . |0a'^ S . /o/3"^ = S . p7"S 
 
LECTURE VII. 435 
 
 may be without difficulty proved to represent an hyperbolic 
 PARABOLOID. In general, the formula, 
 
 M^hereyis used as the characteristic of an arbitrary (but sca- 
 lar) FUNCTION, represents an arbitrary surface of revolu- 
 tion round the axis /3 ; and the circular sections of fehis surface 
 are changed to a corresponding system of ellipses.) when the 
 equation is changed to the following : 
 
 where a is still supposed to make some acute or obtuse angle 
 with j3. If, on the contrary, we were to assume a in the same 
 direction as j3, but different from it in length, then the equations 
 lately found, and involving a, j3, p, would come to represent an 
 ellipsoid, a double-sheeted hyperboloid, a cone, a single-sheeted 
 hyperboloid, and a paraboloid, which would all be surfaces of re- 
 volution, like the sphere, &c., from which they might still be geo- 
 metrically derived, although not without a modification of that 
 process of deformation which has been employed in recent arti- 
 cles ; while their equations in quaternions would retain the same 
 forms as before. 
 
 441. It was shewn by the late Professor Mac Cullagh, that 
 a surface of the second order, generally, may be regarded as 
 the locus of a point, whose distance from a given point, or focus, 
 bears b, given modular ratio to the distance of the same variable 
 point from a given right line, or directrix : this /a^ife;* distance 
 being measured parallel to a given directive plane. Let us 
 now seek to express by quaternions this method of modular ge- 
 neration : and for that purpose, let us place the origin o of 
 vectors on the given directrix, and denote by a the given focus 
 corresponding, supposing also that b is another point on the di- 
 rectrix, and that the line oc is perpendicular to the given direc- 
 tive plane ; let also p denote a variable point of the surface, and 
 s the point where the directrix is crossed by a plane through p, 
 drawn parallel to the directive plane ; finally let the modular 
 ratio be that of m to 1, and let us write for abridgment, as we 
 have often done before, 
 
 2 f 2 
 
436 ON QUATERNIONS. 
 
 OA = a, OB=j3, 0C = 7, OF=p, OS = 0-. 
 
 Then one form for the equation sought is evidently the foliow- 
 
 T{p-a) = mT (p-a); 
 
 in which, however, we must seek to express a, in terms of the 
 variable vector p, and of the constant vectors (5, y, by the help 
 of the two conditions, 
 
 ff 11/3, p-(T X y- 
 
 The latter of these two conditions shews that the two variable 
 vectors p and a must have one common projection on the line 
 y, or (by 424) that 
 
 S . -yo- = S . yp. 
 
 The former condition shews (compare 430) that cr must be of 
 the form rrjS, where x is some scalar coefficient ; and therefore 
 (by 410) that 
 
 (TS.7j3 = (^i3S.7j3=)i3S.7(7. 
 
 Hence the required expression for a, in terms of /S, y, p, is, 
 
 o- = j3 S . YjO -f- S . y(5- 
 
 Now it is easy to see, by a simple use of similar triangles, that 
 any difference of two vectors is multiplied by a scalar, when 
 each vector separately is" multiplied thereby, and the difference 
 afterwards taken ; for example, in fig. 88, if a line were drawn 
 from the middle point of ob to the middle of oa, this line would 
 have for its immediate expression ia-^/3, while it would be 
 equal in all respects to the line ca, which has been seen to have 
 ^ (a - j3) for its expression. Hence 
 
 mT (p - a) = T . m (p - o-) = T (ijip - ma) 
 
 where nothing hinders us to assume 
 
 m = S . 7j3, 
 
 because we may multiply the line /3 or y by any constant scalar, 
 without violating the conditions of the construction. Mac Cul- 
 lagh's method of modular generatio7i of surfaces of the second 
 
LECTURE Vll. 437 
 
 order may, therefore, in the present calculus, be expressed by 
 the equation : 
 
 T (p-a) = T (pS . 7/3 - /3S . yp) ; 
 
 or by this other, 
 
 (p-ay={pS.(3y-(5S.yp)\ 
 
 It will be found that the equation thus obtained may also be 
 written as follows : 
 
 T(p-a) = TYiyY.(5p); 
 or, 
 
 {p-ay={W.yV.(5py: 
 
 and in fact we may already see that the two symbols, 
 
 V.yV.jSp, andpS.(5y-(5S.yp, 
 
 as applied to the geometrical generation above mentioned, agree 
 with each other, and with the product m (p - a), in representing 
 each a vector, which (by the beginning of art. 429) is at once 
 perpendicular to y, and coplanar with /3 and p ; being also mul- 
 tiplied by any scalar coefficient x, when p is multiplied thereby ; 
 and remaining unchanged, when the extremity p of p moves pa- 
 rallel to the given directrix, namely to the line jS or ob. Ano- 
 ther known method, which has been named the method of wnbi- 
 licar generation of surfaces of the second order, is expressible 
 with even greater ease, by the notations of the calculus of qua- 
 ternions. 
 
 442. The symbol, 
 
 V(V.a/3.V.y8), 
 
 denotes (by the lately cited art. 429) a line, which is at once per- 
 pendicular to V. a/3 and to V. yS ; and is therefore (by the same 
 article) at once coplanar with the two lines a, j3, and with the 
 two lines 7, S ; or is a line situated in the inte?'section of the tivo 
 planes of a, |3, and of 7, S, if all these vectors be conceived to 
 diverge from one common origin. If then six such diverging 
 lines be denoted by the symbols, 
 
 ctj a 5 a ; a 5 a J a ] 
 
438 ON QUATERNIONS. 
 
 and if three others, diverging still from the same origin, be de- 
 duced from them by the three formulae, 
 
 /3 = V(V.aa'.V.a'"a""), 
 
 ^' = V(V. a' a". V. «""«'""), 
 /3"=V(V.a"a'".V.a'""a); 
 
 these three new lines will be respectively the intersections of 
 three pairs of opposite faces of the hexahedral angle^ whose edges 
 are the six former lines : and if we then establish the equation 
 
 = S.j3V./3'i3", 
 
 it will express (by 430) that these three lines /3j3'j3" are in one com- 
 mon plane. Hence by an easy application of the celebrated Theo- 
 rem of Pascal, respecting a hexagon in a plane conic ; namely, that 
 its opposite sides meet by pairs on three points which are on one 
 straight line (at a finite or infinite distance), and conversely that 
 if the sides so meet, the hexagon can be inscribed in a conic; we 
 may infer that the equation last written, which will be found to 
 admit of being reduced to the following still simpler form, 
 
 expresses the condition for the six lines, a, a, . . . a"", being 
 sides q/oNE common cone of the second degree (a cone with a 
 plane conic for its base). On this account I have been induced 
 to call this equation, namely 
 
 = S./3V.j3'i3", orO==S./3i3'j3", 
 
 the equation of homoconicism, relatively to the six lines 
 a, . . a"" : and when this equation is not satisfied, or in other 
 words, when the scalar function S . j3V. j3'j3"does not vanish, in 
 consequence of the six lines o . . not belonging to any one cone of 
 the s^ond degree, I have been led to call this scalar the aconic 
 function of those six aconic lines (using the alpha privativum), 
 or of those six heteroconic vectors. And this aconic function has 
 again served me to form a sufficiently simple expression, by quater- 
 nions, for what I call the adeuteric function often vectors, 
 a, a', . . . a'^, for the case when those ten diverging lines do not 
 terminate on any one surface of the second order ; and then to 
 
LECTURE VII. 439 
 
 express the case where the ten vectors do so terminate, or to 
 form what may by analogy be named the equation of homo- 
 DEUTERisM, or the condition for ten points being situated on 
 ONE COMMON SURFACE OF THE SECOND ORDER, by simply equat- 
 ing the adeuteric function to zero. 
 
 443. But it is time that we sliould proceed to consider, ye^ie- 
 rally, the ""operation of addition of quaternions ; or to assign 
 what, in the presentCalculus, is to be regarded generally as the in- 
 terpretation OF A SUM. And for this purpose, we shall find that 
 it is only necessary to introduce a very slight and obvious exten- 
 sion of principles which have already been employed by us, near 
 the beginning of the present Lecture, for the addition of a scalar 
 to a vector. In short, we have only to continue to apply the 
 notion of a common operand. But it may not be useless, pre- 
 viously, to examine whether and how this notion adapts itself to 
 those easier cases of addition, what had been earlier considered ; 
 namely, to the case of the addition of a scalar to a scalar^ and to 
 the case of the addition of a vector to a vector. 
 
 444. With respect, indeed, to the addition of one scalar y to 
 another scalar x, it can scarcely at this stage require to be for- 
 mally proved, that the received and usual algebraical sum, 
 y + X, of these two scalars, satisfies the general condition, 
 
 {y + x)a = ya + xa, 
 
 whatever vector the letter a may denote : and that thus any arbi- 
 trary line a may be assumed as the common operand, and the 
 symbol y + xhe then, consistently with received usage, interpreted 
 (compare 405) by the formula, 
 
 y + X = (ya + Xa) -r- a. 
 
 In fact it is clear that whatever rectilinear step in space may be 
 denoted (art. 18) by the symbol a, and whatever positive or ne- 
 gative numbers (whether integral or fractional, and whether com- 
 mensurable or incommensurable) may be denoted by x and y, it 
 will always be true that x such steps, folloived by y such steps, 
 are on the whole, equivalent to a positive or negative number of 
 steps of the same sort (each = a), which resultant number may be 
 denoted by the symbol of the algebraical sum, y + x. Three for- 
 
440 
 
 ON QUATERNIONS. 
 
 ward steps, followed by Jive backward ones, are on the whole 
 equivalent to two backward steps, of the same common length, 
 and on one common axis ; and this very simple conclusion may 
 be expressed by writing (as usual), 
 
 - 5 + 3 = - 2, or more fully, - 5a + 3a = - 2a ; 
 
 so that the algebraical sum -5 + 3, may be interpreted (if we 
 think fit) by the help of the identical formula : 
 
 - 5 + 3 = (- 5a + 3a) -T- a. 
 
 And generally, we see already, by writing /3 and y for the lines 
 xa and ya, that 
 
 (y _^ a) + (/3 -- a) = (y + ^) 4- a, if jS |1 a, 7 1| a. 
 
 445. It is not quite so obvious, on the principles of the pre- 
 sent Calculus, so far as they have been hitherto laid down, that we 
 must have also, 
 
 (7 4- a) + (/3 -^ a) = (7 + ^) ^ a, when j3 _L a, 7 ± a ; 
 
 under which conditions of perpendicularity, of the common divi- 
 sor line a to the two dividend lines j3 and 7, we know (122) that 
 the two quotients to be added, namely (3 -^ a and 7 -^ a, repre- 
 sent, in this calculus, lines. Yet there is little difficulty in prov- 
 ing, for this case also, that the lately written formula of addition 
 still holds good. Conceive, for example, that, in the annexed 
 figure 94, the sides ob and oc of the 
 parallelogram bocd are the two vec- 
 tors /3, 7, and therefore (by 100) that 
 the diagonal od is the sum 7 + /3 ; and 
 because the vector a is to be perpendi- 
 cular to both j3 and 7, let us conceive 
 it to be constructed by a line oa, which 
 shall be erected at the point o, at right 
 
 angles to the plane of the figure Suppose also (to fix our con- 
 'ceptions), that this plane is horizontal, and that the line a is di- 
 rected upwards; and let its length be double the unit of length : 
 we shall then have this particular value for the divisor line, 
 
 Fig. 94. 
 
 O = OA = 2;^, 
 
LECTURE VII. 441 
 
 while the two proposed dividend lines, as also their sum 7 + /3, 
 will be horizontal. Then, by the principles explained in art. 
 122, we^shall have the two following quotients, 
 
 /3 -1- a = £ = OE, y -^ a = Z = OF, 
 
 if we suppose that the vectors e and ^, or the lines oe and of, 
 are sides (as in the figure) of a new parallelogram eofh, which 
 is derived from the former parallelogram bocd, by turning that 
 former one round o, right-handedly, through a right angle, and 
 halving each of the sides. But, in this process, the diagonal od 
 is also made to turn in the same direction, and through the same 
 amount of rotation, and is also halved in length, in becoming the 
 diagonal oh. Denoting therefore these two diagonals by S and 
 ■)], so that 
 
 7 + j3 = S = OD, ^+£ = t} = 0H, 
 
 we have (see again 122) the quotient, 
 
 S -7- a = }j ; 
 and therefore, by substituting the values of S and ?j, 
 
 (7 + /3) -^ a - ^ + £ = (7 -^ a) + (/3 -^ a). 
 
 The proposed formula of addition is therefore verified for this 
 example; and it is evident that an exactly similar construction 
 would prove it to be true, for every other case where a was per- 
 pendicular to /3 and 7. We see, at the same time, that because 
 (with the recent significations), 
 
 /3 = £Xa, 7 = ^xa, 7 + /3 = S = r)xa=(^+£)xa, 
 
 we may also write, 
 
 {Z+ e)a = Za+ ea, when a _L £, a ± Z- 
 
 446. The|two connected formulae, 
 
 (7^a) + (/3-4-a) = (7 + j3)-4-«, 
 r+q = {ra + qa) -7- a, 
 
 are therefore true for the two cases, where 
 
 1st, a II i3, a II 7 ; or, 2nd, a ± /3, a _L 7 ; 
 
 that is, for the two cases where (see 407, 412) we have, 
 
442 ON QUATERNIONS. 
 
 1st, V^=0, Vr=0; or 2nd, Sq=0, Sr=0. 
 
 The same two formulse hold good also (by 405) for two other 
 cases of addition, namely, the case where, 3rd, a scalar is added 
 to a vector, and that where, 4th, a vector is added to a scalar: 
 or, in symbols, where 
 
 3rd, a ± (5, a\\y; or 4th, a || /3, a±y; 
 
 or for the cases where 
 
 3rd, S^ = 0, Vr = 0; or 4th, V^ = 0, Sr = 0. 
 
 In all these various cases, we have had the two products qa and 
 ra equal to two lines, namely, to those denoted above by |3 and 
 y I or in symbols, we have had, so far, 
 
 S . ga = ; S . ra = 0. 
 
 If then we now establish, as a definition, of the operation of 
 the ADDITION OF QUATERNIONS, that whenever a non-evanescent 
 and COMMON operand line, a, can he found, which shall satisfy 
 these two last conditions; or shall give two lines, |3 and y, as 
 \he results of the two separate multiplications of the line a by 
 the two proposed quaternions, q and r, then the sum (j + (5) of 
 these two separate product-lines, divided by the original operand 
 line (a), shall be regarded as the sum of the two proposed 
 quaternions, or as equal to r + q : if, in a word, we establish 
 now the formula that (a denoting still some non-evanescent 
 vector), 
 
 r+q = (ra + qa) -r- a, when S .qa=0, S . ra = ; 
 
 or (which comes to the same thing) if we now agree to define that 
 the distributive principle of multiplication, 
 
 (r+q) a = ra + qa, 
 
 holds good whenever the two partial products, qa and ra, are 
 LINES : we shall have established a definition of addition, 
 which embraces every case that has been hitherto considered in 
 these Lectures ; and which will be found to give, in every 
 other case, tvithout ambiguity, a value for the sum of any two 
 quaternions : while the distributive form of the equation is ob- 
 viously consistent with the results and usages of common algebra- 
 
LECTURE VII. 443 
 
 447. It may be well however to offer here a few remarks, for 
 the purpose of making more clear the universal applicability of 
 the foregoing definition of the addition of quaternions, and the 
 perfect unambiguousness of the results. Consider then the gene- 
 ral case, where neither of the two quaternions to be added reduces 
 itself to either a scalar or a vector : and let us also suppose, for 
 the sake of additional generality, that their axes are not parallel 
 to any common line. Constructing them then by two biradials 
 (art. 93), with their common vertex at some assumed origin o of 
 vectors, i\xe\Y planes will necessarily intersect each other along 
 some right line, of which any finite portion OA may be taken for 
 the vector a, and employed as the common operand, to give ge- 
 nerally (compare 108, 309, 310) two transformed ox prepared bi- 
 radials, such as AOB, Aoc, and thereby two new lines, 
 
 §'a = j3 = 0B, ra = 7=oc, 
 
 in the respective planes of the two proposed summand quater- 
 nions, q and r : after which it will only be necessary to complete 
 the parallelogram, bocd, and to draw the diagonal, od or 3, in 
 order to obtain a third biradial, aod, which shall represent the 
 required sum, namely, 
 
 r^-g' = g-^-a = 0D-7- oa, 
 
 in virtue of the general definition of a sum of two quaternions, 
 adopted in the preceding article. Conversely, in order that a 
 line a may be properly assumed as the common operand, in the 
 process of that article, it must be taken in or parallel to both the 
 planes of the two proposed summands; and consequently, when 
 transported to the assumed origin of vectors, it can only differ from 
 the lately assumed line oa in length, or by its having an exactly 
 opposite direction : but the new parallelogram, constructed with 
 reference to this neiu line a, vvill have its new diagonal S altered 
 at the same time, in the same (positive or negative) ratio. In 
 other words, the only permitted variation in the recent construc- 
 tion will consist in multiplying each of the four lines, o, /3, y, S, 
 by some common scalar coefficient, such as x ; but this will not 
 alter the quotient of any two of them, and we shall have still, by 
 the definition of a sum, given in the last article, the value, 
 
444 ON QUATERNIONS. 
 
 r + q = xd -r- xa = S -7- a. 
 
 In the less general ease, indeed, where the planes of the two 
 proposed summands are parallel to each other, so that they coin- 
 cide when transferred to the assumed origin, the recent rule fails 
 to assign any one determinate position for the line a, regarded as 
 the intersection of those two planes ; but in this case it is allowed 
 to assume, for the common operand a, any line in the common 
 plane, and to use it in constructing a parallelogram, on the same 
 general plan as before ; and no ambiguity can result, because if a 
 be turned about through any angle in the plane, or in any man- 
 ner lengthened or shortened, the parallelogram will at the same 
 time turn through exactly the same angle and towards the same 
 hand, while the length of each side and diagonal will be changed 
 in the same ratio. And similar remarks apply to the case where 
 one of the two summands reduces itself to a scalar, and may 
 therefore be regarded as having an indeterminate plane, in which 
 case any line a may be assumed, that is in or parallel to the plane 
 of the o^Aer summand. In every case, therefore, the rule of 
 THE COMMON OPERAND, as laid down in the foregoing article, is 
 applicable without ambiguity. 
 
 448. The sum of any two proposed quaternions having thus 
 a perfectly definite and known signijication, may be expected 
 also to have discoverable properties, and to be adapted to become 
 the subject matter of ^Aeore^w*. (Compare again the analogous 
 remarks on products^ in arts. 108, 309, 310.) And accordingly, 
 in the first place, because (by art. 100) we have 
 
 7 + /3 = /3 + 7, ov, ra + qa = qa + ra, 
 
 when a is, as above, so chosen that qa and ra are lines, we have 
 therefore, as a corollary from our definition of the sum of two 
 quateo^nions, combined with an earlier result respecting the sum 
 of any two lines, this simple but useful property : 
 
 r + q = q+ r; 
 
 or in words, the addition of two quaternions is always a commu- 
 tative OPERATION. Again, if the two sides j3, 7, and the dia- 
 gonal S, of the parallelogram in the recent construction, be sup- 
 posed to be projected on a into three other lines, j3', 7', S', or ob', 
 
LECTURE VII. 445 
 
 oc', Od', by letting fall the perpendiculars bb', cc', dd' on the in- 
 definite line through the points o and a, then the Jour points 
 o, b', c', d', will be arranged on that line in a way analogous to 
 the four points a, b, c, d of fig. 20, art. 97, and we shall have the 
 relation, 
 
 od' = oc' + ob', or, h' = j' + j3'. 
 
 We shall therefore have also, by our recent definition of a sum of 
 two quotients, 
 
 where, by the construction in art. 407 for the scalar of a quo- 
 tient, 
 
 /3' -7- a = S (|3 -T- a) ; 7' -f- a = S (y -f. a) ; S' -^ a = S (8 -7- a) : 
 
 but also, because B is here equivalent to y + jS, we have 
 
 S^a = (7--a) + (i3-«); 
 
 where (by what has been lately shewn) the quotients j3 -f- a and 
 y -r- a may represent any two quaternions, q and r. We have 
 therefore generally the formula, 
 
 S (r+ q) = Sr+ Sq ', 
 
 or in words, the scalar of the sum of any two quaternions is equal 
 to the sum of the scalars. Again, if we let fall perpendiculars, 
 bb", cc", dd", from the three points b, c, d, on the plane which is 
 drawn through o at right angles to the line oa, we shall obtain 
 those three other components of the vectors /3, 7, 8 which are 
 perpendicular to a, namely 
 
 j3" = 0B", 7"=oc", S"=od", 
 
 and the projected parallelogram b"oc"d" in this new plane will 
 give the relations, 
 
 8" = 7" + jS", S" - a = (7" - a) + 03" - «), 
 
 where (by 407), 
 
 /3"-f-a = V(j3-4-a), 7"--a = V(7^a), §" -^ « = V (8 - «) : 
 
 the vector of the sum of any two quaternions is therefore equal 
 to the su7n of the vectors, or in symbols 
 
446 ON QUATERNIONS. 
 
 V (r + q) = Wr + Yq. 
 
 And hence, by the formula 
 
 K=S-V, 
 
 of art. 408, or more immediately by reflecting the parallelogram 
 BOCD, with respect to the line oa (compare fig. 32, art. 186), we 
 may infer that 
 
 K (/• + ^) = Kr + Kg : 
 
 or in words, that the conjugate of the sum of any two quaternions 
 is equal to the sum of their conjugates. 
 
 449. It can give no trouble now to extend these results, from 
 the case of two summands, to the more general case where it is 
 required to accomplish the addition o/any number of quater- 
 nions. We can easily prove, for example, that the addition of 
 three quaternions is always an associative operation, or that 
 
 (* + r) + 5- = s + (r + g), 
 
 by shewing that each of the two processes of summation here in- 
 dicated conducts to one common quaternion, whereof the scalar 
 part is the sum of the scalars, and the vector part is the sum of 
 the vectors, of the three summand quaternions, q, r, s. In general, 
 for ajiy number of summands, the addition of quaternions, like 
 that oi lines (see 100), on which it has been found in great part 
 to depend, is in all respects subject to the associative and com- 
 mutative laws : for example we have, as in algebra, 
 
 (s + r) + q = s + (q + r) = (q + s) + r; 
 
 t + s + r+ q = r + s + q + t, 8ic. 
 
 We may also verite, generally, 
 
 SS = SS, VS = SV, KS=SK, 
 
 using S as the characteristic of the operation of taking the sum 
 of any number of proposed summands, which are here supposed 
 to be quaternions. With respect to the subtraction of one qua- 
 ternion from another, you anticipate, of course, that this is to be 
 eifected by adding the quaternion from which the subtraction is 
 to be made, to the negative of the subtrahend : or that the diffe- 
 rence r- q is interpreted, in this calculus, by the identity. 
 
LECTURE VII. 447 
 
 {r-q) + q = rf or r-q==r+ {-q). 
 
 This operation, therefore, requires no special rules : yet it may be 
 worth while to note here, what you can have no difficulty in 
 proving for yourselves, that 
 
 S(r-q) = Sr-Sg; Y {r-q)=\r -Vq ; K {r- q) = Kr-Kq; 
 
 or more concisely, using A as the characteristic of the operation 
 of taking a difference, that 
 
 SA=AS; VA = AV; KA = AK. 
 
 The sum of any two conjugate quaternions is the double of their 
 common scalar, and their difference is the double of the vector 
 part of one of them (see 408) ; thus 
 
 i (a/3 + i3a) = S . aj3 = S . j3a, i (a/3 -M = V. «j3 = - V. j3a, 
 
 whatever two lines may be denoted by a and j3 ; and in fact 
 
 1 was accustomed to employ these symbols, ^ (aj3 + j3a) and 
 
 2 (aj3 - j3a), to denote respectively the scalar and vector parts 
 of the quaternion product aj3, before I ventured to introduce the 
 notations S and V- 
 
 450. I shall take this occasion to remark that a quaternion, 
 generally, may now be seen, more clearly perhaps than at any 
 former stage of the present Course, to admit of being expressed by 
 
 the QUADRINOMIAL FORM, 
 
 q = tv ^- ix \jy 4 kz ; 
 
 where the sum of the three terms ix,jy, kz composes (compare 
 407) the vector -part, while the remaining term w denotes the 
 scalar part of the quaternion : so that we may write, in con- 
 nexion with the recent form, 
 
 ^q = W', Vq = ix +jy + kz. 
 
 Indeed this quadrinomial form for a quaternion, which may 
 (compare 111) be regarded as an expansion of the shorter form 
 w + p, where p denotes a vector, was communicated by me, so 
 long ago as 1843, to the Royal Irish Academy, along with the 
 values above assigned (in arts. 394, &c.) for the squares and pro- 
 ducts o{ i, j, k ; and it has been referred to by anticipation, in 
 this Course, so early as at the close (art. 78) of the Second Lee- 
 
448 ON QUATERNiaNS. 
 
 ture. But the signification of this quadrinomial form may be now 
 more fully understood, in consequence of the recent remarks on 
 Slims of several summands. We may now see, for instance, by the 
 associative property (449) of such summation^ that although we 
 may interpret this quadrinomial form as simply equivalent to the 
 binomial form w + p^ or number plus line, to which in an earlier 
 part of the present Lecture a quaternion was proved to be redu- 
 cible ; and may with that view write the expression for q as fol- 
 lows : 
 
 q=W + (ix +jy + kz) ; 
 
 yet we may also otherwise combine the four terms, w^ ix^ jy, kz^ 
 into partial groups, writing, for example, 
 
 q={w + ix) + {jy + kz), 
 
 where the partial sum w + ix is itself sl certain quaternion, which 
 is to be added, according to the general rule of arts. 446, 447, to 
 the linejy + kz. Again, if we write, as the analogous quadrino- 
 mial expression for another quaternion, 
 
 q =w' + ix +jy' + kz, 
 
 we shall have no difficulty now in establishing the following ex- 
 pressions for the su7n and difference of these two quaternions : 
 
 q + q = w' + w -^ i {x + x) +j (y +y) + k{z'+ z) ; 
 q'-q = w'-w + i (x - x) +j {y -y) + k(^z' - z). 
 
 The FOUR SCALARS, w, X, y, z, are called (78) the four consti- 
 tuents of the quaternion w+ix+jy+kz; and a quaternion q 
 cannot va7iish, or become equal to zero, without each of these 
 four constituents separately vanishing : that is, in symbols, 
 
 [{ q = 0, then w = 0, x = 0, y = 0, z = 0. 
 
 In fact, if a be any actual divisor line, the quaternion q, regarded 
 as the quotient j3 -r- a, cannot be considered as vanishing, so long 
 as the dividend |3 is an actual (or non-evanescent) line ; but when 
 jS vanishes, its two components j3'and j3" (see fig. 85, art. 406), 
 respectively parallel and perpendicular to a, must also vanish : 
 so therefore do the two partial quotients, obtained by dividing 
 these two components by a. In symbols, 
 
LECTURE VII. 449 
 
 if ^ = 0, then S^ = 0, Yq=0; 
 
 but the scalar S^- has been above denoted by w, and a vector 
 such as Yq, or ix+jy + kz, cannot vanish, without its three pro- 
 jections, on any three rectangular axes (such as the axes of i,j^ k), 
 all vanishing together, that is, without our having separately, 
 
 t'x = 0, jij=0, kz = 0; ora;=0, p = 0, z = 0. 
 
 For the same reason, the difference q'-g cannot vanish, except 
 by our having the four separate evanescences, 
 
 w -w = 0, x - x = 0, y -2/ = 0, z - z=0 ; 
 
 or, as we may otherwise state the same result, 
 
 \iq-q, then w = w, x = x, y' = y, z'-z. 
 
 An EQUATION BETWEEN TWO QUATERNIONS is therefore equiva- 
 lent to a SYSTEM OF FOUR EQUATIONS BETWEEN SCALARS ; Or in 
 
 other words, two quaternions cannot be equal, unless each consti- 
 tuent of the one be equal to the corresponding constituent of the 
 other. The importance therefore of the number Four in this 
 whole theory, from which indeed (compare 91, 106, 107, 120) 
 the present Calculus derives its wa?/2e, exhibits itself here again. 
 451. T^e distributive principle, or property, of the multipli- 
 cation of quaternions, has (in the present Lecture) been in part 
 already established by definition, and has been used as the chief 
 element (446) in the general interpretation of a sum: just as 
 the associative property of multiplication of quaternions had 
 been previously established, in these Lectures, to some extent^ 
 by definition, for the sake of interpreting a product (compare 
 309, 310). We have lately defined that 
 
 {r + q) a = ?a + qa, 
 
 as we had at an earlier stage defined that 
 
 rq . a^^r . qa, 
 
 whatever two quaternions may be denoted by q and r, provided 
 that the symbols a, qa, and ra denote three lines. But pre- 
 cisely because we are thus enabled to give now (see 447) a defi- 
 nite interpretation to the symbol of a sum, r + q, of any two sum- 
 
 2 G 
 
450 ON QUATERNIONS. 
 
 mands^ as we could earlier give (see 108) a definite interpretation 
 to the symbol of a product, r x q, ov r . q, oi rq, of any two fac- 
 tors, we are not now at liberty to assume, without proof 
 that the general distributive principle, 
 
 {r-\- q')s = rs + qs, 
 
 holds good, for three arbitrary quaternions, q,r,s: just as we were 
 not at liberty to assume, without proof the general associa- 
 tive PRINCIPLE of multiplication of any three quaternions, 
 
 s .rq = sr . q, 
 
 which has already been discussed in former parts of this Course, 
 but of which we have promised to give, in the present Lecture, 
 a new and independent demonstration, founded on an independent 
 proof of that other or distributive property, to the general and 
 rigorous examination of which it is necessary that we should now 
 proceed. 
 
 452. An important case in which we can already prove with 
 ease the truth of the lately written distributive formula, 
 
 {r+ q)s = rs + qs, 
 
 is the case where the planes of the three proposed quaternions 
 q, r, s contain, or are parallel to one common line, such as a. For 
 in this case we can find three other lines, such as j3, 7, 8, in those 
 three planes, so as to satisfy the three equations, 
 
 q - ^ -=r a, r = J -r- a, s = a -r- e ; 
 
 and then if (as in 447) we denote 7 + j3 by S, and employ the ge- 
 neral formulae of multiplication and addition (arts. 49, 446), 
 
 (7-^)x(/3^a) = 7-a, 
 (7 -f- a) + (/3 ^ a) = (7 + /3) -^ a, 
 
 we shall have the values, 
 
 r+ q = d -^ a, qs = (5 -7- e, rs = y ^ e, 
 and therefore 
 
 (r + ^) s = g -7- f = (7 -f- e) + (|3 -f- e) = '"* + $'•*• 
 But the condition for the three planes of q, r, s being thus pa- 
 
LECTURE VII. 451 
 
 rallel to one common line, a, is the same with the condition for 
 the coplanarity of their three axes, or of their vector parts^ or 
 with the following : 
 
 Ys 111 V^, Vr. 
 
 We know, therefore, already, that whenever this condition of co- 
 planarity is satisfied, the distributive formula 
 
 {r + q) s = rs + qs 
 
 holds good, whatever it may yet be found to do in other cases, 
 ^ow the vector part of a scalar is a null line (compare 407), 
 which may be regarded as having an indeterminate direction 
 (compare 149, 153, 166,167, 447) ; it may therefore be considered 
 as coplanar with any two lines. And hence, or more directly 
 by choosing a so as to be perpendicular to both of the two re- 
 maining vectors, and reasoning then as in the present article, we 
 can prove that the recent distributive formula holds good, when 
 any one oi the three quaternions, q, r, s, reduces itself to a scalar. 
 For example, let 
 
 q = p, r=w, or let 85- = 0, Vr = ; 
 
 then whatever scalar, vector, and quaternion may be respectively 
 denoted by w, p, s, we shall have 
 
 (w + p) s = ws + ps : 
 
 which is already a more general result than that of art. 405, 
 where instead of s was written a, and a was supposed to denote 
 a vector perpendicular to p. 
 
 453. Again we know (by 448) that the conjugate of a sum is 
 the sum of the conjugates, and (by 190, 222) that the conjugate 
 of the product of any two factors is equal to the product of their 
 conjugates, taken in an inverted order. Hence, at least if we still 
 retain the recent condition of coplanarity of axes, and denote the 
 conjugates of the three quaternions q, r, s, by q', r, s' respec- 
 tively, we shall have the equation 
 
 s {r + q) = s'r + s'q ; 
 
 or by omitting the accents, which here involves no loss of gene- 
 rality, 
 
 2 G 2 
 
452 ON QUATERNIONS. 
 
 s(r + q)=sr + sq, if Vs \\\ Vq, Yr. 
 
 This condition of coplanarity will again be satisfied by supposing 
 q a vector, such as p, and r a scalar, such as w ; and thus we 
 may obtain the formula, 
 
 s (w + p) = sw + sp. 
 
 It is easy hence to infer that for any two scalars a, 6, and any 
 two vectors a, |3, we have, as in algebra, 
 
 {b + j3) (« + a) == ^« + &a + j3« + j3a ; 
 
 where (by 83) [Ba=a(5, and ba = ab, as well as ba = ab; but 
 where (by 78, 89, &c.), (5a is wo^ generally = a j3. And hence 
 again we may infer that 
 
 S.(b + (5) {a + a) = ba + S.(5a', 
 V.(6 + j3) (« + a) =«/3 + &a + V.j3a; 
 
 or that the product of any two quaternions, q and r, may have 
 its scalar and vector parts expressed separately as follows : 
 
 S.rq=SrSq + S.YrVq; 
 V. rq = Yr Sq +VgSr + V. Yr Yq. 
 
 454. Another important case, in which we can easily esta- 
 blish the truth of the distributive principle of multiplication, is 
 that where we have to deal with vectors only. In fact, the for- 
 mula above established for the addition of two quotients, (5 -i- a 
 and y ~- a, may be written as a formula for the addition of two 
 products, by the help of the properties of reciprocals of vectors 
 (see 117, 118), as follows: 
 
 (7 X a-') + ()3 X a-i) = (7 + /3) X a-i ; 
 
 or more concisely thus, 
 
 7a + jBa = (7 + j3) a, 
 
 since a"^ may represent any vector. This result is more general 
 than that given at the end of art. 445, because no condition of 
 perpendicularity is now assumed : and by taking conjugates (as 
 in the foregoing article), we may already infer from it that 
 
 ay + a/3 = a (7 + j3). 
 
LECTURE VII. 453 
 
 whatever three vectors may be denoted by a, /3, 7. Hence for 
 any four vectors a, j3, y, S, it follows easily that 
 
 (3 + 7) (j3 + a) = gj3 + Sa + 7i3 + 7a. 
 For example, 
 
 (/3 + ay = j3^ + j3a + ajS + a% 
 (^-a)^ = i3--j3a-a/3+a^-; 
 
 or more concisely (see the end of art. 449), 
 
 (/3±a)2 = i32+a^±2S.j3a. 
 
 As another example, we have 
 
 (j3 + a) (j3 - a) = j3- - jSa + aj3 - a^ ; 
 
 and therefore (see again art. 449), 
 
 S.(j3 + a) (i3-a) = /3^-a^; 
 V.(/3 + a) (j3-a) = 2V.aj3. 
 
 And these symbolical results will be found to admit of simple 
 geometrical interpretations. 
 
 455. We know now (by 453) that in the multiplication of 
 any two quaternions^ each factor may be distributed into its own 
 scalar and vector parts; and we have just seen (in 454) that in 
 the multiplication of any two vectors^ each factor may again be 
 in any manner distributed into two partial or component vectors, 
 whereof it is the geometrical sum. A vector may also, by si- 
 milar parallelograms, be distributed into such par^mZ vectors, 
 when it is to be multiplied by or into a scalar : see, for example, 
 art. 441, where we had 7n (p- or) = mp - ma. It is still more easy 
 to see, as in 444, that a scalar may be distributed, as a factor, 
 into any parts of which it shall be the algebraical sum, when it 
 is to be multiplied by or into a vector. And the permission so 
 to distribute scalars, when they are multiplied among themselves, 
 is manifest from common algebra. There remains, therefore, no 
 difficulty in establishing, as we proposed to do, the distributive 
 principle generally, for any multiplication of two sums of quater- 
 nions. Resuming with this view the comparison of the product 
 (r+q) s and of the sum rs + qs, we may employ the decomposi- 
 tions, 
 
454 ON QUATERNIONS. 
 
 qs=SqSs+Sqys+Yq Ss + YqVs, 
 
 rs = S?^ S^ + Sr \s + Vr S5 + \r V*, 
 
 (r + 5") 5 = S (r + 5-) S5 + S (r + 5) V5 + V (r + 5) S* + V (r + ^) V5 ; 
 
 and we see that the last of these three expressions is the sum of 
 the two preceding it, because 
 
 S(r+^)S5 = (Sr+S(^)S5=SrS5+S^ S*, 
 
 ^{r + q) V5 = (Sr+ Sg) Vs = Sr V5 + Sg V*, 
 
 \{r+q)^s = ( Vr + V5) S* = Yr Ss + Vg S5, 
 
 V (r+ ^) Vs = (Vr + V^) Ys = Yr Ys + Yq Ys ; 
 
 it is then proved, as was required, that,j^r any three quaternions, 
 we have 
 
 {r+q)s = rs-\-qs : 
 
 the conjugate of which general equation gives (on the plan of 
 453) this other and analogous formula : 
 
 s (?' + q) = sr + sq. 
 
 By combining these two results, or more immediately by decom- 
 posing the factors into scalar and vector parts, and then proceed- 
 ing as above, we find that for any four quaternions, q, r, *, t, the 
 analogous formula of distribution, 
 
 (r +q) (t + s)= rt + rs + qt -\- qs, 
 
 holds good; and indeed it is obvious now that the distributive 
 PRINCIPLE holds good generally, in the multiplication of any 
 TWO SUMS of quaternions, whatever the number of the sum- 
 mands may be, into which either factor is distributed. In other 
 words, the product of the sums will still, as in algebra, be equal 
 to the sum of the partial products : or in symbols, 
 
 S/- . Sg- = S . rq. 
 
 With respect to some of the notations recently used, it may be 
 remarked that the symbols, 
 
 SrSq, SrYq, Yr Sq, YrYq, 
 
 are designed to be respectively equivalent to ih.e products, 
 
 Sr.^q, Sr.Yq, Yr.Sq, Yr.Yq-, 
 
LECTURE VII, 455 
 
 whereas the symbols 
 
 S.VrV^and V.Vr V^ 
 
 denote respectively the scalar and vector parts of the last of these 
 four products, and are equivalent to 
 
 S(Vr.V^) andV(Vr.V^). 
 
 456. I need not now delay to point out the instances which 
 have already occurred to us, containing, by a sort of anticipation, 
 some part at least of what is involved in the general principle 
 recently established ; for example, the equation, 
 
 {W + p) {w-p)=W'^- |0^, 
 
 which was proved on other grounds in art. 409, and which en- 
 ables us to express the tensor of a quaternion, in terms of the 
 scalar and the vector (compare 432, 436). But it may now be 
 proper to shew how the general distributive principle, or even so 
 much of it as was established in art. 454, with respect to the 
 multiplication of vectors, enables . us to effect some ^/aws/orwaa- 
 tions of equations, which have already been proved from geome- 
 trical considerations to be valid, without its having yet been 
 shewn how to accomplish them by any process of calculation. 
 Take, with this view, the three following equations, 
 
 S.a/>"^=1; S . (a-p) jo'^ = ; T {p - ^a) =\ Ta', 
 
 which are already known (by art, 414) to represent one common 
 spherical locus for the extremity of the variable vector p, but 
 which it is now required to exhibit as equivalent formulcB in this 
 calculus. The passage from the first to the second of these forms 
 cannot cause a moment's difficulty at this stage ; for we know 
 now that 
 
 S . (a- jo)jo'^=S (a/)'^- 1) = S .ap~^- 1: 
 
 but in order to transform the third of the above written equations, 
 it is convenient to proceed as follows. Squaring both members, 
 we have, by 111 , 
 
 -(p- ^af = - (ia)= : or, (p - la)^ = ia\ 
 Developing the square of the binomial by 454, we find, 
 
456 ON QUATERNIONS. 
 
 (jo - I a)- = p" - S . ap + ja^ ; 
 so that the equation to be transformed becomes, by transposition, 
 
 p^ = S . ap ; or, S . ap~^ = 1 : 
 
 which latter form is thus shewn, as was required, to follow by 
 calculation from the third form written above, or from the equa- 
 tion between tensors, 
 
 T (|0 - ia) = i Ta, 
 
 without reference to any conception of a spherical surface or 
 locus, 
 
 457. Again, let us take the following equation of art. 415, re- 
 presenting a certain other sphere, 
 
 r[,-^J±yr{^).. 
 
 and let us seek to transform it, by calculation alone, into that 
 other form of the equation of the same locus, which was given in 
 the same article, namely, 
 
 s — ^ = 0- 
 
 Taking again the negatives of the squares of the tensors, we 
 have, by 454, 
 
 where (by the same art. 454), 
 
 hence 
 
 = p^-S.{a + (5)p + S.a(3 
 
 = S(|o2-ap-pj3 + aj3) 
 
 = S.(p-a) (p-(B), 
 
 = T(p-(5rS.{a-p){p-(5y\ 
 
 and the required transformation is effected. We see at the same 
 time that the following equation holds good, as an identiti/, for 
 any three vectors, a, (5, p : 
 
 4S . (p-a) (p - (5) = {2p -a- (3y- {a- py, 
 
LECTURE VII. 45-7 
 
 which may, by principles already laid down, be interpreted as 
 expressing (compare fig. 89, art. 415), that if c be the middle of 
 the base ab of any plane triangle apb, as in 
 the annexed figure 95, then, 
 
 S (ap . BP) = cp2 - CA^ ; 
 
 or, in a notation more received, 
 
 AP . BP . cos APB = CP^ - CA% 
 
 where the symbols ap, bp, cp, ca, marked for distinction with 
 upper bars, denote merely the lengths of certain lines, or the 
 numbers expressing those lengths, and therefore their squares 
 are (as usual) positive. Accordingly this last equation is a known 
 result of elementary principles: but in comparing it with the 
 quaternions, it is proper to remember that (see 111) the lengths 
 ap, &c., which thus have positive squares, are ivith us merely the 
 tensors of the corresponding vectors, ap, &c., of which last, 
 when regarded as directed lines in space, the squares with us are 
 NEGATIVE. Thus, in the present calculations, we pass from the 
 first to the second of the two equations last written, hy changing 
 the signs of all the terms : or by employing the relations, 
 
 S (ap . Bp) = - AP . BP . COS APB, 
 CP-=-CP-, CA^ = -CA-. 
 
 On the same plan, the equation, 
 
 (a-j3)'^ = a^-2S.aj3+i3^ 
 
 of art. 454, is equivalent to the well-known 2iX\& fundamental for- 
 mula of plane trigonometry^ 
 
 B A^ = OA^ - 2oA , OB COS AOB + OB^ ; 
 
 where o, a, b may denote any three points of space. 
 
 458. Some other known and elementary theorems, respecting 
 centres of mean distances, may be expressed, and might be 
 proved, by equally easy processes in this calculus. For exam- 
 ple, whatever three scalars and four vectors may be denoted by 
 a, b, c, a, j3, y, p, we have identically. 
 
458 ON QUATERNIONS. 
 
 where, 
 
 and 
 
 a{p-af-¥b{p- ^y +c(p-yy = 
 tp^-2S.Tp + u==t{p~ny + t-H, 
 
 t = a + h-\-c, 
 T = aa-\- bj3 + cy. 
 
 ^ = 1 
 
 T aa+bl3 + cy 
 
 t a + b + c 
 
 v==fu-T^ = ab(^- ay+bc (y - (5y+ca (a-y)^ 
 
 Thus for any four points a, b, c, p, and any three coefficients 
 a, b, c, we have 
 
 a.AF^ + b. Bp2 + c.cf'^- (a + b+c) mp^ = 
 {a + b+ c)"^ (ab . ab^ + be . bc^ + ca . ca-), 
 
 if M be the point which satisfies the equation, 
 
 a . AM + b . BM + c . CM = 0, 
 
 when directions of lines are attended to ; but this is precisely 
 the essential property of the central point above alluded to, or of 
 what is called in mechanics the centre of gravity of the system 
 of the weights a, b, c, placed at the points a, b, c, respectively. 
 And it is evident that analogous results would be obtained on 
 the same plan, for any number of given points of space a, a', &c., 
 with the same number of given coefficieyits, «, a', &c. ; or in 
 symbols, that we should find, in like manner, 
 
 S {a . AP-) - Sa . MP- = S (ad . aa'^) -t- S«, 
 
 if M be a point such that 
 
 S (a . am) = 0, 
 
 while p is an arbitrary point. For we should have, 
 
 2 . a ((O - a)-= (p- - 2S . p^) S« + S . aa\ 
 
 = {p- liy^a-v'^. aa^ - fx^ Sa, 
 
 if ;U = S . oa -^ S«, or = S . a (a - ^) ; 
 
 while Sa S . aa- - (S . aay =^.aa! (a - of. 
 
 459. Apollonius found, and the ancient result has acquired 
 
LECTURE VII. 459 
 
 fresh interest in our own days by a remarkable application of it 
 to electricity, that the locus of a point whose distances from two 
 given points are in a given ratio of inequality, is (in the plane) a 
 circle. To investigate this locus by quaternions, let the two 
 given points be o and a, and the variable point p; also let the 
 ratio of ap to op be that of n to 1, and suppose n>\ : then, 
 making oa = a and op =p, the equation of the locus is, 
 
 i:{p-a) = nTp, ov {p-ay = n'p\ 
 Developing, transposing, &c., we find successively, 
 (w2-l)p2+2S.ap-a^ 
 
 T {{n^-1) p-Va] = nTa, 
 and finally, ^ig- 96. 
 
 T(^-/3) = c, 
 if we make, for abridgment, 
 ,, — a nTa 
 
 so that 
 
 ^-a = n% c^ = -n'[5' = (5{a-^). 
 
 Hence follows this construction, which agrees with known re- 
 sults. Cut the given line ao externally at b, in the duplicate 
 of the given ratio of the sides, so as to have ab = n^oB ; take bc 
 a geometrical mean between the segments bo, ba ; and with cen- 
 tre B, and radius bc, describe a sphej'ic surface; it will be (in 
 space) the required locus of all the points p, for which 
 
 AP = w . op. 
 As a verification, let c - b = y, p - b = <t ; we shall have 
 
 A-B=ny, 0-B = W"i'y, V - X = (j- Uy, p-o = o--n"^7; 
 it ought then to turn out that 
 
 T((7-W7) = T(w(r-7), ifT(T=T7; 
 
460 ON QUATERNIONS. 
 
 and accordingly, 
 
 (a- - Wy)- = {na - 7)- = (w2 + 1 ) 72 _ 2wS . ytr, if (T^ = -y^. 
 
 It is evident from elementary geometry that the fixed locus of p, 
 constructed as above, cuts perpendicularly the circle circum- 
 scribed about the variable triangle aop, or that its radius bp is a 
 tangent to this circumscribed circle : and this result also might 
 be confirmed by calculation with quaternions, if we chose to use 
 here the conclusion of art. 198, respecting the construction by a 
 tangential vector, of the continued product of the three sides of 
 a triangle inscribed in a circle. 
 
 460. As another example of the present processes of calcula- 
 tion, let us investigate the intersections of the right line and 
 sphere^ whose equations are respectively (see 430, 421), 
 
 V.pa = V./3a; p^-+c2=0. 
 
 The latter equation gives (by principles lately employed), 
 
 C- Ta^ = -c2a^ = |0^ a- = (T . pa^^ (S . paY - (V. pa)' ; 
 
 and therefore the former equation gives, 
 
 But pa = S . joa + V . pa (by 407) ; therefore the required expres- 
 sion for the vectors of intersection is the following : 
 
 p = V . j3a . a-i ± {c^ Ta^ - (T V. /3a)^}* uK 
 
 If for abridgment we write 
 
 p = /3" + p", 
 
 the part p", independent of the ambiguous sign +, is equal (by 
 429) to that component of the given vector jS, which is perpendi- 
 cular to a ; or to the vector ob" in fig. 91, art. 427, where dbb'p 
 represents (by 430) the indefinite right line V.pa = V./3a, of 
 which it was required to find the intersections with the sphere, 
 of radius c, described about the origin o: and accordingly this 
 foot b" of the perpendicular ob", must evidently (by elementary 
 geometry) be the middle point of the intercepted and finite chord. 
 We have also, for the other part p", or for the semichord itself, 
 by the expression recently found for p, 
 
LECTURE VII. 461 
 
 Vp"=\Ja, V=(c=-T/3"^)*; 
 
 and accordingly it is clear that these expressions, when inter- 
 preted in conformity with our notations, agree with elementary 
 results. The value of p" or of T^" shews also, as was to be ex- 
 pected, that the problem h geometrically impossihle, or imaginary, 
 or that the line does not really meet the sphere at all, if the ra- 
 dius be shorter than the perpendicular, that is, if c< Tj3": or, as 
 our symbols allow us to express the same condition, 
 
 if c"' + j3"- < 0, or if c^ a^ + (TV. j3a)^ > 0. 
 
 In fact, for any two real vectors a and p, representing any two 
 actual lines in space, we have, in this calculus, the identity, 
 
 {TY. pay- p~a^ = -(S. pay ^0. 
 
 461. The calculation may be usefully varied by taking, from 
 art. 430, this other form of the equation of the secant line, p = (B 
 + wa, and seeking to determine the scalar coefficient x. Sup- 
 posing for simplicity that a is an unit- vector, or that a'- = - 1, we 
 have now, 
 
 c^ = - jo" = - (/3 + xay = X- - 2xS . aj3 - /3- ; 
 
 and therefore, by the ordinary theory of quadratic equations, 
 
 x=S.a(3 + [c- + j3- + (S . a(5y}h 
 Here 
 
 /32 = _ a^ j32 = _ (T . aj3)^- = ( V. a^y - (S . aj3)% 
 and 
 
 /3 + aS . a/3 = a (-a/3 + S . aj3) =- aV. aj3 ; 
 therefore 
 
 p = -aV.aj3+ {c-+(V.aj3)2)*a: 
 
 and this expression for p agrees perfectly with that which was 
 found in the foregoing article, when we suppose, as we now do, 
 that 
 
 Ta=l, G" = — 1, a = -a'^. 
 
 In fact we found, in 429, that the symbols, 
 
 a-iV.a/3 and V. jSa.a-i, 
 
 were equally fit to represent that component j3"of jS, which is 
 
462 ON QUATERNIONS. 
 
 perpendicular to a. Whichever method we employ, we see that 
 the equation, 
 
 cnV=(TV.j3a)^ or c' a' = {Y . (3ay, 
 
 expresses the limiting condition, which the directio7i of the secant 
 line, or of the line a to which it is parallel, must satisfy, in order 
 that the two points of intersection may coalesce into one point of 
 coiitact. If then we multiply by x^, and change xa to p- (5, ob- 
 serving that 
 
 V./3(p-i3)=V(/3p-^0=V./3p, 
 
 because )3^ is a scalar, we find the following form for the equation 
 of the enveloping cone, which is the locus of all the tangents that 
 can be drawn to the sphere p^-\- c'^= 0, from the extremity of the 
 given vector j3 : 
 
 This is a simpler form of the equation of the enveloping cone 
 than that which was found in 425, and which becomes, by chang- 
 ing a and o- to c and j3, 
 
 {S./3(^-/3))""=(c^ + /30(^-i3)^ 
 
 Yet the two equations agree : for we now see that 
 
 {S./3(p-/3)}^-/3Hp-/3p={V./3(p-/3)}'- = (V.i3^)^ 
 
 462. Each of the two preceding articles conducts to the ex- 
 pression, 
 
 /o = /3-a"^S . o|3, 
 
 for the vector of the point of contact; in connexion with which, 
 it may be well to note that (by 424, 429) we have, for any two 
 vectors a, j3, the equation, 
 
 because the two terms of the second member denote the two com- 
 ponents of j3 which are respectively perpendicular and parallel to 
 a. But also, for the tangents, 
 
 (S .(3ay= i3 V -f ( V . (3ay = (c'- + j8^) a' ; 
 
 therefore each vector p of contact must satisfy the equation, 
 
LECTURE VII. 463 
 
 S.j3|0=i3'-a-2(S./3a)2 = -c'; or S.(5p + c'=0. 
 
 This equation of the polar plane agrees with art. 423 ; and we 
 may now propose to shew by calcuhition that it involves the well- 
 known harmonic property of the plane which it denotes. For 
 this purpose we may employ the following form of the equation 
 of a secant of the sphere drawn still from the extremity of j3 : 
 
 p=/3+2/-ia; 
 
 and may propose to substitute for y the semi-sum iz) of its two 
 values^ as given by the quadratic equation, 
 
 = c^ + (/3 + y-^aY, or, y-" {c"^ + j3~) + 22/8 . «j3 + a^ = 0. 
 
 In this manner we find 
 
 and consequently, 
 
 The polar plane therefore cuts harmonically (as it is very well 
 known to do) every secant from the pole : or in other words the 
 pole (whose vector is j3), and the point of intersection with the 
 polar plane (of which the equation is S . j3|0 = - c-), are harmonic 
 conjugates, with respect to the two points in which the secant 
 (p =j3 + 2/~^a) intersects the sphere (jO^+ c^= 0). 
 
 463. In general it may be said, in conformity with the re- 
 ceived 7iotion of harmonic progression, that the harmonic mean 
 between any two vectors, such as aa, ca, which have one com- 
 mon direction, or opposite directions, is = ba, if 6"^ = i (a'^ + c'^) ; 
 and I think that we may with convenience extend this notion of 
 the harmonic mean in geometry, by establishing, as a more gene- 
 ral definition, that the harmonic mean between any two vectors, 
 a and J, is a third vector, j3, which satisfies the analogous condi- 
 tion, 
 
 whether the vectors be or be not parallel to any common line. You 
 
464 
 
 ON QUATERNIONS. 
 
 Fig. 97. 
 
 will easily find that if oa and oc be any two diverging lines 
 (a and -/), between which it is re- 
 quired to insert a third line^ ob 
 or j3, which shall, in this new or 
 extended sense of the words, be 
 their harmonic mean, the problem 
 may be thus constructed. Circum- 
 scribe a circle about the three 
 given points aoc ; prolong the 
 chord AC to meet in d the line od ^ 
 which touches the circle at o ; 
 and draw the other tangent db, 
 * and the chord of contact ob. 
 Quaternions offer many modes of proving the correctness of this 
 construction, for the reciprocal of the semi-sum of the reciprocals 
 of two diverging vectors : one of the most elementary, as regards 
 geometrical principles, consists in cutting, as in fig. 97, the three 
 chords OA, ob, oc, or rather their prolongations, by a transversal 
 a'b'c', parallel to the tangent od, and then shewing that b' bisects 
 a'c', and that the rectangles aoa', bob', coc' are equal. In the 
 same construction, the two points o and b may be said (by an 
 analogous extension of received language) to be harmonically 
 conjugate to each other, ivith respect to a and c : and it is not 
 difficult to prove that a and c are in like manner harmonic con- 
 jugates with respect to o and b : so that the four points oabc 
 may conveniently be said to compose a circular harmonic 
 GROUP. In symbols, if j3 be, in the sense above assigned 
 the harmonic mean between a and y, then - /3 is in the same, 
 sense the harmonic mean between a - j3 and 7 - /3 ; 7 - a between 
 - a and j3 - a ; and a - 7 between - 7 and j3 - 7. The rectangles 
 under opposite sides of the inscribed quadrilateral, oabc, are 
 easily proved to be equal; and the diagonals, ob and ac, are 
 related as conjugate chords, each passing through the pole of the 
 other. 
 
 464. The same harmonic relation between a, /3, 7 may also 
 be expressed by writing, as in algebra, 
 
 /3-^-i3-^-a-^ 
 
LECTURE VII. 465 
 
 where, if the rectangle AOA'in the recent figure be unity, we have 
 the following geometrical constructions, 
 
 )3-i-a-i = B'-A'; 7-i-j3-i=c'-B'; 
 
 so that the difference ^'^ - a'^ of the reciprocals of any two diverg- 
 ing vectors, a, jS, considered as two co-initial chords, oa, ob, of a 
 circle oab, is a vector which has the direction of the tangent, do, 
 or od', to that circle, drawn at their common origin o. We may also 
 say (compare 131, 198), that this direction is that of the tangent 
 at o to the segmeiit oab, rather than to the alternate segment of 
 the circle. As regards the length of this tangential vector, which 
 thus constructs the difference of the reciprocals of a and j3, it is 
 easy to prove by similar triangles that, in the recent figure. 
 
 ab -f- ab = oa -^-ob = ob -f-OA; 
 
 or with our symbols, that 
 
 T(j3-i-a-0 = Ta-iTj3-^T(a-j3). 
 
 In fact, without referring to the figure, we have 
 
 /3-i-a-i = /3-^ (l-i3a-0=/3-^(a-j3)a-S 
 
 whence the recent expression for the tensor follows. We see also, 
 by taking the reciprocals, that 
 
 (/3-i-a-i)-i=a(a-i3)-^.j3; 
 
 or that the reciprocal of the difference (3'^- a'^ of the reciprocals 
 of any two vectors, is, both in length and in direction, the fourth 
 proportional to the negative (a - (3) of the difference (5- a of those 
 two vectors themselves, and to the same two vectors, a, j3. The 
 difference of reciprocals, /3"^-a'^ itself, has therefore the oppo- 
 site direction ; or in other words it has the direction of the fourth 
 proportional to a- (5, -a, and j3 ; or in fig. 97, to ba, ao, and 
 OB. Accordingly we know that this fourth proportional to three 
 successive sides of a triangle bao inscribed in a circle must have 
 the direction of the tangent at o to the segment bao, or oab ; as 
 appears from art. 131, by changing in that article, or in fig. 26, 
 the letters c and a to a and o. It is equally easy to shew in 
 connexion with art. 463, that 
 
 2 H 
 
466 ON QUATERNIONS. 
 
 if f = I (7 + a) = OE, the point E being thus supposed to bisect the 
 chord AC in fig. 97; so that the harmonic mean, j3, between 
 any two diverging vectors, a and y, is still, as in algebra, the 
 FOURTH proportional to their arithmetical meati, or semi-sum, 
 c, and to the two vectors themselves ; or in other words, the 
 triangles eoa and cob (in fig. 97) are similar ; a result which 
 may be confirmed by elementary geometrical reasonings. 
 
 465. The geometrical interpretation of the sum and diffe- 
 rence of the reciprocals of two vectors being thus suflficiently 
 knovv'n (although they suggest several inquiries of interest, on 
 which we cannot enter now), let us resume the last form given 
 in art. 436, for the equation of an ellipsoid, namely : 
 
 or (because 
 
 TK=T, K = S-V, S.a/3 = S.i3a, V. a|3 = - V. j3a), 
 this slightly modified equation, 
 
 T(S.a-> + V.i3-» = l; 
 in which (by 449), 
 
 S.a-V = i(a-V + pa-0; V. jS' V = i(/3-> - |o/3-0- 
 Make, for conciseness, 
 
 a' = i(a-i + i3-0; i3' = i(a-i-i3-0; 
 
 the last equation of the ellipsoid takes then this very simple 
 form : 
 
 T(a> + /,i3')=l; 
 
 where p is the variable vector of the surface, while a and j3' are 
 two constant but otherwise arbitrary vectors, of which, however, 
 we can prove that a is longer than j3', if we continue to suppose, 
 as in fig. 92, that the angle between a and j3, or that the verti- 
 cally opposite angle between a'^ and |3"^ is acute: because we 
 shall then have, 
 
 IV^- Tj3'^ = i3"- - a'^- = - S . a-i jS-^ > 0, Ta'> T/3'. 
 
LECTURE VII. 
 
 467 
 
 It may also be observed, that if we still suppose, as in fig. 92, 
 Ta> Tj8, we shall have (by 454), 
 
 4S.a'/3' = a-2-i3-^>0; a'^'>|; 
 
 so that the angle between the two new lines, a', j3', will be, on this 
 supposition, obtuse. Make also. 
 
 — 'iK = 
 
 and therefore 
 we shall have 
 
 p — a p~ - a~ 
 
 K^-i^ = (i3'^-a'0-^>0, Ti>Tk, iK>-^; 
 
 and the equation of the ellipsoid will acquire the form, 
 
 T (ip + pK) = K^ - L^ ; 
 
 which is indeed not quite so short as the form last assigned in the 
 present article, but has the advantage of a greater homogeneity, 
 and lends itself with ease to the purposes of geometrical inter- 
 pretation and construction, as, for example, in the following 
 way. 
 
 466. From any assumed point c draw two right lines, ca, 
 CB, as in the annexed figure 98, to repre- 
 sent the vectors k, i of the foregoing ar- 
 ticle, in such a manner as to have 
 
 CA=jc, CB=t, CB > CA, ACB>-; 
 
 Fig. 98. 
 
 and with c for centre, and ca for radius, 
 conceive a sphere to be described, cutting 
 AB in g; so that 
 
 j^2 _ ^2 _ 'p^a _ 'p^2 ^ ^g2 _ ^^2 ^3^ £(j 
 
 Let e be supposed to denote some vari- ^'i 
 able point on the ellipsoid, of which the 
 equation is (by the last article), 
 
 2 H 2 
 
468 ON QUATERNIONS. 
 
 T (tjO + pK) = K^- t% 
 
 and let the fixed origin of the variable vector p be placed at the 
 point a; let d denote the second point where the line ae meets 
 the sphere ; finally let us conceive the lines bd, cd, to be drawn, 
 and denote the latter by cr: so that we shall have 
 
 AE = |0, CD = cr, DB = I - a. 
 
 Then (7 may be regarded as the reflexion of that fixed radius of 
 the sphere which is the prolongation of AC, and which may 
 therefore be denoted by - k, this reflexion being- performed with 
 respect to another and variable radius which has the direction of 
 + p ; and hence it follows, by reasonings similar to those of art. 
 429 respecting the equation ^a= aj3, even without here assuming 
 the knowledge of what was shewn in the preceding Lecture re- 
 specting the symbol ypy'^ (arts. 290, 291), or the connected 
 symbol -yaj~^ (art. 332), that 
 
 ap = p (— k), p(c = -(7|0, ip + pK = {l - a) p; 
 
 and therefore the equation of the ellipsoid becomes 
 
 T (i - (j) Tp = K' - i^ ; 
 that is 
 
 BD . AE = BA . BG = BD , Bd', 
 
 or simply, 
 
 AE = BD', 
 
 if d' be the second point where the secant bd meets the sphere. 
 Conversely, if any secant bdd' (or bd'd) be drawn to the sphere 
 round c from the external point b, and if from the siiperflcial 
 point A of that sphere there be taken, on the guide-chord ad, 
 or on that chord either way prolonged, a portion ae which in 
 lenf/this equal to bd', the /ocus of the point e, constructed thus, 
 is an ellipsoid. This very simple mode of generating that im- 
 portant surface is due (so far as I am aware) to the quaternions, 
 and was communicated as such to the Royal Irish Academy in 
 1846, having been deduced nearly as above from an equation pre- 
 viously exhibited in 1845, which agreed substantially with that of 
 art. 436, namely, with the following, 
 
 (S.p„-)=-(V.p/3-)-l. 
 
LECTUUK VII. 469 
 
 The same ellipsoid will evidently be the locus of the points f, f', 
 if the diameter ff' coincide in position with the conjugate guide- 
 chord ad', and if 
 
 AF = af'= bd. 
 
 467. The equation ae = bd' of the ellipsoid is very fertile of 
 geometrical consequences, a few of which may properly be stated 
 here. First, then, it shews that (as indicated in fig. 98) the 
 point B is itself Q. point on the ellipsoid; because when the 
 GUIDE-POINT D takes the position g, then the connected Doint 
 d'j which may in this construction be called the covjug ate guide- 
 point, comes to be placed at a; so that bd' becomes ba, and this 
 length of one side of the generating triangle abc is to be set 
 oft from the centre a of the ellipsoid, either in the direction of the 
 side AB itself, or else in the opposite direction : but one of these 
 two modes of setting oft" that length conducts to the point B. 
 Secondly, if we draw, as in the figure, from b through c, a secant 
 bkck', to the sphere which is described round c through A, and 
 which from its relation to the ellipsoid whose centre is at A may 
 be called the diacentric sphere, then the length ae of the 
 semi-diameter of the ellipsoid, as being by our equation always 
 equal to bd', will become a maximum when d' coincides with k', 
 and therefore D with k; if then we set off a line al in the direc- 
 tion of ak, and conceive another line al' to be set off in the op- 
 posite direction, these two opposite lines al, al' will be the major 
 semi-axes of the ellipsoid; or in other words, the points L, l' 
 will be the two jnajor summits of that surface. Thirdly, to find 
 the minimum value of the semi-diameter, we must evidently 
 place the guide-point d at k', and the conjugate guide-point D'at 
 K; that is, we are to set off from a, on the guide-chord ak', two 
 opposite lines am, am', whose common length is bk: and then 
 these lines will be the two minor semi-axes, and the points m, 
 m' the two minor summits of the ellipsoid ; while the angle in 
 the semicircle, kak' (or lam'), exhibits the well-known perpendi- 
 cularity of the minor axis mm' to the major axis ll'. Fourthly, 
 let the ellipsoid be cut by any given concentric sphere, of which 
 the radius ae is intermediate in length between bk and bg, or 
 else between bg and bk'; the length of bd' will then (by our 
 
470 
 
 ON QUATERNIONS. 
 
 equation) be given, and so will therefore the length of bd, and 
 this latter length will be different from ba; hence the locus of d 
 will be a circle of the diacentric sjohere, in a plane perpendicular 
 to BC, which plane will not pass through the point a: the cur- 
 vilinear locus of E on the ellipsoid will therefore be (as is other- 
 wise known) a spherical conic, since it will be contained at 
 once on the given concentric sphere, and on the cone which has 
 the centre a for vertex, and the circular locus of the guide-point 
 D for base : and the construction shews (compare 420) that the 
 two cyclic planes of this cone are the two planes through a, which 
 are perpendicular respectively to the two sides cb, CA(oriand k) 
 of the generating triangle abc. Fifthly, these two diametrical 
 planes themselves cut the ellipsoid in circles, or 2ixe cyclic planes 
 of that ellipsoid ; for if d move in the circle which has ah' for 
 diameter, in the larger figure 99 annexed, and is perpendi- 
 cular to the plane of that 
 figure, as being perpendi- 
 cular to the side bc of the 
 triangle, the conjugate 
 guide-point d' will move 
 in that other and parallel 
 circle which has gh in the 
 same figure for its diame- 
 ter ; so that the length of 
 bd', and therefore also (by 
 the equation) the length 
 of ae, will remain constant 
 
 and ==BG, and e will de- ^ 
 scribe a circle on the ellip- Q' 
 sold, whose diameter in f 
 fig. 99 is qq' : and again, 
 if D approach indefinitely 
 to A in any direction on 
 the sphere, d' will at the 
 same time approach inde- 
 finitely too, and the length 
 bd' or ae will tend to be- 
 come BG, and a circle de- 
 
LECTURE VII. 471 
 
 scribed with this radius, in the tangent plane at a to the diaeen- 
 tric sphere, of which plane the trace in fig. 99 is the line nn', 
 will be the intersection of that plane with the ellipsoid. Sixthly ^ 
 the sphere with a for centre, and with a radius = bg, cuts the 
 ellipsoid in the system of these two circles, which are thus 
 a sort oi limit of the spherical conies recently considered; and 
 this sphere may be conveniently called the mean sphere, be- 
 cause if we conceive a perpendicular to the plane of the figure 
 (answering to the line oc' of art. 435), which shall be equal in 
 length to BG, and therefore intermediate in length between the 
 greatest and least semi-axes lately determined, but, like them, a 
 semi-diameter normal to the surface^ this normal semi-diameter 
 will be one of the two mean semi-axes, and its termination will 
 be one of the two mean summits of the ellipsoid. Seventhly, if 
 we denote (as is often done) by a, b, c the lengths of the major, 
 mean, and minor semi-axes, we can express, in terms of these, 
 the lengths of the sides of the generating triangle, as follows: 
 
 Bc=-^(a + c); CA = ^(a-c); ba = acb'^] 
 because ^ 
 
 a = BK', C = BK, b = BG. 
 
 Eighthly, since 
 
 bd.ae = bd .bd'=bg.ba, 
 
 while the angle adb is not in general right, the double area of 
 the triangle aeb is in general less than this last rectangle, and 
 the perpendicular distatice of e from ab is in general less than 
 bg; but for a similar reason this distance is equal to bg, for the 
 particular system of those points E of the ellipsoid, which answer 
 to those points D of the diacentric sphere for which adb is a right 
 angle; draw therefore as in fig. 99, the diameter acr of that 
 sphere, and the secant brr', and conceive a circle described on ar' 
 as diameter, in a plane perpendicular to that of the figure ; this 
 circle will be the intersection of the diacentric sphere with another 
 sphere whose diameter is ab, and will therefore be the required 
 curvilinear locus of those points d, for which the angle adb, like 
 ar'b, is right ; and the corresponding points E of the ellipsoid 
 will be at once situated in the plane of this, new circle, and on 
 
472 ON QUATERNIONS. 
 
 the cylinder of revolution which has ab for axis, and bg tor ra- 
 dius ; they will therefore be situated on an elliptic section of this 
 cylinder, whose major axis is tt' in the fig-ure ; and every other 
 point E will fall within the cylinder : that is to say, the ellipsoid 
 is enveloped, along this ellipse on tt', by the cylinder whose axis 
 is the side ab of the generating triangle abc, and whose radius 
 is equal to the mean semi-axis (b) of the ellipsoid; so that the same 
 cylinder envelopes also the mean sphere, namely, along a circle, 
 whose diameter in fig. 99 is ss'. (The ellipsoid and mean sphere 
 have also another common enveloping cylinder, of which, in the 
 same figure, the axis of revolution is pp'; the angle bap being 
 bisected by the major semi-axis of the ellipsoid, al.) 
 
 468. The foregoing account by no means exhausts the geo- 
 metrical (nor even the easy) consequences of the equation 
 
 AE = bd'; 
 
 which must indeed be conceived to admit of being developed, so 
 as to conduct to every possible property of the ellipsoid. We may 
 for instance, apply that equation to deducing the difference of 
 the squares of the reciprocals of the semi-axes of an arbitrary 
 diametral section, and the law of the variation of that difference, 
 in passing from one such section to another. Conceive for this 
 purpose, that the ellipsoid and the diacentric sphere are both cut 
 by some plane ab'c^ ; B^ and c^ being the projections on it of the 
 points B and c. The guide-point d thus moves along a circle 
 with the projection c'' for its centre, and passing through the 
 
 point a; and because ae varies inversely as bd, we are to seek 
 
 the difference of the squares of the extreme values of bd, or of 
 
 b'd, since bb^ is given, and 
 
 bd2= bb'- + B^D'^ 
 
 Let b'c^ cut the circular locus of d in two points Dj, D2, the one 
 nearer to b' being Dj ; the last-mentioned difference of squares 
 is then, 
 
 b'd2'^ - b'di'- = 4 b'c' . c\\ ; 
 
 it is therefore equal io four times the rectangle under the projec- 
 
LECTURE VII. 473 
 
 tions of the two sides bc, ca of the generating triangle on the 
 plane of the diametral section of the ellipsoid. And because 
 
 4bc . CA = a--c-, andBD.AE = «c, 
 
 while BC and ca are perpendicular respectively to the two cyclic 
 planes of the eliip'soid (and we now see that there are no more 
 than two such planes^, the expression for the difference of the 
 squares of the semi-axes of a diametral section is found by this 
 method to be of the known form, 
 
 aEo"^ - AEf - = (c- - «'-) sin V sin v ; 
 
 El, Ea being the points which correspond to Di, Dj, and v, ?;' being 
 the inclinations of the cutting plane to the two cyclic planes. It 
 may be proper to note that the same construction exhibits, in a 
 very elementary manner, the known mutual rectangular ity of the 
 two extreme diameters of a section; because aEi, AEg have the 
 directions of AD3, aDj (or the opposite directions), and DiADj is 
 an angle in a semi-circle. The^c^ and the law of the gradual 
 diminution of the semi-diameter of a section, in passing from its 
 greatest to its least value, might also easily be put in evidence, by 
 following out the same method of construction. 
 
 469. But however simple may be these geometrical deduc- 
 tions from the equation ae = bd', yet many of the same and other 
 consequences may be obtained with even greater ease by calcu- 
 lation ivith quaternions. To shew, for example, that the ellip- 
 soid is cut in circles by the two diametral planes perpendicular 
 to CBj ca, or to t, jc, that is, by the two cyclic planes whose equa-" 
 tions are, 
 
 S.;p = 0, S . (cp = 0, ov ip = — pt, pK = -Kp, 
 
 we have only to substitute these last values for ip and pK in the 
 equation T [ip + pK) = k^ - i^, and we find that each of the two 
 planes cuts the surface in a curve, which is contained on the 
 mean sphere, whose equation is 
 
 K- - i^ 
 
 To = b, where b = ,=^rz ^ = etc T (t - k)"S 
 
 1 (i-k) 
 
 if we make for abridgment, 
 
474 ON QUATERNIONS. 
 
 a=Ti+ T/c, c = Tt - Tk, 
 
 so that 
 
 and it admits of being shewn, by calculation with quaternions, 
 that the a and c, thus determined, are respectively (as in 467) 
 the greatest and least semidiameters of the ellipsoid, or the max- 
 imum and minimum values of Tp. To shew that b is a point 
 upon the ellipsoid, it is sufficient to shew that its vector ab or 
 i - (c may be substituted for p in the equation of the locus ; 
 which appears from the identity, 
 
 t (t - fc) + (i - k) K = - (k- - t'^), 
 
 because the tensor of a negative scalar is (byl09,l 13) the positive 
 opposite thereof. One form of the equation of the cone of semi- 
 diameters p, which have a given and common length =r, inter- 
 mediate between a and b, or between b and c, is the following, 
 
 T {i + pK.p'^) = acr-^; 
 
 and the corresponding spherical conic on the ellipsoid may be 
 expressed by combining this equation of the cone with the equa- 
 tion, 
 
 Tp = r, 
 
 of the sphere on which the conic is contained. This conic con- 
 sists in general of two separately closed and diametrically oppo- 
 site branches ; but when the radius r = b, that is, when we cut the 
 ellipsoid by the mean sphere, the conic takes (as we have seen) 
 the limiting form of a system of two circles. In fact it will be 
 found that the equation 
 
 T{i + pK.p-')=T{t-K), 
 
 or the following, which is a transformation of it, 
 
 S . { (jOK . jO"^ + k) = 0, 
 
 may be still farther transformed, as follows : 
 
 S . <|0 . S . (cp = ; 
 
 and therefore that it represents the system of the two cyclic 
 planes, which system is thus a sort of limit of the cone. 
 
whence 
 
 LECTURE VII. 475 
 
 470. It may have been noticed that the ellipse and concentric 
 circle in fig. 99 are precisely the same as those in the earlier 
 figure 92 (art. 434), although new lines and letters have been 
 employed in the more recent of these two diagrams, and a dia- 
 centric circle introduced. Accordingly, this agreement was de- 
 signed, and it may be useful to shew how it was attained, by 
 means of the relations of art. 465, which connect the two new 
 vectors t, ic, with the two old vectors a, (3, through two other 
 constant and auxiliary lines, a, j3'. The article just cited gives, 
 by elimination of a, j3', 
 
 -^-^ ^ -i3 . 
 
 - a'^ — a 
 
 such then are the expressions for the two vectors t - k and t + k, 
 or AB and rb of fig. 99, considered as functions of a and j3, that 
 is, of the two vectors oa and ob of fig. 92. These expressions 
 give, 
 
 S.(i-K)a-' = -l = S.{i+K)(3-'; 
 
 V.(t-K)i3-i = = V.(i + K)a-^ 
 
 whence it was easy to infer, by combinations of plane and recti- 
 linear loci, on the plan of former articles, that i-k and -(i + k) 
 were equal respectively to the lines of' and oa' in fig. 92, if a' be 
 supposed to denote, in that figure, the intersection of oA and bc. 
 I therefore placed the new a and b of fig. 99 at the points o and 
 f' of fig. 92, and the new point c at the middle of the old line 
 aV (after inserting a' as just now explained); because, in figs. 
 98, 99, the origin of p is a (not o), and ab, ac are (in these lat- 
 ter figures) the vectors l-k and -k: and then proceeded as 
 above. I shall not delay you by proving here that a given ellip- 
 soid may be constructed in more ways than, one, by means of 
 diacentric spheres; and that it is not indispensable to the con- 
 struction to have the fixed point b external to the sphere 
 
476 ON QUATERNIONS. 
 
 471. Since Kp + pa is a scalar, we have, as an identity in this 
 calculus, holding good for any three vectors, the equation, 
 
 . / Kp + pK\ 
 
 ip + pK = {i - K) I p 1. 
 
 Introducing therefore a new and variable vector X, determined 
 by the expression 
 
 A = (/C|0 + pK) (fC - t)"S 
 
 the equation of the ellipsoid takes the forn), 
 
 T (p-X) = b, because 6 = (k^ - i-) T (i- k)'^; 
 where 
 
 X = h(i-K), if/i = 2S.Kjo.T(t-K)-^ 
 
 If we assign any give7i scalar value to this co-efficieiit A, we get 
 on the one hand a given value for the vector X, 
 
 X= AL = A . AB, 
 
 where Ij h a, new and variable •point, situated on the indejinite 
 line AB, and not now (as in figures 98, 99) a major summit oi ihe 
 ellipsoid; and on the other hand we ohtsan di given plane, per- 
 pendicular to K or to AC, as one locus of the extremity e of p; 
 while the recent equation, 
 
 T {p -X) = b, or LE = b, 
 
 shews that another locus for the same point e is a given sphere^ 
 with centre L, and with radius b. If then this plane intersect 
 the ellipsoid at all, that is, if the value which it gives for S . Kp 
 be not too great numerically (by h being assumed too large), 
 the curve of intersection will be a circle. It follows then that 
 indefinitely many circles can be traced on the ellipsoid, with their 
 planes parallel to one of the two cyclic planes through the cen- 
 tre : a well-known theorem, indeed, but one which it seemed 
 worth while to reproduce by the foregoing calculation with qua- 
 ternions. 
 
 472. Again let fx be another new variable vector expressed 
 as a function of |0 by the formula, 
 
 fi= (ip + pi) {i - (c)'^ = h' {k - l), where h' = 2S . ip .T (t- k)'^. 
 
LECTURE VII. 477 
 
 Then, because 
 
 ip + pK = {ip + pi)- p(i-k) = (p- p) (i - k), 
 the equation of the ellipsoid will take this new form : 
 
 T(p-^) = b; 
 
 and to each assumed value of the scalar coefficient h', which is 
 not numerically too great, will answer a plane perpendicular to 
 I, or parallel to the other cyclic plane of the ellipsoid, and cut- 
 ting- that surface in another circle, contained upon another sphere, 
 which has the same radrus b, but has a different centre from the 
 sphere of the last article : namely, a new point M on the same 
 indefinite line ab as before, which point is the variable extre- 
 mity of the new vector n (and is not now a 7ninor summit of the 
 ellipsoid) ; so that 
 
 AM = fX = - h! . AB, ME = 6. 
 
 The ellipsoid is therefore (as is well known) the lociis of two 
 distinct systems of circles, whose planes are parallel to the two 
 cyclic planes drawn through the centre ; and we see that the 
 planes of these circles 2L\e perpendicular to the two sides, ca, cb, 
 of the generating triangle abc, in the construction of art. 466. 
 
 " 473. Any tico such circles, belonging to different systems, or 
 as vpe may by analogy say (compare art. 420), any two sub-con- 
 trary and circular sections of the ellipsoid, are known to be con- 
 tained upon one common spheric surface; and accordingly it can 
 easily be shewn by quaternions, that whatever two subcontrary 
 circles may be thus selected, with their own corresponding values 
 of the scalars h and A', those two circles (Ji, h') are both contained 
 upon that new sphere whose equation is 
 
 T (|0 - I) = n, or WE = n, 
 
 where the new point n, the vector ^, and the scalar n, are such that 
 
 AN = ^ = At + A'(C = -2 (f - K:)"-(tS . kp + kS . ip), 
 
 and 
 
 n = ^[b'^~{]i + h') {he + A'k^) } : 
 
 and where it is important to observe that n is situated in the 
 
478 ON QUATERNIONS. 
 
 plane abc, because ^||| t, k. In fact, this new sphere, with cen- 
 tre N and radius n, may have its equation thus expanded : 
 
 {i= {p-%y ^n^ = p'~-2{h^ . tp + KS .Kp)-hh' {L-K)'-\-b'; 
 
 and this condition is satisfied, whether we suppose that p satis- 
 fies the equations of the^>5^ circle {h), which may be written 
 thus : 
 
 = p2 _ 2A S . <|0 + 2/iS . Kp + A- {i - kY + b% 
 
 0={/i + h'){2S.Kp + h{i-Ky}; 
 
 or the equations of the second circle (A'), under the forms, 
 = p2 - 2h'S . Kp + 2h'S . ip + h'' (t - kY- + b\ 
 
 474. If these two circles, in planes perpendicular respectively 
 to K and I, be supposed to intersect each other on their common 
 sphere in any one point E of the ellipsoid, it is clear that they 
 must also intersect each other in another point Ei of that surface, 
 which point is such that the common chord eEi is perpendicular 
 to both K and t, or to the plane of the triangle abc ; this chord is 
 also evidently bisected by that plane in a point e\ which is the 
 common projection of the two points e, Ej, thereon ; because this 
 plane contains, by the foregoing article, the centre n of the 
 sphere (which is not to be confounded with any of the points so 
 marked in recent figures). It is evident also that this sphere 
 round n is doubly tangent to the ellipsoid, touching it both at e 
 and at Ei ; because, at each of those two points, the sphere and 
 the ellipsoid have two rectilinear tangents in common, namely, 
 the tangents to the two circles (Ji, h'). Hence the radii ne, nEj, 
 of the sphere must be normals to the ellipsoid, at the points e 
 and El respectively ; or, in other words, the point n is the com- 
 mon Jbot of the two normals en, EiN, which are. drawn to the 
 ellipsoid at those two points, and are continued to meet the plane 
 of ABC. With regard to the common length of these two normals, 
 since it is equal to the radius of the new sphere, it is expressed by 
 the recent radical, n ; while the normal en thus drawn to the 
 ellipsoid at e, and continued till it meets the plane of the gene- 
 rating triangle, that is (by art. 467) the plane of the greatest and 
 
LECTURE VII. 479 
 
 hast axes, is expressed, both in length and in direction^ by the 
 formula, 
 
 EN = ^-p, 
 
 where ^ has its recent value (assigned in art. 473). Operating 
 by S .|0, we find, 
 
 S . jO (^ - |o) = - jo^ - 4 (t - (c)"^ isi . ip S . Kp = b^, 
 
 because, by 471, 
 
 b' = -(p-xy=-p'+2S.p\-x\ \'=4(i-k)-' (s.fcp)^ 
 
 2S .|oX= 2A(S .ip-S .-Kp) = -4 {i-k)'^ S . KjO (S . t|0- S . Kjo) ; 
 or because, by 472, 
 
 b- = - (p - jnY = - p- -i- 2S . pfx - jit-, ju^ = 4 (< - k)"^ (S . tpy, 
 2 S • pfji= 2A' (S.jc^-S.tp) = -4(t-K)"^S.t|o(S.K/>-S. t^). 
 
 If therefore we now introduce a new vector v, determined as a 
 function of jo by the equation 
 
 ^-p = b"v, 
 
 or (see the values already found for b and 5), 
 
 (k^ — t^)^V= (< — kYp + 2 (iS . KjO + kS . tp), 
 
 this vector v will at once be perpendicular to the plane which 
 touches the ellipsoid at e, and will satisfy this very simple con- 
 dition : 
 
 S . V|0 = 1 . 
 
 And we see, at the same time, that the equation of the ellipsoid 
 may be put under this new form, 
 
 (O^ + 6- = \p, 
 
 where X, p are those two functions of p which were so denoted 
 in 471, 472; whence we perceive anew that the mean sphere, 
 whose equation may be thus written, 
 
 p'' + ¥=0, 
 
 intersects the ellipsoid in the system of those two circles which 
 are contained in the two diametral planes, 
 
 X = 0, |L( = 0; orS.Kp^O, S.t|O = 0. 
 
480 ON QUATERNIONS. 
 
 475. The vector i;, thus lately introduced, is an important 
 one in the theory of the ellipsoid. Suppose, for example, that 
 we wish to circumscribe about that surface a cylinder (not gene- 
 rally of revolution), with its generating lines in the direction of 
 some given vector t? ; to find the curve of contact we have im- 
 mediately the equation, 
 
 S . -sjv = 0, because v _L -sr; 
 
 the normal to the ellipsoid, at any point of this sought curve, 
 being normal also to the enveloping cylinder, and the normal to 
 a cylinder being everywhere perpendicular to the common direc- 
 tion of all its rectilinear generatrices. And then, on substituting 
 for V its value as a function of jo, we obtain the condition, 
 
 = (t - k)^ S . -m^ + 2 (S . -?7t S . (CjO + S . -sjK S . (p). 
 
 Let us write, for abridgment, 
 
 v==(p{p'), or simply v = <pp, 
 
 using (p as Si functional sigw, we shall have, in like manner, 
 
 w = <p (^), or w = (p7s, 
 if to be a new vector such that 
 
 (k" - i^y 0w = (k' - i-y a> = (t - k)- -ST + 2 (iS . fctrr + kS . tw) : 
 
 and then the recent condition of contact with the cylinder be- 
 comes simply, 
 
 S . pto = 0. 
 
 The curve of contact is therefore plane and diametral (as indeed 
 it is otherwise known to be) ; and we see that the perpendicular 
 to the plane of contact has the direction of the vector w, or 0w, 
 determined by this easy calculation. 
 
 476. If we introduce for conciseness another functional sym- 
 bol, f{p, w), defined by the equation, 
 
 fipi -st) = S . p0?^, 
 or more fully, 
 
 (k^ - t^Yfip, to) = (t - k)- S . |OCT + 2 (S . <jo S . (COT + S . K'p S . tra-), 
 
LECTURE VII. 481 
 
 we see, on the one hand, that this new function is symmetric 
 with respect to the two variable vectors, p and rs, or that 
 
 /(^5P)=/(/>j^); 
 
 and on the other hand that when w has, as above supposed, the 
 given direction of the sides of a cylinder enveloping the ellip- 
 soid, the equation of the plane of contact takes the form, 
 
 If we farther agree to write for conciseness, 
 f{p,p)=f{p)=fp, 
 
 whatever vector p may be, then, because v = (j)p, and S .|0v = 1, 
 the equation of the ellipsoid reduces itself, in this notation, to the 
 form, 
 
 477. These functions and^, which are respectively equal 
 to a vector and to a scalar, are of great utility in calculations 
 concerning the ellipsoid ; and indeed analogous functions present 
 themselves usefully in investigations with quaternions, respect- 
 ing other surfaces of the second order; and even in some more 
 general inquiries. The vector function ^ (from which the scalar 
 function f is formed) has, relatively to the vector p on which it 
 depends, the distributive character expressed by the for- 
 mula, 
 
 ^ (p + p) = (l>p + <^p\ or, A^p = 0(Ap), 
 
 if A be still the sign of the operation of taking a difference : 
 connected with which is the property, that if x be any scalar co- 
 efficient, 
 
 (p (oCp) = X(pp. 
 
 It follows hence that the scalar function j^(io, ■nr) is distributive^ 
 with respect to each separately of the two vectors on which it 
 depends ; or that 
 
 f(p + p, zs + ot') =/(/), zs + to') +/(/o', zs + to') 
 
 and that 
 
 2 I 
 
482 ON QUATERNIONS. 
 
 Abridging therefore, as above, the symbol /(p, p) to f(p), or to 
 fp, we find that 
 
 f(xp) = ^'fp; 
 and that 
 
 f(p + p) =fp + 2/(p, p) +fp : 
 which last equation may also be thus written, 
 Afp = 2f{p,Ap-)+f(Ap). 
 
 It is easy to foresee, that when a theory of differentials of 
 QUATERNIONS shall have been established, but before these Lec- 
 tures close I hardly hope to give even a sketch or beginning of 
 such a theory, there will result an expression of the following 
 form for the differential of the function /: 
 
 dfp=2f(p,dp) = 2S.^pdp. 
 
 478. Without yet introducing differentials, let o- + t and g-t 
 denote two different directed semi-diameters, or two values of p 
 for the ellipsoid; so that o- is ihe vector of the middle point oi 
 some (rectilinear) chord; while t denotes one of the two directed 
 semi-chords, or a vector equal thereto. Then, by 476, 
 
 l=/(,T + r)=/(a-r); 
 
 and therefore, by 477, 
 
 l=>+/^ + 2/(«T, r); 
 
 The semi-sum of these two equations gives the relation 
 
 1 =/<T +/t ; 
 and their semi-difference conducts to this other formula, 
 
 0=/(.T, r): 
 
 which last may be called the equation of conjugation, be- 
 tween THE two directions of the two vectors, a and r ; 
 namely, between the directions of a diameter of the surface, and 
 a chord which is bisected by that diameter. In fact it is usual 
 to say that two such directions are conjugate, with respect to the 
 
LECTURE VII. 483 
 
 ellipsoid, or other surface of the second order, for which this re- 
 lation of bisection exists : and as regards the known reciprocal 
 character of the relation, it is expressed in our symbols by the 
 formula (see 476), 
 
 /(r,<r)=/(.T,r). 
 
 Or we might observe that, by 477, 
 
 and therefore that if we suppose, as in the present article, 
 
 l=/((r + r)=/(<r-r), 
 we shall have also 
 
 l=/(r + (T)=/(r-a), 
 
 when a and r have been interchanged. Our symbols might 
 therefore in this other way serve to remind us, that if a diameter 
 in the direction of o- bisect a chord of the ellipsoid parallel to r, 
 then reciprocally the diameter in the direction of r bisects a 
 chord parallel to o-. 
 
 479. We are not pretending to offer here a systematic trea- 
 tise, nor even an elementary essay, on the properties of the el- 
 lipsoid themselves ; but rather are employing, in parts of this 
 Lecture, a few of those properties, without much concerning our- 
 selves whether they be already known, or in some cases new, in 
 order to illustrate the method of quaternions. The known and 
 familiar character of some of these conjugate relations need not 
 therefore prevent us from discussing them a little farther here, 
 in connexion with the present calculus. Thus we may notice, 
 that since the equation of conjugation between directions, as- 
 signed in the foregoing article, namely, 
 
 0=/(c7,r), orO=/(r, a), 
 becomes, by 476, 
 
 = S . T^(7, 
 
 it follows that the diameter in the direction of a bisects all the 
 chords which can be drawn across it, parallel to (or contained 
 in) a given diametral plane, to which the normal has the direc- 
 tion of 0(7. Hence this diameter in the direction of a may, con- 
 
 2x2 
 
484 ON QUATERNIONS. 
 
 sistently with usage, be said to be itself conjugate to this diame- 
 tral plane; and by comparing this conclusion with that of art. 
 475, we should arrive in a new way at the known result, that the 
 axis of any cylinder, circumscribed about an ellipsoid, is conju- 
 gate to the plane of contact. It would also be easy to prove, by 
 our formulae, that a chord, parallel to a given diameter, is bisected 
 by the diametral plane which is conjugate thereto. 
 
 480. The equation of 478, 
 
 shews that while the abscissa a, as measured from the centre on 
 a given semi-diameter p, increases from to p, the ordinate t at 
 the same time diminishes (in length) to 0, according to a law 
 easily assigned, from the value which it had when it at first co- 
 incided with some given and conjugate semi-diameter p of the 
 ellipsoid, which new semi-diameter p thus satisfies the two con- 
 ditions (see 476, 478), 
 
 /p = 1 ;/(/,, p') = 0. 
 
 In fact if we make 
 
 (j = xp, T = yp\ 
 
 where x and y are scalar coefficients, we shall have, by the equa- 
 tion of the ellipsoid, and by the properties of the function/", 
 
 \=f{xp + yp') 
 
 =f(^p) + V{^p> yp) +f(i/p) 
 
 = xfp + ^xyfip, p) + y^f (/)') ; 
 or simply, 
 
 1 = a?- + ?/" : 
 
 so that while x increases from to 1, y decreases from 1 to 0. 
 More generally, let p, p\ p" be any three conjugate semi-diame- 
 ters, so that 
 
 1 =fp =fp'=fp"i 
 
 o=f(p,p)-f{p,p")=f(p%p); 
 
 and let w denote any other semi-diameter: we can always con- 
 ceive this vector w decomposed by projections, so as to take the 
 form, 
 
LECTURE VII. 485 
 
 u) = xp + yp + zp" ; 
 
 and then the equation of the ellipsoid will give, by calculations 
 of exactly the same form as those just now made use of, this very 
 simple relation between the three scalar coefficients^ which agrees 
 with known results, although the scalars a;, y, z which it involves 
 are not precisely the same as the usual co-ordinates of the ellip- 
 soid : 
 
 1 = aj2 + 2/2 + 2'^. 
 
 (Compare the equation satisfied by the point p', in art. 435.) 
 
 48 1 . The foregoing results might be employed to prove anew, 
 in various ways, by limits, the known theorem that the tangent 
 plane, at the extremity of any given semi-diameter p, is parallel 
 to the diametral plane, which is conjugate to that semi-diameter : 
 and consequently that the normal to the ellipsoid, at the extre- 
 mity of |0, I's, perpendicular to both of the two conjugate semi- 
 diameters, p and p", lately considered. But 
 
 =/ (p", |o) = S . p"^p ; 
 
 this common perpendicular, or normal, must therefore have the 
 direction oi±(pp. And accordingly, we had, in 475, the equa- 
 tion 
 
 v= (pp; 
 
 where v, by 474, was a vector perpendicular to the plane which 
 touched, at the extremity e of p, a sphere which there touched 
 the ellipsoid. If then we denote by ot, the vector drawn from 
 the centre a of the ellipsoid to any point p of the tangent plane 
 at E, so that -z^ - p is (or is equal to) a tangential vector at e, and 
 is therefore X v, we shall have on this account the condition, 
 
 S . V ("J? - jo) = 0. 
 
 But also we have, by 474, 
 
 S . vjo = 1 ; 
 
 hence the equation of the tangent plane, with zs for a vari- 
 able (while V is a. fixed) vector, is found to take this simple form : 
 
 S . vzj = 1 : 
 
486 ON QUATERNIONS. 
 
 or if we choose to write it so, 
 
 s. v(^-v-o = o. 
 
 And hence again it follows, by the principles of the present Lec- 
 ture, thsit the reciprocal v'^, of the foregoing normal vector v, 
 represents, in length and direction, the perpendicular let fall from, 
 the centre of the ellipsoid upon the tangent plane. On this ac- 
 count I have been led, in imitation of a phraseology of which a 
 happy use has been made by Sir John Herschel, in connexion 
 with other researches, to call the vector v itself the vector of 
 PROXIMITY of the ellipsoid : because it serves to mark, by its 
 direction and its length, the direction and the nearness (to the 
 centre) of the superficial element of the ellipsoid, or of the tan- 
 gent plane; since it is the reciprocal of the perpendicular let fall 
 on that plane from the centre. 
 
 482. The equation of the tangent plane, assigned in the last 
 article, may, by the value v = ^p, and by the relation between 
 the functions and /, be also written thus : 
 
 zs being still the variable vector, terminating at a variable point 
 p on the plane, and p being the fixed vector, terminating at the 
 given point e of contact. But let us now conceive that an ex- 
 ternal point p, with vector zs, is given, and that we wish to find 
 the point of contact e, or to find its vector p. For this purpose 
 we may still employ the last written equation ; and it gives now 
 a plane locus for the point of contact, which plane evidently 
 must be precisely that one which is called the the polar plane 
 of p, with respect to the ellipsoid (compare 422, 423). Every 
 point on this plane is said to be conjugate to the point v, with 
 respect to the given ellipsoid; and the form of the function ^ 
 shews (by 476) that this relation between two conjugate points 
 is (as it is known to be) a reciprocal one (compare again 423). 
 We may therefore say that the equation 
 
 expresses the condition necessary in order that the two vectors 
 p and zs (both drawn from the centre) may terminate on ttvo 
 
LECTURE VII. 487 
 
 conjugate points : and for the same reason we may call this for- 
 mula the EQUATION OF CONJUGATION BETWEEN THE TWO VECTORS, 
 
 p and OT, or between their terminations, e and p. If we change 
 zs to pw, where ^ is a scalar coefficient, the equation of conjuga- 
 tion is changed to the following : 
 
 and then by supposing the number p to increase without limit, or 
 the point p to go off" to infinity^ the equation takes the form, 
 
 0=/(p,^): ; 
 
 which was found by a different process in art. 476, as the equa- 
 tion of the p/a«e of contact of the ellipsoid with an enveloping 
 cylinder, whose generating right lines have the direction of Z3 ; 
 or as the condition for the tangent plane at the extremity of the 
 semi-diameter p being parallel to that given vector w. Accord- 
 ingly, this last equation, =f{p, ts), or at least one of the same 
 form, was assigned in 478, as expressing a relation of conjuga- 
 tion between two directions, and not between two points, at 
 least if the points be supposed to be both 2X finite distances from 
 the centre. 
 
 483. An external point p being given by its vector t?, we 
 may propose to find the equation of the cone of tangents 
 to the ellipsoid, which can be drawn from this point p (compare 
 425, 461). \i p be still the vector of a point e of contact, we 
 shall have the conditions, 
 
 and if in these we make 
 
 p=-7!3 -Vtr, 
 
 where ^ is a scalar, and r a vector drawn in the direction of one 
 of the tangents from p, we find 
 
 1=/ot+^/(w, r); 
 and therefore also (subtracting, and dividing by /), 
 =/(.., r) + (/r. 
 
488 ON QUATERNIONS. 
 
 Eliminating t between the two last equations, we get 
 
 /(^,r)^=(/t.-l)/r; 
 
 and this is one form of the equation of the cone, with the vertex 
 taken for the origin of the variable vector r : because r in it 
 may be changed to tr, each member being then multiplied by t"^. 
 Changing, therefore, r to p - •sr, and observing that 
 
 /(ot, p-ts)=f{p, ZD-) -fzs, 
 f{p-z3) =fp +fz; - 2f(p, zy), 
 
 the lately written form becomes, after a few very easy reductions, 
 
 {f(p,^)-l}^=(fp-l)(f^-l); 
 
 such then is another form of the equation of the enveloping 
 CONE, with the origin at the centre of the ellipsoid; the given 
 vector of the vertex being zs, and p being the variable vector of a 
 point upon the conic surface. 
 
 484. Another mode of obtaining the same equation of this 
 enveloping cone, is to change p to •57 + / (|0 -■ct), or io tp-^uzs, 
 where t+u = \, in the two first equations of the foregoing article ; 
 and then to eliminate t, or to eliminate ut'^, between the two re- 
 sulting equations, 
 
 f + 2tu + u'= ffp + 2^M/(p, zy) + ufzs, 
 t + u = tf(p, 7s) + uf-ss ; 
 
 which give, by easy combinations, 
 
 ^(/(p,^)-l} + M(/^-l) = 0, 
 u[f{p,zy)-\]+t{fp-\) = 0: 
 and therefore, as before. 
 
 By changing zs, as in the last article, to /)ot, and then supposing 
 p infinite, the enveloping cone becomes an enveloping cylin- 
 der, whose generating lines are parallel to zs: and the equation 
 of this cylinder is thus found to be, 
 
 /(p,^)^=(/p-i)A 
 
LECTURE VII. 489 
 
 Accordingly we know (by 476) that the curve of contact along 
 which this cylinder envelopes the ellipsoid, has for equations, 
 
 /(p,..) = 0;/p = l; 
 
 as, for the curve of contact with the cone, the equations were, 
 
 f(p,zy)^l, fp = l. 
 
 485. As veriiScations of these results, let us suppose the ra- 
 dius Tk of the diacentric sphere, in the construction of art. 466, 
 to vanish; the ellipsoid will evidently then degenerate into a 
 sphere, with Tt for its radius : and accordingly the equation of 
 art. 465, 
 
 T {ip + jOk) = k^ - t^, 
 
 reduces itself to 
 
 lip = Ti, when k = 0. 
 
 Under the same condition, the equation which determines v in 
 art. 474 as a function of p, or which assigns the form of <^jO in art. 
 475, becomes 
 
 i*i; = i^p, or V = (pp = I'^p ', 
 
 hence by 476, we have (if k still = 0), 
 
 f(p, zy) = lY S .pzy; fp = r^ p"" ; 
 
 and the equation /p = 1 of the ellipsoid becomes that of a sphere, 
 
 l=fp = r'p\ or, p^ = tK 
 
 The equation of the cone enveloping the ellipsoid becomes, when 
 we thus pass to the sphere, 
 
 {S.pzy- i^y = (p^ - t^) {zs^ - L% 
 or 
 
 {S . pz^y - p^ zj-" = - i" {p^ + Z5^ - 2S . pzj) ; 
 
 that is (compare 460), 
 
 (v.pz;y=-iHp-zy)s 
 
 which coincides with one of the equations in 461, when we 
 change zs to j3, and i^ to -c^ For the cylinder enveloping the 
 sphere, we should find by recent methods the equation : 
 
 {V . pz!y = -i^z}^j or TV. |OOT = Tt .Tto; 
 
490 ON QUATERNIONS. 
 
 and 'accordingly we saw, in 431, that the equation, 
 
 TV. pa = a, 
 
 represented a cylinder of revolution, with the vector a for its 
 axis, and with aTa'^ for its radius. 
 
 486. The equation of conjugation between two directions, 
 assigned in 478, or the formula 
 
 y*(o-, t) = 0, becomes S . or = 0, when k = ; 
 
 and thereby reproduces the known result that any two directions 
 which are conjugate relatively to a sphere are rectangular with 
 respect to each other ; while the more general equation of con- 
 jugation between two vectors p and tj, or between the two points 
 where those vectors terminate, which was assigned in 482, 
 namely, 
 
 y*(jO, •st) = 1, becomes S . pw = i'^ : 
 
 and therefore agrees with the equation 
 
 S . joo- = - a^, 
 
 of art. 423, when we change zy to <r, and denote the radius T« 
 by a. And if we wish to shew by calculation^ from the proper- 
 ties of the function^ that the harmonic section by the polar 
 plane holds good (as it is well known to do) not only for the sphere 
 but for the ellipsoid, we have only to imitate the process of art. 
 462, by making 
 
 and then substituting for t the semi-sum of the two roots of 
 the following quadratic equation in x : 
 
 =^ + 2a;-i/(w, r) + a?-2/-, 
 or 
 
 X^{fis-1) + 'Ixfip, r) +/- = 0. 
 
 For this semi-sum is evidently 
 
 t=f{p,T){\-f^y\ 
 
 and therefore the vector p of the point of harmonic section of a 
 
LECTURE VII. 491 
 
 variable secant of the ellipsoid, drawn from the extremity of the 
 given vector ot, is (if the centre a be still the origin of p), 
 
 p = w+T (l-fzs)f{zs,T)-'^; 
 
 but if we operate on this expression by the functional characte- 
 ristic, f{zs, ), or by the characteristic of operation, S.zj<p, we 
 obtain (by 476, 477) the result, 
 
 /0>» ^) =/(^> p) =/^ + ( 1 -f^) = 1 : 
 
 that is, by 482, we obtain the equation of the polar plane. 
 
 487. The expressions in 471, 472, 473, for X, ju, ^, give the 
 equations : 
 
 ^-X K-fi X-ju 
 
 K 
 
 = h + h'i 
 
 where X, ju, ^ are the vectors of the three corners, l, m, n, of a 
 certain variable triangle, in the plane of the fixed triangle abc. 
 If then we observe that 0, i- k, and - k are (by 466) the vectors 
 of the three corners, a, b, c, of that fixed or generating triangle 
 which was described in our construction of the ellipsoid, when the 
 centre a is still made the common origin of vectors, we shall see 
 that the equations, 
 
 NL -4- CA = MN -f- BC = LM -H AB = - (^ + A'), 
 
 hold good ; and that therefore the new and variable triangle lmn 
 is SIMILAR to the old UlU^ fixed triangle abc ; while it is also 
 SIMILARLY SITUATED, in One commou plane therewith, namely, 
 in the plane of the greatest and least axes of the ellipsoid ; the 
 sides LM, MN, NL of the one triangle being parallel and propor- 
 tional to the sides ab, bc, ca, of the other; while it follows from 
 471, 472, that the two variable points l and m are situated on 
 the same indefinite straight line as the two fixed points a and b : 
 that is, on the axis of that circumscribing cylinder of revolution, 
 which has been considered in former articles. The two vectors 
 AD, AE, of the two points d, e, in the same construction of the 
 ellipsoid, being, by 466, respectively equal to o- - k and p, where 
 ap=- jOK, and therefore 
 
 {(T - k) p = — pK - Kp = - 2S . Kp ; 
 
492 ON QUATERNIONS. 
 
 we have, by 471, 
 
 {<t-k)p = X{i-k) = h(i- kY- 
 
 But in general if two pairs of co-initial vectors, as here c- jc, /o, 
 and X, L-K, give, when respectively multiplied together, one com- 
 mon scalar product, they terminate m four concircular points : 
 the four points d, e, l, b, are therefore contained on the circum- 
 ference of one common circle : and consequently the point l, of 
 recent articles, may be found by an elementary construction, de- 
 rived from this simple calculation with quaternions : namely, as 
 the second point of intersection of the circle bde with the straight 
 line AB, which is situated in the plane of that circle. 
 488. Again, by 471, 472, we have 
 
 T(p-\) = T(p-f,) = b; 
 
 therefore the point e of the ellipsoid is the vertex of an isoceles 
 triangle, constructed on lm as base ; and the point m may thus 
 be found as the intersection of the same straight line ab (or al) 
 with a circle described round the point e as centre, in the plane 
 of ABE, and having its radius equal to the mean semi-axis of the 
 ellipsoid. When the two points l and m have thus been found, 
 the third point n can then be deduced from them, in an equally 
 simple geometrical manner, by drawing parallels, ln, mn, to the 
 sides AC, bc of the generating triangle abc, from which the ellip- 
 soid itself has been constructed. It is clear, from what has been 
 already shewn, not only that these tivo sides ln, mn, of the new 
 and variable triangle lmn, are parallel to the two cyclic nor- 
 mals of the ellipsoid, but also that they are portions of the axes 
 of the two circles which are contained upon the surface of that el- 
 lipsoid, and pass through the point e on that surface; l and m being 
 points on those two axes, because they are the centres of two 
 spheres, vi\\\ch. contain the two circles respectively; while the point 
 N of intersection of those two axes has been seen to be the centre 
 of that common sphere (473), which contains upon itself both 
 those two circular sections, and is doubly tangent (by 474) to the 
 ellipsoid, namely, at the two points of intersection of the two cir- 
 cles. Some of these results, with others yet to be established, will 
 be illustrated by a new diagram (figure 100), which is reserved 
 for a future article (art. 493). 
 
LECTURE VII. 493 
 
 489. In the present Lecture we have not as yet assumed 
 the Associative Principle of Multiplication, although it has been 
 several times alluded to ; but there will be found no difficulty 
 now in proving anew that associative property, as we have pro- 
 mised to do, with the help of the distributive principle. For this 
 purpose, let us make 
 
 then 
 
 = V« = V5 =Vc= Sa = Sj3 = Sy; 
 
 « . rg' = (c + 7) . (6 4 j3) (a + o) 
 
 = (c -f 7) . {ba + 6a + j3a + j3a) 
 
 ~c .ba + c .ba + c . j3a + c . j3a 
 
 + 7 . 6a + 7 . 6a + 7 . j3a + 7 . j3a ; 
 
 and in like manner 
 
 5r . g' = c6 . a + c6 . a + cj3 . a + cj3 . a 
 + 76 . « + 76 . a + 7j3 . a + 7/3 . a ; 
 
 where c .ba = cb .a hy algebra, because «, 6, c are scalar s; and 
 for the same reason, by comparatively easy principles of this cal- 
 culus (see the Third Lecture), we have c .ba = cb . a, c . j3a = 
 c^ ■ a, c . ^a = cfS . a, y . ba = jb . a, y .ba^yb .a, y . (3a = y(B . a. 
 It remains then only to prove the associative formula for the 
 MULTIPLICATION OF THREE VECTORS, namely the equation, 
 
 7 . (3a = 7j3 . a ; 
 
 which has indeed already been discussed at some length in the 
 Fifth Lecture, in connexion with spherical constructions, but 
 which we now desire to establish anew, independently of figures 
 on a sphere. Make for this purpose, as in art. 406, 
 
 i3 = i3'+i3", jS'lla, i3"±a; 
 
 make also, as we are evidently allowed to do, by projections on 
 three rectangular lines, 
 
 / . " , ni I II II II /J" /// I /// , r\ii 
 
 7 = 7+7+7, 7 II a, 7 II p, 7 ±a, 7 ±p; 
 we shall have, by the distributive principle, 
 
494 ON QUATERNIONS. 
 
 7 . j3a = 7' . /3'a + 7' . j3"a + 7" . j3'a + 7" . j3"a + 7'" . |3'a + 7'" . (5"a, 
 7j3 . a = 7'j3' . a + 7'/3" . a + j"(^' . a + 7"i3" . a + 7"'/3' . a + 7'")3" . a ; 
 
 and are to shew that each term of the one expression is equal to 
 the corresponding term of the other; in which comparison of 
 term with term, we may obviously introduce or suppress any 
 scalar coefficients, and so may assume, without any real loss of 
 generality, the values, 
 
 7=p=a, 7 =p =a, 7 =aat la= la; 
 
 d being a new line perpendicular to a, in the plane of a and j3. 
 We may even conceive that the system of three rectangular lines, 
 a, a', ad^ coincides with the system z,^, k (compare art. 77); 
 and then the six equations to be proved are seen to be true, 
 under the forms, 
 
 i . it = - i= II .i; i .ji = — ik = ki = ij . i, 
 
 j . a = - J = - ki =ji .i;j .ji = -jk = -i= jj . i ; 
 
 k .ii = -k =ji = ki .i; k .ji = -kk = \=-ii = kj . i. 
 
 It was nearly thus that I was originally led to perceive the truth 
 of the associative principle of multiplication of quaternions, after 
 having established as definitions (though not as wholly arbitrary 
 ones) the fundamental formulae respecting the multiplications of 
 ijk^ and having assumed (as I at first did) from algebraical analo- 
 gies, the truth of the distributive principle ; although I found my- 
 self compelled to reject the commutative property of multiplica- 
 tion, as not generally true for quaternions. 
 
 490. It was shewn, in the two preceding Lectures, that the 
 investigation and employment of the associative principle of 
 multiplication, without the distributive, led to many inte- 
 resting inquiries and results, especially as regarded spherical 
 geometry : and the present Lecture may have already sufficed to 
 shew that many other geometrical inquiries of interest may be 
 suggested and assisted, by the distributive principle, without 
 the associative, for instance, as regards the generation of the 
 ellipsoid. The Calculus of Quaternions would, however, be ex- 
 tremely incomplete, if it were permanently deprived of the use of 
 either of these two important principles : and indeed the combi- 
 nation of both is essential^ in many of its more advanced applica- 
 
LECTURE VII. 495. 
 
 tions. Without entering at present on any question which could 
 seem to you difficult, I shall resume the discussion of the equa- 
 tion of the ellipsoid, employing both principles freely. 
 
 491. Resuming therefore the equation of art. 465 for the el- 
 lipsoid, namely, 
 
 T {ip + /ok) = k^ - L^i 
 
 let us introduce two new constant vectors i' and k, connected 
 with the two former constant vectors t, k, by the relations, 
 
 which give 
 
 IK = i!k = T . IK\ 
 K — L'^T . IK= — Tk . Ui, 
 
 t' = k' ^ T . tK = - Ti . Uk, 
 
 t'^ = l?i K^ = K^, tV = Kl, 
 
 Substituting for t, k their values in terms of i, k, namely 
 
 l = K 1 . IK, K = l M . t/C, 
 
 we find 
 
 ip + Pk=(k'~^P + Pi''^) T . IK =Tk {k~^ p + pi'^) Tt; 
 T (tp + jO(c) = T . K (k'"^ jO + pi~^) t' = T (pi' + k'p) = T (t'p + jok) ; 
 
 the above cited equation, 
 
 T (ip + pk) = k^ - t^, 
 
 acquires therefore, by these substitutions, the new but analogous 
 form, wherein we see that t and k have merely taken the places 
 of I and K : 
 
 T (I'p + jOk') = k'^ - l'^. 
 
 The PERFECT SIMILARITY OF THESE TWO FORMS ofthe equation 
 of the ellipsoid renders it evident, that all the conclusions, which 
 have been deduced from the one form, can, with suitable and 
 easy modifications, be deduced from the other also. Thus if we 
 still regard the centre a as the origin, and treat i- k and -k as 
 the vectors oi two new fixed points, b' and c', we may consider 
 ab'c' as a new generating triangle; and may derive from it 
 the SAME ELLIPSOID as before, by a geometrical process of gene- 
 ration or construction, which is similar in all respects to the pro- 
 
496 ON QUATERNIONS. 
 
 cess already assigned, but which employs (compare the end of 
 art. 470) a new diacentric sphere, whereof the centre is at 
 the new point c' while its radius (= Tk'= Tk) has the same 
 length as in the former construction. For instance, the two new 
 sides, b'c' and ac', or - l and - k', which indeed have (by the 
 present article) the same directions as k and t, or as the two old 
 sides CA and cb, must have (like them) the directions of the two 
 cyclic normals : and the third new side, ab' or l - k', must be the 
 axis of a second cylinder oj" revolution, circumscribed round the 
 same ellipsoid, and enveloping also the mean sphere. In fact 
 this new side ab' is that semi-diameter of the ellipsoid which 
 was denoted by ap in fig. 99, art. 467 ; and it was remarked, at 
 the end of that article, although only by a sort of anticipation, 
 now justified, that the diameter pp', in that figure, was thus the 
 axis of revolution of a second cylinder, enveloping both the mean 
 sphere and the ellipsoid. It may be noticed here, that the tiew 
 generating triangle ab'c' is simply the reflexion of the old gene- 
 rating triangle abc, with respect to the major axis. 
 
 492. If we determine, on this new axis ab', two new points 
 l' and m', with two new vectors, X' and ju', analogous to the lately 
 considered vectors X and ju, and assigned by similar equations, 
 namely by the following, 
 
 X' (k - C) = k'p + jOK, fx {i - fc') = I'p + pi'i 
 
 we shall have results analogous to those of articles 471, 472, 
 namely, 
 
 where b still denotes the length of the mean semi-axis of the 
 ellipsoid. Again, the relations between t, k, i, k\ give 
 
 tS . /c|0 + kS . t/> = (T . iKy{K-'S. I'-^p + I'-'S. k'-'p} 
 = t'S . k'p + k'S . I'p, 
 because 
 
 {T.i'Ky=i'K"; 
 
 one of the expressions for ^ in 473 becomes therefore 
 
 an = ^ = - 2 (t'- k'Y' (i'S . k'p + k'S . i'p), 
 
 ^ being still the vector of the same point n as before, namely (by 
 
LECTURE VII. 497 
 
 474) the jfoot of the normal to the ellipsoid, which is drawn at 
 the extremity of p. But by the recent values of X', ix, we have 
 
 consequently 
 
 {i - k'Y \' = -2 (i - k) S . k'p, 
 {i-KyfjL=+2{i'-K')S.ip; 
 
 ^ - X' ^ - ju' _ X' - ju' _ 
 
 : — = ; — — — : r — Z, 
 
 K I I — K 
 
 if we make for abridgment, 
 
 2S . (t'+ k) p 
 
 z = ■ 
 
 T(i'-k'/ 
 
 and hence it is easy to infer, by reasonings similar to those of 
 art. 487, that the new variable triangle l'm'n is similar to the new 
 fixed triangle ab'c', and similarly situated in one common plane 
 therewith ; namely in the common plane of the old and new ge- 
 nerating triangles, which is also that of the greatest and least 
 axes of the ellipsoid. We have also, by the equations last esta- 
 blished, combined with the analogous equations of 487, and with 
 the relations (491) between i, k, i', k, the following formulae : 
 
 which may also be thus written, 
 
 ' ?-iU ' ^-X 
 
 where the symbol 
 
 v-io 
 
 may represent any scalar : as the analogous symbol, 
 
 s-^o, 
 
 may represent any vector. We have therefore equations of 
 the forms, 
 
 ^-X' = a^(?-iu); l-p: = y{l-\); 
 
 where x and y are scalars : in fact, with the recent meaning of 
 the scalar z, we have (by the articles just cited), 
 
 2 K 
 
498 ON QUATERNIONS. 
 
 Zk z k — 2: _i k 
 
 K- fx h + Ii I h + h I 
 
 Zl Z I - Z ry I 
 
 ^ ~ ^-X ~ h + h' K " h + h' ~K 
 
 Now the quaternion quotient of the two vectors ^-X and 
 %- fi could not reduce itself to a scalar, if those vectors were not 
 parallel to each other, or to some common line (compare 122, 
 407); the recent equation, 
 
 shews therefore that the three co -initial vectors, X, fi, s, must 
 terminate upon one common right line, or that their three ex- 
 treme points, L, M, N, are collinear. In like manner the equa- 
 tion, 
 
 shews that the terminations, l, m', n, of the three vectors X, jjl, ^, 
 are situated on one straight line : so that the two straight lines, 
 l'm, lm', or their prolongations, must cross each other in the 
 point N. Indeed, if it had not been designed to exemplify some 
 processes of calculation, we might have more rapidly inferred the 
 fact of this intersection from the parallelisms, 
 
 LN 11 AC 11 c'b' II nm', and mn I bc || c'a || nl'. 
 
 But the two lines, lm', ml', may be regarded as the diagonals 
 of a certain quadrilateral inscribed in a circle ; namely, the 
 plane quadrilateral lmm'l', of which the four corners are, by what 
 has been already shewn, at one common and constant distance 
 = b, from the variable point E of the ellipsoid. (Or the concircu- 
 larity of the four points L, M, m', l', might be established on the 
 plan of 487, by means of the equation, juX = )<fx = p~ + b-.) If 
 then we here content ourselves with assuming it as known, that 
 when a straight line af (= b~v = en) is drawn from the centre a of 
 an ellipsoid, so as to be in direction opposite, and in length reci- 
 procally proportional, to the peiyendicalar let fall from the same 
 centre a on the tangent plane at e, this line must terminate in a 
 point F on the surface of another ellipsoid; which new sur- 
 face is concentric with, and is (in a certain well-known sense) 
 
LECTURE VII. 
 
 499 
 
 RECIPROCAL to that former ellipsoid, which contains the point b 
 itself (or the termination of the vector p) : we may combine the 
 recent results, so as to obtain the fo\lo\^'ing geometrical co?isfruC' 
 Hon, which serves to ge?ierate a system of two reciprocal 
 ELLIPSOIDS, by means of a moving sphere. 
 
 493. Conceive then a sphere, with constant radius = h, but 
 variable centre e, of which e^ represents the projection, on the 
 plane of the annexed figure 100 ; let this sphere be supposed to 
 move, so that it always 
 
 intersects two fixed ^'^' ^'^^• 
 
 and mutually inter- 
 secting straight lines, 
 AB, ab', m four points 
 L, M, l', m', of which 
 L and M are on ab, 
 while l' and m' are on 
 ab'; and let it farther 
 be supposed that one 
 diagotial, lm', of the 
 inscribed quadrilate- 
 ral lmm'l', is con- 
 stantly parallel to a 
 third fixed line ac, 
 
 which will oblige the other diagonal l'm of the same quadrilate- 
 ral to move parallel to a fourth fixed line ac'. Let n be the 
 point in which the diagonals intersect; and conceive a line af 
 so drawn as to be equal in length and similar in direction to en; 
 or so that aenf shall be a parallelogram, projected into ae^nf^ 
 in the figure. Then the locus of the centre e of the moving 
 sphere is one ellipsoid; and the locus of the opposite corner f of 
 the parallelogram is another ellipsoid reciprocal thereto. These 
 two ellipsoids have a common centre a, and a common mean axis, 
 which is equal to the diameter (2b) of the moving sj^here, and is 
 a mean proportional between the greatest axis of either ellip- 
 soid and the least axis of the other ; of which two last-mentioned 
 axes the directio7is coincide. Two sides, ae, af, of the parallelo- 
 gram aenf, are thus two semi-diameters which may be regarded 
 
 2 K 2 
 
500 ON QUATERNIONS. 
 
 as mutually reciprocal, one of the one ellipsoid, and the other of 
 the other; but because they fall at opposite sides oi the princi- 
 pal plane (containing the four fixed lines and the greatest and 
 least axes of the two ellipsoids), it may be proper to call them, 
 more fully, opposite reciprocal semi-diameters ; and to call the 
 points E and f, in which they terminate, opposite reciprocal 
 points. The two other sides, en, fn, of the same variable pa- 
 rallelogram, are the normals to the two ellipsoids, meeting each 
 other in the point N, upon the common principal plane. In that 
 plane, the two former fixed lines, ab, ab', are the axes of two 
 cylinders of revolution, circumscribed about the first ellipsoid ; 
 and the two latter fixed lines, ac, ac', are the two cyclic normals 
 of the same first ellipsoid : while the diagonals lm', ml', of the 
 inscribed quadrilateral in the construction, are the awes of the 
 two circles on the surface of that first ellipsoid, which circles 
 pass through the point e, that is, through the centre of the 
 moving sphere ; and the intersection n of those two diagonals is 
 the centre of another sphere, which cuts the first ellipsoid in the 
 system of those two circles ; all which is easily adapted, by suit- 
 able interchanges, to the other or reciprocal ellipsoid, and flows 
 with facility from the quaternion equations above given, and 
 from the remarks that have been made in recent articles. 
 
 494. If we introduce five new vectors, X,, fx^, X', /.i', E,y, of 
 five new points l, m^, h', m^', h, connected with those lately con- 
 sidered by the relations : 
 
 X = AL = LE =10 - A ; IJI = AM^=ME=p - /n; 
 
 X/ = al/ = l'e = jO - X' ; /x/ = am/ = m'e = p-p! ; 
 
 ^ = AH = NE = p - ^ (= - 6-1; = fa) ; 
 
 then, by 471, 472, 492, 
 
 T\=T^^=TX;=T^;=6; 
 -X, _X^ 
 
 t - K 
 
 P-l±s^±_ = k = \-^Q; 
 
 P-^^, 
 
 K— I K - t 
 
 -J^=h'=V-'0; 
 
 -X, X ^ _v- 
 
 fX^ fX 
 
 h' 
 

 LECTURE VII. 
 
 and because 
 
 
 501 
 
 we shall have, by 487, 492, 
 
 
 = h + h' = Y-'0 
 
 X/ - ^^ ju/ - ^^ ft/ - x/ 
 
 t 
 
 whence again it follows, by 491, that 
 because 
 
 Hence, on the plan of recent articles, we may infer that the five 
 new points are all situated in one common plane, which is parallel 
 to i\\Q principal plane (493), and contains the point e of the ori- 
 ginal ellipsoid ; while h is the point reciprocal to e, upon the 
 second or reciprocal ellipsoid, and is diametrically opposite to 
 the point r thereon. In fact, so much as this might at once be 
 inferred from the circumstance, expressed by the five equations, 
 
 AL^= LE, AM^= ME, AL ' = l'e, AM/ = ME, AH = NE, 
 
 that the five lines ll , mm^, l'l', m'm', nh, bisect and are bi- 
 sected by the line ae ; or that alel^, &c., are parallelograms. 
 The equations above written also shew that the four new points 
 L, m^, l/, m/, are situated on one common circle of the mean 
 sphere, namely, its intersection with the above-mentioned paral- 
 lel plane ; that the lines l m^ and l 'm' are parallel respectively to 
 the lately considered lines ab, ab', and intersect each other in the 
 point E of the or?^i?2«/ ellipsoid ; and that the lines lm' and l'm 
 are parallel respectively to ac, ac', and cross in the correspond- 
 ing point H, of the reciprocal ellipsoid. And hence we may 
 derive the following method oi generating a system of two reci- 
 procal ellipsoids by means of a fixed sphere, which seems to 
 
502 ON QUATERNIONS. 
 
 possess some advantages over the process lately given, for the 
 generation of such a system by means of a moving sphere, but is 
 intimately connected therewith. 
 
 495. In the fixed sphere (of which the centre is A, and the 
 radius 6), inscribe a plane quadrilateral, l m l 'm/, of which the 
 four successive sides, lm^, m^l', l'm', m/l, shall be respectively 
 parallel to four fixed right lines, ab, ac', ab', ac; and then pro- 
 long, if necessary, the first and third sides till they meet in a point 
 E, and denote by h the intersection of the second and fourth sides. 
 Then these two points of intersection, e and h, of the two pairs of 
 
 opposite sides of this inscribed quadrilateral {which sides move pa- 
 rallel to themselves^, will be two reciprocal points on two recipro- 
 cal ellipsoids : namely, the same system of ellipsoids which was 
 otherwise generated in 493, if the centre a, the radius (or com- 
 mon mean semi-axis) h, and the directions of the four fixed lines, 
 be the same in the two constructions. The relation of recipro- 
 city between the two ellipsoids, which was before assumed as 
 known, is made very evident by the present process ; being seen 
 to be connected with the passage from onepair of opposite sides 
 of an inscribed quadrilateral to the other pair. The same con- 
 sideration shews also clearly (what however is otherwise known), 
 that the cyclic normals ac, ac', of the first ellipsoid are the axes 
 of the cylinders of revolution circumscribed about the second; 
 and that, conversely, the axes ab, ab', of those two cylinders of 
 revolution, which have been seen to envelope the original ellip- 
 soid, are the normals to the two cyclic planes of the second or 
 reciprocal surface. 
 
 496. Another mode of generating the original ellipsoid is 
 easily derived from the relations established in some of the re- 
 cent articles. Conceive two equal spheres to slide within 
 TWO cylinders of revolution, whose axes intersect each other, 
 in such a manner that the right line joining the centres of the 
 spheres shall be parallel to a fixed right line; then, the locus 
 OF THE varying CIRCLE in ivhich the two spheres intersect each 
 other will be an ellipsoid, inscribed at once in both the cylin- 
 ders, so as to touch one cylinder along one ellipse of contact, 
 and the other cylinder along another such ellipse. And ihesame 
 ellipsoid may be generated as the locus of another varying cir- 
 
LECTURE VII. 503 
 
 cle, which shall be the intersection of two other equal spheres 
 sliding ivithin the same two cylinders of revolution^ but with a 
 connecting line of centres which now moves "parallel to another 
 fixed right line; provided that the angle between these two fixed 
 lines, and the angle between the axes of the two cylinders, have 
 both one common pair of (internal and external) bisectors, which 
 will then coincide in direction with the greatest and least axes 
 of the ellipsoid : while the diameter of each of the foiir sliding 
 spheres is equal to the mean axis. In fact, we have only to con- 
 ceive (with the recent significations of the letters), that four 
 spheres, with the same common radius, = h, are described about 
 the points l, m', and l', m, as centres ; for then the first pair of 
 spheres will cross each other (if they cross at all), in one circular 
 section of the ellipsoid ; and the second pair of spheres will cross 
 (if at all) in another circular section of the same surface. We 
 might also conceive an arbitrary curve on the ellipsoid to be de- 
 scribed by the vertex E of an isosceles triangle lem' (or l'em), 
 the common length of whose two equal sides is constant, and 
 = b, while the base lm' (or l'm) varies indeed in length, but 
 moves parallel to one fixed right line ac (or ac'), and is con- 
 stantly inscribed in a given angle bab', l (or m) moving along 
 the given right line ab, and m' (or l') moving along another 
 given right line ab'. Or, we might conceive the two equal sides 
 of the triangle to be two adjacent sides of a rhombus of constant 
 perimeter, of which one diagonal moves parallel to itself within 
 a given rectilinear angle, while the plane of the rhombus turns^ 
 according to an arbitrary law, and the extremities of the other 
 diagonal describe two curves on the ellipsoid, each separately 
 arbitrary, but not entirely unconnected with each other. 
 
 497. With the recent significations of the letters, w-e have, 
 by 492, 491, 472, 
 
 = {ip + pi) (« (k- <) /c"^}'^ - {ip 4 jOi) K ((c- t)"^ r^ 
 
 = - A'k (k - t) r 1 = K (k - K.h' 1) ; 
 and 
 
 \X — hb{K.-l). 
 
504 ON QUATERNIONS. 
 
 If then we make for abridgment, 
 
 and employ two new fixed vectors, ij and 6, defined by the equa- 
 tions, 
 
 which give 
 
 I - (c= T (i- K)U(t - k) = rjT , 
 
 and also (compare 464), 
 
 along with other analogous or connected expressions, some of 
 which will offer themselves to our notice afterwards : we shall 
 have the values, 
 
 H = gr}; y = g9. 
 
 Hence the equations, 
 
 T{p-f.c)^b, T(p-X) = b, 
 
 of one of the two pairs of sliding spheres, may be made to assume 
 the forms : 
 
 T{p-gn) = bl T(p-ge) = b; 
 
 between which it remains to eliminate the scalar coefficient g, in 
 order to find in a new way an equation of the ellipsoid, regarded 
 as the locus of the circle in which the two spheres intersect each 
 other. And it will be useful here to effect this elimination, both 
 as an exercise in the present Calculus, and for the sake of the 
 results to which it leads. 
 
 498. Squaring for this purpose the two last written equa- 
 tions, we find, for the two sliding spheres, the two following 
 more developed equations : 
 
 = 62 ^ p2 _ 2^ S . i7|0 + g- rf ; 
 Q = b-' + p''-2g^.ep+g''Q\ 
 
 Taking then the difference, and dividing by g, we find the equa- 
 tion. 
 
LECTURE VII. ■ 505 
 
 which, relatively to p, is linear, and may be considered as the 
 equation of the plane of the varying circle of intersection of the 
 two sliding spheres ; any one position of that plane being dis- 
 tinguished from any other by the particular value of the variable 
 coefficient g. Eliminating therefore that coefficient by substitu- 
 ting its value, namely, 
 
 we find that the equation of the ellipsoid, regarded as the locus 
 of the varying circle, may be presented under either of the two 
 following new forms : 
 
 ^[p :;^:rw — j = ^- 
 
 And we may verify that these two last equations of the ellipsoid 
 are consistent with each other, by observing that the semisum of 
 the two vectors under the signs T is perpendicular to their semi- 
 difference (as it ought to be, in order to allow of those two vec- 
 tors themselves having any common length, such as h) ; or that 
 the condition of rectangularity, 
 
 {Q + n)^.{9-n)p , . 
 
 is satisfied : which may be proved by shewing (compare 454), 
 that the scalar of the product of these two last vectors vanishes. 
 We may also verify the recent forms of the equation of the ellip- 
 soid, by remarking that they concur in giving the mean semi- 
 axis b, as equal to the length Tp of the radius of that diametral 
 and circular section, which is made by the cyclic plane having 
 for equation, 
 
 S.(0-r,)p = O; 
 
 this plane being found by the consideration that rj - has the di- 
 rection of the cyclic normal i, because (by 497), 
 
506 ON QUATERNIONS. 
 
 = - 1-' {ic'-r) = Ut Tt-' . bT (i-k), 
 
 so that 
 
 ri-e = bUi: 
 
 or by making the coefficient g = 0, in the linear formula of this 
 article. 
 
 499. If we observe that 
 
 6^ - r]"^ = K^ - i^ = ac> 0, 
 and that 
 
 T(r,-0) = 6, 
 
 while the vector expression (0^ -rf) p- 2?}S . (6 - ij) p is equal to 
 its own vector part ; we shall easily see that the first of the two 
 lately obtained equations of the ellipsoid may be successively 
 transformed as follows : 
 
 T{n-e)(Q''-yf) = b{Q''-yf) 
 
 = T{{e'--n')p-2nS.{e-v)p} 
 = TV{(0— I,'-) p - 2^S . (9 - 1}) p} 
 
 = TY {(6'^- r)p-niO-v)p-vp {9- r,)] 
 = TY[e'p-n{Op + pe) + npn} 
 =^TY{{9-n)ep-vp{B-v)}- 
 
 But 
 
 y.(9-n)0p=Y.pe{6-',i), 
 
 because in general for atiy three vectors a, jS, 7 (compare 317), 
 the following relations hold good, 
 
 ai37 = -K.7j3a, S . a[iy = - S . y[5a, y.a[5y = +Y.jl5a; 
 
 hence 
 
 (0^- - r) T(n-e) = TV. (pO - np) (B - v) 
 = TY. (np-p9){rj-e); 
 or, more concisely, 
 
 TV. {np - p9) U („ -9) = 9' - »,'- : 
 and the same transformation may be obtained with equal ease. 
 
LECTURE VII. 507 
 
 from the second form of the equation of the ellipsoid, which was 
 deduced in the foregoing article. Again, the versor of every vec- 
 tor has, in this calculus, a negative square (see 113); we have 
 therefore, in particular, 
 
 and under the sign TV, as under the sign T, it is allowed to di- 
 vide by -1, without affecting the value of the tensor; it is there- 
 fore permitted to write the equation of the ellipsoid under the 
 form : 
 
 jy W-P^ 
 
 U (77 - 0) 
 
 -rj^ 
 
 and this form seems to me to be deserving of attention, on ac- 
 count of the simple and remarkable geometrical relations to the 
 surface, which the two fixed vectors, tj, 0, will be found to possess. 
 500. The last form of the equation of the ellipsoid, which 
 may also be thus written, 
 
 ™^ rip- pQ _ Q'^-yf 
 n-e ' T(„-0)' 
 
 may be deduced in another way, as follows, from the equation, 
 
 T [ip + pk) - k~- l^, 
 
 of articles 465, &c. : and the deduction will be an useful exer- 
 cise. Writing the cited equation thus, 
 
 ' rp {ip + pk) (i-k) rj. . . 
 
 T ^^-^^ = TO-k), 
 
 we may observe that while the denominator of the fraction in the 
 first member is a pure scalar, the numerator is a pure vector; 
 for the identity, 
 
 t|0 + p/C = S . (t + k) jO + V. (t - (c) jO, 
 
 gives 
 
 S . (ip + pK) (< - /c) = S . (t - k) V. (« - (c) p = ; 
 
 because generally, for any two vectors a and /3, 
 
 /3±V.)3a, S./3V.j3a = 0: 
 
508 ON QUATERNIONS. 
 
 indeed we may easily now see (compare 442), that for any three 
 vectors, a, j3, 7, we have the identity, 
 
 S . 7 V. j3a = S . -yjSa ; 
 
 which last expression reduces itself to 0, when 7 = j3, because 
 j3'a is a vector. We may therefore change T to TV, as operating 
 on the last written fraction ; and, under the sign V, may substi- 
 tute (jl-k) pt for ip (f - k), on the principle referred to in the last 
 article ; namely, that the vector part of the product of any Mree 
 vectors remains unchanged, although the scalar part of it changes 
 sign, when their order is reversed : which principle indeed is 
 easily seen to hold good for any odd number of vectors, because 
 the new product, thus reversed, is the negative of the conjugate 
 of the old product. (Compare again art. 317 ; see also 408, 
 410.) Again, it is always allowed in this calculus to divide 
 (although not generally, to multiply) both the numerator and de- 
 nominator of a quaternion fraction by any common vector or 
 quaternion (different from zero) ; that is, to multiply both numera- 
 tor and denominator into the reciprocal oi?,\xch. common vector or 
 quaternion : namely, by writing the symbol of this new factor, 
 or reciprocal, to the right (but not generally to the left) of the 
 symbols of numerator and denominator, above and below the 
 fractional bar. Dividing therefore thus above and below by i, or 
 multiplying into rS after that permitted transposition of factors 
 which was just now specified, and after the change of T to TV, 
 we find that the last written equation of the ellipsoid assumes the 
 form, 
 
 i V — — T'-lx-^ J- (t - fc) ; 
 
 (t - k) + (;c - K^ L ^) 
 
 the new denominator indeed at first presenting itself under the 
 form K^ c^ - L, but being changed for greater symmetry to the de- 
 nominator just now written, which we are allowed to do, because 
 under the sign T, or under the sign TV (though not under V 
 itself, nor under S, U, or K), we may multiply by negative 
 unity. Substituting finally for i-k and /c-k^T^ their values 
 given near the beginning of art. 497, and suppressing, above and 
 below, the common factor T . (t - k) rS we find as a transformed 
 equation of the ellipsoid: 
 
LECTURE VII. 509 
 
 r] - tf 
 where 
 
 T{i-k) = b-' (,c- - 1') = (0' - ri") T{n- e)-\ 
 
 The form written at the commencement of the present article is 
 therefore deduced anew. 
 
 501. The geometrical construction already mentioned (in art. 
 496), of the ellipsoid as the locus of the circle in which two 
 sliding spheres intersect, shews easily (see art. 497) that the sca- 
 lar co-efficient g, in the continued equation, 
 
 of that pair of sliding spheres, becomes equal to the number 2, 
 at one of those limiting positions of the pair, for which, after cm^- 
 ti?ig, they touch, before they cease to meet each other. In fact, 
 if we thus make $' = 2, the values /i = ^i), A' = gO (see the last cited 
 article) of the vectors of the centres of the sliding spheres will 
 give, for the interval between those two centres, the expression, 
 
 T(^-X')=^T(r,-0)^26; 
 
 this interval will therefore be in this case double of the radius of 
 either sliding sphere, because it will be equal to the mean axis 
 of the ellipsoid, and the two equal spheres will touch one another. 
 Had we assumed a value for g, less by a very little than the num- 
 ber 2, the two spheres would have cut each other in a very s?nall 
 circle, of which the circumference would have been (by the con- 
 struction) entirely contained upon the surface of the ellipsoid ; 
 and fhe plane of this little circle would have been parallel and 
 very near to that other plane, which was the common tangent 
 plane of the two spheres, and also of the ellipsoid, when g re- 
 ceived the value 2 itself. It is clear, then, that this value 2 of 
 g corresponds to an umbilicar point on the ellipsoid ; and 
 that the equation, 
 
 S.{0-v)p = 0"'-r,S 
 
 which is obtained from the more general equation in 498, of the 
 plane of a circle on the ellipsoid, by changing g to 2, represents 
 an UMBILICAR tangent plane, at which the normal has the di- 
 
510 ON QUATERNIONS. 
 
 rection of the vector r]~6 : and accordingly it has been seen that 
 this last vector has the direction of the cyclic normal i ; in con- 
 nexion with which circumstance it may be remarked that the vec- 
 tor 6'^-r}~^ has the direction of the other cyclic normal, k. In 
 fact, it is not difficult to prove from the expressions in 497, that 
 
 from which, or immediately from the expressions just cited, it 
 follows (compare 469) that 
 
 T}} = Ti = ^(a-\- c) ; T6=^Tk = ^{a- c). 
 
 The lengths of the three semi-axes of the ellipsoid admit there- 
 fore of being very simply thus expressed, in terms of the neiv 
 fixed vectors^ i^, 9 : 
 
 « = Tjj + T0 ; Z» - T (i, - 0) ; c = Ti, - T0. 
 We have also the formulae : 
 
 \Jt-VK=V(n- 6) + U (r' - 0-') \\Un+V9; 
 
 Vi+UK==V{ri-e)-\J{r,-'-e-')\\Vr,-Vd; 
 
 the members of the first formula having each the direction of the 
 greatest axis of the ellipsoid, and the members of the second for- 
 mula having each the direction of the least axis; as may easily 
 be proved, for the first members of these formulae, by the con- 
 struction with the diacentric sphere, already given in articles 
 466, &c. 
 
 502. The recently obtained equation of an umhilicar tangent 
 plane may also be verified by observing that it gives, ft^the 
 length of the perpendicular (p) let fall from the centre or the 
 ellipsoid on such a plane, the expression 
 
 p = {9' - r,-) T (,, - 9)-' = acb-' ; 
 
 which agrees with known results. And the vector w of the um- 
 hilicar point itself must be the semi-sum of the vectors of the cen- 
 tres of the two equal and sliding spheres, in that limiting posi- 
 tion of the pair in which (as above) they touch each other; this 
 UMBILICAR VECTOR o) is therefore expressed as follows : 
 
LECTURE VII. 511 
 
 because this is the semi-sum of ^ and X', or of grj and gO, when 
 g = 2. As one verification we see that rt+ 9 may be substituted 
 for p, without violating the equation of the ellipsoid, because 
 this substitution gives, 
 
 r}p - p9 = T)^ - 6'^ ; 
 
 and as another verification, we may observe that the same ex- 
 pression 7] +6 for (u conducts to the following known value for 
 the length (u) of an umhilicar semi-diameter of the ellipsoid : 
 
 u = Tw = T (}/ + d) = ^ (a-- b- + C-) ; 
 because for any two vectors ?j, 6, we have the identity, 
 
 T (r, + ey + T (r, - ey = (t,7 + Toy + (Xr, - Toy. 
 
 503. By similar reasonings it may be shewn that the expres- 
 sion, 
 
 (o'^T^Ve + TeUn, 
 which may also be thus written, 
 
 represents another umbilicar vector ; in fact, we have, 
 
 (I)'- = (jj -t dy — (i)', Taj'= Tco, 
 and 
 
 a,-a,'=(T7,-T0) (Vn-Ve); 
 
 so that the vectors w, to are equally long, and the angle between 
 them is bisected by Utj +U0, or by U(t-K:) + U(i'-K), that is 
 by the direction of the axis major of the ellipsoid; while the 
 supplementary angle between w and - w' is bisected by U?j -UO, 
 or by U(i - K:)-U(t'-K:'), and therefore by the axis minor. It 
 is evident that - w and - to are also umbilicar vectors; and it is 
 clear, from what has been shewn in former articles, that the vec- 
 tors 7] and 9 have the directions of the axes of the two circum- 
 scribed cylinders of revolution. 
 
 504. A few additional remarks may assist to render evident 
 the utility, and to illustrate the significations, of the two fixed vec- 
 tors ij, 0, although our remaining time will not allow us to enter 
 
512 ON QUATERNIONS. 
 
 largely into the subject. And first we may observe that the va- 
 lues for abc, in terms of tj, d, give 
 
 (a2 _ c2)i = 2T V nB, (6'- - c^)* = 2S V^; 
 
 in obtaining which expressions we have employed these other 
 values : 
 
 «2 = (Trj + TQf = Tr,' + 2Tr, TO + Td' 
 = -V" + 2T.r,9-0^; 
 
 c'= {Tr,-Tey = - r,''-2T . nd- 9'; 
 and 
 
 b' =T (rj - Oy = - (v- Oy = - T + 2S . nO - 6' ; 
 
 observing also tha.tjhr miy quaternion, such as here 
 
 q= /^, 
 we have 
 
 q^ = {Sq +yqy = Sq^ + 2 Vq Sq +Yq\ 
 S . q'^= Sq'^+Yq^, 
 Y.q^=2VqSq, 
 
 T . 5^ = Tq'^ = S^^ -Yq^', 
 
 2 (S . g2 + T . g^) - 48^^- - {2Sqy I 
 
 so that generally the scalar of the square root of any qua- 
 ternion q" (in the present instance, rj0), which square root (by 
 152) is considered as hemg genexdMy ?c[x acute-angled quaternion, 
 admits of being expressed by the formula, 
 
 S y/ q = V {l^q + ^Tq'y 
 
 And here it maybe noted that this is only one out of a vast 
 number o/ general transformations, with which the present 
 calculus abounds : and which may be deduced, with more or less 
 facility, from the laws of the symbols, S, T, U, V, K, by the 
 principles already laid down. 
 
 505. If then, retaining the centre as the origin of vectors, we 
 change at once B to tO, and rj to ^"^rj, where t is any positive sca- 
 lar, since we shall not alter thereby any one of the three functions, 
 
 we shall leave unaltered the three following things, namely : 1st, 
 the directions of the axes of revolution of the two circumscribed 
 
LEC'l'URE VII. 513 
 
 cylinders; 2nd (in connexion with these), the directions of the 
 three priticipal axes of the ellipsoid ; and 3rd, the differences of 
 the squares of the semi-axes^ a, b, c. To those then who are at 
 all acquainted with the theory of the focal conics, oy focal 
 curves, which have in modern times been made to play so impor- 
 tant a part in the theory of surfaces of the second order, and who 
 have attended also to the foregoing calculations with quaternions, 
 it will be evident that these simultaneous changes of 
 
 7j and B, to t~'^r] and tB, 
 
 can merely cause a passage to a confocal surface : leaving the 
 FOCAL ELLIPSE, and the focal hyperbola, unchanged. The 
 latter curve (the focal hyperbola), which is known to have the 
 axes of the cylinders for its asy^nptotes, and to cut the ellipsoid 
 (perpendicularly) in the four umbiUcar points, will be found to 
 be adequately represented, in our calculus, by the single equa- 
 tion, 
 
 Y.np.V.pB^{Y.yiBy. 
 
 For the former curve (the focal ellipse^, it is convenient to em- 
 ploy a system of two equations : the first of which may be that 
 of its plane (perpendicular te the minor axis of the ellipsoid), 
 namely, the equation, 
 
 while the second may be at pleasure either of two equations, re- 
 presenting two cylinders of revolution, with a common radius 
 = (6^ - c^)^, on each of which cylinders the focal ellipse is situated ; 
 namely, either of the two equations following, 
 
 TV.pUr, = 2S\/^, 
 and 
 
 Ty.p\]B = 2^V'^. 
 
 The foregoing will perhaps be considered as expressions suffi- 
 ciently simple for these two known and important conics, and 
 for their connexions with a system of confocal surfaces. 
 
 506. It may, however, appear strange that in this species of 
 
 SYMBOLICAL GEOMETRY OF THREE DIMENSIONS it should be 
 
 said, that a curve in space, as here the focal hyperbola, may 
 
 2 L 
 
514 ON QUATERNIONS. 
 
 admit of being adequately represented by a single equa- 
 tion, such as the equation, 
 
 whereas we have repeatedly seen, in the present Lecture, that a 
 curve may be not more than adequately expressed by a system 
 OF TM^o equations, representing a system of two surfaces^ For 
 example, the focal ellipse of the last article was represented by 
 the system, 
 
 which denoted separately a plane and a cylinder; the spherical 
 conic of art. 421 by the system, 
 
 TjO=c, S . joa'^S . j3,o'^ = 1, 
 
 representing separately a sphere and a cone ; its cyclic arcs were 
 each represented, in the same article, by a system of two equa- 
 tions, denoting a plane and a sphere ; an analogous system served 
 to I'epresent the circle of contact in 422; the ellipse of art. 433 
 was represented by the two equations, 
 
 S.|oa"^ = a, TV. |0j3"-^ = ^, 
 
 denoting again a plane and cylinder ; while another plane, com- 
 bined with the same cylinder, was used to express a circle in 
 432 ; a plane and sphere gave in art. 417, the equations 
 
 S.joa"^=l, S.|3jO"^=l, 
 
 which jointly represented the circular base of a cone; and the 
 major axis of the same cone, in art. 426, when regarded as an 
 indefinite vight line, had its position expressed by the two equa- 
 tions, 
 
 S . ap = 0, S . j3jO = 0, 
 
 which, separately taken, denoted the two cyclic planes. Nor could 
 we, in any one of these examples, which might easily have been 
 made more numerous, have rightly contented ourselves with re- 
 taining one alone out of the two equations, although the system 
 might in each case have been varied. 
 
 507. But it is to be observed that, in all these cases, each 
 separate equation has been of scalar form, and therefore quite 
 
LECTURE VII. 515 
 
 analogous^ in this new symbolical geometry, to the usual Carte- 
 sian expression for a surface, by an equation between its co-or- 
 dinates x^ y, z^ which with us are regarded as three scala?'s. 
 In general, if p be still regarded as a variable vector, and if^ 
 denote a7ii/ scalar /unction of it (whether this function be of the 
 second or of any other dimension), then, on substituting for p its 
 value ix+Jy + kz (101, &c.), the equation 
 
 fp = 0, ory(0 = constant, 
 
 where the constant is still a scalar, will take, by the rules of this 
 calculus, the form of an ordinary algebraic equation between 
 X, y, z, and may be interpreted as expressing a surface, on the 
 usual plan of the Cartesian co-ordinates. Thus if we did not 
 otherwise know (by 168, &c.) the signification, in the present 
 Calculus, of the equation 
 
 p2 + 1 = 0, 
 
 as representing the unit-sphere round the origin, or if we had 
 forgotten that signification, or desired to deduce it anew, we 
 might write the equation under the form, 
 
 (ix +jy + kzy +\ = 0, 
 
 and then perform the operation of squaring the trinomial as fol- 
 lows : 
 
 ix +jy + kz 
 ix +Jy + kz 
 
 - x^ + kxy -jxz 
 -y"^ - kyx + iyz 
 
 - z^ +jzx - izy 
 
 -x^ -y'^- z^- {ix +jy + kzy ; 
 
 the three lines here added up being respectively the products of 
 ix+jy + kz, multiplied by ix, hy jy, and by kz. For thus the 
 proposed equation p^ + 1 = would take the ordinary form, 
 
 = 1 - x^-y"^- z'^, 
 
 and would be seen anew to represent the unit-sphere. 
 508. Again, suppose that we meet the equation 
 
 S . OjO = 0, 
 
 2 L 2 
 
516 ON QUATERNIONS. 
 
 where a is a given and p a variable vector. Here, instead of em- 
 ploying the principles of articles 413, 420, 421, we might write, 
 
 a = la +jb + kc, p = ix +jy + kz, 
 
 and should then find, by distributive multiplication, 
 
 ap = {ia +Jb + kc) (ix -{-jy + kz) 
 
 = - ax + kay -jaz 
 
 -by - kbx + ibz 
 
 - cz -vjcx - icy 
 
 = - (ax + by+ cz) 
 
 + i {bz - cy) -vj (ex - az) + k (ay - bx) ; 
 
 this product is therefore seen anew to be a quaternion^ as in the 
 Third Lecture it was otherwise shewn to be : because it is now 
 found to be reducible by actual multiplication to the standard 
 quadrinomial form of arts. 450, &c., namely, to the form, 
 
 w + ix ■\-jy + kz. 
 
 At the same time the scalar and vector parts, taken separately, 
 of this quaternion product a/a, are seen to be, 
 
 S . ap = - (ax -\-by + cz), 
 V. ap = i (bz - cy) +j (ex - az) + k (ay - bx) ; 
 
 to assert then the evanescence of the scalar function S . ap, is 
 equivalent to establishing the following ordinary equation be- 
 tween X, 2/, 2r, 
 
 ax + by + cz = ', 
 
 and thus a person familiar with the usual method of co-ordinates 
 might recover for himself the interpretation of the equation of 
 this Calculus, 
 
 S . ap = 0, 
 
 as denoting a plane through the origin perpendicular to the line 
 a, b, c : namely, to the line drawn from the origin (0, 0, 0) to 
 the given point (a, b, c). 
 
 509. Again, let it be proposed to interpret, by the assistance 
 of co-ordinates, and by the relations between the symbols i,j, k, 
 without using the transformation S . a'ap = S . a' V. ap of art. 500, 
 
LECTURE VII. 517 
 
 or the condition of coplanarity assigned near the end of 430, 
 this other scalar equation : 
 
 S . aap = ; 
 
 in which we may suppose that 
 
 d = id ^jb' + kc\ 
 
 while a and p are still expanded into the two trinomials which 
 were substituted for them in the preceding article. The actual 
 process of multiplication gives immediately, on the plan recently 
 employed, the following developement for the ternary product of 
 vectors, at present under consideration, 
 
 dap = - a {hz - cy) - b' (ex - az) - d {ay - bx) 
 
 - {ia' +jb' + kc) {ax + by^- cz) 
 + i {b'{ay - bx) - c' {ex - az) } 
 +j {c'{bz- cy) - a' {ay - bx) } 
 
 + k { a {ex - az) - b' {bz - ey) } . 
 
 The scalar and vector parts admit therefore of being respectively 
 and separately expressed as follows : 
 
 S . dap = a (cy - bz) + b' {az - ex) + c' {bx - ay) 
 = cc {be' - cb') + y {ed - ac) + z {all - bd) 
 = a {b'z -e'y) + b {dx - dz) + c {dy - Ux) ; 
 
 V . dap = {ia +jb + kc) {dx + b'y + dz) 
 
 - {id +jb' + kd) {ax -\-by-\- cz) 
 
 - {ix -^^jy + kz) {da + b'b + dc). 
 
 To establish the equation S . dap = 0, is therefore equivalent to 
 establishing that ordinary equation between x, y, z, which (as is 
 well known to all persons familiar with the method of co-ordi- 
 nates) expresses the coplanarity of the three lines xyz, abc, db'd, 
 or the condition for the variable point (cc, y, z) being situated 
 somewhere upon the plane which is drawn through the origin 
 (0, 0, 0), and through the two other given points, (a, 6, c), and 
 (a, &', d). 
 
 510. We see, at the same time, that the ^cdX^iV function 
 S . dap admits of being expressed, in the modern notation of de- 
 terminants, as follows : 
 
618 , ON QUATERNIONS. 
 
 S . a'ap = 
 
 a, b, c, 
 a, h', c', 
 
 and that thus (as also in other ways) there exists a connexion 
 between the theories of quaternions and of determinants ; or of 
 ELIMINANTS, as soMie prefer to call them. In the recent question, 
 or example, this connexion of the proposed equation, 
 
 S . aap = , 
 
 with an elimination^ might easily have been foreseen. For, with- 
 out the use of co-ordinates, by principles of the present calcu- 
 lus above cited, we might have seen that this equation is a for- 
 mula OF COPLANARITY for the three vectors a, a, p; and that it 
 is therefore equivalent to a system of three perpendicularities, 
 since, 
 
 p 111 a, a', gives A _L a, X _L a', X _L p, 
 
 if X be a vector perpendicular to the plane of a, a. The pro- 
 posed equation might therefore thus have been seen to be equi- 
 valent to the system of the three following, 
 
 S.Xa = 0, S.Xa'-O, S.X|O = 0, 
 
 and to be conversely derivable from them, by some process of 
 elimination of X. And if we now introduce co-ordinates and 
 i,j, k, making, 
 
 X = il +jm + kn, 
 
 and employing for o, a, p the same three trinomial expressions 
 as before, we see that this process must answer to eliminating the 
 three scalars I, m, n, or their ratios, between the three following 
 equations of the 1st degree, 
 
 la + mb + nc = 0, la + 7nb' + nc = 0, Ix + my + n2 = : 
 
 which conducts to the lately mentioned determinant. Indeed, it 
 will be found that processes more peculiarly belonging to the cal- 
 culus of quaternions give, generally, for any four vectors, a, 
 /3, 7, p, the two following identities, which are frequently useful 
 in the applications : 
 
LECTURE VII. 519 
 
 pS . yl3a = aS . 7j3p + j3S . ypa + 78 . jOj3a ; 
 pS . 7j3a = V. 7/3 . S . ap + V. a7 . S . )3p + V. j3a . S . 7p ; 
 
 and hence, without any use of xyz, or Z/'^, we might infer that if 
 p be supposed to denote any vector different from 0, its elimina- 
 tion between the three equations of either of the two following- 
 systems, 
 
 1st, S . 7j3|0 = 0, S.7pa = 05 S.joj3a = 0, 
 
 or 2nd, S.ap = 0, S . j3p = 0, S . yp = 0, 
 
 conducts alike to the final equation, 
 
 S.7j3a = 0, 
 as the result. 
 
 511. We may take this opportunity to remark that the geo- 
 metrical sigjiifications not merely of equations^ but also of Junc- 
 tions in this calculus, may be investigated (if not otherwise 
 known) by the same or similar transformations with co-ordinates: 
 and that on the other hand a person who was already familiar with 
 quaternions might conveniently employ them to deduce ov recover 
 many of the most important formulae in the method of co-ordi- 
 nates, by introducing (as above) trinomial forms for the vectors, 
 and employing the properties of the symbols ijk. As an exam- 
 ple of this last sort of process, if it were required to find an 
 expression for the distance of the point (xyz) from the origin 
 (000), or more generally from the point {abc), we should have 
 (by 111, 507) the transformations, 
 
 Tp = V{-p') = {x' + y' + z')^; 
 T(p-a)-{-(p-ay}i^{(x-ay+(:y-by+{z-cy}i; 
 
 and thus the known results would be reproduced. Again let it 
 be required to express the rectangle under the two lines from 
 the origin to the points (abc) (xyz), multiplied by the cosine of 
 the angle between them ; this product would be, by 423, 508, 
 as by other and more usual methods, 
 
 -S.ap = ax+by+ cz. 
 
 Again, if it were required to find the co-ordinates of the extre- 
 mity of a line drawn from the origin, so as to be perpendicular to 
 
520 ON QUATERNIONS. 
 
 the plane of the two lines drawn to the points {abc) (xyz), and 
 numerically equal (in a well-known sense) to the area of the pa- 
 rallelogram under those two lines ; while the rotation round this 
 sought perpendicular from the first to the second should be re- 
 quired to have the same character as the rotation round +z from 
 + X to + y ; we should only have (by 427) to take the coefficients 
 of i, j, k, in the recent developement (508) of V. ap ; and thus the 
 required co-ordinates, or the three co-ordinate projections of the 
 area of the parallelogram, on the planes perpendicular to x, y, z, 
 would be found in a new way to have the well-known values, 
 
 bz - cy, ex - az, ay - bx ; 
 
 while the area itself, considered as a magnitude, would be de- 
 noted by TV. ap, and would be seen anew to be equal to the 
 square root of the sum of the squares of these three last ex- 
 pressions. Finally, to find, hy the help of quaternions, that 
 function of the co-ordinates (aha) {a'b'c') (xyz) of three points, 
 which expresses the vo/z«;?e of the paraUelepipedo7i, having for 
 three of its edges the lines a, a, p, which are drawn to these 
 three points from the origin, we might first denote this volume, 
 as being the product of base and altitude, by the scalar pro- 
 duct of the two parallel vectors V. ap, and S . a'V. ap -=- V. ap, 
 whereof the latter denotes (by 430) the component of a which is 
 perpendicular to the plane of a and p ; and then we should find, 
 for the required volume, the expression S . a'V. ap, or simply (by 
 500), S . a'ap : and this last expression, thus deduced without co- 
 ordinates, might then be transformed, by the process of 509, 
 510, into the determinant lately considered. 
 
 512. In this way we should also be led to see that the deter- 
 minant (or eliminant) just cited, or the expression S . aap of 
 which it is an expansion, represents a positive or a negative 
 volume, according as the rotation round d from a towards p is 
 opposite or similar in character to the rotation round z from x 
 to y. And thus we might perceive, what we can, however, other- 
 wise prove, that the scalar of the product of three vectors changes 
 sign, ichen any two of its factors are interchanged : or that 
 
 S . y/Sa = - S . aj5y - S . /Say = - S . (3ya = S . ay(5 = - S . yo/3. 
 
LECTURE VII. 521 
 
 In fact, we saw in 499 that S . j(5a = - S . ajSy, and in 500 that 
 S . 7j3a = S . 7 V. j3a ; which last transformation gives also, 
 
 S . 7j3a = S (V. j3a .7) = S . j3ay = - S . ya(d, &C. 
 
 If we take any four vectors a, j3, 7, S, the scalar S . dy(3a of 
 their continued product may be decomposed into two parts, of 
 which one vanishes, by decomposing the product 7j3a into its 
 own scalar and vector parts ; thus 
 
 S . g7i3a = S . gV. 7j3a = S {Y. y(5a . B) = S . yfiad ; 
 
 the same scalar is therefore also equal to S . (3aSy, and to 
 S . aSy^ ; and a similar process shews that in general, under the 
 sign S, a7iy number of vector factors may have their order cycli- 
 cally altered. The same cyclical permutation is therefore 
 also permitted, for any number of quaternion factors, under the 
 same sign S, because each quaternion may be treated as the pro- 
 duct of two vectors : we have therefore generally 
 
 S . srq = S . rgs = S . qsr, 
 S . tsrq = S . srqt = &c.. 
 
 where q, r, s, t, represent quaternions arbitrarily chosen. 
 
 513. We have seen (507, 508, 509) that a scalar equation, 
 such as //3 = constant, gave generally a surface as the locus of 
 the extremity of jo. But let us now suppose that we meet a vec- 
 tor equation, such as 
 
 <i>p = Aj 
 
 where is supposed to be the characteristic of a vector func- 
 tion, such asV. a'ap, &c., of the first or of any other dimension, 
 while A denotes a constant and given vector. If we here change 
 again p to ix+jy + kz, and develope by the rules of this calculus, 
 the 07ie proposed vector equation will generally break up into 
 three scalar equations, which are in general sufficient (theoreti- 
 cally speaking) to determine, or at least to restrict to a. finite va- 
 riety of (real or imaginary) values, the three co-ordinates x, y, z, 
 and therefore also the vector p. For instance, if, with the recent 
 values of the symbols, the vector equation, 
 
 V. aojo = A, 
 
522 ON QUATERNIONS. 
 
 were proposed, it would be found to give, by comparison of the 
 coefficients 2,j, h, the following system of three scalar equations 
 of the first degree : 
 
 l = -x {act! + hy + cc') + y {all - bd) - z {cd - ac), 
 m = ~y {ad + hh + cc) + z {be - cb') - x {ab' - bd)^ 
 n--z {ad + bb' + cc') + x {cd - ac) - y {be - cb") ; 
 
 which might be treated by ordinary elimination, so as to give ex- 
 pressions for X, y, z^ and therefore also for ix +jy + kz. I regard 
 it, however, as an inelegance and imperfection in this calculus, 
 or rather in the state to which it has hitherto been unfolded, 
 whenever it becomes, or seems to become, necessary to have re- 
 course, in any such way as this, to the resources of ordinary 
 algebra, for the solution of equations in quaternions. 
 Indeed, very much remains still to be done towards the attain- 
 ment of anything approaching to perfection in the establishment 
 oi general methods for such solutions of equations, and for qua- 
 ternion elimination generally. But so far as regards equa- 
 tions OF THE first DEGREE in quaternions, 1 have been for 
 some years in possession of what appears to me to be such a ge- 
 neral method of solution. 
 
 514. Without entering at this moment on the exposition of 
 that ge?ieral method, I may remark, that it is allowed to write the 
 last proposed equation as follows, 
 
 Y.qp = X, or gp+Y.yp = \i 
 
 if we make for conciseness 
 
 q = a'a, g=Sq, y=Yq. 
 
 Operating by the characteristic of operation S . y ( ), or more 
 concisely by S . 7, that is to say, multiplying by y, and taking 
 the scalar part of the product, we get (compare 500), 
 
 ^S . -yp = S . yX, S . yp =^"^S . yX ; 
 but (by 407), 
 
 S . yp + V. yp = yp ; 
 hence 
 
 {g + y)p = \ + g-'S.yX; 
 
LECTURE VII. 523 
 
 SO that, ivithout the use of co-ordinates, the solution of the pro- 
 posed equation is obtained, under the sufficiently simple form : 
 
 Hence also, in this example, 
 
 a'Vp = Tf .p = (^^-72)p = (^-7)(X + ^-^S.7A) 
 
 = gX- 7X+ S .7X- ^"^78". 7X 
 
 =^"^(^^X- 5?'V. 7A-7S . 7X) 
 
 = ^-H(^^-70X-(^-7)V.7X)} 
 
 =5'"MX(^-- 7') -V. 7X . (^+ 7)) ; 
 
 and therefore 
 
 ff-y 
 
 that is, re-introducing the quaternion q^ 
 
 pSq = \ + q-'Y.\yq. 
 Accordingly, if we operate on this equation by V. q, or more fully 
 ^Y^'9 i )> we get , 
 
 Sq .V. qp =Y. qX + V. X V^ =V{(S^+V^)X) -V(Vg . X) = S^ . X, 
 and therefore V. qp = X, 
 
 as was required. I leave it to yourselves to verify the agreement 
 between the results of this and the preceding article. When you 
 shall have acquired a little practice in the use of the notations of 
 this calculus, and in the applications of its principles, you will 
 find, of course, that fewer steps of quaternion transformation will 
 suffice. 
 
 515. As respects notation, I take this opportunity to re- 
 mark, that I have frequently found it convenient to employ a 
 new SYMBOL, not yet introduced in these Lectures, to denote the 
 quotient of the vector part divided by the scalar part of a qua- 
 ternion; which quotient is evidently (by our principles) itself a 
 vector: and is quite as important and useful, in the applications 
 of this calculus, as the function tangent is, in trigonometry , 
 with which indeed it has a very close connexion. This new 
 symbol is the following : 
 
 %=yq-^^q. 
 
524 ON QUATERNIONS. 
 
 On the same plan I write, 
 
 tq = Sq-i-Yq; — q=TVq ^Sq; kc; 
 and thereby obtain the general transformations, 
 
 TV S 
 
 — o' = tanZg'; — q = cotan Z q. 
 s ^ ' TV ^ ^ 
 
 I do not lay so much stress on these notations as on others 
 already mentioned, but must repeat that I have often found them 
 useful. If they shall come to be adopted by other writers, it will 
 
 be necessary to distinguish between the symbols ^ and S'S ^^^ 
 
 similarly in other instances. In fact, I do not see why trigonome- 
 
 tricians might not have agreed to denote the secant oi x by the 
 
 , 1 , , sin , , cos J 
 
 symbol — x ; the tangent by — x ; the cotangent by -r- a; ; and 
 
 so forth, without the slightest prejudice to the modern mode of 
 denoting the inverse functions, cos"^r», &c., of which x is the 
 cosine^ or other direct function indicated. In this mode of nota- 
 tion, the vector equation of the foregoing article, V. qp = \, would 
 have its solution expressed as follows : 
 
 ^ hq ^ s 
 
 516. Again, let there be proposed the following vector equa- 
 tion ofthefrst degree, 
 
 V.j3p7 = X. 
 As this is of the form, 
 
 V. dpa = A, 
 
 it would be easy to break it up, on the plan of 509, 513, by in- 
 terchanging a and p, or {ahc) and {xyz), into three scalar equa- 
 tions of the first degree, between the three co-ordinates of p, 
 which might then be treated by ordinary elimination. We might 
 also see, by the developements already effected in art. 509, that 
 generally, for any three vectors, the following identity holds 
 good : 
 
 V. (tap = a'S . ap - aS . a'p + joS . a a ', 
 
LECTURE rii. 525 
 
 and therefore that, in the present question, 
 
 A = j3S .yp - pS . /By + yS . jSjo. 
 Hence, 
 
 S.(5\ = (5'S.yp, S.y\ = y'S.(5p; 
 
 S.yp = S.(5-'\, S.I5p = S.y-'X; 
 
 pS . jdy = (5S . (5-'X + y^ . y-'X-\; 
 
 and finally (by 449), the required expression for p, or the solu- 
 tion of the equation proposed in the present article, may be 
 written under the form : 
 
 ^~ ^y + 7^ ■ 
 
 517. This last symbolical expression admits of a very simple 
 geometrical interpretation, which it may be worth while briefly 
 to consider. Suppose, to fix the conceptions, that the angle be- 
 tween j3 and y is acute ; suppose also that j3 and y are unit lines, 
 and make a = jo"S UX=S. Then, 
 
 jSy + 7j3 = - 2 cos jSy < ; 
 Va = -lJp=V{(5S(3-' + ydy-'); 
 y.fia-'y = X; VV.(3a-'y=B. 
 
 Reflect the unit-vector S, separately and successively with re- 
 spect to y and j3, into two positions, £ and ^, such that 
 
 we shall then have 
 
 the line a will therefore bisect the angle between the two unit 
 lines, a and ^. Now this result exactly agrees with the conclu- 
 sions of the Fifth Lecture (art. 224, &c.), respecting the direc- 
 tion of the axis S, of the quaternion which is the fourth propor- 
 tional to three given lines, a, j3, y. In fact, if in fig. 40 (of the 
 article just cited) the points b, c, d were given, and a sought, we 
 might first double the arcs dc, db, and then bisect the arc ef. 
 The direction of the vector p, as determined by the last formula 
 of art. 516, agrees therefore with earlier results. 
 
526 ON QUATERNIONS. 
 
 518. With respect to the length of the same vector p, the 
 same formula gives, with our recent notations, the expression, 
 
 Tp = TA . 2^ll^ ; and XTa =VU. jSa-^Y ; 
 cos jSy 
 therefore, 
 
 T^VTT ^ -1 T \ T^ COS^V C0Sj3y 
 1 V U . pa ^ 7 = 1 . a A = 1 - = x = 7- ; 
 
 P cos ^ £^ cos ae 
 
 whence (by 227, 411) we may derive the following theorem of 
 spherical trigonometry, in connexion with fig. 40 : 
 
 . . ,^ ^ -.-Ts cos BC cos CA cos AB 
 
 ^ml(D-^E + F) = = = . 
 
 ^ ^ cos AE cos BF COS CD 
 
 In fact, in that figure, the arc ab is equal (by 224) to the hypo- 
 tenuse LM of the right angled triangle lnm, while cd (by 225) is 
 equal to the base ln of the same triangle, and the altitude mn 
 (by 258) represents the semi-area, or the semi-excess, of the tri- 
 angle DEF. 
 
 519. This appears to be a convenient opportunity for offering 
 a few remarks, on some general transformations of scalars and 
 vectors of products, and on their connexion with spherical trigo- 
 nometry. 
 
 Since, by 317, the conjugate of a product of any number of 
 quaternions is equal to the product of the conjugates taken in an 
 inverted order, a principle which we may agree to denote con- 
 cisely by writing the formula 
 
 Kn = n'K; 
 
 and since the symbolic equations of 407, 408, 
 
 1=S+V, K = S-V, 
 
 give, with analogous interpretations, these other general for- 
 mulae, 
 
 S = i(l + K), V=i(l-K); 
 
 we may write, on the same plan, the following abridged but ge- 
 neral equations : 
 
 sn = in + in'K; vn-in-in'K. 
 
LECTURE VII. 527 
 
 More fully, we have, for any set of quaternion factors, qi^qz^ . . qn^ 
 the two identities, 
 
 (S -Y){qn ... ^3 qi}= (S^i -Vg-i) (S^2 -Vg-s) . . . (S^„ -V^„) ; 
 
 by taking the semisum and semidifference of which, expressions 
 can be obtained for the scalar and vector of a product of any 
 number of quaternions. For example, 
 
 V. q, q, = Sq.Yq^ +Yq, Sq, + i {Yq.Yq^ -Yq.Yq,). 
 
 520. As a case of the application of the foregoing general 
 method, let there now be proposed any number of vectors, 
 a\, az, ■ • . an, and let us investigate expressions for the scalar and 
 vector parts of their continued product. Here (see again 317), 
 
 Kai = — ai, K . 0201 = + aia2, K . OsOaai = - «ia2a3, &C. ; 
 
 and therefore the formulae 2S = 1 + K, 2 V= 1 - K, give 
 
 2S . ai = Ui — a = ; 2 V. oi = Oi + ai = 2ai ; 
 
 2S . 0201 = a2«l + 0102 ; 2 V. 0201 = 0201 — 0102 J 
 
 2S . 030201 = 03O2O1 — OiOaOs ; 2 V. 03O2O1 = 030201 + 010203 ; 
 &c. &c. 
 
 results of which the law is evident, and of which the few first (or 
 others equivalent to them) have been already found, in 407, 449. 
 The formula just obtained for the scalar part of a ternary pro- 
 duct of vectors gives evidently the transformation, 
 
 . S . 7/3a = ^ (y^o - a|3y) ; 
 
 and thus, as we may now perceive, a comiexioti is established be- 
 tween tivo forms for the equation of coplanarity of three lines 
 (c, A, fx, which were separately and independently deduced in 
 former articles : for we had found in 195, that 
 
 /xAk: = KkfXy when |U [H A, /c ; 
 
 and knew also, by 430, 500, or by 511, that 
 
 S . 7j3a = 0, when Y 111 |3, o. 
 
 And the recent formula respecting the vector of a ternary pro- 
 duct gives. 
 
528 ON QUATERNIONS. 
 
 = h 0« + «^) - i (7« + ay)(d + ia {y(5 + /By) 
 = 78 . |3a - j3S . 7a + aS . jSy ; 
 
 an expression which obviously agrees with one already used in 
 516, but which is here deduced (compare 513) without any refe- 
 rence to co-ordinates, or any use of ijk. 
 
 521. Another mode of investigating a transformation equiva- 
 lent to that last written, and like it extensively useful in the ap- 
 plications of the present calculus, is the following. We are 
 allowed to write, generally, for any three vectors, a, a,d\ 
 
 V (V. aa. . a") = \ (V. ad . a — a"V. ad) = 2" (a« • d' ~ a", ad^ 
 = ^a\aa +aa^ — 2 \a.vi -^^ a a) a = a<i . aa — ao.aa; 
 
 whence also generally (compare 441), 
 
 V. a" V. da = aS . dd- a'S . ad' . • 
 
 Thus we have the two equations, 
 
 V(V.y/3.a)-7S./3a-/3S.7a, 
 V.7V./3a = aS./37-i3S.a7; 
 
 and by adding respectively to these the two identities, 
 
 V (S . 7)3 . a) = aS . 187, V. 7S. j3a = 7S . a^, 
 
 the recent formula of transformation for V. 7j3a is, in two ways, 
 reproduced. 
 
 522. Let there be r\ovi four proposed and arbitrary vectors 
 a, a', a", d". Operating by the characteristic S . d", on the iden- 
 tity, 
 
 V. d'da = aS . da — a'S . da + a S . ad^ 
 
 we obtain the expression : 
 
 S . d"d'da = S . da • S . dd — S . d"d . S . da + S . d"d . S . ad. 
 But 
 
 / o ' , \T ' . I" n Ci in " , \J III II , 
 
 aa = o-aa + \.aa, a a =k5.a a + \ . a a , 
 therefore 
 
 S . a" a" da = S . d"a" .S . da + S ( V. d"a" .^. da). 
 
 Comparing then these two expressions for S . a'd'da, we obtain 
 
LECTURE VII. 529 
 
 the following general expression for the scalar part of the pro- 
 duet of the vectors of any two binary products of vectors : 
 
 S (V. a'a .\ . da) = S . a" a . S . aa— S . a"'d . S . a"a ; 
 
 which may be also otherwise deduced, and is occasionally useful. 
 523. The vector part of the same product of vectors is easily 
 found, by similar processes, to admit of being expressed in either 
 of the two following ways : 
 
 V (V. a'a .^ . a a) = a"S . a" a a — a"S . a" da 
 
 Sill II I ICi lit II 
 
 • a aa— ao.a a a\ 
 
 of which the comparison conducts to one of the identities men- 
 tioned (without proof) towards the end of article 510 ; or to this 
 general expression for any fourth vector |0, in terms of any three 
 given vectors a, a', a", which are not parallel to any one common 
 plane, the laws (512) of permutation of three vector factors under 
 the sign S being remembered : 
 
 pS . a da = aS . a dp + o'S . a'joa + a'S . pda> 
 
 And if we here suppose that 
 
 a" = V. a'«, 
 we shall have 
 
 S . a'o'a = ( V. doif = a"^ ; 
 
 and after dividing by a"^, the recent formula will become, 
 
 p = alo — ;r + a O — 7, + 
 
 dp /Q P" S . a"p 
 
 a 
 
 whereby an arbitrary vector p may be expressed in terms of any 
 two given vectors a, a', which are not parallel to any common 
 line, and of a third vector a", which is perpendicular to both of 
 them. 
 
 524. If, in the last equation of 522, we change o, a', a", d" to 
 7, j3, j3, a, we find that, generally, for any three vectors a, /3, 7, 
 the following equation holds good : 
 
 S (V. ajS .V. (5y) = (5'S .ya-S.a(5.S.(5y. 
 
 To shew the geometrical meaning of this formula, let us divide 
 both members by T . 18-70, and transpose ; it then becomes, 
 
 2 M 
 
530 ON QUATERNIONS. 
 
 -SU.ya=SU.a/3.SU./B7 + S(VU.aj3.VU.^7); 
 
 or simply, 
 
 -S.ya = S.aj3S./37+S(V.a/3.V./37), 
 
 if we treat a, /3, 7, as unit vectors, which may be conceived to 
 terminate at three points a, b, c upon the unit-sphere. Here, 
 by the principles established in the present Lecture for the in- 
 terpretation of the scalar and vector parts of the product of any 
 two vectors, we have the values, 
 
 S . 7a = - cos b, S . aj3 = - cos c, S . j37 = - cos a, 
 
 if «, b, c denote the arcs or sides of the spherical triangle abc, 
 respectively opposite to the points a, b, c. By the same princi- 
 ples, 
 
 TV. a/3 = sin c ; TV. (5y = sin a ; 
 
 while UV. aj3, UV. j37, are vector units directed respectively 
 towards the positive poles of the rotations ab, bc, and are there- 
 fore inclined to each other at an angle which is the supplement 
 of the spherical angle abc, or B ; so that the scalar of the pro- 
 duct of these two last vector units is the cosine of that angle 
 itself, 
 
 SU(V.a^.V./37) = + cos5, 
 and 
 
 S (V. a/3 .V. fty) = sin c sin a cos-S. 
 
 The equation to be interpreted takes therefore the form, 
 
 cos b = cos c cos « + sin c sin a cos B ; 
 
 and thus is seen to coincide, as regards its signification, with a 
 well-known and fundamental formula oi spherical trigonometry. 
 525. More generally, if we divide the expression lately found 
 for the scalar part of the product of the vector parts of two binary 
 products of vectors, by the tensor of the product of the four pro- 
 posed vectors themselves, we obtain the equation, 
 
 S (VU . a'V. VU . a'a) = SU . a"'a . SU . a'a" 
 — S U . d"a. SU. a a ; 
 
 which signifies, when interpreted on the same principles, that 
 
LECTURE VII. 531 
 
 sin ad. sin a a", cos (ad ad") = COS aa". cos da" 
 - COS ad", cos da" ; 
 
 where the spherical angle between the two arcs from a to d and 
 from a" to d" may be replaced by the interval between the poles 
 of the two positive rotations corresponding. The same result may 
 be otherwise stated as follows : If l, l', l", l'" denote any four 
 points upon the surface of an unit-sphere, and A the angle which 
 the arcs li/, l"l"' form where they meet each other (the arcs 
 which include this angle being measured in the directions of the 
 progressions from l to l, and from l" to if" respectively), then 
 the following equation will hold good : 
 
 cos ll". cps l'l'"- cos ll'". cos l'l" 
 = sin ll'. sin l'l'". cos A. 
 
 Accordingly, this last equation has been given, as an auxiliary 
 theorem or lemma, at the commencement of those profound and 
 beautiful researches, entitled Disquisitiones Generates circa Su- 
 perficies Curvas, which were published by Gauss at Gottingen 
 in 1828. That great mathematician and philosopher was con- 
 tent to prove the last-written equation by the usual formulae of 
 spherical and plane trigonometry ; but, however simple and ele- 
 gant may be the demonstration thereby afforded, it appears to 
 me that something is gained by our being able to present the re- 
 sult under the form recently assigned (at the end of art. 522), as 
 an identity in the quaternion calculus. 
 
 526. The following is a still easier way than that adopted in 
 art. 524, of deducing from quaternions the fundamental formula 
 which expresses the cosine of the side of a spherical triangle, in 
 terms of the two other sides, and of their included angle. Taking 
 the scalars of both sides of the identity, 
 
 7-« = (7-/3)x04-a), orl = ^.2, 
 
 we find at once, by this calculus, the equation (compare 519, 
 520), 
 
 a p a pa 
 
 2 M 2 
 
532 ON QUATERNIONS. 
 
 where, by our principles of interpretation, 
 
 S -^ = cos a, S - = cos 6, S — = cos c, 
 p a a 
 
 TV^ = sin«, TV@ = sinc, 
 p a 
 
 SU.V^3 v2 = cos5; 
 p a 
 
 so that we still arrive, as before, at the well-known result, 
 
 cos b = cos a cos c + sin a sin c cos B. 
 
 It may be added that, with the same meanings of the symbols, 
 the following equation in quaternions holds good, and admits of 
 being extensively applied to questions of spherical trigonometry : 
 
 V. yjS . V. j3a = sin a sin c (cos + j3 sin) B ; 
 
 where it is understood that 
 
 (cos + j3 sin) -B = cos ^ + j3 sin 5 : 
 
 and the rotation round j3, from a towards y, is supposed to be 
 positive. If, on the contrary, the rotation round jS from^y towards 
 a were positive, we should then be obliged to change the sign of 
 j3 (or of B) ; for we have generally, by 523, 512, 
 
 V ( V. 7j3 . V. ^a) = - i3 S . yj3a = i3 S . ajSy, 
 
 and this last scalar factor S . aj3y would be negative (by 512) in 
 the case last considered. At the same time we see that we may 
 write, subject to this last condition respecting a change of sign, 
 
 S . ajSy = sin c sin a sin B, 
 
 which expression for the scalar part of the product of three unit 
 lines might be employed to reproduce (by 511) a known value of 
 the volume of cm oblique parallelepipedon. We find also the 
 following expression for the trigonometric tangent of an angle of 
 a spherical triangle, in terms of the vectors of the three corners, 
 
 tan a^7-tan^-/3-^-(V.Yi3.V.j3a). 
 
 527. Another fundamental connexion of quaternions with 
 spherical trigonometry may be more clearly understood after a 
 
LECTURE VFI. 533 
 
 few observations on their connexion with plane trigonometry, or 
 rather with that well-known doctrine of functions of angles, 
 which some writers have named goniometry. 
 
 Suppose then that we had not yet heard of the functions 
 cosine and sme, but had in other respects acquired a knowledge 
 of the principles of the present calculus, as hitherto set forth in 
 these Lectures : and let a, j3, 7, . . . t, denote any unit vectors, 
 and t any scalar exponent (positive or negative). Thepoiuers a*, 
 j3*, . . . are seen (by the Third Lecture) to be all versors, and 
 by the symmetry of space t/ieir scalar parts must be equal ; thus 
 we may write, 
 
 s.a'=s.j3*=s.y = . . . = s.,'=/(o, 
 
 f{t) denoting here some scalar function of t. In fact, by articles 
 86, 407, if 
 
 A = i*K = X'+X", where i ± k, A' || k, A"± i, A"_L k, 
 we have 
 
 S.l' = \'k-\ V.t'=AVM 
 
 and the scalar quotient A' -f- /c depends only on the angle 
 {-tx 90°) through which A has revolved from k in a plane per- 
 pendicular to i, and not at all on the plane of this rotation, 
 nor on the initial direction of the line. We see at the same 
 time that because t, k, A" compose a rectangular system, or be- 
 cause the rotation from k to A has been performed round t as an 
 axis, we must have 
 
 Hence 
 
 V. t*^i = tS . i\ V. t'= tS . I*-' = if{t-\)', 
 
 and we have the general transformations, 
 
 '.'=fit)-^if{t-\), a^=f(t) + af(t-\),&c. 
 
 Also, by 89, t* and r* are conjugate versors, and by 408, K = S- V ; 
 hence 
 
 r'=f{t)-f(t-l). 
 
 Thus/is an even function, 
 
534 ON QUATERNIONS. 
 
 as indeed its geometrical nature as the quotient X' -f- k might at 
 once shew ; also because t° = 1 , i^ = i, t'^ = - 1 , we have 
 
 /(0) = l,/(l) = 0,/(2) = -l; 
 
 and more generally 
 
 f{2 + t)=f(2-t) = -f{t); 
 
 it is therefore sufficient to know the system of the positive and de- 
 creasing values of the function yj from ^ = to ^= 1 ; or even from 
 / = to t = ^, because by multiplying together the two conjugate 
 versors i\ i'^, or by taking the t&nsor of either of them, we are 
 conducted to the functional relation, 
 
 {/(0)^+{/(^-i))^=i. 
 
 But again, if m be any other scalar, we have, by 117, 150, t"i' = 
 t"*', and therefore the two functional equations hold good, 
 
 f(u + t) =fiu)f{t) -f{u - l)f(t - 1), 
 f{u + t-l)=f{u)fit-l)+f(u-l)ft, 
 
 of which indeed the latter can be derived from the former, by the 
 consideration thatj'(t - 2) = -f{t). Hence 
 
 f{2t)^[f{t)Y-[f(t-l)Y. 2{/(0J^ = l+/(20; 
 
 and, therefore, at least within that range which gives a positive 
 value to/f- 
 
 /(0 = {i + i/(O}*- 
 
 Thus, from y(2) = - 1, we might infer /(I) ^^0, as before ; and 
 thence, 
 
 /a)=vi,/(i)=v/a+ivi), &c, 
 
 and might so calculate and tabulate a system oi approximate nu- 
 merical values of the function : in doing which we might assist 
 ourselves by many artifices, not necessary to be stated here. 
 And thus the function ^(^), or S . t*, would come to be numeri- 
 cally known. You will easily see that the same principles give 
 expressions for functions of multiples, analogous to the usual 
 formulae for cosines and sines of multiple arcs : the principle 
 
LECTURE VII. 535 
 
 being here that at least for any whole value of n (compare the 
 Fourth Lecture), (t*)"= i"', and therefore 
 
 528. If the increment u of the exponent t be treated as a 
 very small angle, the geometrical consideration of the small ro~ 
 tation answering to the versor t" would give the two following 
 limits : 
 
 lim. M-i(l-S.t«) = 0, and lim. m-^V. t«=^t; 
 
 where tt denotes as usual the semi- circumference of a circle of 
 which the radius is unity. Hence 
 
 lim. M-i(t"^'-i') = lim. u-^{i!'- l).t'= Jt'"'; 
 
 or in the notation of differentials, 
 
 d.t* = |i*^id^ 
 
 Taking the scalars and vectors of the members of this formula, 
 we have the two following separate equations, of which indeed 
 the one includes the other : 
 
 and because /(^ + 2)= -/*(^), we have this differential equation 
 of the second order, 
 
 ^ /"(0 + (|)V(0-o, 
 
 with the initial conditions, 
 
 /(0)=l,/'(0) = 0: 
 from which might be inferred the developements, 
 
 s.,.=/(o=i-/^Yi:+(^Y_ii--&c., 
 
 2/ 2 V2y 2.3.4 
 
 &C. 
 
 '-v-.'=/(^-i)=-'-/'»=lT-(|)'r|-3- 
 
 If then we suppose it known from algebra (by an investigation 
 
536 ON QUATERNIONS. 
 
 conducted without any use of trigonometry), that for every real 
 value of a;, of the ordinary algebraical kind (any positive or nega- 
 tive number or zero), the series 
 
 is equal to the a;* power of the base jp (0), or of the known con- 
 stant, 
 
 ^ 2.3 
 we may thus be led to establish, by analogy, and as a definition^ 
 the equation 
 
 where the second member is merely employed as a concise ex- 
 pression for the developement, 
 
 1 + (iTT^o + i {^iruy + ^ {h-^uy + &c. 
 
 And to effect a complete agreement between the results of the 
 investigation thus sketched, and the usual language of trigono- 
 metry, it would only be necessary to write (compare 411), 
 
 S..*=/(0 = cos^^, r^V..*=/(^-l) = sin^S 
 or, 
 
 i' = COS — + i sm — . 
 
 529. Consider now the formula of article 280, 
 7^/3^a^" = - 1 , or 7 2 - ^ = j32/„^. 
 
 Making, as in that article, 
 
 we have the transformations, 
 
 a^ = cos ^ + a sin A, (^y = cos B + (5 sin B, 
 and 
 
 y ~-^ = cos (tt- C) + y sin (tt-C) ; 
 
 the formula becomes therefore the following : 
 
LECTURE VII. 537 
 
 COS (tt - C) + 7 sin (tt - C) = (cos 5 + j3 sin B) (cos A+ a sin A) ; 
 
 and is now seen to include (as it was earlier stated to do) the 
 whole doctrine of spherical trigonometry. In fact, if we merely 
 take the scalar parts, and remember that S . aj3 = - cos c, we 
 obtain the equation, 
 
 - cos C= cos A cos B - cos c sin A sin B^ 
 
 from which everything else could be deduced. The formula 
 however gives also, by taking the vector parts, 
 
 y sin C= a sin ^ cos B + (5 cos A sin B +V. (5a . sin A sin B : 
 
 from which it follows that if three vectors be drawn from the 
 centre of the sphere, one towards the point a, with a length 
 = sin A cos B, another towards the point b, with a length 
 = sin B cos A, and the third perpendicular to the plane of the 
 arc AB, and on the same side of it as the point c, with a length 
 = sin A sin B sin c, and if with these three lines as edges we 
 construct a parallelepipedon, the intermediate diagonal will be 
 directed towards the point c, and will have its length = sin C. 
 The addition as well as the 7nultiplication of quaternions, and 
 the distributive as well as the associative character of such mul- 
 tiplication, may also be illustrated generally by spherical trigono- 
 metry, and may be employed to furnish theorems therein. 
 
 530. Perhaps it may not be improper here to mention the 
 process by which, so long ago as in October, 1843, I was con- 
 ducted to results substantially agreeing with those of the fore- 
 going article, but obtained in a quite different way. 
 
 At that time I had been led, by a train of speculation too long 
 to be here described, to establish : 1st, The fundamental 5'M«(/n- 
 nomialform of the quaternion (see art. 450, &c.), 
 
 q = iv-\- ix -Yjy + Jcz, 
 
 with the geoTYietrical interpretation of the trinomial part, ix -vjy 
 + kz, as denoting (see arts. 17, 101, &c.), a directed right line in 
 space ; 2nd, the squares and products ofi^j, k (see articles 75, 
 76, &c.), which maybe collected as follows in a symbolical mul- 
 tiplication table, and illustrated, as regards the cyclical character 
 of the products^ by a diagram, fig. 101, as follows : 
 
538 
 
 ON QUATERNIONS. 
 
 
 
 MULTIPLICAND. 
 
 
 
 i 
 
 J 
 
 k 
 
 
 
 i 
 
 -1 
 
 k 
 
 
 -3 
 
 
 Hi 
 
 
 
 
 
 
 Id 
 O 
 
 o 
 
 O 
 
 J 
 
 -k 
 
 -1 
 
 i 
 
 
 
 
 
 
 
 k 
 
 J 
 
 - i 
 
 
 -1 
 
 
 Fig. 101. 
 
 each symbol, i oxj or k, when multiplied ^/^^o the one which cy- 
 clically follows it, giving a product which /o//o2^^5 the multipli- 
 cand, in the same cyclical succession, but the sign of the product 
 being changed, when the order of the factors is reversed ; 3rd, 
 the distributive principle of multiplication of quaternions (see 
 arts. 455, &c.), which gave (compare art. 489) the associative 
 pj'inciple also, because this latter principle was seen to hold good 
 for the multiplications oii,j,k, among themselves ; but 4th, I 
 had found it necessary (as already abundantly illustrated) to re- 
 ject the commutative property of multiplication, except as be- 
 tween the ordinary reals of algebra, such as the four constituents 
 w, X, y, z, of the quaternion (or between the old and ordinary 
 imaginaries of algebra, which however I did not then employ), or 
 as between such a real and any 07ie of my new imaginaries (as 1 
 then called them, on account of their squares being each equal to 
 negative unity), namely the three symbols of my new system 
 ijk; ^o i\\a.i xy = yx, oxiA xi = ix, although in this new calculus 
 
 ji = -V- 
 
 531. With these preparations, it was easy to conclude that 
 the product, q . q', of two quaternions, was equal to a third qua- 
 ternion, q", such that if 
 
 q = w + ix +jy + kz, 
 q'= w' + ix +jy' + kz, 
 q" = lo" + ix" -vjy + hz, 
 
 then (compare 508) the four following relations between the 
 twelve constituents hold good : 
 
LECTURE VII. 539 
 
 w' - WW - XX - yy - zz!, 
 x" = wx + xw + yz - zy\ 
 y" = wy + yw + zx - xz, 
 z" = wz' + zw' H- xy' - yx. 
 
 These gave, by ordinary algebra, the equation, 
 
 w"'^ + x'"^ + y""^ + z"~ = (w^ + X' + y'^-\- 2~) (^f;'2 + a;'^ + y'^ + 2;'-) ; 
 
 which, as a decomposition of a sum of four squares into two fac- 
 tors^ of which each is itself the sum of four squares^ had been 
 (I believe) anticipated by the illustrious Euler, although 1 
 had not then heard of its being known, nor have I since met 
 with the paper, or passage, in which the theorem was given by 
 him. This opened a connexion between quaternions and the 
 THEORY OF NUMBERS, by mcans of sums of squares, which was 
 soon happily followed up by my friend John T. Graves, P^sqr, 
 with whom I had long been engaged at intervals in a corres- 
 pondence on the subject of imaginaries, and to whom I had re- 
 cently communicated my results respecting quaternions. He 
 found, for sums of eight squares, and for certain octaves, or octo- 
 nomial expressions, connected with a system of seven distinct 
 imaginaries, results which he sent to me in return, about the end 
 of 1843, and beginning of 1844, as a sort oi extension of my own 
 theory, in letters of which 1 have elsewhere placed the substance 
 upon record. But it is impossible for me here to attempt to do 
 any kind of justice to the talents and candour of the many able 
 and original mathematical writers in these countries, who have 
 been pleased to acknowledge that some subsequently published 
 speculations of theirs, on subjects having some general connexion 
 with or affinity to the present one, were, more or less, suggested 
 or influenced by the quaternions. 
 
 532. Resuming the account of my own investigations, I may 
 mention that I was led, by the lately mentioned relation between 
 sums of squares, to assume a system of expressions for the consti- 
 tuents of a quaternions of the forms, 
 
 w = fx cos 0, 
 
 x = fx sin B cos (p, 
 
540 ON QUATERNIONS. 
 
 «/ = ju sin sin (p cos xp, 
 ^ = iusin sin (p sin t/-, 
 
 and to call ju the modulus, the amplitude, the colatitude, and 
 ^// the longitude, of the quaternion zi? + ea; +J2/ + ^•^r. The words 
 " modulus" and " amplitude" were suggested by the correspond- 
 ing phraseology of M. Cauehy, respecting the ordinary imagina- 
 ries of algebra ; 1 have since come to use habitually, as in this 
 Course, these other names, " tensor," and "angle." With re- 
 spect to the two angular or spherical co-ordinates, <j> and ip, which 
 mark the direction of the axis of the quaternion, or of the vector 
 part ix +jy + kz, the motives for calling them as I did are evident. 
 The suggestion of calling the four reals, w, x, y, z, " consti- 
 tuents" of the quaternion, I took from Mr. Graves : the interpre- 
 tation of the three co-efficients oii,j, k, as co-ordinates, was one 
 which, from the first conception of the theory, occurred to my- 
 self Thus the modulus (or tensor) was the square root of the 
 sum of the squares of the four constituents ; and the relation be- 
 tween such sums of squares came to be expressed by the follow- 
 ing very simple formula, 
 
 // t 
 
 fi =fXfJ., 
 
 which I called the law of the moduli. It has presented itself 
 in these Lectures (see arts. 188, 208), under the form of the 
 theorem that the " tensor of the product is the product of the ten- 
 sors" as expressed by the formula, Tn = IlT: for, by 409, 507, 
 
 Tq =T {w+ ix +jy + kz) = {w~ + a;^ + ?/^ + 0-)^. 
 
 533. Introducing the recent expressions for the constituents 
 of^, with analogous expressions for those oi q and q, and divid- 
 ing by fifx or by fx', the expression for w" (in 531) gave me, 
 
 cos Q" = cos d cos Q' - sin sin 6' { cos (p cos (ft' 
 
 + sin sin 0' cos {xp - \p') ] . 
 
 But also the expressions (in same art. 531), for w", x", y", z", 
 gave 
 
 w'w" + x'x" + y'y" + z'z" = w {w"^ + x'" + y"^ + z"), 
 IV w" + xx" + yy" + zz" =w' {w'^-\- x- + ?/- + a;-) ; 
 
 and therefore 
 
LECTURE VII. 541 
 
 COS = cos B' cos 9" + sin 0' sin 0" { cos (f)' cos ^" 
 
 + sin (j)' sin ^" cos (?//' - i//") } , 
 cos 6' = cos 0" cos + sin 0" sin (cos 0" cos (p 
 
 + sin 0" sin ^ cos (t//" - i/') ) . 
 
 And hence, by usitig as knoum the two equations of spherical 
 trigonometry, 
 
 cos b == cos c cos a + sin c sin a cos B, 
 - cos C= cos A cos jS - sin ^ sin 5 cos c, 
 
 (which, in M?^ Lecture, have been on the contrary deduced from 
 quaternions, in articles 524, 526, 529), I concluded that if 0, 
 ■(p were regarded as the spherical co-ordinates of one point r on 
 the unit sphere ; 0', ip', as those of a second point r' ; and 0", ;//" 
 as those of a third point r"; which three points r, r', r" might 
 be called (compare 225, 264, 361, &c.) the representative points 
 of the three quaternions q, q\ q" : then, in the spherical triangle 
 rr'r", the angles were respectively equal to the amplitudes of the 
 two factors, and to the supplement of the amplitude of the -pro- 
 duct : or that in symbols (compare 265), 
 
 E = e, R'==Q', R"=Tr-Q": 
 
 the rotation round r from r' towards r" being also found to be posi- 
 tive (272). At the same time, or rather indeed a little earlier, I per- 
 ceived that the three relations between the nine angles 6, ^, ip, 0', <p', 
 \p', 6", (p", \p", might be interpreted, on similar principles, as signify- 
 ing that if, with the amplitudes, 0, 9', 9', of any two factors and 
 their product, as sides, we construct a spherical triangle, the 
 angle opposite to the amplitude of the product will be the 
 supplement of the inclination of the factors (or of their axes, or 
 vector parts) to each other ; and that the angle opposite to the 
 amplitude of either factor will be the inclination of the other 
 factor to the product. These and other connected results were 
 communicated by me to the friend already mentioned (Mr. J. T. 
 Graves), in letters of October 17th and October 24th, 1843, 
 which have since been printed in the Supplementary Number of 
 the Philosophical Magazine, for December, 1844, and in a note 
 appended to the Essay, entitled " Researches respecting Qua- 
 ternions, First Series," in the Second Part of the Twenty-first 
 Volume of the Transactions of the Royal Irish Academy. (The 
 
542 ON QUATERNIONS. ^ 
 
 theorem last stated may be illustrated by inspection of the tri- 
 angle KLM, in figure 51, article 266.) 
 
 534. Another early and more general result of this Calculus, 
 connected with spherical polygons, was obtained nearly as fol- 
 lows. Let R, r', r" be any three points on the sphere, for which 
 the rotation round r from r' towards r" is positive, and may be 
 denoted by R. Then the rotation R round r would bring the 
 arc rr' to coincide in direction with the arc rr"; and the supple- 
 mentary rotation, tt-R, round the same pole r, would bring the 
 prolo7igation of the arc r"r to coincide in like manner with the 
 arc rr' in direction ; or would bring the positive pole p' of the 
 arc r"r to coincide with the positive pole p" of the arc rr'; that 
 is, the pole p' of the preceding side of the triangle r"rr' to coin- 
 cide with the pole p" of the folloiving side. Hence it was easy to 
 infer, that if Zr, i^', ip", denoted the three unit-lines, drawn from 
 the centre of the sphere to the points R, p', p', we must have the 
 equation, 
 
 ip' . ip" = cos R + ia sin R ; 
 
 the amplitude of the quaternion product of any two such unit-lines 
 having been previously seen to be the supplement of the angle 
 between them (compare 87) ; and the axis of the same product, 
 or the part of it involving i,j, k, having been also seen to be di- 
 rected towards the positive pole (in this case r), of the arc drawn 
 from the representative point (p') of the multiplier line, to the 
 representative point (p") of the multiplicand line (compare again 
 87). In like manner, if rr'r"r"' . . r'"'^^ be any spherical /*o/y- 
 gon, and if the positive poles of its n successive sides rr', r'r", 
 . . . r'"-'^) r^'^"^^, r("-i^ r be denoted by p", p'", . . . p, p', while the 
 angles R, R' . . and 72^""^' denote respectively the rotations at 
 the corresponding points, from rr' to rr'^'^^, from r'r" to r'r, . . 
 and from r^^-^^r to r^"-i) r^"-^)^ which rotations may be conceived 
 for simplicity to be each positive and less than two right angles; 
 then the same reasoning shews that, besides the lately deduced 
 equation, we have also these others, 
 
 ipn «>'= cos R' + V sin R\ , . . ip v= cos R'-'"'" + 4(»-i) sin R^"-^) ; 
 
 and therefore, by the associative principle of multiplication, 
 
LECTURE VII. 543 
 
 (cos R + ?"ji sin R) (cos R' + v sin R') . . . (cos i2(«-i) 
 + e,(„.x) sin i2("-^) ) = (-!)", 
 
 because i\ = ^V = i^p^/ = ... = - 1 . 
 
 535. We have assisted our conception of the foregoing pro- 
 cess and result, by supposing that the n rotations, R, R\ &c., 
 are each positive, and less than tt; but it is not difficult to inter- 
 pret the formula above obtained, when those conditions are not 
 satisfied. Thus, for a spherical triangle, the theorem is, that 
 
 (cos R + 4 sin R) (cos R' + ?h.' sin R') (cos R" + i^>' sin R") = - 1 ; 
 
 where if we change R", R', R to A, B, C, and the corresponding 
 unit-lines i^", v, 2r to a, j3, y, the formula becomes : 
 
 (cos C+y sin C) (cos 5 + j8 sin B) (cos ^ + a sin ^) = - 1 ; 
 
 the rotation round y from j3 to a being here supposed positive, so 
 that we fall back on the case of figure 56, art. 280, and through 
 such transformations as those of art. 529, on the formula, 
 
 But if we suppose that a, j3, y take the places of ?r, v, iw, in 
 the formula of the present article, the rotation round y from j3 to 
 a being still positive, and therefore that round a from j8 to -y 
 being negative, we must substitute, for the rotations, R, R\ R", 
 either values greater than two right angles, such as 
 
 R=27r-A, R' = 27r-B, R" = 2ir-C; 
 
 or else negative values, such as 
 
 R = -A, R'=-B, R" = -C, 
 
 R still denoting the rotation round the point r from rr' to rr", 
 &c. Thus, in this case, the general formula becomes, 
 
 (cos A - a sin A) (cos 5 - j3 sin B) (cos C- 7 sin C) = - 1, 
 or 
 
 a-^ l^-y y-^=- 1; 
 
 but these last equations are equally true with the foregoing, and 
 are indeed consequences of them. When the theorem has been 
 in any manner established for a triangle, it is easy to extend it 
 to a polygon, by breaking up that polygon into triangles, having 
 
544 ON QUATERNIONS. 
 
 any common vertex on the sphere; and in fact it was thus. that I 
 was first led to perceive it. 
 
 536. With the same sort of use oi scalar exponents, and of 
 powers of unit-lines, we may express the general theorem as 
 follows : 
 
 a°."i' . . .a2°'-ai'^^ a" = (-!)»; 
 
 where the scalars a, «i, . . an-i, represent the positive or nega- 
 tive numbers of right angles contained in the respective rotations, 
 round a from a„.i towards Ai, round Ai from a towards Aj, &c., 
 and finally round a„.i from a„_2 towards a. It is not difficult to 
 findapoto' transformation of the theorem, in viih\c\y supplements 
 of sides shall take the place of angles : nor again to transform the 
 result so obtained into another involving the sides themselves, 
 which also holds good for any spherical polygon, and may be 
 otherwise and more immediately deduced from the identity of ar- 
 ticle^345, or from the following : 
 
 a an -I a2 Oi _ , 
 
 On-i an-2 «! a 
 
 In fact, if we make 
 
 and 
 
 i3=UV^,j3,= UV^, . . . j3,.,= UV-^, 
 
 a ai a„-i 
 
 , 2 ai , 2 ao 
 
 b=-L—, bi= - L—, &e. 
 TT a IT ai 
 
 where a, ai, ao, . . . may be conceived to be n unit vectors, 
 terminating at the corners a, Aj, Aj, . . of a polygon, of which 
 the sides aai, AiAj, . . contain respectively b, bi, . . quadrants, 
 while 3, j3i, • . are n other unit-lines, terminating at the positive 
 poles of those n successive sides, we shall have the transformations, 
 
 a ai 
 
 and finally the equation : 
 
 |36„-i . . . j3*2 j3;^j3'=i. 
 
 Indeed an equation with the same geometrical signification 
 might have been obtained from the first formula of the present 
 article, by transforming it as follows : 
 
LECTURE Til. 545 
 
 ^2-a _ 2-a 
 
 But I leave it to yourselves, as an exercise, to demonstrate this 
 agreement of meaning. 
 
 537. All the POWERS that have been hitherto considered in 
 these Lectures have had scalar exponents, with the single ex- 
 ception of the power in article 528, which had e for its base, and 
 a vector, namely, ^-rrti, for its exponent. But if we now define 
 that for the same base, e, and for any quaternion, q, as expo- 
 nent, the symbol e« of the jooi^er shall be interpreted as acow- 
 cise expression for the series, 
 
 „ = r(,)=I+f + ^^.^3 + &e. 
 
 we shall not violate any conditions hitherto established, but shall 
 on the contrary be able to give useful extensions to results 
 already obtained. It may be proper however here to shew that 
 this series, so well known in the algebra of ordinary reals and or- 
 dinary imaginaries, is, in this calculus likewise, convergent ; and 
 that it gives an absolutely definite quaternion as its value, 
 or as the limit to which it tends, when continued indefinitely far, 
 the quaternion q being supposed given. In other words, if, in- 
 stead of the infinite series above written, we consider the finite 
 dev elopement, 
 
 
 m. 
 
 it is to be shewn that, for sufficiently large and increasing values 
 of the number m, the function F (5) is very nearly equal to a 
 certain definite limit, which may be denoted by Y <»{jq) or by 
 F {q) ; or that the scalar, vector, and tensor, of the variable qua^ 
 ternion F {q) - F,„ {q), where F {q) is a cevtdiin fixed quaternion, 
 converge each separately to zero : in such a manner that 
 
 S(Fg-F^^)and Y{Fq-F,,q), 
 
 may be made, respectively, as small a number and as small a 
 line as we may desire, by taking for in a sufficiently large whole 
 number. 
 
 2 N 
 
546 ON QUATERNIONS. 
 
 538. Let there be any two quaternions, q and r, and let us 
 seek the tensor of theii' sum. By principles of transformation 
 already explained, we have 
 
 T {r + qY ={r-¥q){Kr+ Kq) = Tr'- + Tq- + 2S . rKq 
 
 = Tr^ + Tq^ + 2TrTq S U . rKq 
 
 ==(Tr+Tqy-2TrTq{\-S\J.rKq) 
 
 = (Tr-Tqy+2TrTq(l + S\].rKq)', 
 
 and the scalar of the versor of a quaternion, being equal to the 
 cosine of its angle, cannot fall outside the limits + 1 ; whence we 
 derive these two important inequalities, 
 
 T (r + q) :j> T^ + Tq^ T (r + q) <[: Tr - Tq. 
 
 In words, the tensor of the sum of any two quaternions cannot 
 be greater than the sum, nor less than the difference, of the tensors 
 of those two quaternions themselves. Hence for any number of 
 quaternions, the tensor of the sum cannot exceed the sum of the 
 tensors; or in symbols, 
 
 TS^:|>ST^. 
 
 539. It follows hence that, in the notation of 537, 
 
 T{ F„,,„(^) - F„, (^) j :^> F,,U (T^) - F^ (T^) ; 
 
 but if we take 
 
 m> 2Tq~ 1, 
 we shall have 
 
 ^ < i, . . -^ < i and F,,,„ (T^) - F„, (T^) < :;— J^^, 
 m+l m + n \ 2/ ^ ^^ 1.2.3.. m 
 
 because 
 
 Again, let a new whole number m" be taken, greater than 2Tq - 1, 
 and let us write 
 
 then for any whole number m' > m" we shall have 
 
 To™' a 
 
 < 
 
 l.2...m' 2™'-"'"' 
 
LECTURE VII. 547 
 
 SO that this term of the series F,„ (T^) will be less than any 
 GIVEN positive quantity, b, however small., if the number m' be 
 so taken as to satisfy the inequality, 
 
 and every following term will evidently be still less, because 
 
 1 — TT < t: -,•> if »« > W. 
 
 1 . 2 . . m 2™;'"" 
 
 Hence, by the arithmetical properties of the series, we have 
 
 Fot+„ (T^) - F,„ (Tg-) <b,iim>m'; 
 
 and therefore, by what was shewn in the foregoing article re- 
 specting the tensor of a sum, and by the inequalities m > m 
 > 'iHq- 1, we have, in passing to quaternions, the inequality, 
 
 T { F^^ „ {q) - F,„ {q) \<b,\i tn > m', 
 
 however lar(/e the number n and the tensor T^^ may be, and 
 however small the given and positive quantity b. Thus if the 
 number m be taken sufficiently great, that is, ifwe take a term 
 sufficiently advanced in the series, but always at a finite distance 
 from, the beginning, the sum of any number (n) of the quaternion 
 terms which follow it will have its tensor less than any given 
 small quantity (b) : and consequently the scalar and vector parts 
 of the same quaternion sum of these n following terms, however 
 numerous, will each separately and independently approach inde- 
 finitely to zero, since we shall have 
 
 S{^r,^.n{q)-^,.{q)]>-b,< + b', T\{Y,,,n{q)-ym{q)]<b. 
 
 The series does therefore converge, as was asserted, to one defi- 
 nite quaternion, F oo {q) or F^, as a limit ; of which quaternion 
 the scalar part SFg, must lie between SF„i {q) - b and SF,,^ {q) + b, 
 and can therefore (theoretically speaking) be calculated within 
 any required degree of numerical accuracy, by calculating 
 SF„, (q) ; while the vector part YFq, of the same quaternion 
 limit, if drawn as a right line from the origin of vectors, must 
 terminate on some point in the interior of a very small 
 
 2 N 2 
 
548 ON QUATERNIONS. 
 
 SPHERE ; the vector of whose centre would be the assignable line 
 VFm (<7), while its radius would be the proposed small quantity, b. 
 540. Consider next the the function, 
 
 Fm (r+ q), supposing rq = qr. 
 
 Under this condition of commutativeness, we shall have as in al- 
 gebra, 
 
 (r + q)^ / rP (f 
 
 1.2.3..m"^V1.2../?'1.2..w 
 
 where the exponents n and p are each <|; o> "^ m^ and p + n = m. 
 Hence, if we write 
 
 F™ (r) ¥m (q) -Fm(r+q) = s,n, 
 
 the difference Sm will be developed into a polynome containing 
 ^m (m + 1) terms of the form just written, but with the conditions 
 that each of the exponents n and p shall now be > 0, :j> m, and 
 that p + n> 7n. By 538, the tensor of this polynome cannot ex- 
 ceed the sum of the tensors of its terms ; and therefore 
 
 T^^ > F,„ (Tr) F^ (T^) - F^ (Tr+Tq), 
 
 because T (rPq^^) = (Tr)P (TqY. Again the developement of 
 F2M {Tr+ Tq) contains all the terms of the developement of the 
 product F^CTr) . F^ (T^), and other positive terms,^in number 
 = m(m+l), besides ; therefore 
 
 T*^ < F,^ {Tr + T^) - F^ (Tr + T^). 
 
 Hence, by the foregoing article, 
 
 Tsm < b, if rd > m' ; 
 
 that is, by the present article, 
 
 T[Fn.{r)F„.(q)-F^ir+q)}<b, 
 
 however small the given and positive quantity b may be, if the 
 number m of the terms in each of the three finite series F„i ($■), 
 Fjn (r), Fm, {r + q), be taken large enough. But the smallness of 
 a tensor infers the smallness of the scalar and vector also; thus, 
 at the limit m = 00, we find, rigorously, for quaternions as for or- 
 dinary algebra, but still subject to the condition of commutative- 
 
LECTURE VII. 549 
 
 ness, that the well-known series above mentioned possesses the 
 EXPONENTIAL CHARACTER : or in symbols, that 
 
 F (r + ^) = F (r) F (q), if rq = qr. 
 
 54 L If this last condition were wo? satisfied, the foregoing 
 process would be inapplicable, and the result would cease to be 
 true. We should find, for instance, 
 
 Yz{r-vq) = \^-r-vq^\{f'^-rq-V qr-\-q^^'i 
 s, = F, (r) F2 (q) -F,(r + q) = i (rq -qr) + i {r'q + r(f) + Jr^^^ ; 
 
 but this expression for the diiFerence ^2 contains a part, namely, 
 
 \(rq-qr)=NNrNq, 
 
 which had not previously presented itself, but which we are not 
 at liberty in general to reject. We cannot therefore say, without 
 restriction^ in quaternions, that 
 
 we must add, as before, the condition, 
 
 if rq = qr, or if \{yr .Nq) = 0. 
 
 It is worth noticing, however, that although the expressions, 
 
 r + q, r^+ 2rq + q'^, r^ + 3r^q + Srq- + q^, &c., 
 
 do NOT generally, in quaternions, form a series of powers of 
 a quaternion^ such as 
 
 {r + q)\ {r+q)\ {r + q)\ kc, 
 (with the exception of the first), yet they are, generally, the co- 
 efficients ofx^, — , - — - &c., in the developement of a certain pro- 
 
 DUCT OF TWO exponentials, namely, the product e^^ e^i, if x 
 be a scalar. Thus, under this last condition, we may write, as 
 in the ordinary differential calculus, for any positive whole num- 
 ber n, if X be supposed to vanish after the differentiations, 
 
 d V n(n-\) 
 
 T- ) . e'^'" e-"^^ = r" + «r"" ' q + — ^ — - r^-'^q^ + . . + q"; 
 
550 ON QUATERNIONS. 
 
 although the second member of this formula is not, in quater- 
 nions, a general expansion for the power {r + qy. 
 
 542. A scalar w being always commutative in multiplication 
 with a vector p, the theorem of art. 540 gives the following gene- 
 ral decomposition of the function F into two factors, 
 
 Y{q)^¥{w + p) = YwYp = Y^q.'P Vq. 
 
 Here the factor FS^- is always a positive scalar (as appears from 
 the ordinary algebra of reals), and is greater or less than unity 
 according as Sq is positive or negative ; in fact, 
 
 FSq^e^i, Sq = \FSq, 
 
 the letter 1 being here used to denote a logarithm of the natural 
 or Napierian kind. On the other hand, because (V^')^^- (TVg')-, 
 the other factor FV^- is always a pure versor : for we have the 
 following scalar and vector parts of its developement, 
 
 SFVq = i-l(Tyqy+^-^^{Tyqy-ko. = cosTyq; 
 
 YFyq=VYq.[TYq-J-{Tyqy + kc.} = lJYq.smTVq; 
 
 whence 
 
 FV^= (cos +UV^ sin) TV^= (UV^)^'^''^^9; 
 so that 
 
 TFV^ = 1. 
 Hence also generally, 
 
 TF^=FS^; UF^-FV^; lTFg=Sg. 
 
 543. The function FNq is 2i periodic one, in the sense that 
 generally, 
 
 F(V^+i7rUV^)=UV^.FVg; 
 which gives 
 
 F(V^ + 7rUV^) = -FV(?. 
 
 In fact UV^- is commutative in multiplication with V^^, and 
 
 F (A ttU V^) = cos ^ + U V(7 sin ^ = U V^. 
 We have then, for any whole number ;?, 
 
LECTURE VII. 551 
 
 F (V^+ nirUYg) = (- 1)« FV^ ; 
 F(V^+2n7rUV^) = FV^. 
 
 We may therefore add or subtract, under the functional charac- 
 teristic F, any even multiple of rrUYq, without making any 
 change, and any odd multiple of the same vector, if we merely 
 change the sign of the result. But by these operations, we may 
 be considered as merely adding some even or odd multiple, posi- 
 tive or negative, of 7r to TVg'. We have also, 
 
 - ¥Yq = - cos TYq-VVq sin T V^ = (cos -U V^ sin) (tt - TYq). 
 
 If, then, any proposed versor, Ur, have been in any manner 
 found, or put, under the form 
 
 Vr=FYq, 
 
 and if the vector Yq do not already satisfy the condition TYq 
 :j> TT, we can always prepare or transform the proposed expres- 
 sion, so as to oblige that condition to be satisfied by a certain new 
 and substituted vector, Yq ; namely, by subtracting tt a sufficient 
 number of times from TV^-, and then subtracting the remainder 
 from TT, if this number have been odd. In this manner we shall 
 have, 
 
 Ur = F\Y, TV(?':j>7r, UV^' = + UV^; 
 
 the upper or the lower sign being taken, according as we have 
 been obliged to assume 
 
 T V^'= TV^ - 2«7r, or = (2^ + 1) tt - TV^. 
 
 And in this prepared state, if not in the proposed one, we are 
 allowed by the foregoing article, and by the definition of the 
 angle of a quaternion assigned in art. 148, combined with the 
 usual reference to a well-known theoretical unit of angle (which 
 gives, as usual, 180° = 7r = 3'14159), to write 
 
 zr = zU/--zFV^'=TV^. 
 
 544. From the periodical character of F V^', which allows us 
 (as we have just seen) to write 
 
 Ur = FV^=FV^, 
 
 without Yq and V^' being equal, it might seem that the inverse 
 
552 ON QUATERNIONS. 
 
 function^ F*^Ur, admits of more values than one, or indeed o* 
 infinitely many values, which would all equally well satisfy the 
 functional equation, 
 
 FF-iUr=Ur. 
 
 And this is true : but for this very reason, I propose to include 
 by definition, m the signification of this inverse function, F'S 
 something more than merely its being obliged to verify the last 
 written equation. And the last article sufficiently explains my 
 motives for making the additional condition to be, 
 
 TF-iU/->7r. 
 
 For thus we may write generally, without violating that definite 
 signification of the symbol L q which was agreed on in the Fourth 
 Lecture, the equation, 
 
 ^r = zUr=TF-iUr. 
 
 Under the same conditions we shall have also, definitely, 
 
 UF-iUr = UVr=Ax.r; 
 
 and therefore (compare 542), 
 
 VF-^r=F-iUr = UVr.zr; SF-ir= F'^Tr^ ITr; 
 
 and finally, 
 
 F-ir = lTr + UVr.zr. 
 
 It will be remembered that the tensor of a quaternion is never 
 negative in this calculus ; and therefore that the recent expression 
 for L r will never give a negative angle : a condition which was 
 in fact required, by the definition in 148. 
 
 545. The function, ¥~^r might be called the imponential 
 of r, because it is the inverse of the exponential functio7i F (or at 
 least an inverse thereof) ; but it may be simpler, and more con- 
 formable to analogy, to call it still, as in 542, the logarithm, 
 or more fully the natural logarithm, of the subject on which it 
 operates, although that subject of operation is now a quaternion; 
 and to write generally, 
 
 F"'7'=logr; or simply, F"^;- = b\ 
 
LECTURE VII. 553 
 
 With this extended notation, the equations of the last article will 
 give, 
 
 Slr=lT/-; UVlr = UVr; TVlr = Zr; 
 
 and thus the logarithm of a quaternion comes to receive (by 
 the foregoing conventions) the following generally definite value : 
 
 \r=\Tr + \]Yr . Lr; 
 
 where it may be observed that 
 
 UVr.Lr=V\r=Wr; and that lr= ITr + lUr. 
 
 Indeed the only exception to the definiteness of this expression 
 may be said to be the case where the quaternion r degenerates 
 into a negative scalai', in which case (as in 149, &c.), its angle 
 is = IT, and its axis has an indeterminate direction ; so that if x be 
 any positive scalar, and r = - x, we have, as in older theories, the 
 formula : 
 
 \r=\(-x) = \x + Trx/{-l): 
 
 but the symbol \/- 1 is here, as in arts. 167, &c., to be interpre- 
 ted as denoting an arbitrary unit-line in space. I am of course 
 aware that logarithms are by many writers interpreted as having 
 generally a certain degree of i7idetermination ; but it has been 
 my object, in the present theory, to preclude, so far as I could, 
 that indeterminateness by definition : as has been done, in some 
 analogous questions respecting ordinary imaginary expressions, 
 by M. Cauchy and Professor De Morgan. And I scarcely count 
 the logarithm of zero as a case of indetermination, because its 
 scalar part is negative infinity, 
 
 SIO = - 00, 
 
 although no doubt its vector part is undetermined. 
 
 546. To exemplify the convenience of this generally definite 
 interpretation of a logarithm, I resume the consideration of 
 powers with scalar exponents, which were discussed in the Fourth 
 Lecture. You will find that we may now write, with the recent 
 signification of the symbols, for any such power, as in algebra, 
 the expression : 
 
554 ON QUATERNIONS. 
 
 In fact 
 
 therefore 
 
 T.e'i'-=e'i^'-=(Try=T.r^ 
 and 
 
 U. e*''" = (cos + U Vr . sin) {t L r) =U.r% 
 
 with that definite meaning of such a power a sr*or g*, which was 
 assigned in the Fourth Lecture. Again, if we treat the positive 
 number e (more often perhaps now written a) as a quaternion 
 with a null angle, and submit it as such to the foregoing gene- 
 ral rules, we shall have z e = 0, le = F'^e = 1 ; and therefore the 
 equation ei = Fg, may now be written as follows : 
 
 Thus all the powers hitherto considered by us are seen to \>q con- 
 sistent with the first formula of the present article : ahd if we now 
 extend that formula by definition^ so as to write, generally ^^ 
 
 we shall hereby violate no condition already established ; and 
 shall be able to interpret every such symbol as q'', or to assign, 
 generally, a definite signification to a power, even when both 
 exponent and base are quaternions. 
 
 547. As an example, if it be required to interpret the symbol 
 f, we have 
 
 T;"=l, ^i = |, UV/'^y, an d therefore l;"=l7r/; 
 whence the required value of the power is, 
 
 ji _ gilj = f,\wij = gjTr/t ^ ^_ 
 
 More generally, if a and j3 be any two rectangular vector units, 
 then 
 
 la = -a, and a/^ = 62 = pa. 
 Again, 
 
 IT 
 
 jt _ Qi\i — Q~~i =jj = ^A_ 
 
 But the results will not usually be so simple as these : and it may 
 suffice to remark here that 
 
LECTURE VII. 555 
 
 U. ^'•= F (z 5 .V. rVWq + Vr . IT^). 
 
 It once occurred to me that the logarithm of the tensor of a qua- 
 ternion might be conveniently called the mensor of that quater- 
 nion, and denoted by the symbol, 
 
 Mg=lTg; 
 
 but I do not desire to introduce any unnecessary innovation of 
 language, nor to complicate the calculations with any new sign, 
 which does not appear to me to be of real and extensive utility. 
 The recent use of the notations F^-, F"^^', for e?, \q, has been 
 merely for temporary convenience. 
 
 548. We have seen (in art. 545) that the logarithm of the 
 versor of a quaternion, which is also the vector of the logarithm 
 of the same quaternion, is the product of axis and angle ; it 
 is therefore the representative arc (namely, by 216, a certain 
 portion of a great circle of the unit-sphere^, rectified, and 
 placed perpendicularly to the plane of the arc. The 
 same construction for the logarithm of the versor of a 
 quaternion has been suggested to me by a certain process of de- 
 finite integration, on which I cannot enter here. I must also 
 suppress all notice in this place, of the developements of loga- 
 rithms of quaternions by series, and of their other transforma- 
 tions. 
 
 549. But it may be proper here to shew how, on the fore- 
 going principles, a definite interpretation may be assigned to such 
 a symbol as log^. 5^; or to the logarithm of a given quaternion, 
 q, referred to a given quaternion base, q. For this purpose, 
 I propose to adopt from algebra the formula, 
 
 log5.g'=l^'-r-lg; 
 
 retaining still the recent and definite significations of the sym- 
 bols Ig, \(^. In fact, if we call this quotient r, we shall have 
 
 qr = e'^i = e"i' = q . 
 
 Indeed it is true that this equation, (f = q', is satisfied, not only 
 by the recent value of the exponent, r, but also by all those other 
 exponoits, r', which are included in the formula, 
 
556 ON QUATERNIONS. 
 
 For if we substitute any such value for r' {n being any whole 
 number), we shall have 
 
 qt' _ grig _ gig'+2W7ruvg' — gig' _ q' 
 
 as before. And if we should content ourselves with establishing 
 
 the formula log. 5^=-, where e^=q, e^'=q, without otherwise 
 
 restricting the exponents s and s\ we should thus obtain, as the 
 general value for the logarithm of a quaternion q, to a quater- 
 nion base q^ an expression of the form, 
 
 "' ,_\q^-2n''iT \JYq 
 
 involving a double indeterminafion, and introducing a pair of ar- 
 bitrary integers, as in the results of Graves and Ohm, respecting 
 the general logarithm of an ordinary imaginary expression re- 
 ferred to an ordinary but imaginary base. I prefer, however, in 
 this calculus, to exclude this indetermination by definition, as in 
 some earlier and easier questions : and therefore Siher fixing (as 
 in 545) the signification of the natural logarithms, \q, \q\ 1 pro- 
 pose to write definitely, as above, 
 
 log5.g'=l^-f. ]q. 
 
 Comparing the two notations* we might also write, 
 
 o 
 
 I0g5.g'=l0gg.^'. 
 
 o 
 
 550. If we adopt as definitions the developements, 
 
 co8^ = l-| + ^^^^^ -&c.; sing=5r_-i_ + &c.; 
 
 and observe that 
 
 -q^={\JYqyq'={qWq)\ 
 
 because q is commutative as a factor with UV5'; we shall easily 
 find that whatever quaternion q may be, the two following ex- 
 pressions hold good, with the recent meaning of the function F : 
 
LECTURE VII, 557 
 
 2 COS ^ = F (^U Vg) + F (- ^U V^) ; 
 
 2smq.Wq=F(q.\jyq)-F{-gljyq). 
 
 These finite expressions suflSce to define the sine and cosine of a 
 quaternion : and on the same plan we may write, as a definition 
 of the tangent of a quaternion, the formula, 
 
 ^,,, F(^UV^)-F(-^UVg) 
 
 with other analogous expressions, on which it seems needless 
 here to delay. 
 
 551. When a quaternion function {fq^, of a sought quater- 
 nion {q), has a given form (f), and a given value (r), so that we 
 have the quaternion equation, 
 
 fq = r, 
 
 we can always breah up, or at least conceive as broken up, the 
 one proposed equation in quaternions, into four equations of an 
 ordinary algebraical kind, involving the four sought constituents^ 
 w, x^y, z, of the sought quaternion q : and may then eliminate, 
 or at least conceive as eliminated, the three scalar co-ordinates, 
 X, y, z, between those four equations, in such a way as to con- 
 duct to one final and scalar equation, involving the one sought 
 scalar, w, or Sq : after resolving which (if we could in all cases 
 do so), we might then proceed to determine x, y, z, and therefore 
 finally q. Or we may conceive that after forming the two sepa- 
 rate equations, 
 
 Sfq = Sr,Yfq = Yr, 
 
 we deduce p = Vq from the second equation, in terms of iv = Sq, 
 and substitute its expression in the first equation, which is then 
 to be resolved with respect to w. Or the first equation may be 
 supposed to he previously resolved for w, and the value of 2^ sub- 
 stituted in the second equation, which thus becomes a vector for- 
 mula, involving one sought vector p. And instead of the single 
 vector equation \fq = Vr, we may, either before or after the eli- 
 mination of w, employ the following system of three scalar 
 equatiofis, 
 
 S . Kfq = S . Kr ; S . \fq = S . Xr ; S . fifq =S . fxr ; 
 
558 ON QUATERNIONS. 
 
 when K, A, n may denote any three assumed vectors, which do not 
 vanish, and are not coplanar with each other. 
 
 552. To fix more fully our conceptions, let the quaternion 
 function /</ be supposed to consist of some finite number of terms, 
 in each of which the sought quaternion q shall enter only as a 
 factor, some finite number of times repeated; and let the highest 
 number of those times be n. The equation y^^ = t* may then be 
 called an equation of the w"* degree in quaternions. For example, 
 
 bqa + b'qa! + b"qa" + &c. = c, or S . bqa = c, 
 
 will be an equation of the first degree, or, as we may agree to 
 call it, from analogy, a linear equation in quaternions, what- 
 ever given quaternions may be denoted by a, a', a!', . . b, b', b", . . 
 and c. Again the formula 
 
 S . «2 qo,\ qa + ^ .biqb- c, 
 or more fully, 
 
 «2 gcii qa + a'2 qa'i qa! + d\ qd\ qa" + . . . + biqb+ b'l qV + b"i qb" 
 
 + . . . = c, 
 
 will represent an equation of the second degree, or a qua- 
 dratic EQUATION in quaternions : and soforth. 
 
 553. Now, upon substituting, on the plan of 551, in that 
 form of the equation of the w"' degree which is described in the 
 last article, for the sought quaternion q, its quadrinomial value 
 w + ix +jy + kz, with analogous values for the given quaternions, 
 a, b, c, &c., we shall evidently break up that 07ie proposed equa- 
 tion into four others, between the four sought scalars, w, x, y, z, 
 and some number of given scalars, which will not generally be 
 identical equations, and will in general be each oi the proposed 
 (n*^) degree. Elimination between them will therefore generally 
 conduct, by known principles of ordinary algebra, to an algebraic 
 equation in w, which has n^ for the exponent of its degree: and 
 such will generally be the exponent also of the degree of the final 
 equation in any one of the three other required scalars, x, y, z. 
 Thus a linear equation in quaternions has generally only one root; 
 but a quadratic equation may be expected to have generally six- 
 teen roots (real or imaginary): a cubic equation in quaternions 
 must, on the same plan, be supposed to have in general eighty-one 
 
LECTURE VII. 559 
 
 quaternion roots : and so on. It is, however, as we shall see, 
 quite possible to meet v/'xih particular equations of these degrees 
 which shall \\di\e fewer quaternion roots, or at least shall appear 
 to have fewer, in consequence of the absence of certain terms in 
 the component scalar equations. Thus the particular class of 
 quadratic equations in quaternions, which is of the form 
 
 q^ = qa + h, 
 
 and which hitherto I have chiefly studied, appears to have only 
 six roots (two real and four imaginary), as will be soon ex- 
 plained : but probably it should be said that the ten missing roots 
 are, for this particular equation, infinite. 
 
 554. Confining ourselves for the moment to linear equations, 
 or equations of the^r-s^ degree, let us resume the general type of 
 such equations assigned in art. 552, namely the form, 
 
 S . bqa = c ; 
 
 where a, b, a, b', . . . and c are given quaternions, but 5' is a 
 sought quaternion. Taking separately the scalar and vector 
 parts, we obtain the two following equations : 
 
 wh + S.r)'p=Sc;wti+Y.(h'+0)p+'2^{yaS.bp + YbS.ap)=yc; 
 
 where 
 
 w=Sq,p^Yq; h='2S .ba,rj = '2Y . ba, r]'='2Y . ab; 
 A' = S (S6 Sa- S . V6 V«)- SS . & K«; 0= S (V6 Sa- SbYa) ; 
 
 in deducing which expression for Yc, we have employed the for- 
 mula (520), with which it is important to be familiar, 
 V . yjSa = 7 S . j3a - j3 S . ya + aS . jSy. 
 
 Eliminating w, and making for abridgment, 
 
 h (Ji + 9) = r, KYc - ?7 Sc = (7, 
 
 we find an equation of the form, 
 
 S./3S.ajo + V.r;o = o-, 
 
 where a, a', . . . /3, /3', . • and a are given vectors, and /• is a given 
 quaternion, but p is a sought vector : and this appears to be 
 the most general possible form for a linear and vector equa- 
 tion (or to include all possible forms of such an equation). We 
 
560 ON QUATERNIONS. 
 
 shall now proceed to resolve it, by means of that general method 
 which was alluded to at the end of article 513. 
 
 555. Operating by S . A, where X is an arbitrary vector, we 
 obtain the result : 
 
 SXc7 = S.X>, if X' = S.aS.j3X + V.5X, ands=Kr. 
 
 In like manner, 
 
 S . juor = S . fxp, if ju' = S . aS . j3/i + V . Sjx. 
 
 Hence, if we so assume X and fx as to satisfy the condition 
 
 V.X^ = (7, 
 
 we shall have 
 
 S . X'/> = 0, S . |u' jo = 0, and mp = Y . Xy, 
 
 where m is some scalar coefficient. Now on developing this last 
 vector of a product, and replacing V . X^u by a, we find, 
 
 V (aa'S . j3XS . j3> + a'aS . /3'XS . /3^) = V . aa'S . /3'/3(T ; 
 
 V (a V . 5^S . jSX + V . 5X . a S . /3^) = V . aV . 5 V . i3(7 ; 
 Y{Y .sX.Y .sii) = ^sY.sa-YsS . sa ; 
 
 which last transformation maybe obtained in various ways, serv- 
 ing as useful exercises in this calculus. For example, we may 
 observe that generally, for any two quaternions q and r, we have 
 
 rq~qr = 2V . YrYq ; 
 and that 
 
 i (sX .Sfx-Sfi. s\) = ^s {Xsfx - fxsX) = I5 (S + V) {\sfx - nsX) ; 
 where (because a- = V.Xju), 
 
 ^S (X*^ - fXsX) = -^S . 5 {fxX - Xfx) = ~S .S(7, 
 
 ^Y (Xsfi - fxsX) = ^Y .X(s+ Ks) iuL = (tSs; 
 so that 
 
 Y(Y .sX.Y.Sfx) = s{a.Ss-S.scT) = Y.scTSs-YsS.s<T, 
 as above. Or we might write, 
 
 Y .sX = sX-S . sX, Y .Sfx = S . Sfx-K .s/x= S .sjuL+fx Ks, 
 and observe that 
 
 V. sXfi Ks = S(tKs, because V .sKs = 0, S .S(tKs = 0; 
 
LECTURE VII. 561 
 
 and that 
 V .sXS.Sfi-y. fxKsS .s\=V.s(XS,Sfx-nS.sX) = sY.gYs, 
 
 it being unnecessary to prefix the sign V to this last expression. 
 For thus the proposed expression would be found to become, 
 
 s((TKs + Y.aYs)^s{S.(TKs+Y.<j{Ks+Ys)}=s((TSs-S.s<T), 
 
 and therefore equal to the expression already written. We have, 
 therefore, by summing the terms, and changing s to Kr, the 
 formula : 
 
 p = m- 1 V . Xy = m-i { SV . aa' S . j3'i3(7 + SV . a V ( V . j3(r . r) 
 
 + SrV.(7r-VrS.(7r); 
 
 and it only remains to determine the scalar coefficient ni, in terms 
 of a, a', . . /3, /3', . . and r, by substituting this expression for p 
 in the linear equation of the foregoing article, namely, 
 
 S . /3S . ap + V . rp = 0-. 
 
 556. Effecting this substitution, with analogous reductions, 
 and employing the first or both of the two identities of article 5 10, 
 of which the latter may be proved to be correct by operating on 
 it separately and successively with the three characteristics S . a, 
 S . j3, S . ■y, the four following transformations are obtained, of 
 which it will be found an instructive exercise to examine and to 
 prove the validity : 
 
 I ., j3S . aaa"S . j3"/3'(T + jS'S . a'aa"S . j3"^(T + (5"S . a'aaS . (3'(5ct 
 
 = (tS . aa'a'S ■ |3"/3'j3 ; 
 II., jSS . aa'Y (V . j3'(T . r) + j3'S . a'aY {Y . (5<t . r) + Y . rY . 
 
 aa'S . (5'(5(T = aS (rV .aa'.Y. jS'jS) ; 
 III., j3 (SrS .acrr-S. arS . ar) + Y . rV . aV (V . j3(t . r) = 
 
 (7 (SrS . raj3 - S . raS . r(3) ; and 
 IV., V . r(SrY . rjr - VrS . ar) = (rSrTr^ 
 
 The coefficient m has, therefore, the following value: 
 
 7w = S (S . aaVS . j3"j3'i3) + SS (rV .aa .Y . /3'j3) + S;-SS . raj3 
 -S(S.raS.r/3) + SrTr^ 
 
 And the recent transformations suffice to prove^ a posteriori, or 
 synthetically, that with this value of m, the linear equation, 
 
 2 o 
 
562 ON QUATERNIONS. 
 
 S.j3S.ajo + V.r|0 = (T, 
 
 of article 554, is, in fact, satisfied by the expression assigned for 
 p in art. 655^ as the analysis of the last-cited article had given 
 us reason to foresee that no other value of p (generally speaking) 
 could satisfy the same linear equation. 
 
 557. It is important to attend, in all such formulae as these, 
 to the notation of points employed ; in virtue of which, we have, 
 for example, in the foregoing article, 
 
 Y.rY. aa'S . /3'/3<T = V [rV {aa'S (|3'j3<t) }] : 
 
 while such symbols as Sr, Vr, Kr, Tr, Ur, &c., when thus writ- 
 ten without poi7its, are treated, in their combination with others 
 or among themselves, as if they were single letters ; so that, for 
 instance, in the last article, the expression SrY . ar does not 
 mean S {rV {(tv)]^ but S?' x V (o-r) : also Sr^ denotes (Sy)-, while 
 S (r^) may be written as S . r^. (See the remarks made at the end 
 of art. 455 ; and the examples of transformation in art'. 504.) Still, 
 from the properties of scalar s^ this plan of notation allows us to 
 write, 
 S . raS . rj3 = S {ra) x S (rj3), and V . raS . r/3 = V (ra) x S {r^) : 
 
 though not, in general, 
 
 S . raV . r/3 = S (ra) x V (rjS), nor V . raV . r)3 = V (ra) x V (rj3). 
 
 A very experienced calculator might, perhaps, safely trust to his 
 recollection of his own meaning, in any particular question, and 
 dispense with some of these precautions : but I do not advise the 
 attempt. The mixture of multiplication with other operations of 
 this calculus might in that case produce a confusion, against 
 which it is prudent to guard, by using a notation exempt from 
 ambiguity, such as I think the one above proposed will be found 
 in practice to be. It is perhaps unnecessary to state, that in the 
 sum S S (rV. aa'. V.j3'/3), each combination of two pairs of vec- 
 tors, a, j3, and a, j3', is to be only once employed ; and that, in 
 like manner, each combination of three such pairs is to be only 
 taken once, in another sum which enters into the expression 
 of m. 
 
 558. To exemplify the general process above given, for the 
 
LECTURE VII. 563 
 
 solution of a linear and vector equation^ let us resume the equa- 
 tion of art. 516, under the form, 
 
 V . /3|oa = (7 ; or, j3S . a/o + aS . /3/0 — /oS . aj3 = 0-. 
 Here 
 
 a'=/3, /3' = a; a"= . . =i3"= . . =0; r = -S .a/3; 
 
 and the general formula of article bbb becomes 
 
 »wp = V . ajSS . a/3a - V (aV . /3(r + jSV . aa) S . ajS + a (S . a/3)^ 
 
 = V . a/3S . a/3(T + (aS . /3(T + /3 S . a(7 - ffS . ajS) S . ajS 
 
 = V . iSaS . a(7^ + V . affjSS . |3a = ^a^/S^ (aaa-^ + jS^r/B '0, 
 
 because in general, 
 
 V^Sr + Vz-S^ = i (^^ - KgKr), 
 and 
 
 K . pa = api K . a(rj3 = — jScra. 
 
 But also in the general formula of 556, we have now, 
 
 SS . aa'a'S . /3"i3'/3 = ; SS (r-V . aa .V . jS'/B) = - S . aj3 (V. ajS)^ ; 
 SrSS . rajS = 2 (S . ajS)^ ; S (S . raS . r/3) = ; SrTr^ = - (S . a/3)^; 
 therefore 
 W = S .«/3 {(S.ai3)''-(V . aj3)^) =a2i3\S .aj3 = ia^/3Ha/3 + /3a). 
 Thus in the present question, our general method gives, 
 
 ^ aj3 + /3a ' 
 
 which may be verified by comparison with the result of art. 516. 
 
 As another verification, we may observe that this expression for 
 
 p gives 
 
 ,j (7aj3 + ajScr 
 
 "/'P ^ — «T73 — 5 
 ap + pa 
 
 and that 
 
 V (aajS + ajSff) = V . (t (a/3 + jSa) = a (a/3 + /3a); 
 
 so that 
 
 V. /3joa = V . ajo/3 = <T, as was required. 
 
 559. Again, let each a and j3 vanish, in the general form of 
 recent articles, so that the linear equation becomes simply, 
 
 V . rp = <T. 
 2o2 
 
564 ON QUATERNIONS. 
 
 The general solution gives then, 
 
 pSrTr'=SrY.cTr-YrS. <rr; 
 
 or, making Sr = g, Yr = y, 
 
 9 {9^ - y^) p = 9^(T~ gy ' y<T -yS . y(T ', 
 
 which agrees with a result already obtained in art. 514, where X 
 and q were written instead of tr and r. 
 
 560. As an example of the general process of art. 554, let 
 there be proposed the linear equation in quaternions, 
 
 bq + qb=c. 
 Here 
 
 a=l,6'=l, a'=b, a"= . .=b"= . .=0, 
 h' = h = 2Sb, n' = r, = 2Yb, e = ; 
 
 and the two equations between which w is to be eliminated 
 
 become, 
 
 wSb+S.bp = lSc, wyb + pSb = ^yc, 
 
 giving 
 
 V, YcSb-YbSc 
 
 p-ao .ap = (T, where a = -b, a = t^tt . 
 
 ' s 2 00^ 
 
 Comparing this last linear and vector equation in p with the ge- 
 neral form of art. 554, we have 
 
 (5 = -a, a = . .=/3'= . . =0, r=l; 
 
 and therefore, by 555, 556, 
 
 p = m-^ ((7 - a V . aa) = (1 - a^)-i ((7 - a V . Off) ; 
 
 an expression forp, which in fact is seen to satisfy the last linear 
 equation, and which gives, 
 
 2p Sb ( Sb' - Yb') = Sb{YcSb-YbSc)- YbY. YbYc 
 = {Sb'- Yb') Yc - Yb {SbSc - S . YbYc) ; 
 
 or because Sb' - Yb' = Tb' = bKb, and SbSc - S . YbYc =S.cKb, 
 
 2pSb = Yc-YbS.cb-K 
 Hence 
 
 2SbS . bp = S.YbYc - Yb'S . cb-^ ; 
 2wSb' ^ SbSc -S .YbYc + Yb'S.cb:' = {Tb'+Yb')S .cb-\ 
 
 and finally, 
 
LECTURE VII. 565 
 
 iv=^S.cb-\ because Tb^+Yb''== Sb\ 
 
 Thus the solution of the proposed equation hq + qb = c (where 9 
 = w + p) may be thus written : 
 
 2gSb^Yc+KbS.cb-\ 
 Accordingly, 
 
 bYc + Ycb = 2SbYc + 2S . YcYb=2€Sb-2S . cKb; 
 and 
 
 (bKb+Kbb) S.cb-'=2Tb'S.cb-' = 2S . cKb; 
 
 so that the expression found for the quaternion q does, in fact, 
 satisfy the linear equation proposed. 
 
 561. Or we might have begun (compare the general remarks 
 of art. 551) by eliminating p instead of w, between the two equa- 
 tions, 
 
 w^b +S.bp = iSc, wYb + pSb = iYc ; 
 
 and thus have found, more rapidly, 
 
 2wTb'=SbSc-S.YbYc = S.cKb,w = ^S.cb-'; 
 
 after which we might at once have inferred that, as above, the 
 linear equation bq + qb = c gives, 
 
 2pSb = Yc-YbS.cb-', 2qSb=Yc+KbS.cb-K 
 
 562, When an equation is so simple as the one last treated, 
 less general methods may often be conveniently employed. As 
 an example, let us take this other linear equation, 
 
 aq\ qb-c, 
 
 where abc are three given quaternions, and §' is a sought one. 
 Multiplying separately by K«, and into b, it gives, 
 
 Kaaq + }Laqb = K.ac ; aqb + q¥ = cb ; 
 
 therefore adding and observing that K«a = Ka . a = Ta% Ka + a 
 = 2Sa, we find, after a division, 
 
 Kac + cb 
 
 And if we here change a to b, we fall back on the equation bq + 
 qb = c, and obtain, as a new form of its solution, the expression. 
 
566 ON QUATERNIONS. 
 
 q = ^^^^, because T6^ + 6^ + 26S6 = b{Kb + b + 2S6) = 46S6. 
 ^ 46S6 
 
 Accordingly, 
 
 i(K^>c + c6) = cS5 + V. Vc V5 =Vc5 + S .cK6 = ( Vc + KiS . c6-i)5 ; 
 
 so that this article, like the two foregoing ones, gives 
 
 2q^h = Vc + K6S . cb-\ if bq-^qb = c. 
 Or, again, we might infer from this last linear equation, that 
 
 bc-cb = h'q - ^62 = 2 V (V . 6^ V^) = 4S6V . V^V^, 
 and therefore that 
 
 whence 2qbSb = cSb+Y . YcYb = ^(cb+ Kbc), as above. And 
 other modes of solution, and forms of expression, may be assigned 
 with nearly equal ease. Of course it is only practice which can 
 render you expert in such transformations as these : of which, 
 however, the principles have all been stated already in the pre- 
 sent Course of Lectures. 
 
 563. The general linear and vector equation of article 554 
 may also be treated as follows. Making, as in 559, S*'=^, Vr 
 = y, and writing, for abridgment, 
 
 S.jSS .a,o + V.y/) = ^/), 
 
 ■where 0p is a new distributive and vector function of p, the equa- 
 tion to be solved becomes 
 
 <j)p + gp = C7, or more coiicisely, {(}> + g) p = (r; 
 
 and we are to seek the form of the following inverse function^ 
 
 P = {<!>+ 9)-' (T = ^p-'<r, iii^ = (j) + g. 
 
 Operating with ^, and making reductions analogous to those of 
 recent articles, we find, 
 
 ^/3 = p'+|oSS.aj3, ifp'= V. 7(0-SV.aV.j3(o; 
 0/>' = p" + p { S S (V . aa' . V . jSjS') + S S . ayjS + 7= } , 
 where p" = SV . aa'S . /B'jS/o - SV . a V . yV . jSp - yS . 7,0 ; 
 and finally 0p"= -np, if we write 
 « = SS . aa'aS . jSjS'iS" + SS (yV . ad .Y . (3(5') + SS . ayS . jSy. 
 
LECTURE VII. 567 
 
 If, then, we also write, 
 
 w'= SS (V . aa'Y . jS'jS) + SS . ajSy - 7^ w" = - SS . ajS, 
 we shall have, 
 
 (j)p = p'- n"p ; (pp = p" - n'p ; <pp" = -np; 
 and therefore, 
 
 0^p = p" - n'p - n"(pp, 0^jO =~-np~ n<pp - ti'^^p ; 
 
 or, abstracting from the operand vector p, 
 = n + n'(j) + n"(p^ + 0^ 
 
 564. Here, then, is a certain symbolic and cubic equa- 
 tion, which t\ie functional characteristic must satisfy : and it 
 is clear that the connected characteristic ^ (= ^ + ^) must satisfy 
 the connected cubic, 
 
 = i//^ - m"ip'^ + m'lp - m, 
 or 
 
 m\p' ^ = m - 7n"\li + ^^ ; 
 where 
 
 m, = g^— n'g"^ + n'g — n ; 
 
 m' = 3g^ - 2n"g + n ; 
 m"=3g-n". 
 
 And thus the proposed linear equation in p is resolved anew, by 
 the assigning of the sought^onw of the inverse function, ip'^ ; or 
 by shewing what the direct operations are, of which that inverse 
 operation is compounded. 
 
 565. The method of the two foregoing articles gives, 
 
 nip = w^^" 1(7 = {m - ni"\p + \p^) cr = a" - ga + g"cr, 
 where (by 563), 
 
 a" = {n + n"^ +<P')^ 
 
 = SV . aa'S . /3'i3tr - SV. aV. 7V. |3(t - yS . ycr ; 
 
 a = {fl" + (p)a = Y.jcx- SV. aV. JBct. 
 
 And accordingly these results agree exactly with those which are 
 obtained from the earlier expressions for tnp and for m, in articles 
 555, 556, when the quaternion r is expanded into g + y. 
 
 566. The recent results of our analysis, respecting the exist- 
 ence of a symbolic and cubic equation in ^, wiiere ipp = '2 . /3S . ap 
 + V. rp, admits of the following geometrical interpretation, which 
 
568 ON QUATERNIONS. 
 
 appears to me to furnish a somewhat remarkable and possibly 
 new THEOREM. ^^Ifby any one fixed mode of linear de- 
 formation (represented here by the operation ^) we pass from 
 a vai'iable vector p to another co-initial and dependent vector \Lp, 
 which may be called the First Derivative; if we then pass by 
 the same fixed mode of deformation, from this first to a Second 
 Derivative, ip^p; and thence, by still the same mode of change, to 
 a Third Derivative, \p^p; and if (by constructing a parallelepi- 
 pedon) we decompose the original line p into three othe?'s, in the 
 directions respectively of these three successive derivatives (or in 
 the opposite directions): then the ratio of each component ^o 
 the correspondijig derivative line, or the ratio of each projection to 
 the line on which it is projected, will be expressed by a constant 
 SCALAR (m'^m\ or -m'^in", or m"^), which depends only on the 
 mode of deformation (or on the Jbrm of the linear and vector 
 function ip), but not at all on the length, nor on the direction, of 
 the original and variable line p, thus operated upon." It is clear 
 that we should equally be permitted to decompose any other of 
 the four lines, p, ipp, ■ip'^p, ^p^p : and that we should still obtain, 
 from the cubic equation in ip, three constant scalar ratios. 
 
 567. If none of the given vectors a, j3, a', /3', . . 7, nor the 
 given scalar g, be infinite, then neither will any one of the three 
 scalar coefficients m, m, m",be so, in the cubic equation of art. 564 ; 
 and because ^//O = 0, ^//^O = 0, we shall have also the formula, 
 
 which will generally give 
 
 ;//-iO = 0; orp=0, ifi/.p=0. 
 
 There is, however, a remarkable exception (or class of excep- 
 tions) to this general result. For if the scalar g be so chosen as 
 to be A root of the cubic equation, 
 
 m = 0, or ^^ - ng"^ + ng - w = 0, 
 
 we shall then not be able to infer that \)i\e factor xp'^O vanishes, 
 from the fact oi the product yniP'^O vanishing; 'dnd values of p 
 different from zero, or, in other words, actual lines, instead of 
 null lines, may in this case satisfy the condition. 
 
LECTURE VII. 569 
 
 i//|O = 0, or (l>p = -gp. 
 
 In fact if we suppose that gi, g^, gz are three distinct scalars, any- 
 one of which, when substituted for g, satisfies the ordinary cubic 
 equation lately written, or renders m = 0, for some given system 
 of values of the vectors a, j3, a, /3', . . and y, and therefore for 
 some given form of ; and if, after assuming any arbitrary vec- 
 tor, 0-, we derive from it three others, pi, p-i, pz, by the formulae, 
 
 j02 = (t" - g%o' + 5^2 V, 
 
 jOs = a-" - gzd' + gia, 
 
 where a, a" are vectors derived from o-, by the formulae of article 
 b^b : we shall then have, by that article, 
 
 i/'ijOi = 1//2/O2 = ^3j03 = wio- = ; 
 where 
 
 In other words, for these three directions, pi, pn, /03, we have, 
 respectively, 
 
 ^Pl = - 9ipi 5 0/02 = -^2/02 ; 0jO3 = - ^3j03. 
 
 This opens a very interesting train of research, analogous to, and 
 including, several known investigations respecting the pinncipal 
 axes of a surface of the second order, and the axes of inertia of 
 a body, on which I cannot enter here. 
 
 568. Although, as already remarked in art. 477, it will not 
 be possible in this Course to do much more than allude to the 
 DIFFERENTIAL CALCULUS OF QUATERNIONS, yet I Cannot forcgo 
 the opportunity of giving here at least some general notion of the 
 connexion of that differential calculus, with such linear equations 
 in quaternions, as have been lately discussed. For this purpose, 
 it is necessary first to define the differential, Afq, of a func- 
 tion OF A quaternion ; and I do so by the following formula : 
 
 Afq = lim. n \f{q + - d^-) -fq ] ; 
 
 ra = 00 " 
 
 where q and Aq are any two proposed quaternions, and w is a po- 
 sitive whole number, which, as the formula expresses, is con- 
 ceived to increase without limit. In fact this formula is evidently 
 
570 ON QUATERNIONS. 
 
 true, in the ordinary differential calculus ; and because it does 
 not involve the commutative principle of multiplication, it is fit to 
 be extended, as a definition, to differentials of quaternion func- 
 tions. (Compare the calculation of d . ^, in art. 528.) 
 
 569. For example, let /5' = j^ Then the general definition 
 gives, for the differential of the square of a quaternion, the ex- 
 pression, 
 
 d . g'^ = lim. n { (g' + - ^qf- (f\ 
 n=(X) ri 
 
 = lim. {gAq + dqq + -\^q^) = qdq + dqq ', 
 
 n=<xi ^ 
 
 where d^- is treated as a simple symbol, or as if it were a single 
 
 letter, denoting an arbitrary quaternion ; so that the symbol Aqq 
 
 is interpreted as being equivalent to this other and fuller symbol, 
 
 Aqy. q: while d^'^ denotes (d^)-. In like manner, the definition 
 
 gives, for the differential of the cube of a quaternion, this other 
 
 expression, 
 
 d . q^= q'^dq + qdqq + dqq^. 
 
 And similarly for the difi'erentials oi other powers of quaternions, 
 with whole and positive exponents. 
 
 570. Again, if a, b, c, . . . be treated as constant quaternions 
 independent of q, so that da = db = dc = 0, then d. aq = adq ; d . qb 
 = dqb ; d . aqb = adqb ; d . aqbqc = aqbdqc + adqbqc, &c. : the only 
 distinction in such cases between these results and those of the 
 ordinary differential calculus, being that each quaternion factor 
 is to be differentiated in its own place (or as we might say, in 
 situ); commutation of factors heing here {as elsewhere m this 
 calculus) not generally allowed. 
 
 571. As one other example of this sort of differentiation, let 
 us seek the differential of the reciprocal of a quaternion, or let 
 us suppose ^5' = 9" ^ Here, 
 
 f{q'^r)-fq={q^r)-^ - q-^ 
 = (q + ry' Iq - {q + r)}q-' = - (q -^ r)'' rq-'; 
 therefore, by the definition in art. 568, 
 
 d . q'^ = -Mm. (q + - dq)'^ dqq'^ = - q'^ dqq'^ ; 
 
 n = CO n 
 
 a result which I have often found useful. 
 
LECTURE VII, 571 
 
 572. It is easy to shew that if we suppose Tr <Tq, we shall 
 have the following developement, in a converging series, for the 
 reciprocal of the sum of two quaternions : 
 
 (M + 0"^ = 9.'^ ~ 9'^^9~^ + q'^rq'^rq'^ - &c. ; 
 in fact 
 
 q (q + r)-^ = (1 + rq-^)-^ = I -rq-^ + (rq~^y - {rq-^y + &c. ; 
 
 the convergence of this last series (in the case proposed) being 
 proved almost as easily as in ordinary algebra, with the help of 
 the principle established in art. 538, respecting the tensor of a 
 sum. Here, then, we have an example of the truth of the follow- 
 ing theorem, which can generally be shewn to hold good for qua- 
 ternions, as well as for algebra, in virtue of the definition recently 
 assigned : " whenever the function y(g' + dg') can be developed in 
 a series, involving terms or parts of successively higher and 
 higher dimensions, with respect to the proposed differential dq, 
 the part of the developement which is of the first dimension, 
 with respect to it, is the required differential, ^fq, of the ^xo- 
 i^o%eA function, fq." Indeed, it has not been uncommon, in other 
 works, to propose this result, or a result of this form, as a defini- 
 tion^ rather than as a theorem. But there are many cases, in which 
 the definition (568) of the differential of a function of a quater- 
 nion can be more easily applied^ than the developement of the 
 function can be found. A case of this sort will after a while 
 be pointed out. I have also other reasons for preferring my own 
 definition. 
 
 573. Meanwhile I may state that the theorem or Series of 
 Taylor may he extended to quaternions (with analogous cases of 
 apparent ^zVwT'e), under the form: 
 
 or more concisely and symbolically, 
 
 f(q+dq)=e^fql 
 
 dfq denoting here that value for ddfq which is obtained by treat- 
 ing dq as constant. For example, xifq-cf, then, by 569, 
 
 dfq = q^q + dqq, d~fq = ^dq^, d^fq = 0, &c., 
 
572 ON QUATERNIONS. 
 
 and 
 
 Again, the value of d . g-^, in the same article 569, gives 
 
 ^d^ . (^ - qAq^ + diqqAq + Aq'^q, ^—5 d? . q^ = Aq^, 
 and 
 {q + Aqy =q^+ (q^dq + qdqq + Aqq~) + {qdq~ + dqqdq + dq^q) + dq^. 
 
 In like manner, by 571, 
 
 |d2 . g-i = + q-^dqq-^dqq-^i 
 
 1 
 ^-^ d^ .q-^ = - q-^dqq-^ dqq-' dqq-\ &c. ; 
 
 and the developement of (g + r)'^, which was given in art. 572, 
 
 might in this way be reproduced. 
 
 574. When a quaternion r'ls, treated as 2i function of a scalar 
 
 (f, r=ft^ then the general definition gives a result of the usual 
 
 form, 
 
 dr=^dft=ff .dt, 
 
 dt appearing here as a simple^c^or (of the usual kind), with a 
 coefficient ft, which may be called (as usual) the derived func- 
 tion, because the differential d^ is here supposed. to be a scalar, 
 and, as such, commutative in multiplication. In particular if a 
 vector (p) be regarded as a given function (0^) of a scalar vana- 
 hle if), so that the extremity of p describes (generally) a given 
 curve in space while the value of t varies, we have an expression 
 
 of the form, 
 
 dp = d^^ = (p't . d^ = p'dt, 
 
 where ^t or p is a 7iew vector, tangential to the curve at the 
 extremity of p, or parallel to such a tangent, and having its 
 length equal to unity, \{ t denote the length of the arc of the 
 curve, measured from some fixed point thereon. In mechanics, 
 if t denote the time, in any motion of a point in space, p may be 
 named the variable vector of position, and p may be called 
 the VECTOR OF VELOCITY ; and when, by another differentiation, 
 we obtain a new result, of the form, 
 
 dp =(^'t . dt = p" dt, 
 
LECTURE VII. 573 
 
 then the new vector p may be said to be the vector of acce- 
 leration. In geometry, if t be still the arc of a curve, p may 
 
 be called the vector of curvature : for p — y, can be shewn to 
 
 P 
 be then the vector of the centre of the osculating circle. 
 
 575. When the equation of a surface is expressed, as in 507, 
 under the form, 
 
 fp = 0, or^ = const., 
 
 where^ is a given scalar function of a variable vector p, we may 
 always, by cyclical permutation (512) under the sign S, express 
 the differential of this function under the form : 
 
 &fp = 2 S . vdp ; 
 
 and if, by a suitable use of an arbitrary scalar coefBcient, we 
 oblige the new vector v to satisfy the condition (compare 474), 
 
 S . V|0 = 1 , 
 
 then, by reasonings similar to those of art. 481, it may be shewn 
 that v"^ represents, in length and in direction, the perpendicular 
 let fall from the origin of vectors on the tangent plane to the sur- 
 face, which is drawn at the extremity of p : and therefore that 
 (in the sense of the last-cited article) the vector v itself may be 
 called the vector of proximity, because it represents the near- 
 ness of the surface, or of its element, to the origin. 
 
 576. Without restricting v to satisfy the equation S . vp = 1, 
 if we merely choose it so as to give 
 
 S . vdp = 0, 
 
 as the diflferentiated equation of the surface, v will still denote a 
 normal vector; and general equations for classes of sur- 
 faces may be formed by the help of this symbol. Thus an ar- 
 bitrary conical surface, with its vertex at the origin, maybe 
 
 denoted by the equation 
 
 S . VjO = ; 
 
 because, for such a surface, vA_p. For an arbitrary cylindric 
 surface, with its generatrices parallel to a, we have v± a; and 
 ihe equation of this family of surfaces is, therefore. 
 
574 ON QUATERNIONS. 
 
 For an arbitrary surface of revolution, with the line /3 
 from the origin as axis, we have the following general equation 
 (because v \\\p, j3), 
 
 S./3vp = 0. 
 
 Now in the problems of forming and transforming such general 
 equations of surfaces as these, so as to prove, for example, that 
 the last-written equation agrees with the formula, 
 
 of article 440, we have the germs of a future Calculus of 
 Partial Differentials in Quaternions, and the indications 
 of future researches, analogous to those of Monge. 
 
 577. To exemplify the possibility of such transformations, 
 let the scalar and vector of the quaternion /o/3"^ be denoted thus, 
 
 so that the formula of 440 assumes the form 
 
 T(7=/5, or (7'^ + (/5)2 = 0. 
 
 Differentiating, and observing that 
 
 d . (7^ = ffdo- + dffo- = 2S . o-dff, 
 
 we obtain the equation, 
 
 S . crd(T +/? .fs . d* = 0, 
 where 
 
 d(T= V.dpj3-S d5=S.djoj3-^ 
 Hence 
 
 S . vdp = 0, if V = ^-^a^^-fsf's. 
 
 But this expression gives, 
 
 the ARBITRARY FUNCTION, /, is therefore in this way elimi- 
 nated, and the equation 
 
 S./3i/|O = 0, 
 
 of article 576, is obtained, as the general representation of a cer- 
 tain class of surfaces^ namely, of those which are of revolution 
 round the axis j3. 
 
 578. Again, let us suppose that this last equation has pre- 
 
LECTURE VII. 575 
 
 sented itself, as the expression of the geometrical property, that 
 the normal to a certain surface, otherwise as yet unknown, inter- 
 sects a fixed vector, /3, or that v is coplanar (see 509, &c.) with 
 j3 and p. To integrate the equation 
 
 which is analogous to an equation in partial differentials, we may 
 first write it under the form, 
 
 V = a;/3 + ^jo, giving a; S . /3dp + ^S . |od|0 = 0, 
 
 where x and y are scalars. Hence the two functions S . j3jo and 
 p^ are together constant, or together variable ; and one must there- 
 fore be a function of the other. That is, we have 
 
 which is accordingly one form of the integrated equation 
 of an arbitrary surface of revolution. To obtain hence the form 
 of article 440, it is sufficient to observe that 
 
 for thus we obtain this other functional equation, 
 
 which was the one required. 
 
 579. The symbol v is useful in many other geometrical in- 
 vestigations, for instance, in those which relate to geodetic 
 LINES, or curves, on any proposed surface. One known and fun- 
 damental property of such a curve is, that its osculating plane is 
 always normal to the surface ; which may be expressed in our 
 notations by the formula (compare 574), 
 
 S . v^p d^jo - 0, or S . vp'p" = ; 
 
 the vector p being regarded as a function of some scalar variable 
 t. If this scalar variable be the arc of the geodetic, then (by what 
 was remarked at the end of the last-cited article), p" is the vector 
 of curvature, which must (by the known property just mentioned) 
 have the direction of the normal to the surface : and therefore in 
 this case we may reduce the formula to the following : 
 
 V. i;d^,o = 0; or Y.vp"=0. 
 
576 ON QUATERNIONS. 
 
 In general, whether the arc be or be not the independent scalar 
 variable, Udp is a tangential vector^ and its differential, dUd^o, 
 is a vector having the direction of the vector of curvature, which 
 is drawn in the osculating plane//-om the proposed point of oscu- 
 lation, towards the centre of the osculating circle : thus, for the 
 geodetic lines on any surface, the general equation may be writ- 
 ten as follows : 
 
 V.vdUd|O=0. 
 
 Accordingly, since Udp = dp -^ Tdp, when we suppose Tdp = 
 constant, we fall back on the less general formula, lately written, 
 
 V. vd>=0. 
 
 580. For a spheric surface, round the origin of vectors as 
 
 centre, 
 
 p2 = const., S . pAp = 0, vllp, V. VjO = ; 
 
 hence, for this surface, the general equation of the geodetic lines 
 becomes, by elimination of v, 
 
 V.pdUd/, = 0; 
 
 therefore, because for any curve on a sphere round the origin, 
 p ± Udp, or because ( VpY = - 1 , and S . |0 Ud|0 = 0, we have 
 
 d . pUd/) = d V . |oUdp = V. dpUdp = -V.Tdp = ; 
 
 and consequently an immediate integration gives, for the geo- 
 detic on the sphere, rs being here an arbitrary but constant 
 
 vector, 
 
 pUdp = ■S7, and S .zsp = 0: 
 
 the curve being thus seen to be (as is very well known) a gi'eat 
 circle. As a verification, we have also 
 
 S.t^Udp = 0, 
 
 of which equation the signification is manifest. 
 
 581. Again, let there be an arbitrary cylindric surface, for 
 which (compare 576) we have the equation 
 
 S . j/a = 0, 
 Eliminating the symbol v, by substituting for it the differential 
 
LECTURE VII. 577 
 
 dUdp, to which (by 579) it is, for any geodetic, parallel, we ob- 
 tain the equation 
 
 S.adUdp = 0, 
 
 which gives, by an immediate integration, 
 
 S . aUd|0 = c = constant, 
 
 and expresses that the geodetic on a cylinder is always a heux, 
 making a constant angle with the generatrices of the surface. 
 
 582. For a geodetic on an arbitrary conical surface (see the 
 lately-cited article 576), with vertex at origin, we have the equa- 
 tion, 
 
 S . vp = 0, and therefore S . pdUdp = 0, 
 that is, 
 
 dS.pUd/>=S.d/>Udp = -Tdp, 
 or finally, 
 
 where c is a scalar constant. This result expresses that the 
 length of the projection of the vector p, on the rectilinear tangent 
 to the geodetic on an arbitrary cone, differs only by a constant 
 quantity c, from the length of the arc of the curve: and hence 
 might be deduced the known rectilinear developement. But the 
 following process is perhaps still more simple. Multiplying the 
 differential equation 
 
 dS.pUdp + Tdp = 0, by 2S.pUdp, 
 it becomes 
 
 = d {(s . p\jdpy+ p'} = d . {v. pV6p)\ 
 
 and gives, by an immediate integration, 
 
 (V.pUd|o)^ = const., or TV. |0 Ud|0= const,, 
 
 so that the length of the perpendicular let fall from the vertex of 
 the cone on the tangent to the geodetic is constant ; or, in other 
 words, the rectilinear tangents to any such curve are tangents 
 also to a fixed sphere, described about the vertex as centre. This 
 gives again the rectilinear developement : and for the case of an 
 Apollonian cone, or cone of the second order, it agrees with a 
 theorem of M. Chasles, namely, that the tangents to a geodetic 
 
 2 p 
 
578 ON QUATERNIONS. 
 
 on a surface of the second order are tangents also to another sur- 
 face confocal therewith. 
 
 583. Again, consider the geodetics on an arbitrary surface of 
 revolution. Here, by 570, &c., we have the equation, 
 
 S .^pv = 0, 
 and therefore by 579, 
 
 = S.i3pdUd^ = dS./3pUd|O, 
 
 because /jd^oUdp = - jSTdp = S'^O. Hence integration gives, 
 
 const. = S . ^pUdp = TV. j3p . SU ( V. j3p . dp) ; 
 
 and thus it may be seen (what indeed is otherwise known) that 
 the perpetidicular distance of a point on the geodetic, /rom the 
 axis of revolution of the surface, varies inversely as the cosine of 
 the angle under which the geodetic crosses a parallel. Or we 
 may interpret the integral as follows: if ^o be conceived to be a 
 function of the time t, then the projected areal velocity, ^S . ^pp\ 
 in a [Aa.ne perpendicular to the axis of revolution, bears a con- 
 stant ratio to the unprojected linear velocity, Tp, where jo' = djO 
 -r- dt, as in 574. In fact it is well known that each of these two 
 velocities would be constant, if a point were to describe the curve, 
 subject only to the normal re-action of the surface, and not ex- 
 posed to any foreign force : and indeed this very illustration, from 
 mechanics, has been elsewhere given by an author whom I should 
 think it an impertinence to cite upon so slight an occasion. It 
 may be noticed that the differential equation S . |3pdUd|0 = 0, is sa- 
 tisfied, 7iof only by the geodetics, but also by the parallels (or cir- 
 cles) on the surface : which fact of calculation is connected with 
 the obvious circumstance, that the normal plane to any such 
 circle coincides with the plane of the meridian of the surface of 
 revolution. 
 
 584. Geodetics furnish perhaps the simplest example of what 
 may by analogy be called the Calculus of Varl-vtigns in 
 Quaternions. We have, by 577, for the differential of the 
 tensor of any arbitrary vector a, the formula, 
 
 dT(T = iTo-M (Ta-) = -ilVM . (7- = - S .Utrda = S .U.T-1 da ; 
 whence we may write, 
 
 S icr = - S . Uo-Scr ; 
 
LECTURE VII. 579 
 
 gTdp =- S .UdpSdp = - S .Udpdg^ 
 = -dS.Ud^gp+S.dUdpgp, 
 
 where dUdp is treated as a simple factor, multiplying dp; and 
 therefore, 
 
 SJTdp = JgTdp = -AS.UdpSp + |S.dUdpV 
 
 Comparing this expression Jbr the variation of the length of the 
 arc of a curve, traced upon any proposed surface, with the varied 
 equation of the surface^ namely (compare 576) with this formula, 
 
 S . vSjO = 0, 
 
 we are conducted, as before, to the general differential equation 
 
 of a geodetic (579), 
 
 V.vdUd^=0, 
 
 and also to the two following equations of limits^ 
 
 S . UdpoS/Oo= 0, S . UdpiSjOi = 0, 
 
 which express that the sought shortest line is perpendicular, at 
 its extremities, to any two given curves upon the surface, between 
 which it is required to be drawn. You see that, in these later 
 articles of this Lecture and this Course, I leave many hints to be 
 unfolded by yourselves, respecting the working of this new Cal- 
 culus, both for the sake of brevity, and because it seems that at 
 this stage I may very safely do so. 
 
 585. Let the surface be an ellipsoid, or more generally a cen- 
 tral surface of the second order, with its centre at the origin of 
 vectors, and having its equation of the form 
 
 yjb = 1 , where fp = 8 . vp, v = (pp ', 
 
 the functions (p and f having those general properties which 
 were treated of in earlier articles (475, &c.) of the present Lec- 
 ture, and which give (compare 477), 
 
 dv = d^p = <l)dp,fdp = S . dvdp, dfp = 2S . vdp, dfdp = 2S . dvd^p. 
 
 Now in general if the length of the arc of a geodetic be assumed 
 as the independent variable, and if the difterentiated equation of 
 the surface be written (as in 576) under the form 
 
 S . vdp = 0, 
 2p2 
 
580 ON QUATERNIONS. 
 
 then, by a second differentiation, and by tiie last formula of 579? 
 we have 
 
 vd^^ + S . dvdp = 0, d> = - v-^S . dvdp. 
 
 For a shortest line on the central surface of the second order we 
 have, therefore, by the present article, 
 
 = ^ + S — , or const. = Tv^ (/Udn) ; 
 4/ap V 
 
 where Tv denotes the reciprocal of the length of the perpendicu- 
 lar P let fall from the centre on the tangent plane to the surface, 
 and ^f{\Jdp) denotes the reciprocal of the length of the semi- 
 diameter D which is parallel to the element dp. We find our- 
 selves then reconducted, by this analysis, to the theorem of 
 Joachimstal for geodetics on an ellipsoid, or other central surface 
 of the same order, expressed by the well-known formula, 
 
 P . D = const. 
 
 586. Consider next a geodetic line on an arbitrary deve- 
 lopable SURFACE. Let s be the arc of its cusp-edge (or of 
 its arete de rehroussement), regarded as a positive scalar, and as- 
 sumed as the independent variable ; and let us make (compare 
 5T4), 
 
 J- ( ) = ( y> that is, more fully, -^ = p\ &c. 
 
 Then if (5), or more concisely ^, be the vector of a point on 
 this edge, we shall have Td^ =As, T0'= 1, ^'2=- 1, S .^y = 0, 
 S . <l>'(p"' = - 0"2 _ X^"2. Let + ^ be another scalar variable, repre- 
 senting the length of a tangent to the edge; then the expression 
 for the vector of an arbitrary point on the developable surface 
 will be, 
 
 p = (j> + t(ji' ; giving |o'= (1 + ^ ^' + i<p"- 
 
 Hence the angle x under which the curve (geodetic or other), 
 whereof p is the variable vector, and whose form and position de- 
 pend on the forms of the vector function 0, and scalar function t, 
 crosses a generating right line of the developable, is determined 
 by the formula : 
 
 tT<h" 
 tan x = - — ^. 
 1 + ^ 
 
 ^ 
 
' LECTURE VII. 581 
 
 We may assume v = 0'0""S whereby the vector v will become in 
 length the radius of curvature of the cusp-edge^ and in direction 
 the normal to the developahle surface: and shall then have 
 
 vp={l+t')<^"-^ + t<l»', 
 because 
 
 But for a geodetic on any surface, we have, by 579, the general 
 equation, 
 
 S . vp'p" = ; 
 
 whence, in the present case, 
 
 Again, we have here, 
 
 /o" = r^'+(l+2^')0"+/0"'; 
 
 whence, by the above written properties of the function ^, 
 
 S . <l>'p" = - i" + tTf^ ; 
 and 
 
 S . p"f-^ = 1 + 2^' + ^S . f >"-! = 1 + t'+ (tT<p"yT<p"-\ 
 
 because S . <p"(^"'-^ = {Tf)'T(^"-\ We are then led to the diffe- 
 rential equation, 
 
 = (1 + r)"-+ (1 + t') {tTf)'Tf-^ - it"+ (tT<t>y ; 
 which, when we multiply by 
 
 and employ the lately-mentioned angle a;, becomes simply 
 
 T^" + x'=Of or I Td0' + x = const. : 
 
 a formula which exhibits the known rectilitiear developement of 
 the' geodetic, because Td^'may here be regarded as denoting the 
 angle between two consecutive generatrices of the developable sur- 
 face, if for convenience we here (as in many other geometrical in- 
 vestigations) treat the differentials as infinitely small quantities; 
 although the definition assigned in art. 568 by no means requires 
 that we should generally do so, in dealing with differentials 
 
 OF QUATERNIONS. 
 
 587. It is quite possible, as I may soon shew, to employ 
 a somewhat similar analysis, so as to deduce anew the very ge- 
 
582 ON QUATERNIONS. 
 
 neral and beautiful theorems of Gauss (published in the Essay 
 referred to in art. 525), respecting geodetic triangles on ar- 
 bitrary surfaces: especially those which relate to what may be 
 called the spheroidical excess (or defect) of such a triangle. 
 But, for the sake of variety, I prefer to indicate briefly here 
 another application of the calculus of variations in quater- 
 nions, whereby we can reproduce some remarkable results of 
 M. Delaunay, respecting the curve which, on a given surface, 
 and with a given perimeter, contains the greatest area; and 
 which curve, from the well-known classical story suggested by 
 its definition, 1 propose to name a Didonia. . Beyond the mere 
 suggestion of this name, and the quaternion analysis of which I 
 proceed to submit to you a rapid sketch, it will (I hope) be clearly 
 understood that I have no claim to make, on the subject of this 
 curious class of curves : of which the following geometrical pro- 
 perties have all, so far as I am aware, been discovered by M. De- 
 launay. 
 
 588. For such a Didonian curve, we have, by quaternions, 
 the isoperimetrical formula, 
 
 JS.U^dpgp + cgjTdp-O, 
 
 where c is an arbitrary and constant scalar : and hence may be 
 deduced, by the rules of variations in this calculus (compare 
 'art. 584), the differential equation, 
 
 c-idp = V.UvdUdp; 
 which shews that geodetics are that limiting case of Didonias, for 
 which the constant c is infinite. On a plane, the Didonia is a 
 circle, of which the equation, obtained by integration from the 
 last-written general form, is 
 
 p — zs-\- cU. vdp, 
 w being the vector of the centre, and c being the radius of the 
 circle. 
 
 589. Operating by S .Udjo, the general differential equation 
 of the Didonia takes easily the following forms : 
 
 c-^Tdp = S (U. vdp . dUd^) ; c-^TAp~ = S (U. vdp . d-^) ; 
 
 c-^Tdp3.S.UvdpdV;c-^ = S-^^. 
 
 U.vdp 
 
LECTURE VII. 583 
 
 But in general (compare 574), the vector w of the centre of the 
 osculating circle to a curve in space, of which the element Tdp 
 is constant, has for expression, 
 
 dp- 
 
 Hence for the general Didonia, 
 
 e-^s^^V=^; T(p-co) = cSuq^. 
 
 U, vdp ^ vdp • 
 
 so that the radius of curvature of any one Didonia varies, in ge- 
 neral, proportionally to the cosine of the inclination of the oscu- 
 lating plane of the curve to the tangent plane of the surface. 
 And hence, by Meusnier's theorem, the difference of the squares 
 of the curvatures of curve and surface is constant: the curvature 
 of the surface meaning here the reciprocal of the radius of the 
 sphere, which osculates in the direction of the element of the 
 Didonia. 
 
 590. In general, for any curve on any surface, if % denote the 
 vector of the intersection of the axis of the element (or the axis 
 of the circle osculating to the curve) with the tangent plane to 
 the surface, then 
 
 S.(^-p)v = 0; S.(^-p)dp = 0; S.(?-p)dV = dp^-; 
 
 and therefore, 
 
 S . vdpd'^p' 
 
 Hence, for the general Didonia, with the same significations of 
 the symbols, 
 
 ^ = p - cJJ. vdp ; 
 
 and the constant c expresses consequently the length of the in- 
 terval p-K, intercepted on the tangent plane, between the point 
 of the curve and the axis of the osculating circle. If, then, a 
 sphere be described, which shall have its centre on the tangent 
 plane, and shall contain the osculating circle to the curve, the 
 radius of this sphere shall be constant, and equal to c. The re- 
 cent expression for ^, combined with the first form of the gene- 
 ral differential equation of the Didonia, gives also 
 
 d| = - c V. dUvUdp ; and therefore V. vd| = 0, 
 
584 ON QUATERNIONS. 
 
 And hence, or from the geometrical signification of the constant 
 c, the known property may be proved, that \fa developable sur- 
 face be circumscribed about the arbitrary surface, so as to touch 
 it along a Didonia, and if this developable be then unfolded into 
 a plane, the curve will at the same time he flattened (generally) 
 into a circular arc, with its radius = c. We might also have 
 written 
 
 + |T . dpSjo, instead of JS .\Jv^p^p, 
 
 in the isoperimetrical formula of art. 588, with the condition 
 S^J-dp, and have then proceeded nearly as above. 
 
 591. It will be admitted that the mechanism of these new 
 calculations is sufficiently simple and rapid : and it can scarcely 
 be expected that, at this nearly closing stage of a long Course, 
 the logic of them should he fully developed. Yet it may be pro- 
 per to say a few words on ^ome fundamental points of the theory 
 of differentials of functions of quaternions. And especially you 
 may expect me to shew distinctly that, without necessarily treat- 
 ing those differentials as small, or their tensors as nearly null, we 
 can yet rigorously deduce a differentiated equation, of the form 
 S . vdp = 0, from an equation of a surface, proposed under the 
 form^ = const. ; and may then infer with certainty (compare 575, 
 576, &c.), that V is a normal vector. From the definition (568) 
 of a differential of a function of a quaternion, we can, no doubt, 
 very easily prove (compare 569, 577), that 
 
 d . p- = p . dp + djO . |0 = 2 S . pdjo ; 
 
 jo- being here regarded as di function of p, and dp being an arbi- 
 trary vector. And again, if the vector p be regarded as & func- 
 tion of a scalar, t, the tangential ch?iV?Lctex (574) of dp, with 
 respect to the curve which is the locus of the extremity of p, may 
 be regarded as an easy consequence (compare 528) of the same 
 general definition. Yet it may not be captious to call for proof, 
 that when p^ is considered as hem^-d function oft, in consequence 
 of its being a function of p, which is itself & function of ^, the 
 differential of this function of a function has still the same 
 form as before. And such a proof is necessary, to justify our in- 
 ferring (for example) that the equation p"= - 1 gives pA.^p,for 
 
LECTURE VII. 585 
 
 any curve upon the unit-sphere : or for proving, by quaternions, 
 that the normals to a sphere are its radii. 
 
 592. I take, therefore, the function of a function^ 
 
 r =f(i,q =fp, where p = cpq, 
 
 and seek its differential, by the definition in article 568. That 
 definition gives, immediately, 
 
 dr =:df^q = \im .n{f(p (q + n'^ dq) -f<pq}- 
 But we have also, by the same definition, 
 
 d02'= lim . n[<p (q + n'^ dq) -^q}. 
 
 If, then, we make, for a moment, 
 
 <l> {q + n'^ dq) = ^q + n''^\}j (n, q,dq) =p + n-^\P„, 
 
 we shall have 
 
 \P = xP{oc,q,dq) = d(l,q^dp; 
 
 and 
 
 dr = dfcpq = lim . n{f(p + u'^ ;//„) -fp] = dfp. 
 
 That is to say, we arrive by the definition at one common qua- 
 ternion, as the value of dr, whether we differentiate it a* a func- 
 tion (/) of the quaternion p, which is itself a function {<p) of 
 another quaternion q ; or whether we differentiate r immediately, 
 as a compound function (f(p), of this last quaternion, q. In sym- 
 bols, we may express this general result by writing 
 
 and we see that it includes the result proposed for investigation 
 in the foregoing article, where the independent variable q was a 
 scalar, t, while <p was a vector function, and /"a scalar function. 
 The first statement of art. 576 has, therefore, been fully justified. 
 And I think that analogous reasonings will convince you that 
 other and connected results have not been stated without war- 
 rant, nor at random, although briefly, and perhaps informally, in 
 recent articles. 
 
 593. To exemplify in a new way the process of differentiat- 
 ing the equation of a surface, let us take the form 
 
 T (jp + pK) = k'^ - t^. 
 
586 ON QUATERNIONS. 
 
 which was assigned in article 465, for the equation of the ellip- 
 soid. Since T^q^ = q}Lqi &c., this equation easily gives 
 
 (k^ - L^y = (tjO + jOk) (jOi + Kp) 
 
 = p^ (t^ + K^) + tpiCjO + /OK/Ot 
 
 = (i^ + K2)p^ + 2S.<jOKp 
 
 = (t - k)^ jO^ + 4S . tjO S . K^ = &c., 
 
 a long series of transformations being allowed (compare 499), 
 on the principles of the present Calculus. Thus (compare 476), 
 we may write the equation of the surface as follows : 
 
 \=fp={K^-L'y'- {(£-K)V'- + 4S.<pS.Kpj. 
 
 Differentiating relatively to p, we find (compare 575), 
 
 0=d/p = 2S.vdp 
 = 2 (k^ - i^)-^ {(<- k)2 S . pdp + 2S . td|oS . Kp + 2S . tpS . Kdpj ; 
 
 and finally, as in 474, 
 
 V = (k- - i')-' { (t - k)'- p + 2iS.Kp-^ 2kS . i^} 
 
 = (k"— t")'^ {(t^ + K") jO + t/3K + KjOt} 
 
 = (k^ - r)'- [ (r + K-) |0 + 2V. ipK} = &c. 
 
 Such then is the expression found, by this process of differentia- 
 tion, for the normal vector to an ellipsoid. 
 
 594. The following very general transformations come natu- 
 rally to be mentioned here. By 568, the differential of the 
 TENSOR OF A QUATERNION is, ifwc make for the moment, dq = r, 
 
 dTq = lim. n{T{q + n'^r) - Tq], 
 
 71=03 
 
 where, by 538, 
 
 T {q + ?i-'r) = V [Tq^- + 2n-'TqTrSU. rKq + n-'Tr-"-] ; 
 
 thus 
 
 dTq=TrSU.rKq = S.rUq-' = S.dqUq-K 
 
 We may deduce from this result an expression for the differen- 
 tial OF THE LOGARITHM OF THE TENSOR (or for the differential 
 of the mensor, 547), of any proposed quaternion ; and may write 
 that expression as follows : 
 
 dlT^ = ^^ = S^ 
 
LECTURE VII. 587 
 
 We may also write, generally, 
 
 dT^ = S . d^KU^ = S . dqVKq. 
 
 595. Again, since q= Tq .\Jq, we have this general expres- 
 sion, for the differential of any quaternion: 
 
 Aq=ATq.\Jq+TqA\Jq. 
 Hence 
 
 d^.U^-i = dT^+T^.dU^.U^-^ 
 
 But it has just been seen (594) that 
 
 ^{dq.\Jq-') = ATq; 
 
 it follows then that 
 
 V. dgU^-i = T^ . dU<7 .\Jq-^ ; 
 
 or that we may write (compare 545), 
 
 {Jq q 
 
 This vector quotient is therefore an expression (compare 548) for 
 the differential of the logarithm of the versor of any proposed qua- 
 ternion, q. There exists no very close connexion between the 
 foregoing general transformations and the following, which yet I 
 may not find any other and more natural opportunity of men- 
 tioning : 
 
 y-i (^2^2)i q-i ^ u (Sr .S^+ Vr V^) = \J{rq + Kr Kq) ; 
 
 where q and r may denote any two quaternions. 
 
 596. To exemplify the general transformation of art. 594, let 
 us resume the equation of the ellipsoid, cited in 593, namely, 
 
 T {ip + pK) = K^- c = constant. 
 Differentiating, we find, by 594, 
 
 = S . (tdp + dpfc) (tp + pk)' ^ ; 
 or, because K (ip + pk)= pi+ Kp, 
 
 = S . (td|0 + dpK) (pi + Kjo) 
 
 = (r + K") S . pdp + 2S . Kpidp 
 
 = S.{{C' + ic')p+2y.Kpi]dp; J 
 
588 ON QUATERNIONS. 
 
 SO that 
 
 (i^ + K~) p + 2V. KpL, or (t^ + K^)p+2Y. ipK, 
 
 is a normal vector as before. 
 
 597. When in any of the ways above explained, we have 
 found for the vector of proximity, v, of the ellipsoid, considered as 
 a function of p, the expression given in 593, or any equivalent 
 expression, we can then, by the general method of articles 555, 
 &c., or even by less general processes, deduce this converse ex- 
 pression for p, regarded as a function of v : 
 
 p =(l^ + K^) V — 2V. IVK + 4 (i - (c)'" V. IkS . IKV. 
 
 And then by substituting this last expression in the equation 
 
 S . Vp = 1 5 
 
 we obtain the following equation of that known and reciprocal 
 ellipsoid, which is the locus of the termination of the vector v, or 
 of the reciprocal of the perpendicular from the centre on the 
 tangent plane : 
 
 1 = (i^ + tc) v' - 2S . LVKv + 4 (i - k)-- (S . iKvY. 
 
 It is to be observed, however, that this latter is not in general 
 coincident with the reciprocal ellipsoid mentioned in 492, 493, 
 494, 495, of which the vector was ^, or b'^v, and of which the 
 mean semi-axis was taken = 6, not b''^. With respect to the known 
 and general relation of reciprocity, for any two surfaces^ of which 
 one is derived from the other by thus taking reciprocals of per- 
 pendiculars, we can easily prove it with our present symbols, by 
 merely remarking that the equations 
 
 S . vp = c, S . vdp = 0, give S . pv = c, S . pdv = 0. 
 
 598. The lately cited equation of the original ellipsoid offers 
 us an useful illustration of that extension of Taylor's Theorem 
 which was mentioned in article 573. For if we treat in it the dif- 
 ferential dp as constant, we shall have d^ = 0, and 
 
 /(p + dp)=/p + d/p + idyp; 
 
 which last equation is accordingly found to be rigorously correct, 
 where for the first differential Afp we substitute its value given 
 in 593, and for dy the derived value, 
 
LECTURE VII. 589 
 
 d^/p = 2 (k^ - i2)-2 { (t - kY dp' + 48. idpS . Kdp } . 
 
 And, in this example, it may be regarded as clear, that nothing 
 whatever is neglected, and that dp is not necessarily swa// (com- 
 pare 591). Thejinite developement recently given for J'{p + dp) 
 is here seen to be absolutely accurate, whether the chordal vector 
 dp be supposed to be short or long. 
 
 599. More generally, let us assume the existence of the fol- 
 lowing developement where :r is a scalar variable, 
 
 f(g + xr) =fo + a/i + x'f2 + &c., 
 
 and seek, on that hypothesis, to determine the law of the forma- 
 tion of the successive terms of the series. We shall have, 
 
 f{q + Or)^fq=fo; 
 
 f{q + Ir) =/o +/i +/2 + &c. ; 
 
 f(q + 2r) =/o + 2/, + 2'f, + &c. ; 
 
 /(^ + 3r)=/o+3/,+ 3y; + &c,; 
 
 Hence, 
 
 f{q + Ir) ~f(q + Or) = l/i + P/, + 13/3 + &c. ; 
 ' f(q + 2r) -f(q +lr) = (2-\ )f, + (2^ - P)/^ + &c. ; 
 fiq + 3r) -f(q + 2r) = (3 - 2)/, + (3^ - 2')/, + &c. ; 
 
 and by pursuing this analysis, it is found, with ease, that, in a 
 known notation, if we make r= Aq, then 
 
 A/S'=/i+/2+/3 + &c.; 
 A'fq = AW.f, + A^03 ./a + &c. ; 
 A'fq = A^O^ ./a + &c., &c. ; 
 and generally, 
 
 If then we make r = dq, and consider that by the very process 
 OF SUCCESSIVE DIFFERENTIATION, as thus extended to quater- 
 nions from common algebra, or from the ordinary form of the 
 differential calculus, the n*'' differentiatial, d^fq, is necessarily 
 that part of the w"^ difference which is of the n^^ dimension, we 
 shall see that we may write, 
 
 _ ciy^=A"Oy:„;or/„=-^- -^^ 
 
 A"0'^ 1 . 2 
 
590 ON QUATERNIONS. 
 
 And hence may be obtained the developement (compare 573), 
 
 /(^ + d9) = (i + d+id^ + 2^d^ + ...)A = eyg. 
 
 600. Another method of conducting the analysis is the fol- 
 lowing. Assuming still the existence of the series, and seeking 
 only its exact ^^rm, we may regard the differential d/(g' + r) as 
 the coefficient of x^ in the developement of/ {(j+r -\- xdiq\ if dr = 0. 
 Making then ?•= d^, and dd^ or 6rq= 0, we shall have ^fi^q + d^) 
 = coefficient of x^, in the developement oi f[q + {\ + x) Aq] ; 
 that is, 
 
 ^f^ + 4/1 + d/2 + • • 4/«-i + &c. =/i + 2/2 + 3/3 + . . + nfn + &c., 
 
 if 
 
 f{q^ x^q) =/o + x^f-^ + x^f^ + . . . + af'fn + &c. 
 
 Comparing then the terms of corresponding dimensions, we find 
 the general relation, 
 
 which gives, 
 
 /i = 4/0 ; /2 = i 4/1 = \ <i!/o ; /s = ^^ ^'f"" '^^'• 
 
 and therefore 
 
 f(q + xdq) = e'^'fq, f (q + dq) = e^q, 
 as before. 
 
 601. The following process may, however, be considered 
 more satisfactory, as not setting out with any assumption re- 
 specting the existence of a developement, and as extending even 
 to cases where, at a certain stage, the successive differentials of 
 the function become infinite. The definition (568) gives the fol- 
 lowing expression for what may be called a differential quotient^ 
 although 1 prefer not calling it generally a differential coefficient, 
 because it is not generally independent o/Udg: 
 
 dq ;r=o xdq 
 
 wliere x is still an auxiliari/ 'dnd scalar variable, but dq, like §', is 
 an arbitrary and givefi quaternion, which may or may ?wt have 
 a small tensor. If then the /m?Y just expressed be /?W2^e (as it 
 
LECTURE VII. 591 
 
 will usually be), and if we assign any small value to x^ which 
 may be said to be of the^r*^ order, we shall have the equation, 
 
 lim . x~ 1 [f{q + xAq) -fq - xdfq ] = ; 
 
 and the expression within the brackets may be said to be small, 
 of an order higher than the first. More generally, let 6:^q=Q, and 
 let the successive differentials oifq, as far as d"/^', be supposed 
 finite ; I say that the expression, 
 
 Sn ^f{q + ^d^) -fq - xAfq - ^x'^d'^fq - ... - ^ ^ ^^ d"/^, 
 
 is small relatively to the small scalar x, of an order higher than 
 the n"'-; or that if we make D = -t-, we shall have not only Sn = 0, 
 
 but 
 
 Dsn = 0, D^ Sn = 0, . . . Z)'^ 5„ = 0, when x = 0. 
 
 In other words, it is asserted that, if x be thus made to vanish 
 after the differentiations, we shall have, 
 
 Df(q + xdq) ^ dfq, D^f(q + xdq) = d^fq, ... 
 and finally, 
 
 D''f(q+xdq) = d''fq. 
 
 In fact the general definition of article 568 gives here, 
 
 in dx 
 
 D/(q + xdq) = lim . -T-{f(q + xdq + — dq) -f{q + xdq) j 
 
 7jl= CD Uct/ Ut 
 
 = lim . y-'{f(q + xdq + ydq) -f{q + xdq) ] ; 
 
 y = 
 
 but by the same definition, this latter limit is also the value of 
 the differential df {q + xdq), if d be supposed to operate only on 
 q, but not on dq, nor on x. With these suppositions, we have, 
 therefore, the equation 
 
 Df{q + xdq) = df{q + xdq) ; 
 
 and consequently (dq being still treated as constant), 
 
 D-f (q + xdq) = d'^/(q + xdq), . . . jD"/(^ + xdq) = d"/ (q + xdq). 
 
 Making then x = after the differentiations, we see that the first 
 n differential coefficients of the polynome Sn, taken with respect 
 
592 ON QUATERNIONS. 
 
 to X, vanish as was asserted, at least if the first n differentials of 
 the function /g are finite : or that this polynome s„ is small of 
 an order higher than the w"", if x be considered as small of the 
 first order : which is one form of Taylor's Theorem as extended 
 in this calculus to quaternions. 
 
 602. From the remarks in recent articles (591, &c.)it appears 
 that the symbol dp may be used in at least two principal senses, 
 in connexion with the theory of surfaces : for it may represent a 
 TANGENT, Or it may represent a chord, according as we choose 
 that it shall be regarded as o. function, ^t, of a scalar variable, t, 
 or as a vector satisfying the differenced (not differentiated) equa- 
 tion of the surface, which may be written thus, 
 
 or thus, 
 
 A//)=0, where A|0 = dp. 
 
 When used in the first sense, we have, rigorously, by the demon- 
 stration in 592, and by our use of the symbol v, 
 
 0=d/p = 2S.vd/); 
 
 and it would be improper to add any other term, by way of m- 
 proving the approximation : for such addition would change the 
 meaning of the symbol, dp, and would prevent it from being truly 
 that which it was designed to be. But, at another time, it may 
 be convenient, after warning given, to use the symbol dp in that 
 second sense, in which it denotes a chordal (and not a tangential) 
 vector, drawn from the extremity of some given vector p, to the 
 extremity of some variable vector p + d^p, these two vectors being 
 here obliged only to terminate each somewhere on the surface, 
 and the second being otherwise arbitrary. And the?i the recent 
 equation of linear form (0 =2S . vdp) will not in general be ac- 
 curate. We must, then, add other terms, more or fewer accord- 
 ing to the degree of approximation required, and obtained from 
 the extended form of Taylor's theorem, or from some other mode 
 of developing the function f{p + dp). Of these new terms, the 
 first, by that extended theorem, may be thus written, with the 
 same signification of v as before : 
 
 id-/p = S . dvdp ; 
 
LECTURE VII. 593 
 
 where dv is a linear function of dp. If we go no farther than 
 this new term of the developement, we shall have the following' 
 equation : 
 
 = 2S . vdp + S . (^v^^p , 
 
 which would be rigorously true (compare 598) with the present 
 sense of dp as a finite and chordal vector, if the surface were one of 
 the second order only. For example, if/p = -p'^ = a^, so that the 
 surface is a sphere round the origin, with a radius = a, we find by 
 differentiation that v = -p, di' = -dp, and the recent formula be- 
 comes, 
 
 = - 2 S . pdp - dp2, or S -—^ = 1 , if Ap = dp ; 
 
 which is accordingly true (compare 414), for any chord Ap or dp 
 whatever of the sphere, drawn from the extremity of p, because 
 the projection of the inward diameter -2p on the chord Ap coin- 
 cides with the chord itself. But if the given surface be of an 
 order higher than the second, then we can only say that it ap- 
 proximately satisfies, by its chords, the equation 
 
 0=2S. vdp+ S.dvdp, 
 
 namely, by those chords which are drawn to points upon the sur- 
 face, not far from the given extremity of p. In rigour, for the 
 given surface itself^ we must add, or conceive added, an " &c." 
 after the term S . dvdp, or must actually append some additional 
 terms, of the third or higher dimensions : all singularities of 
 form being at present kept out of view. 
 
 603. It is not difficult to see, however, that when dp is thus 
 treated as a finite vector, drawn from the extremity of p, the last 
 written equation represents an osculating surface of the se- 
 cond ordF':r, which has contact of that order v!\t\\ the proposed 
 surface, «■?« every direction, at the same termination of p. Indeed, 
 if it be only required to secure this sort of osculation^ or this 
 complete contact of the second order ^ we may introduce, at plea- 
 sure, as follows, another arbitrary term into the equation, and 
 may write it thus: 
 
 2S . vdp + S . dvdp = S . vdpS . wdp ; 
 
 2 Q 
 
 1 
 
594 ON QUATERNIONS. 
 
 where ot is any arbitrary, but constant, vector. Accordingly, in 
 co-ordinates, the nine disposable constants, or coefficients of the 
 equation of a surface of the Second order, are not all fixed by the 
 six conditions of the contact recently considered : there still re- 
 main three constants of the ordinary (or scalar) kind disposable, 
 which are here all included in the one vector constant, t?. 
 
 604. The given and osculating surfaces being seen to have, 
 relatively to each other, the same curvature in eve?"y direction^ 
 we may proceed to inquire what this common curvature is, for 
 any one proposed direction. Dividing, for this purpose, the 
 double of the perpendicular distance from the tangent plane, by 
 the square of the length of the chord, and taking the limit of the 
 quotient, we find, 
 
 curvature of section =lim (- 2S .Uvdp -^ Td^o-) 
 
 1. 2S.vdp ,. /- S . dvdpY "^ 
 
 = hm . -^ -i; = iim 7r—^ i* 
 
 i V . dp \ 1 V / 
 
 But also, if ff denote the vector of the centre of the osculating 
 circle, for any proposed and normal section of the surface, we 
 have, 
 
 curvature of section = . 
 
 a-p 
 
 Comparing these expressions for the curvature, of which each is 
 positive or negative, according as the deviation from the tangent 
 plane, for any near point of the supposed normal section, has the 
 direction of + v or of-v, we arrive at the following formula, 
 which appears to me an important one, 
 
 /O — (T dp 
 
 the second member being understood to denote a limit, if dp still 
 denote a chord. 
 
 605. The following is another way of arriving at the same 
 result. The equation, 
 
 0-2S. vdp+^dp2, 
 
 may represent any sphere, touching the given surface at the given 
 point, by a proper choice of the scalar coefficient g, regarded as 
 
LECTURE VII. 595 
 
 an arbitrary constant. If we now inquire in what directions 
 does this tangent sphere cut the given surface, or its osculatrix 
 of the second order, we are conducted to the equation, 
 
 gdp^=S . dvdp, or g= S-T-, 
 6p 
 
 with the condition that dp is ultimately a tangential vector. This 
 last equation may be regarded as immediately determining a cone 
 of the second degree; and the two (real or imaginary) directions, 
 in which this cone is cut by the plane 
 
 S . vdp = 0, 
 
 that isj'by the tangent plane to the given surface, are precisely 
 the two (real or imaginary) directions of intersection of the 
 sphere with the surface, or the two directions of osculation of 
 that sphere. Conversely, if the sphere be required to osculate 
 in a given direction, Ud^o, we have only to deduce the value of ^, 
 by the recent formula, as 2i function of Ud|0, and then substi- 
 tute the g, thus found, in the equation of the sphere, which may 
 be written thus, 
 
 0=2S— +^; 
 Ap 
 
 AjO being here used, for the sake of greater clearness, to denote 
 a chord of the sphere, drawn from the point of osculation. Eli- 
 minating in this way the coefficient g^ we obtain the following 
 equation of the sphere : 
 
 Ajo dp 
 
 And by then making Ap = 2 (o- - p), to express that Ap is a dia- 
 meter of the sphere, a being still the vector of its centre, we are 
 again conducted to the important and general formula, 
 
 p - (T dp 
 
 in which the second member is generally a function of Udp, and 
 so depends on the direction of osculation. 
 
 606. To exemplify this formula, for the case of a given ellip- 
 soid, or other central surface of the second order, let its equation 
 
 2q2 
 
596 ON QUATERNIONS. 
 
 be/(|o) = l, where v = (p(p), &e., as in several former articles. 
 Then (see 585) dv = (dp) ; S . dvdp =^f{dp) = Tdp^f{Udp) ; 
 
 and the general formula becomes =/(Ud|o), giving a- p = 
 
 a-p 
 
 V 
 
 j7tJJ~\- ^"^ (^^^ again 585) we have Tv = P'S /(Ud|o) = 
 
 Z)"^; therefore the radius of curvature of a normal section 
 = T {a- p) = D"^ . P'^ : that is, it is, as is well known, the square 
 of the semi-diameter parallel to the direction of osculation, di- 
 vided by the perpendicular let fall from the centre on the tan- 
 gent plane. 
 
 607. In general, for any surface, it may be shewn by one 
 process, that one member, and by another process that the other 
 member, of the equation 
 
 dS.dvdp = 2S.dvddp, 
 
 is the coefficient of x^y^ in the developement of the function, 
 
 f(p+xdp + yddp). 
 
 It follows therefore that these two members are equal, or that 
 we have, for any surface, the equation, 
 
 S . Uvdp = S . dvldp. 
 
 It is necessary to observe, as concerns the notation employed, 
 that the vector v being regarded as a function of p, its diffe- 
 rential dv becomes a linear and vector function of dp, which may 
 however involve p also : but that in passing to the variation Sdv, 
 of this differential of v, we here conceive the symbol S to operate 
 only on dp, and not on p. Thus having found, 1st, dfp = 2S . vdp, 
 as in 575 ; 2nd, from this^ an expression of the form v = <f>p; 
 and 3rd, dv = \p (dp, p); the plan of the notation, and the linear 
 form of the function \p, so far as it depends on dp, enable us 
 to write, 4th, Bdv = ip {ddp, p). And then the theorem of the pre- 
 sent article is, that 
 
 S . dp^ (Sdp, p) = S . gdp;// (dp, p) ; 
 
 or that for any two vectors, a and r, and for any form of the sca- 
 lar function, f, the vector function j/* must satisfy the condition, 
 
 S . r^// ((T, p) = S . a-^ (r, p). 
 
LECTURE VII. " 597 
 
 In the example of the ellipsoid, (pp was itself a linear function of 
 |0, so that -ip (dp, p) was = ^pdp ; and accordingly, for this surface, 
 we found, in 476, a formula which may be written thus: 
 
 S . T(p(T = S . (T(f)T =y(cr, t). 
 
 608. By operating, as above, with B only on dp, and on dv so 
 far as it involves dp, but not as it may involve p also, we find, 
 with the help of the general formula of the last article, 
 
 dp^ gS ^ = S . dv (gdpdp - dpddp) dp ; 
 
 remembering that (compare 571), by the analogy of the opera- 
 tions d and S, the variation of the reciprocal of a quaternion is, 
 generally, 
 
 SO that we have here, 
 
 Updp - dpldp := 2 V. Idpdp = 2dp^ V ^; 
 
 B.dp-^=-dp-^.Up.dp'\ 
 But 
 
 Sdp 
 
 therefore (permuting cyclically under S, and dividing by dp^) we 
 have 
 
 gS^=2S.dp-dvV^. 
 dp '^ dp 
 
 It may be noted that (compare 595), 
 
 and that therefore the recent formula may be thus written, 
 dv 
 
 609. To interpret these results, I observe that because v is 
 perpendicular to both d|0 and Bdp, therefore V. ddpdp'^ must have 
 the direction of + v ; and that consequently the supposition 
 
 SS -r- = 0, gives = S . vdvdp. 
 Of these two formulae, the former, by 604, expresses the condi- 
 
 gS^ = -2Tdp-iS.dvSUd|0, because dp-^i:dp=-Tdp-K 
 
598 ON QUATERNIONS. 
 
 tion for the osculating sphere being the gtxatest or least possible : 
 or, more accurately, for the centre of that sphere attaining for a 
 jnoxaewt 2i stationary position, v^hWe the direction of osculation 
 varies. The latter formula expresses that dv, or that v + dv, is 
 coplanar with v and with dp ; or that two near normals intersect. 
 And thus is reproduced the well-known theorem, that ihe greatest 
 and least spheres which osculate to a surface, do so in the direc- 
 tions of the LINES OF CURVATURE. We might derive the same 
 interpretation from the formula, 
 
 = S.dv8Udp, 
 
 by considering that the tangential vector SUdp is perpendicular 
 at once to the normal v, and to the tangent Udp ; since then it is 
 perpendicular also to dv, we must have 
 
 dv 111 V, dp, 
 as before. 
 
 610. The form recently found, for the differential equation of 
 the lines of curvature, namely, 
 
 = S . vdvdp, gives dp ± V. vdv ; * 
 
 and thereby reconducts to a theorem of Dupin, that the tangent 
 to a line of curvature is perpendicular to its conjugate 
 TANGENT. For, in general, the vector V. vdv, as being perpen- 
 dicular both to V and to v + dv, has the direction of the intersec 
 tion of the two consecutive tangent planes, whose points of con- 
 tact with the given surface have for vectors p and dp ; or in other 
 words, it has the direction of the rectilinear generatrix of 
 THE circumscribed DEVELOPABLE, which touchcs the surface 
 along the element dp : it has, therefore, in Dupin's phraseology, 
 the direction of the tangent conjugate to this element, or to the 
 corresponding tangent, Udp. It may be noted here, that the 
 curve of the second order, which has been called by the same 
 eminent geometrician the indicatrix oi the curvature of a given 
 surface, at a given point, may be expressed, in our symbols, by 
 the system of two equations, 
 
 S . vdp = 0, S . dvdp = constant. 
 The differential equation of the lines of curvature may also be 
 thus written. 
 
LECTURE VII. 599 
 
 = V.dpdUv; 
 
 and, under this last form, it is easily seen to contain a theorem 
 of Mi". Dickson, namely, that if two surfaces cut each other 
 along a common line of curvature, they do so under a con- 
 stant angle: for the differential of the cosine of this angle is 
 
 dSU . vi.'= S . U.;dUv'+ S . dUvUv'= 0, 
 
 each term here separately vanishing. 
 
 611. In obtaining (see 602) by the extension of Taylor's se- 
 ries, the term S . dvdp, of the developement off(p + dp), as the 
 half of the differential of the preceding term 2S . vdp, we treated 
 dp as constant, according to the general rules of articles 573, &c. 
 But vphen this term has been thus obtained, it is allowed to trans- 
 form it as follows, treating p now as the vector of a curve upon 
 the surface, or as a function of a scalar variable (compare 574, 
 591): 
 
 = dS . vdp = S . dvdp + S . vd^p ; S . dvdp = - S . vd'^p. 
 
 The formula (605) for the centre of an osculating sphere comes 
 thus to be transformed as follows : 
 
 V ^ vd-p V 
 
 = & -J-— = o ; 
 
 a—p dp u) — p 
 
 if b) be (as in 589) the vector of the centre of the osculating cir- 
 cle to the curve in which p terminates, and which may be here 
 conceived to be a plane and oblique section of the surface. The 
 logic of this very simple process oi calculation might deserve, and 
 would support, a stricter scrutiny. For the present 1 content 
 myself with observing that the result is an expression for the 
 theorem of Meusnier, referred to in the article last mentioned ; 
 since it shews, on multiplying by the scalar (o--p) v'S that 
 
 1 = S ^, = S , a-- (u_LfL» -/o, 
 
 U) - p (1) - p 
 
 and therefore that the centre of the osculating circle (to the ob- 
 lique section) is the projection of the centre of the osculating 
 sphere (to the surface), on the absolute normal to the curve. 
 
 612. The formula of 604, or 605, for the curvature of any 
 
600 ON QUATERNIONS. 
 
 normal section, may be verified, and might have been derived, by 
 the following geometrical considerations. It is permitted, in that 
 formula, to change v to tiv, where n is any scalar multiplier ; be- 
 cause S . vdndp'^^0, if dp be a tangential vector. We may 
 therefore dispose of the length of v at pleasure, provided that we 
 retain its normal direction; and, for the purposes of the present 
 inquiry, we may transport it, parallel to itself, to any position we 
 choose. Thus, we may suppose v to denote here that portion of 
 the normal which terminates at the surface, but begins at any as- 
 sumed transversal plane, and the formula of 604 will still hold 
 good. Now let this plane be drawn through the centre c of the 
 sphere which osculates at a given point p, in the given direction 
 of an element pp'; and let it be parallel to the tangent plane at 
 p. Let also the normal to the surface at the near point p' of the 
 section be cut by this transversal plane in the point c', near to c. 
 Then, considering the differentials as infinitesimals, or suppress- 
 ing what must disappear at the limit, and denoting by a + dV the 
 vector of c', as g in the formula denotes the vector of c, we shall 
 have 
 
 V = cp = jo - (T, dv = dp' - cp = pp' - cc' = dp~ d'a ; 
 
 therefore, with this construction for v, the formula becomes, 
 
 p.-(x dp \ dp/ ap 
 
 and shews that 
 
 d'o- J_ dp, or cc'± pp'. 
 
 But we have also, by the construction, 
 
 cc'i. CP ; therefore cc'j_ cpp'; 
 
 that is, the point c is the projection of the point c', and the line 
 cp' is the projection of the line c'p', on the plane cpp'. In other 
 words, this interpretation of the formula shews, that " if the nor- 
 mal to the surface at a near point (p') of the section be projected 
 
 ON THE GIVEN NORMAL PLANE (cPP'), this projection (CP') wHl 
 
 cross THE GIVEN NORMAL (cp) in the Centre (c) of the sphere 
 which osculates in the direction of the section." Now this result 
 might have been foresee?!, by a very simple geometrical reason- 
 ing. For if, at any point p', 7iear or far, upon the section, we 
 
LECTURE VII. 601 
 
 draw, 1st, the tangent to that section; 2nd, the normal to that 
 curve in its own plane; and 3rd, the normal to the surface, then 
 these two latter normals will both be perpendicular to the tan- 
 gent, and therefore their plane will be so; and the normal to the 
 surface, when projected on the plane of the section, will become 
 the normal to the curve. Hence, it is easy to see that when p' 
 is infinitely near to a given point p of the same section, the nor- 
 mal to the surface at p' intersects the axis cc of the circle ivhich 
 osculates to the section at p ; or that its projection crosses the 
 normal cp in the centre c of that circle. Conversely if we had 
 begun by seeing, geometrically, that this projected and near nor- 
 mal thus crosses the given normal in this centre, we might have 
 inferred that, in the notation of the present article, cc'_L pp', or 
 dV J_ dp, and thence have obtained the formula of 604, at least 
 for the case when v is supposed to be bounded as above. But 
 this restriction would be removed by changing v to «v, as before. 
 The formula might therefore in this way have been proved to be 
 generally true. I shall not delay you by pointing out the man- 
 ner in which it may be employed, to assign the known law of the 
 variation of curvature in passing from one section of a surface to 
 another. 
 
 613. Suppose now that the vector of the given surface is ex- 
 pressed as follows : 
 
 p = xP{x,y); 
 
 namely, as some known vector function of some two scalar varia- 
 bles, X and y, which may or may not be the two rectangular co- 
 ordinates, usually so denoted. We shall then have expressions of 
 the forms, 
 
 dp = p'dx + p^dy, dp = p"dx + p'dy, dp^ = p'dx + p,,dy, 
 
 p, p^, p", p,', p^^ being five new vectors, of which the two first are 
 tangential to the surface, so that we may write, 
 
 V = V. p'p^, S . vp = 0, S . vp^= 0. 
 Hence 
 
 d^p = p"da;^ + 2p^dxdy + p,,dy^ + p'd^x + pd^y, 
 
 d'^x and d^^ being introduced, to express that x and y are consi- 
 dered as being, for any one curve upon the surface, functions of 
 
602 ON QUATERNIONS. 
 
 some one independent variable, which may (if we think proper) 
 be supposed to be the arc of that curve. Operating by S . v, we 
 find, 
 
 S . vd-|0 = S . vp" . da;- + 2 S . vpl . 6.xAy + S . vp^, . dy^y 
 d^x and d^y going off. Making then 
 
 0--/0 
 
 so that R is, by 604, the radius of curvature of a normal section, 
 and is positive when the deviation of a near point of that section 
 from the tangent plane has the same direction as v ; and observ- 
 ing that, by the present article, 
 
 dp'^ = p"^ dx^ 4 2 S . pp^dxdy + p; dy^ ; 
 we find that the formula of 611, or the following, 
 
 R-^dp''=S.\Jvd^p, 
 becomes 
 
 = Adx"" + 2Bdxdy + CdyS 
 where A= R-^p'--^ . p"\Jv, B = R-^S . p'p,- S . p;Uv, 
 
 For the lines of curvature, 
 
 Adx + Bdy = 0, Bdx + Cdy = 0; 
 
 and, therefore, to determine the extreme curvatures Ri'^, Ri^, 
 we have the quadratic equation, 
 
 B^-AC=0. 
 
 Hence what is called by Gauss the measure of curvature of 
 the surface, namely, the product of the reciprocals of its two ex- 
 treme radii of curvature, being the product of the roots of this 
 quadratic equation, has for expression, in our present symbols, 
 
 Rf^ Ri^ = v-ms . p;Uvy - s . p'UvS . pVv} -, 
 
 because 
 
 v^ = ( V. p'p)- = (s . p'py - p'^' p/. 
 
 We may also write, with equal generality, because v~^ = - Tv'% 
 this still more simple expression. 
 
LECTURE VII. 603 
 
 V V \ V 
 
 614. To exemplify this general process, and to compare it 
 with known results, let us take the expression for p which has so 
 often occurred already, namely, p = ix +jy + kz, in which xyz de- 
 note three rectangular co-ordinates, and z is now regarded as a 
 function of x and y. Then making, as is commonly done, 
 
 dz = pdx + qdy, dp = rdx + sdy, dq = sdx + tdy^ 
 
 we find for the five vectors, p . , p^^ the expressions : 
 
 p'=i + kp, p^ =j + kq ; p" = kr, p^ = ks, p^^ = kt. 
 
 Hence, by the foregoing article, 
 
 V = V. p'p^ = k-ip-Jq; v-^ = ( 1 + p^ + q^~)-^ (ip +Jq - k) ; 
 
 *j — = ■: :: :; ? O — — ■; :^ „ ? tJ — = ' 
 
 V l+p'^ + q^^ V l+p^ + q"^^ V l+p'^-Vq"' 
 so that we are conducted finally to the known value, 
 
 ^' ^' -{l+p^+qY 
 
 615. The general formula of article 613 may be thus written : 
 
 - v'R,-^Ri' = (S . vp;y - S . vp'S . vp„ ; 
 
 where if we make for abridgment, 
 
 e = - P^f= - S • p'p,, 9^- pf, 
 
 and denote the partial differential coefficients of these three sca- 
 lars, taken with respect to x and y, on a plan similar to the fore- 
 going, as follows, 
 
 e = - 2 S . p'p% /' = - S . p'pl - S ; p"p^, ^' = - 2 S . p^p'^^ 
 e=-1^.pp',,f = -^.pp'-^.p'p,,,g=-2^.pp^,, 
 
 we shall have, by the general priiiciples of this calculus, because 
 V =V. p'p,, the transformations : 
 
 2 (S . vp'y = 2v^p;' -eS. vp,p' + g'^. vpp] ; v"" =p - eg ; 
 2 S . v/)"S . vp^, = 2i;2S . pp^^ + {g- 2/) S . vpp + ^,S . vpp ; 
 2 S . vp, p', = ge^ -fg' ; 2 S . vpp[ =/e - eg ; 
 
604 ON QUATERNIONS. 
 
 2S . vp,p" = ^e'+/(e -2/); 2S . vp'p" =^fe' -v e{e^-1fy, . 
 and finally, 
 
 if, by the same analogy of notation, we write, 
 
 - e„ = 2S . p>; + 2p;\ - g" = 2S . pp," + 2p;\ 
 and -// = S . p'pj + S . p^p" + p/^ + S . p"p^^. 
 
 It follows then that the measure qfcurvature, 22f ^i?2"S depends 
 ONLY ON THE THREE SCALARS, e,f, g, which enter as coefficients 
 into the following expression for the square of the length 
 or A linear element, 
 
 Tdjo'^ = eda;^ + 2/cl^dz/ + gAy"^, 
 
 and on their PARTIAL, differential coefficients, of the first 
 and second orders (namely, on all of the first, but only three of 
 the second order), taken with respect to the two independent 
 and scalar variables, x and g: that is, altogether, on the twelve 
 scalar s, 
 
 c» / ^ ; e', /', g' ; e,, X, g, ; e,^, //, g". 
 
 And thus is reproduced, in a diiFerent notation, and by adifi'erent 
 method, but with perhaps sufiicient simplicity, regard being had 
 to the difficulty of the subject, what has been justly called by 
 Gauss, a most important theorem {theorema gravissimuvn) : 
 namely, that Theorem which was discovered by himself, respect- 
 ing the constancy of what he has named (as above) the mea- 
 sure OF curvature of any surface^ at any point, vvhen the sur- 
 face is treated as an infinitely thin, and flexible, but inex- 
 tensible solid, and is conceived to be unrolled, or otherwise 
 transformed, as such; each linear element of the surface 
 retaining its length during the process. The letters e, /, g, of the 
 present article, answer to the symbols E, F, G, in the notation 
 of the Memoir referred to : in which also the two independent va- 
 riables are denoted by p and q, instead of x and y. 
 
 616. Conceive now that x denotes the length of the geodetic 
 line drawn to the end p of p, from some fixed point a upon the 
 surface ; and let y be the angle which the line so drawn makes, at 
 that fixed point, with a fixed tangent to the surface there ; the 
 
LECTURE VII. 605 
 
 suggestion of these two scalar co-ordinates being taken from the 
 Memoir of Gauss. By retaining?/ unchanged, but infinitesimally 
 altering x, we move along the geodetic line ap, through a linear 
 element, p'dx, of which the length =d:r; thus 
 
 Tp'^l, f)'^ = -l; e = l, e'=0,e = 0, e^=0; 
 
 and p is seen to be an unit vector, in the direction of the last- 
 mentioned element. Again, by infinitely little altering y, without 
 making any change in a;, we move from p along a trajectory 
 which cuts perpendicularly the various geodetics issuing from a, 
 through a linear element jo d?/, of which the direction is perpen- 
 dicular to that of the element p'dx ; thus 
 
 P,J_p', S.p>,= 0;/=0,/' = 0,/ = 0,/>0; 
 
 and instead of the expression v=V.pp^, we may write simply 
 V = p'p,. As a verification we have now, 
 
 = S . p'p" = S . jo'jo/ = S . p"p^ ; p" X pi p" _L p,i p" II V ; 
 
 and finally, 
 
 Y.vp"=0, 
 
 as, by the supposed geodetic character of the lines for which y is 
 constant, and the constant length of the element p'dx, we ought 
 (by 579) to find. Now, without any restriction on e, /, g, or on 
 their partial differential coefficients, the calculations of the pre- 
 ceding article give this equation (differing only in notation from 
 the formula obtained by Gauss), to determine the measure of 
 curvature : 
 
 4 (eg -fy Ef' R,-^ = e {g'^ - 2gJ' + g,e) 
 
 +/ (^>, - ey - 2e,/ - 2g'f + 4/X) 
 
 + g {e; - 2eX + eg) - 2 (eg -p) (e„ -2// + /). 
 
 Introducing then the values of the present article for e, /, &c., 
 and making also 
 
 g = rn?^ g' = 2mm', g" = 2mm" + 2m'-, 
 
 we find that the measure of curvature comes to be expressed as 
 follows (agreeing again substantially with an important result of 
 Gauss) : 
 
 Ri-' R.-' = (^) -^=-m-'m", where ?w = To. 
 \2gJ 2g 
 
606 ON QUATERNIONS. 
 
 The same conclusion might of course have been more rapidly 
 obtained, by using- earlier the special system of co-ordinates em- 
 ployed in the present article. 
 
 617. With the recent significations of x and p, let us now 
 conceive that those two scalar co-ordinates belong to a variable 
 point of some new geodetic curve on the same surface, 7iot pass- 
 ing through the given point a ; and let she the arc of that curve, 
 measured from some assumed point b thereon. Then, by 613, 
 if we write, 
 
 da; = x'ds, dp = yds, d^5 = 0, d^a; = x"ds\ d^y = y'ds^, 
 
 we shall have 
 
 d'~p = (p"x' + 2p>V' + P,y'' + p'x" + py") ds"' ; 
 
 where by 579, 613, 
 
 d^p II V ± jo', and therefore S p'd'^p = ; 
 
 but we have now, 
 
 p'^^-i, s.p'p=o, s.p>"=o, s.p>;=o, 
 
 S . p'pj= - S . pfi' = mm ; 
 
 thus the general differential equation of a geodetic on the surface 
 becomes 
 
 x" = mm'y"^, or v' =- my', 
 
 if we write, as we may, 
 
 x = cos V, y -m~^ sin v, x =-v sin v, 
 
 where v is the angle apb or qpp', between the direction of the 
 element pp' or d* of the geodetic curve bp prolonged at the 
 point P, or (a;, ?/), and the element pq or da; of the other geodetic 
 line ap, prolonged at the same point. We may also express the 
 last result as follows : 
 
 dv = - ir^dy ; or thus, Sv = - ni'Sy, 
 
 if we employ the symbol S to denote the passage from the first 
 geodetic line (?/) to a near geodetic line (y+Sy), and reserve d 
 to signify motion along the line ap or (y) itself. In whatever no- 
 tation the result may be expressed, it is essentially equivalent to 
 one which Gauss obtained, by an entirely different process of cal- 
 
LECTURE VII. 607 
 
 culation, in the Memoir already referred to : which was pre- 
 sented, in 1827, to the Royal Society of Gottingen, and has re- 
 cently been reprinted, with very valuable comments and addi- 
 tions, by M. Liouville (Paris, 1850), in the Second Part of a 
 work, entitled " Application de 1' Analyse a la Geometric;" the 
 First Part of the work- being, in fact, a Fifth Edition of the ce- 
 lebrated Treatise of that name by Monge. 
 
 618. To see clearly the geometrical signification of the re- 
 sults of the two last articles, let us conceive that np and pq are 
 two smali, successive, and equal elements of the geodetic line 
 AP; and that nNi, pPi, qQi, are three small geodetic perpendi- 
 culars to that line {y), erected at the three successive points n, 
 p, Q, and continued to meet, in Ni, Pi, Qi, a near geodetic line 
 {y + Si/), which issues from the same fixed point a. Then 
 
 and the expression found in article 616 for the measure of curva- 
 ture becomes, 
 
 NNi + 2pPi - QQi 
 
 R{'Ri^^ 
 
 NP . PQ. PPi 
 
 it being understood, of course, that the ultimate value of this quo- 
 tient is to be taken. Again, with respect to the last formula of 
 617, we may conceive that pp' is an element of the new geodetic 
 considered in that article, intercepted between the lines (?/) and 
 {y + dy); and then, if pq be still an element (dx) of the line ap 
 or (y) prolonged, the theorem expressed by that formula is, that 
 
 Qpp'- Ap'p = (qQi - pPi) -j- PQ ; 
 
 the recent significations of Pi and Qi being retained. With qua- 
 ternion symbols, the two results may be denoted as follows : 
 
 ^' ^' "dp'-TSp'^"^ Tdp ' 
 
 where d still refers to motion along the original geodetic line 
 AP, and S to passage /rom that line to a near one. The results 
 may also be interpreted as relating to two near normal sections 
 of a surface, npq and Ni Pi Qi, considered as cut^ in p and p', 
 
608 ON QUATERNIONS. 
 
 by a third normal section, or new normal plane to the surface. 
 And there are other modes of illustrating and even of deducing 
 the same results geometrically^ on which it is impossible here to 
 delay. 
 
 619. Conceive now that qq' is another transversal and geo- 
 detic element, intercepted between the lines {y) and (j/ + Sy), and 
 very near to pp' : so that pqq'p' is a little geodetic quadrilateral, 
 whose opposite angles are almost, but not quite, supplementary. 
 If we denote those angles at its corners simply by the letters 
 P, Q, Q', P', we shall have by the foregoing articles, 
 
 P'+P=7r-Sv = 7r + 7nSy, 
 Q'+ Q = '7r + Sv + dEv = TT - {m' + m"da:) By ; 
 
 and the spheroidical excess of the quadrilateral (compare 
 587) is therefore expressed as follows: 
 
 P+Q+Q+P'-2Tr = ddv=- in"dxdy ; 
 
 at least if we neglect all terms of the third and higher dimen- 
 sions. But, to the same order of accuracy, the area of the same 
 quadrilateral is 
 
 pPi .PQ = mBy .dx. 
 
 If, then, the spheroidical excess of this (and therefore of any other) 
 small figure he divided by the area, the quotient is ultimately 
 equal to the measure of curvature of the surface; or in symbols, 
 
 — ^r — 7- = - mm' ^ = P,{ ^R^' \ 
 mdydx 
 
 But again, either by observing that, with the notations of the last 
 few articles, we have the expression, 
 
 Uv = m'^pp^, 
 
 or by using the less general formulae of article 614, it may be 
 shewn that 
 
 V. dUv8Uv = i2f ^ Bi'V. dpdp ; 
 
 and therefore that the measure of curvature of any surface at any 
 point, multiplied into the area of any infinitely small figure on 
 that part of the surface, gives, as its product, what has been 
 
LECTURE VII. 609 
 
 named by Gauss) the total curvature of that superficial ele- 
 ment: namely, the area of the corresponding portion of the unit- 
 sphere, this correspondence consisting here in the parallelism of 
 the radii (Uv) of the sphere, to the normals {y) of the surface. 
 Hence the total curvature of any such quadrilateral element as 
 has been considered in the present article, and therefore also the 
 total curvature of any geodetical triangle, or indeed oiany closed 
 figure on any surface, Abounded by geodetic lines, is equal to its 
 SPHEROIDICAL EXCESS: in such a manner that if ab, bc, ca, be 
 geodetic lines, then, y^ + 5 + C - tt = total curvature of geodetic tri- 
 angle ABC = area of the corresponding triangle on the unit-sphere; 
 which latter triangle will not in general be what is called a sphe- 
 rical triangle, because it will not generally be bounded by arcs of 
 great circles. In applying this very remarkable and beautiful 
 theorem of that great mathematician. Gauss, whose name we have 
 so often mentioned lately, we are to remember that (as he pointed 
 out) the elements of area on the unit-sphere must be supposed to 
 change their algebraic sigfi, when the measure of curvature passes 
 from being positive to negative, that is, when the surface changes, 
 (if it anywhere change) from being convexo-convex like an ellip- 
 soid, to being concavo-convex like a single-sheeted hyperboloid : 
 also that all singular points, like the vertex of a cone, are excluded 
 from those portions of the surface to which the investigation 
 refers. 
 
 620. These specimens of the application of the differential 
 calculus of quaternions to geometrical investigations might easily 
 be greatly multiplied : but perhaps they are already too nume- 
 rous. Were it not for this apprehension of being tedious on the 
 subject, I might shew you that a variety of problems respecting 
 the osculating and normal planes, and the torsions, evolutes, &C.5 
 of curves of double curvature, in space or on a surface, may be 
 treated by processes analogous to those which have been already 
 explained. For example, what is called by M. Liouville the ra- 
 dius of geodetic curvature of a curve upon an arbitrary surface 
 may be expressed, in our notations, by any one of the values 
 which were assigned, in article 589, for the constant c of the 
 curve there called a Didonia. But I prefer to mention here a 
 
 2 R 
 
610 ON QUATERNIONS. 
 
 peculiar application of the fundamental symbols, i,j^ k^ of this 
 calculus, which seems likely to become, at some future time, ex- 
 tensively useful in many important jo%52ca/ researches. Intro- 
 ducing, for abridgment, as a new characteristic of operation^ a 
 symbol defined by the formula, 
 
 . d . d , d 
 Ax "^ dy Az . 
 
 which is to be conceived to operate on any scalar, or vector, or 
 quaternion, regarded as a function of the three independent sca- 
 lar variables, x^ y^ z -, we shall have generally, by such calcula- 
 tions as those of art. 508, the formula 
 
 , ^ ( dt du dv\ 
 
 ./dv Au\ .(At Av\ /du dt\ 
 '^ \d^~ di)'^-^ [Tz~ '^J '^ [d~x~ d^J' 
 
 where t, u, v may denote any three functions of those variables 
 X, y, z. And if we conceive that x\ y\ z' are three new and in- 
 dependent scalar variables, and introduce the analogous symbol 
 of operation, 
 
 , . d . d , d 
 ^^^^d^'-'^d^'-'^di-" 
 
 then we shall have this other formula, 
 
 ~x^'^ dy^ dz) \ d«i ^"^ dy' ^ ' dz' ) 
 
 _ / d^ d^ d^ \ 
 
 \da;daj' dydy dzdz'J 
 
 \dydz' dzdy' j \dzdx' dxdz J \dxdy' dydx 
 
 the subject of operation being here any arbitrary function of the 
 six independent and scalar variables, x, y, z, x',7/', z'. The same 
 sort of calculation with the symbols i^j, k, gives (compare art. 
 507) this other general transformation, which was communicated 
 by me to the Royal Irish Academy in July, 1846, and was sub- 
 
■ LECTURE VII. 611 
 
 stantially reprinted (with the foregoing formulie of this article) 
 in the Philosophical Magazine for October, 1847 : 
 
 ^ dx -^ dp dz' \dx'' dif dzy' 
 
 so that, if u be any scalar or vector or quaternion function of the 
 three independent and scalar variables x, y, z, we have this im= 
 portant formula : 
 
 d^y d~v d'^v 
 
 dx'^ dy^ ' dz'^ 
 
 The bare inspection of these Jhr7ns may suffice to convince any 
 person who is acquainted, even slightly (and I do not pretend to 
 be well acquainted), with the modern researches in analytical 
 PHYSICS, respecting attraction, lieat^ electricity, magnetism, &c., 
 that the equations of the present article must yet become (as 
 above hinted) extensively useful in the mathematical study of na- 
 ture^ when the calculus of quaternions shall come to attract a 
 more general attention than that which it has hitherto received, 
 and shall be wielded, as an instrument of research, by abler hands 
 than mine. Meanwhile I may remark that if v denote the tem- 
 perature of the point whose rectangular co-ordinates are x, y, z, 
 in a solid body, then the symbol - <1w may denote the flux of 
 HEAT at that point. Again, '\iv be what is called the potential 
 of a system of attracting bodies (with the Newtonian law), or the 
 sum of their masses divided respectively by their distances from 
 a variable point xyz, then <^?' is a vector which represents the 
 amount and the direction of the accelerating force at that 
 point, produced by the actions of these bodies. And if we simply 
 consider v as some scalar function of the three rectangular co- 
 ordinates X, y, z, then the symbol + <j y denotes a normal vector 
 to the surface, of which the equation is 
 
 V = constant ; 
 
 in which latter view, we have also this symbolical equation, 
 
 <l = -(S.dp)-id. 
 
 621. Since I have been led to mention physical applications, 
 I shall devote an article or two to some methods of expressing 
 
 2r2 
 
61-2 ON QUATERNIONS. 
 
 by quaternions the attraction of the Sun upon the Earth, and 
 the disturbing force of the Sun upon the Moon, or of a superior 
 on an inferior planet, which occurred to me in 1845, and were in 
 part communicated to the Royal Irish Academy in that year, but 
 more fully in the two years following. 
 
 If we conceive an unit of mass to be concentrated at any 
 fixed or moveable point, from which the vector to some other 
 physical point is a, then the accelerating attraction which this 
 mass exerts on this latter point, according to the Newtonian law, 
 is represented, in length and in direction, with the notations of 
 the present calculus, by the symbol, 
 
 ^ (a) = a"^Ta"^; 
 
 which vector-function, (a) or ^a, 1 for this reason propose to 
 call the TRACTOR, corresponding to the vector of position^ a; or 
 more concisely, the tractor of a. With this signification oi (pa, 
 if we now suppose that the two points compose a binary system, 
 with a sum of masses denoted by M, the equation of the relative 
 motion of the latter about the former may be thus written : 
 
 a = 3I^a ; 
 
 where a" is the second difierential coefficient of a with respect to 
 the time t, and therefore (by 574) the vector of relative accelera- 
 tion, while the first differential coefficient a is the vector of rela- 
 tive velocity. An immediate integration, containing the laws of 
 constant jt?/awe and area, is obtained by observing that the re- 
 cent equation gives, 
 
 V. aa" = 0, and therefore V. aa= y, 
 
 where 7 is a constant vector, 'perpendicular to the plane of the 
 orbit, and representing the doubled areal velocity. Again, the 
 tractor is a function which, in virtue of its meYeJbr7n, and inde- 
 pendently of any physical supposition, admits of being thus ex- 
 pressed : 
 
 <pa=dVa ^ V. ada = (Ua)'-^ (V. ««') ; 
 
 one way, among many, of obtaining which transformation, is to 
 observe that, by 595, 
 
LECTURE VII. 613 
 
 dUa = dl Ua .Ua = V (da . a') .Ua ^UaV. a 'da 
 = aTa'^y. a'da = a'Ta'Y. ada = 0a .V. ada. 
 
 For the relative orbit of the binary system we have, therefore, 
 this other integral, 
 
 a + My~'\Ja = constant, or Ua + M'^ya = e, 
 
 E hei'e denoting a second constant vector. Thus, in the undis- 
 turbed motion of a planet or comet about the sun, the whole va- 
 rying tangential velocity, a', may be decomposed into two partial 
 velocities. My'' e, and - My'^JJa, of which both are constant in 
 magnitude, while one of them is constant in direction also. The 
 component velocity (-My'^Ua), which is constant in magni- 
 tude, but not in direction, is perpendicular to the heliocentric 
 vector (a); the other component {My'^i), which is constant in 
 both magnitude and direction, is parallel to the velocity at peri- 
 helion ; and the fixed component bears to the revolving one, in 
 amount, the ratio of Ts to 1, where Te is the excentricity of the 
 orbit. For if we operate by S . a on the integral equation last 
 obtained, and observe that 
 
 S . aUa = — Ta, S . a^ya' = — S . -yaa' = — y~, 
 
 we find, as the completely integrated equation of the relative or- 
 bit, the following : 
 
 = Ta+ S . as. + M~'y'^, or r'=^p'' {\+e cos v), 
 where 
 r = Ta, p = M-'Ty\ e = Tf, ^J = TT - ae, so that c^ = Mp, if c = Ty ; 
 
 the well-known character of the orbit as a conic section, with the 
 sun as one focus, being in this way reproduced with ease. At 
 the same time we see that if from the sun, or other point taken 
 as origin, we draw a series of vectors a to represent the heliocen- 
 tric velocities, and give the name of Hodograph to the curve 
 -which is the locus of their extremities, this curve will always be 
 (with Newton's law) a circle ; of which the vector of the centre 
 is the constant component of velocity, My'h ; while the radius is 
 the constant 7nagnitude Mc', =cp'', of the component which 
 varies in direction, namely, the sum of the masses divided by 
 
614 ON QUATERNIONS. 
 
 the constant of double areal velocity ; or the constant c divided 
 by the semiparameter p ; or the square root (ik/p"')i of the quo- 
 tient obtained, when the same sum of masses is divided by the 
 semiparameter of the relative orbit. But I cannot enter here 
 into the details of that theory of the Law of the Circular Ho- 
 DOGRAPH, which was communicated to the Royal Irish Academy 
 about the end of 1846, with some additions shortly subsequent, as 
 printed in the Proceedings of the body ; from which (for March, 
 1847) I shall merely extract the following theorem of hodogra- 
 phic isochronistn, equivalent virtually to a celebrated theorem of 
 Lambert, but presenting itself under a different form, and ob- 
 tained by a quite different process: " If two circular hodographs^ 
 fmvifig a common chord, which passes through, or tends towards, 
 a common ce^itre of force, be both cut perpendicularly by a third 
 circle, the times ofhodographically describing the intercepted arcs 
 will be equaV 1 am anxious to acknowledge here, that in the 
 genenil conception of connecting by some curve or line (by me 
 called as above the hodograph) the terminations of lines drawn 
 from one common point to represent the varying velocities of a 
 body, I have found myself anticipated by Moebius, who has in- 
 troduced that conception (but not, so far as I have noticed, the 
 theorems above referred to), in his clear and valuable book on the 
 elements of physical astronomy, entitled "Mechanik des Him- 
 mels" (Leipzig, 1843). The inverse curve, which connects the 
 extremities of what may be called the vectors of slowness, or 
 the locus of the extremity of the rectilineal vector a'"S has also 
 been the subject of some researches of my own, and 1 have ven- 
 tured to propose for it the name oi anthodograph, or, more con- 
 cisely, that of Anthode. 
 
 622. Suppose now that a is the heliocentric vector of the 
 earth, and j3 the geocentric vector of the moon; also let AI now 
 denote the mass of the sun alone. Then, because /3 + a denotes 
 the moon's heliocentric vector, the accelerating actions of the 
 sun on the earth and moon are, respectively, in the notation of 
 the foregoing article, 
 
 M(l) (o) and M(j) (/3 + a) ; 
 from which it follows that the disturjbing fokcEj exerted by the 
 
LECTURE VII. 615 
 
 sun upon the moon, in her motion about tlie earth, is represented 
 by the expression, 
 
 M(l> (j3 + a) - M(^a, or MA^a, if we make /3 = Aa : 
 
 that is, the sun's disturbing force is the difference of the two he- 
 liocentric tractors, multiplied by the mass of the sun. It be- 
 comes therefore an object of great importance, in the applications 
 of quaternions to physical astronomy, to develope this difference of 
 tractors, A«^a, which might perhaps be named the turbator. 
 An obvious mode, but not in this case the easiest one, of effect- 
 ing this developement, is to differentiate the tractor, ^a, regarded 
 as a function of the vector of position a, and to employ the ex- 
 tended form of Taylor's series (arts. 573, 599, &c.). A first dif- 
 ferentiation of this function gives, when we make da =(3, 
 
 d0a = d . a-i Ta"^ = - a-'daa'' Ta'^ - a'^ Ta'^ dTa 
 
 = (aj3 + S . a/3) . a-' Ta'^ = - (a'^jS + S . a'^ j3) . ^a ; 
 
 and a second differentiation, after a few analogous reductions, 
 would be found to furnish the expression, 
 
 so that we have thus the terms of the first and second dimen- 
 sions relatively to [5, or those which are of the same order as 
 /3a '^, j3^o'*, in the required developement of the new tractor 
 (a + /3), or of the disturbing force A^a. But the following 
 process is, in this question, simpler, and conducts to results 
 which are more easily and interestingly interpretable. We have 
 
 <^ (i3 + a) = T (j3 + a)-^ (jS + a)-' = {- (/3 + af]'i (j3 + a)'' 
 
 = {- a^ (1 + a-^^) (I + /3a-0}-^ {a(l + a''^)]-' 
 
 -(l + jSa-i)-* (l+a-ii3)-ta-i(-aO-4 
 
 = (1 + 9-)-* (l + ^')-*(^a, 
 
 where 
 
 q = (5a'\ q' = a'^(5= Kq. 
 
 But, as in ordinary algebra, we have the developements, 
 
 {\ + q)-i=l-^q + §q^-..., 
 
 (i + q'yi = i-W+W- ■■ •; 
 
 whence we may write, 
 
616 ON QUATERNIONS. 
 
 (/3 + a) = Sjo n' f^ni n'i 
 
 where 
 
 0«, «'= W«, «/ (i3a)" (aj3)"' a-^ (- aO"^-"-^ 
 
 1.3... ( 2n-\) 3.5...(2n^+l) 
 2.4... (2n) 2.4... (2w ) 
 
 Supposing therefore still that Tj3 < To, we see that the attrac- 
 tion (/3 + a), which a mass-unit, situated at the beginning of the 
 vector /3 +a, exerts on another mass-unit situated at the ewe? of the 
 same vector, is thus decomposed into an infinite but convergent 
 series of other forces, (^„, „■, of which the intensities are determined 
 by the tensors, 
 
 while the dii^ections of the same partial forces are determined by 
 the versors, 
 
 U0„,,.= (U.i3a)"-«'Ua-^= (- U^)«-'U(-a), 
 
 of the expressions recently given. Let a, h, denote the lengths, 
 or tensors, of the vectors -a and +/3, and let C be the angle be- 
 tween them ; so that, in the astronomical example lately men- 
 tioned, a and h are the geocentric distances of sun and moon, and 
 C the geocentric elongation of one of those two bodies from the 
 other; then 
 
 angle from - a to component force ^m, „/ is = (w - n')C: 
 and intensity of same partial force = »?„, „, (6«"^)"*"'a"^; 
 
 where w„, „' is the same numerical coefficient as before. 
 
 623. Let A, B, c, denote respectively the positions in space 
 of the centres of the moon, the sun, and the earth ; so that 
 
 a = BC, /3 =^ CA, a + /3 = BA ; a = BC, 6 = ca ; 
 
 then the sun's disturbing force on the moon, if his mass be still 
 treated as unitj'", may be, by the foregoing analysis, decomposed 
 into a series of groups of smaller and smaller forces, of which 
 groups it may here suffice to consider the two following. The 
 symbol ^o, o denoting here the sun's attractive force ^a on the 
 earth, the first and j)rincipal group consists of the tivo disturb- 
 ing forces, (j)i, and 0OJ 1 ; ancj of these the frsi is purely ablati- 
 
LECTURE VII. 617 
 
 tious, or is directed along the prolongation of the side of the tri- 
 angle ABC, which is drawn from c to a, and it has its intensity 
 denoted bj' the expression ^ ba'^ ; since we have for this force, and 
 for its tensor and versor, the expressions 
 
 ^i,o = ii3(-a^)-t; l>i,o = |6«-^ U0i,o=U/3. 
 
 The secowc? disturbing force, o^ thh first group, has for expres- 
 sion, 
 
 where aj3a"^ denotes (by 290, 429) the reflexion of the line j3 
 with respect to a, or to - a ; its intensity is exactly triple of that 
 of the former force, being represented by f6«"^; and its direction 
 is the same as that of a straight line drawn from c to a', if a' be 
 a point such that the line aa' is perpendicularly bisected by 
 the line bc (prolonged through c if necessary). Of these two 
 principal disturbing forces, in the case here considered of our own 
 satellite, the first may therefore be said to be directed towards 
 the geocentric place of the moon ; while the second is directed 
 towards what may be called a fictitious moon, namely, to a point 
 in the heavens which is to be conceived to be 2^% far from the sun 
 on one side, as the actual moon is on the other side, but in the 
 same great circle ; so that it may be imagined to be a sort of 
 reflexion of the moon, with respect to the sun. If we now ex- 
 tend the same conception and phraseology, so as to imagine a 
 similar reflexion of the sun with respect to the moon, and to call 
 the point in the heavens so found the first fictitious sun, the 
 moon being thus imagined to be seen midway among the stars 
 between the actual and this fictitious sun ; and if we farther ima- 
 gine a second fictitious sun, so placed that the actual sun shall 
 appear to be midway between tliis and the first fictitious sun ; 
 we shall then be able to describe in words the directions of the 
 three disturbing forces of the second group, and to say that those 
 directions tend respectively, for the case of our own satellite, to 
 these three (real or fictitious) suns. For these ihxQe forces will 
 have, for their respective expressions, the three corresponding 
 terms of the developement of the tractor assigned above, namely, 
 the three following terms : 
 
618 ON QUATERNIONS. 
 
 01, i = |/3-a (-a-)-*; ■ 
 f/>oj 2 = ^4aj3a(5a^ ( - <r)-i; 
 of which the intensities are, respectively, 
 §b'a-'; |6-a-^; V'^^a"*; 
 so that they are exactly proportional to the three whole numbers, 
 1, 2, 5 ; while they are directed, respectively, to the^r*^ fictitious 
 sun, the actual sun, and the second fictitious sun. In fact the line 
 U. j3aj3, =U. j3(- a) j3 S has the direction of the sun's geocentric 
 vector ( — a) reflected with respect to the moon's geocentric vector 
 ()3) ; U . /3^a, = U (- a), has the direction of the sun's geocentric 
 vector itself ; and the line U . a(5a[Ba^ has the direction of the re- 
 flexion of U . /3a/3 with respect to + a. The disturbing force of 
 a superior planet, exerted on an inferior one, may be developed 
 or decomposed into a series of groups of lesser disturbing forces, 
 the intensities of the several forces in each group being con- 
 stantly proportional to whole numbers, in an exactly similar way; 
 nor does the application of the principle and method ofdevelope- 
 ment thus employed terminate here. Nothing depends, in the 
 foregoing investigation, on any supposed smallness of excen- 
 tricities or inclinations : the actual (and not the meari) dis- 
 tances of the points b and a from c are those denoted above by 
 a and b; and the great circle in which the above-mentioned re- 
 flexions, and all the subsequent ones which would be found by 
 taking higher terms of the developement of 0(|3 + a), are per- 
 formed, is the actual or momentary plane of the three bodies, with- 
 out any reference to an approximate or momentary orbit. 
 
 624. 1 have made several other applications of quaternions to 
 various departments of mechanical or physical science, of which 
 applications some have been published. Among them, I shall 
 just mention here, that it was shewn to the Royal Irish Aca- 
 demy in 1845, that the known integrals of the equations of mo- 
 tion of a system of bodies, attracting according to Newton's law, 
 or of the system of equations included in the following formula 
 (where the recent notation is employed), 
 
 ^ = S.«20(a-or), 
 
LECTURE VII. 619 
 
 the accent here referring- to the passage from one body to another, 
 might easily be deduced, by the principles of the present calcu- 
 lus; and that a formula including those differential equations, 
 which becomes with our abridged notations, 
 
 = S . ^« S . 8a ^" + gS . mm'T {a - a)', 
 
 might (theoretically speaking) be integrated by an adaptation of 
 that general method in dynamics, which had been previously 
 published by me in the Philosophical Transactions of the Royal 
 Society of London, for the years 1834 and 1835 ; and which de- 
 pend on a peculiar coinbination of the principles of wanWiows and 
 partial differentials, already illustrated by me, in earlier years, 
 for the case of mathematical op}ics. It was also shewn to the 
 Royal Irish Academy, in 1845, that the general conditions of 
 equilibrium of a rigid system admit of being concisely expressed 
 
 by the formula, 
 
 S . ajS + c = ; 
 
 where a is the vector of application of a force denoted by the 
 other vector j3 ; and the scalar, - c, which is thus equal to the 
 sum of all the quaternion products, aj3, a'j3', &c., is, in the case 
 of equilibrium, independent of the position of the point from which 
 all the vectors a, a', . . are drawn, as from a common origin, to 
 the points of application of the various forces j3, /3', . • . In 
 fact this independence requires the existence of the two separate 
 equations of condition {each of which is equivalent to three equa- 
 tions, when translated into ordinary algebra), 
 
 2)3 = 0, SV.a/3 = 0; 
 
 whereof the former expresses that all the applied forces would 
 balance each other, if they were all transported, without any 
 change of length or of direction, so as to act at any common 
 point, such as the origin of the vectors a ; and the latter equa- 
 tion expresses that all the statical couples (in Poinsot's sense of 
 the word), arising from such transport of the forces, j3, or from 
 the introduction of a system of new and opposite forces, - j3, all 
 acting at the same common origin, would also balance each 
 other : the axis of any one such couple being denoted, in mag- 
 
620 ON QUATERNIONS. 
 
 nitude and in direction, by the symbol V. aj3. When either of 
 these two vector sums, 
 
 S/3 and SV. aj3, 
 
 is different from 0, the system cannot be in equilibrium, at least 
 if there be no fixed point nor axis; and in this case, the qua- 
 ternion quotient, which is obtained by dividing the latter of these 
 two vectors by the former, has a remarkable and simple signifi- 
 cation. For it was shewn to the Royal Irish Academy, in 1848, 
 that the scalar part of this quaternion quotient, 
 
 S(SV.aj3--Si3), 
 
 represents the quotient obtained by dividing the moment of the 
 principal resultant couple by the intensity of the resultant force; 
 with the direction of which force the axis of this principal couple 
 is known to coincide, being the line which is distinguished (in 
 Poinsot's justly celebrated theory) by the name of the central 
 axis o^ihe system. And the vector part of the same quaternion 
 quotient, namely, the line 
 
 V(SV.aj3--Si3), 
 is the vector of the foot of the perpendicular, let fall from the as- 
 sumed origin, on that central axis of the system. But I cannot 
 enter here into any further account of any such applications of 
 quaternions. I shall merely state that I have found these new 
 methods of calculation appear to work well, as applied to some 
 other problems oi physical astronomy, and also oi physical op- 
 tics : and even to a practical subject of so excessively dissimilar 
 a kind, as the construction of skew bridges in engineering. In- 
 deed it is obvious that if the method of quaternions be fitted to re- 
 place (though perhaps not in every instance with advantage) the 
 Cartesian method of co-ordinates, the one method must, like the 
 other, be available in every case of the application of calculation 
 to geometry; and therefore to all those mechanical or physical 
 sciences to which geometry itself can be applied. 
 
 625. It appears to be proper and almost necessary to say a 
 few words here, but they must be very few, on the subject of de- 
 finite INTEGRALS IN QUATERNIONS. VVherevcr we meet with 
 an expression of the form, 
 
LECTURE VII. 621 
 
 J to 
 
 where to, ti are scalars, and F{t) is a given quaternion function 
 of a scalar variable, t, which we shall suppose, for simplicity, to 
 vexndixn finite, while t varies from ^o to ^i, there is no difficulty in 
 interpreting the symbol, in conformity with well-known analo- 
 gies, as equivalent to the following limit of a sum: 
 
 R = \\m.^.*±:^F{to + -{t,-to)]; 
 n= "^ m n n 
 
 the summation relatively to m extending from m = to 7n = n-l, 
 or, if we choose, from m= 1 to m = n. Or we may write this 
 other formula, which expresses a slightly more symmetric sum- 
 mation: 
 
 rti m = n 
 
 F(t) d^ = lira . S . n' (t, -to)F{to + {m- ^)n-' (t^ -to)}. 
 
 Jto n=^ m=i 
 
 Thus the symbol jTdp, of 582, 584, 588, would come, as in those 
 articles, to be interpreted as denoting the length of an arc, s, of 
 the curve which was the locus of the extremity of the variable 
 vector p, regarded as a function of a scalar variable t: for we 
 might thus transform it, 
 
 JTdp = pTp/d^; 
 
 Jto 
 
 and might then regard it as the ultimate value of the sum of an 
 indefinitely great number (n) of indefinitely small elements of 
 length, of which the general expression would be 
 
 w"i (^1 - ^o) T|o/, where t=to+ {m -^) n-'^ (ti - ^o)- 
 
 In fact, if the arc (s) be itself the independent and scalar varia- 
 ble, then (compare 574) Tp'= 1, and n-'^^ti-to) becomes the 
 little element of arc : or if (see again 574) t denote the time, in 
 the motion of a point, then Tpt denotes the velocity ; and, 
 when multiplied into the time-element n'^{ti-to), gives still 
 a product which is ultimately the element of arc. On the other 
 
 hand the symbol jdp, or p'd^, would denote the chord of the 
 
622 ON QUATERNIONS. 
 
 same curve. Ap=pi- po, because this chord is ultimately the 
 vector sum of all the directed or vector elements {tangential^ 
 while n is finite, but at last chordal), which are of the form 
 n-'^ {ti - to) pt, and are taken between the two proposed limits 
 of integration. And similarly in other eases, where the proposed 
 expression of the definite integral is given, or can be prepared, so 
 as to have, in a known way, the differetitial of a scalar under 
 the sign of integration, although with a vector or quaternion for 
 its coefficient : all difficulties from singular forms, or injinite va- 
 lues of that coefficient, being for simplicity kept out of view. 
 
 626. But when the dijBFerential yac^or under the sign of inte- 
 gration is itself, essentially, the differential of a quaternion, 
 then difficulties arise, of a sort which seems to be quite neiv, and 
 which do not appear to offer themselves in the usual differential 
 and integral calculus. Take even the following very simple form 
 of a definite integral, 
 
 Q=r^dg, 
 
 where q^ and q^ denote some two given quaternions, and Q a 
 variable quaternion. What quaternion is this integral Q to be 
 conceived to be ? It seems to me that this must depend on the 
 assumed form of the function which the variable quaternion q is 
 supposed to be, of some independent and 5c«/a/* variable t, which 
 changes value from some t^ to some t^, while q, as depending in 
 some way upon it, changes from q^ to ^i. The simplest of all 
 such laws of dependence appears to be the following linear form: 
 
 q=qQ+ t {qi- qo), with the values, ^o = 0, ^i = 1. 
 
 With this assumed law, or functional^;v;2 oi q^ we find 
 
 Q-t{{^-t)q.-^tq,]{q,-q,)dt 
 
 = i (?i + ^o) (g-i - (Zo) = i (^r- - ^oO + i (Ml - ^i^o). 
 But we may also assume a different law, for example, the fol- 
 lowing: 
 
 q=qo-^t{q^-qo) + t{\ -t)p, 
 
 p being here an arbitrary qziaternion, which may be supposed to 
 be constant : the limits of the scalar variable t being still and 1. 
 And then we have, 
 
LECTURE VII. 623 
 
 and the definite integral acquires this neiv value : 
 
 where Q denotes the /orMe/- value of the integral, but 8Q is the 
 following new quaternion : 
 
 SQ=ip(qi- qo) - i (qi-qo)p = iy.YpY (q, - qo) ; 
 
 the term involving /)2 going off, because the us?ml theory of defi- 
 nite integrals gives, 
 
 r^(l-c)(l-20d^ = 0. 
 
 627. More generally, if we make 
 
 C9i 
 Q = \ fqdq, 
 
 !qo 
 
 where fq denotes some given and finite function of the variable 
 quaternion q, we may interpret this integral in various ways, con- 
 ducting to different results, according as we attribute one form 
 or another to the supposed dependence of this quaternion q on an 
 assumed and variable scalar t, in order to accomplish the definite 
 integration, on the plan of 625. For let this quaternion func- 
 tion of t be more fully denoted by qt, and let it receive some 
 small variation dqt, which vanishes for each of the two extreme 
 values of t, so that if these be still and 1, we shall have 
 
 ^qo = 0,dq,=^l. 
 
 Then the original and the varied integrals become, 
 
 ;d/, 
 
 Q.=ffqtgtc 
 
 Q + SQ^Q + ^'^Sfqtqt'dt + ^yqtdqt'di. 
 
 But 
 
 Eqtdt = dSqt; 
 
 therefore, integrating by parts, and attending to the limiting va- 
 lues of dq, we find that 
 
624 ON QUATERNIONS. 
 
 ffqtBqtdt = -^\fqt)'Sqtdt. 
 
 Hence we obtain the following formula for this new sort of va- 
 riation of a definite integral: 
 
 Jo 
 
 or more concisely, 
 
 ^Q = U¥qdq-<^fqSg); 
 
 an expression which, as here interpreted, does not in general va- 
 nish. In the example of the foregoing article, 
 
 fg=q = {l-t)qo + tqi, Sfq = dq = t{i- t)p, 
 
 and the recent formula becomes, 
 
 dQ=^J{l-t){p{q^-qo)-(qi-qo)p}dt 
 
 = i{p(q^-qo)-(qi-qo)p}, 
 
 as before. 
 
 628. More generally still, if F(q, ?-) denote any function of 
 the two quaternions g and r, which is distributive with respect 
 to the latter, so that 
 
 F(q,r-i-s) = F{q,r) + F(q,s), 
 
 we are naturally led to adopt the following transformation, 
 
 Q = ^l'^F{q,dq) = ^y{qt,qt')dt, 
 
 with an interpretation for the latter of these integrals, of the 
 kind assigned in 625 ; but when we come to apply this expres- 
 sion, we shall still, in general, be conducted to different values, 
 according to the different forms, which may be assumed for the 
 function qt, even if this function remain always finite, between the 
 two given quaternion limits of integration. For if we write 
 
 ^F{q,r)-F{q,^r)=\F{q,r), 
 and similarly, 
 
 dF{q,r)-F{q,dr)=d^F{q,r), 
 we shall have 
 
LECTURE VII. 625 
 
 ^Q=S\''F (q, dq) = P d,F (q, q) At + ^F {q, Bq') dt ; 
 
 JQo Jo Jo 
 
 where 
 
 F (q, dq) dt = F (q, dq'dt) = F (q, ddq) 
 = F{q,ddq) = dF(q, Sg) - d,F (q, Sg) ; 
 
 but F (q, 0) = 0, and therefore F (q, dq^) = 0,F (^, Sq^) = 0, 
 
 because the limits of integration, qo^ qi, are not supposed, in this 
 investigation, to vary; hence, with these limits, 
 
 lF(q,Bq)dt = -ld,F{q,dq); 
 
 and the recent formula becomes, 
 
 g {''F(q, dq) = hd,F(q, dq)-d,Fiq, Sq)], 
 J?o Jqo 
 
 an expression which does not generally vanish. As an example, 
 making F(q, r) =f(q) r, we recover the formula of the foregoing 
 article; and by supposing F{q, r) = rfq,v/e obtain this analogous 
 formula, 
 
 Jqo 
 
 629. There is, however, an extensive case in which this new 
 variation of an integral does vanish, the limits being still given, 
 and the function being still known and finite, namely, as might 
 have been expected, the case where the subject of the integration 
 is an exact differential of some function of a single quaternion. 
 In fact if we suppose, in the last article, 
 
 F((i, dq) = d/7, and therefore F{q, dq) = Sfq, 
 
 then, by the deffiition of a differential in 568, combined with the 
 analogous definition of a variation of a function, namely, 
 
 Sfq = lim . m { f(q + m'^ Sq) -fq } , 
 
 we shall have 
 
 dqdfq = lim . 7nn{f{q + m'^dq + n'^dq) -f(q+ n~^dq) 
 
 n= » 
 
 -f{q + m-'Sq)+fq}, 
 dqSfq = \im . nm{f{q + n'^dq + m'^dq) -f(q + ?n~^dq) 
 
 n = X 
 in = CO 
 
 2 s 
 
626 ON QUATERNIONS. 
 
 -f(q + n-'6g)+fq} 
 and, therefore, with these significations of the symbols, 
 
 d,dfq=c\dfq, , 
 
 whatever the form of the quaternion function /may be. Hence, 
 with the form of the function P considered in the present article, 
 we have 
 
 and, therefore, with this form ofF, we have also, 
 
 gf^i^(g,d^) = 0. 
 
 For example, if F (q, dq) = d . q^- = q . dq + dq .q, then, by the two 
 foregoing articles, 
 
 dlqdq=l i^qdq - dq^q) ; gjd^g = J {dqdq - dqdq) ; 
 
 and although these two integrals do not separateli/ vanish in this 
 calculus, yet their sum does, so that 
 
 d{'^\qdq+dqq) = 0. 
 
 J 10 
 
 Thus, by whatever law we conceive q to vary from qo to qi, re- 
 ceiving always finite values, we have, in quaternions as in al- 
 gebra, 
 
 r (qdq + dqq) == q,'' - q,~. 
 
 630. You will conceive that analogous interpretations may 
 be assigned for double (or triple, &c,) dejinite integrals in quater- 
 nions; or that such an expression as 
 
 ' \ F{q, r,dq,dr). 
 
 I go 
 
 where the function F is distributive with respect to each of the 
 differentials dq, dr. can be treated generally as the limit of the 
 result of two successive summations. But besides all difficulties 
 arising from infinite values of the function to be integrated, there 
 would be found, in this calculus, new sources of indetermination or 
 variation, arising from the non-commutative character oimul- 
 
LECTURE VII. 627 
 
 tiplication, and analogous to those considered in the few preced- 
 ing articles, but on a more extensive scale, in consequence of the 
 doubly (or triply, &c.) arbitrary mode of passage, from one given 
 system of limiting values of the varying quaternions, to the other 
 given limit-system. If this difficult subject shall be pursued, it 
 will probably be useful, or even necessary, to consider it in con- 
 nexion with the important researches of M. Cauchy, on definite 
 integrals taken between imaginary limits, when those imagina- 
 ries are of the ordinary kind. 
 
 631. When I began (in article 568) to speak of the differen- 
 tial calculus of quaternions, I had no expectation of being led to 
 enter into it at so great length, although you cannot fail to per- 
 ceive that only the merest sketch (compare 477), of that calculus 
 and of others allied with it, has been given. But I was anxious 
 to point out (see again 568) the connexion between this differen- 
 tial calculus and linear equations in quaternions, or equations of 
 thefirst degree, such as were discussed in articles 554, &c. Let us 
 consider, with this view, the problem, to differentiate the square 
 root of a quaternion. Let r and dr be any two given quater- 
 nions, from the former of which its own square-root q^r^ can in 
 general be definitely inferred, by the rules of the Fourth Lecture ; 
 then the present question is to deduce from these another quater- 
 nion d^', by the application of the definition in 568, which gives 
 
 df/ = d . r^ = lim .n[(r + n' ^d?-)^ -rl]', 
 
 or, 
 
 q = lim . n{(r + n''^ r')h - r^] , \i q = ^q, r' = ^r ; 
 
 or finally, 
 
 q' =p^ =lim .pn, if (^' + w"^r')^-H = w"^/>„, 
 
 This last equation gives, 
 
 r' = 7^{(^•5 + ?^^lp„)2_r} = r*j^„+/>J^r5+ n-'^pn^-, 
 and therefore, at the limit, where n is infinite, 
 r = qq + qq ; or, dr = ^'d^' + 6.qq. 
 
 In fact, we might at once have obtained this last equation, 
 
 2 s 2 
 
628 ON QUATERNIONS. 
 
 by differentiating one which is supposed to connect q and r, 
 namely, r^^q^; for this simple process would have given (com- 
 pare 569, 592), 
 
 6.r = qAq + Aqq. 
 
 Now the recent formulae are equations of the first degree^ rela- 
 tively to the differential, dq or q', considered as a sought quater- 
 nion; and more particularly, they are of the form discussed in 
 articles 560, &c., namely, 
 
 hq + qb = c : 
 
 and consequently are soluble as such, so as to conduct to a great 
 variety of forms, for the required Differential of a Square 
 Root. One form, for instance, is the following (see again 560) : 
 
 d^ = d . H = 1 S(/ 1 ( Vd/- + Kg S . drg-i) ; 
 
 where (compare 455, 504, 557), the symbol ^q~^ is treated as 
 equivalent to this fuller symbol, (S^')"^ 
 
 632. With the same mode of notation, we have also (compare 
 562), these other forms, which might be further multiplied, for 
 the double of the differential of the square root, q, of a quater- 
 nion, r : 
 
 2dg = 2d . r* = 1 (dr + KqArq- ') Sq-' = ^ {dr + q' ^dr Kq) Sq- ^ 
 = (drq + Kqdr) q-^ {q +' Kq)-^ -= {drq + Kqdr) (r + Tr)'^ 
 
 dr+Uq-^drVq-^ drUq+Uq-^dr q'^{Uqdr + drUq-^) 
 ^ Tq{lJq+JJq-') ^ q{lJq+Uq-') Uq+\Jq-^ 
 
 q-^ (qdr+ Trdrq'^) drUq + Vq'^dr _ drKq-^ + q-^dr 
 Tq{Uq+ Uq-') ^ Tq{l+Ur) l+Ur 
 
 = [dr + Y{Ydr-q)}q''= {dr-Y {Vdr- . q-^)}q-^ 
 
 s s 
 
 dr dr v dr dr v 
 
 = — + V ( V g) = V ( V .q M 
 
 q q s ^^ q q s 
 
 = drq-' + V (V. q-KYdr) (1 +- • q-'). 
 
 For some of the foregoing forms 1 have found geometrical inter- 
 pretations and applications ; for instance, in connexion with an 
 
LECTURE vir, 629 
 
 investigation, on which 1 cannot here delay, of the angle of the 
 following quaternion product of square roots, 
 
 and which led me, by a process quite different from that of the 
 Fifth and Sixth Lectures, to perceive that this angle represents 
 (compare 258, and the formula given at the end of 595) the semi- 
 excess (or semi-area) of a certain spherical triangle def, the vec- 
 tors of whose corners are, respectively, §, e, ^ : but the recent 
 expressions are at present offered only as examples of transfor- 
 mation in this calculus, which may serve also as exercises therein. 
 
 633. In general, if we are given an equation of the form, 
 
 i^((/,?-)=0, 
 
 where q and r are two variable quaternions, and -Fis a function 
 of known form, we may regard one of these two quaternions, r, 
 as an implicit function of the other, q, of which the differential 
 d;- may be had, by first differentiating the equation, and then re- 
 solving the result, as an equation of the first degree, on the gene- 
 ral plan of articles 554, &c. (Compare again the reasoning in 
 592.) For example, to differentiate the reciprocal of a quater- 
 nion, we may differentiate the equation, rq= 1, and thus obtain, 
 
 Arq + rAq = 0, d/* = d • q'^ = - q'^dqq-^, 
 
 as in 571. Again, to differentiate a cube-root, r = qi, we may 
 employ the equations (compare 569), 
 
 q = r^, (\q = r-dr + rdrr 4 drr^, 
 
 and resolve the latter as a linear equation in dr: a process which 
 will be found to lead, after reductions, to this among other forms : 
 
 dr= d . qi =p + (V. r^ + rYr) Yq-'^ {rp -pr) ; where p = ^r'^dq. 
 
 634. The following is a theorem of some generality, respect- 
 ing differentials of functions of quaternions. 'Letfx denote a 
 power, or other ordinary and scalar function, of an ordinary and 
 scalar variable, x ; and let the differential coefficient of this sca- 
 lar function be denoted (compare 574) by /"'a;. Then, supposing 
 5' to be a quaternion, and the functions ff to retain the same 
 
630 ON QUATERNIONS. 
 
 forms as before (so that if, for instance, fq = ^-, then/'^ = 2^), 
 we shall have the expression, 
 
 ¥q =-fq .^q + TVfq. dUYq, if dq = Sdg + S {dqVq-') Yq ; 
 so that 
 
 Aq-^q=Y^^Yq = TYq .AUYq, 
 
 = that par^ of dg' which is a vector perpendicular to V^*. Our 
 time will not admit of entering into the investigation of the 
 general theorem, enunciated in the present article. I can only 
 observe here, that one of the many transformations of expression, 
 of which the theorem admits, is easily seen (by what has been 
 already observed) to be the following : 
 
 ¥l =fqAq + ( TV/^ -f'qTYq) d U Vg ; 
 
 and that one of the chief elements in the investigation is supplied 
 by the relation, 
 
 V.YqY/q = 0, or UY/q = ± UYq ; 
 
 combined, for simplicity, with the supposition that the zipper 
 sign is adopted, or that the axes of the quaternions q andfq 
 have similar (and not opposite) directions. One general corol- 
 lary is, that 
 
 ■^ ^ Yqdq + dqYq ' 
 
 For example, when fq = q-, fq = 2q, the general formula be- 
 comes, 
 
 _ Yqd.q' + d.q\Yq 
 ^ Yqdq + dqYq ' 
 
 a result which may easily be verified by shewing that 
 
 Yqd . (f = 2qYqdq -Yq {Yqdq - dqYq), 
 d . q^ .Yq= 2qdqYq+Yq (Yqdq-dqYq), 
 
 635, The process by which, in 631, we calculated the diffe- 
 rential of a square root of a quaternion, did not require (com- 
 pare 572) any previous developement in series ; nor did it even 
 assume the existence of any such developement, for the square 
 root of a sum of two quaternions. But if we now propose to 
 
 ■A 
 
LECTURE Vll. 631 
 
 ourselves to develope such a square root, we may proceed as fol- 
 lows. Assuming that 
 
 (6^ + C)i = Z> + 5'i + 5'2 + 5-3 + 5-4 + &C., 
 
 and supposing that Tc is small, with respect to TZ»^, we may de- 
 termine successively the various quaternion terms of this series, 
 by means of a corresponding series of linear equations, namely, 
 the following, which are ail of the form considered and resolved 
 in 560: 
 
 bqi + qj)-=c; 
 
 bq2 + q2b = - qi^ ; 
 
 bqz + qzb =^ - qiq. - q^qi', 
 
 bqi + qj) = - q^qz - qi - q^q^ ; &c. 
 
 It is evident that the square-root of a polynomial, such as (b'^ + c 
 + e + f. . .y, may be developed on a similar plan, the question 
 of the co7ivergence or sign of the series being not at present dis- 
 cussed : and that a great variety of more general problems, re- 
 specting DEVELOPEMENTS OF FUNCTIONS OF POLYNOMES, is in 
 
 like manner reducible to the successive solution of a series of 
 equations of the first degree, on the principles of former articles. 
 In practice such a process of developement would be, it may be 
 admitted, a tedious one ; nor had even the notion of so develop- 
 ing the square root of a sum occurred to me, when I found and 
 applied^ some years ago, on the plan of article 631, an expres- 
 sion for the differential, d . g*, of the square root of a varia- 
 ble quaternion : although, no doubt, if any shorter or other way 
 of effecting the developement of (g +dg)4 shall be hereafter dis- 
 covered, it will then be possible to calculate in a new way that 
 differential of q^, by selecting the term or terms of the first di- 
 mension relatively to Aq. (Compare again the remarks of article 
 572.) 
 
 636. Let there be now proposed a quadratic equation in 
 quaternions, of the form mentioned in art. 553, namely, 
 
 q^ = qa-\-b', 
 
 where a and b are two given quaternions, and g- is a sought qua- 
 ternion. Writing 
 
 g = i (« + m; + p), 
 
632 ■ ON QUATERNIONS. 
 
 where w and jO are supposed to denote the scalar and vector parts, 
 not here oi q^ but of the new quaternion, 2q-a', making also, 
 for conciseness, 
 
 V« = a, S {a" + 4b) - c,' V (a^- + 4h) = 27 ; 
 
 the proposed quadratic becomes, 
 
 (w +py + ap - pa = C+ 2y; 
 
 and breaks up into the two following equations, which are re- 
 spectively of scalar and vector forms (c loeing here agiven scalar, 
 and a, 7 being two given vectors) : 
 
 w^+ p'^ = c; y. {w + a) p = 7. 
 
 The latter equation, so far as relates to p, is of the form consi- 
 dered in 514 (or in 559), and gives, with the present symbols, 
 
 Wp = y + {w + a)'^V. ya = (w + a)"^ {tVy + S . ay) ; 
 
 whence, after a few reductions, it is found that 
 
 wy = 72 - {w^ - a^y (V. 07)^ = {w^ - a^)-! { wY~ - (S . 07)-} . 
 
 Substituting for p'^ its value in terms of w, namely, the value 
 p'^ = c- w"^, we are led to the following scalar equation of the 
 SIXTH DEGREE in w, which is, however, only of cubic form, 
 
 =f{w'') = {w^ - a^) {w^ - CW^ + 7^-) - ( V. ayY ; 
 
 or, as it may be also written, 
 
 =f{w^) = w^w'-{c+ a^) W^ + ca} + 7-} - (S . 07)^ 
 
 And when a scalar root w of this equation has been found by or- 
 dinary algebra, we may then in general easily determine the 
 corresponding value for the vector p, by the linear expression 
 assigned above : after which it will only remain to substitute 
 these values in the formula above written, namely, 
 
 q = ^{a + w + p), 
 
 in order to obtain a quaternion q, which shall satisfy the pro- 
 posed quadratic equation, 
 
 q^ = qa + b. 
 
LECTURE VII. 633 
 
 637. Now because -y^=-T7^<0, the ordinary quadratic 
 equation, 
 
 X^-CX + 'y'^= 0, 
 
 has two real roots, one positive, suppose = + ^\ and the other ne- 
 gative, suppose --W, where g and h are reals, of the ordinary 
 and scalar kind. Hence, making 
 
 Ta = 1, TV. ya = m, 
 we have 
 
 fix) = (« - g"^) {x + h^) (x + P) + m^; 
 
 so that, in general, 
 
 f(9') =f (- ^^0 =/(- ^0 = "^^ > ; and/(0) = - (S . ya)^ < 0. 
 Since theny(-oo) = -oc, it is clear that the cubic equation, fx = {i, 
 has in general Tnn^^ real and unequal roots : namely, one 
 root (iCi), which is positive and <^^; another {x^, which is nega- 
 tive, but algebraically greater than each of the two negative 
 numbers - A- and - P ; and a //«>rf {x^ also negative, and alge- 
 braically /e^5 than each of those two numbers. The algebraical 
 equation of the sixth degree in w has therefore tivo real andfour 
 imaginary roots (+ \/ Xi, ± \J x^, ±^^3)5 to each of which may in 
 general be considered as cor/'fi5powflfe/2^, AT least symbolically, 
 by formulse given above, one determined value of p, and thence 
 also one determined value of q. Thus (compare 553) the pro- 
 posed quadratic equation in QUATERNIONS, q^ = qa + b, is 
 proved to have in general six roots; of which, however, only 
 TWO (suppose qi, q.^ are real quaternions, such as have hi- 
 therto been considered in these Lectures : while the other four 
 roots (§'3, qi, q^, q^ may be said, by analogy and contrast, to be 
 four imaginary quaternions. For although ihe^efour latter 
 expressions symbolically satisfy the proposed quadratic equa- 
 tion, as well as the two former ones, yet the parts which by 
 analogy are to be called their scalar parts are not any real num- 
 bers (positive or negative or null) ; nor do those other parts of 
 these new roots, which must be called their vector parts, repre- 
 sent in general any real lines in space. 
 
 638. To illustrate this distinction between real and imaginary 
 quaternions, and generally to throw additional light on the pre- 
 
634 ON QUATERNIONS. 
 
 ceding investigation, let it be now supposed that the two vectors 
 a and y of art. 636 are rectangular ; so that 
 
 S.a7 = 0,/(0) = 0. 
 
 At this limit, one of the roots of the cubic equation {fx= 0) va- 
 nishes; and therefore two roots of the equation in w vanish also. 
 The general and linear expression for p in terms of w becomes in 
 this case illusory ; but on going back to the two original equa- 
 tions between w and p, and making «t? =0, we find that they give 
 here, 
 
 p2 = c ; V. ap = 7 ; 
 
 and that therefore (compare 460) they conduct to the two follow- 
 ing values of the vector p : 
 
 where ^ is a scalar, namely, 
 
 ^ = S . ap = {ca- + 7")2. 
 
 The two corresponding values of the quaternion q are in this case, 
 
 ^1 = i (a + pi) ; ^2 = i (« + /02) ; 
 or more fully, 
 
 639. To shew, a posteriori, that these two values of q do in 
 fact satisfy the proposed quadratic equation, which may be writ- 
 ten thus, 
 
 {2q - a)- + 2 {aq - qa) = ar + 4b, 
 
 or thus, on account of the values (636) of a, 7, c, 
 
 (2q -ay + a {2q -a) - (2q -a) a = c+ 27, 
 
 we are to shew that this equation is satisfied by the substitution, 
 
 2q-a= a~^y + a'^t, where t- = ccr + 7- ; 
 
 a and 7 being treated as two rectangular vectors, but c and t as 
 two scalars, so that 
 
 ay - — 7a, but ai = + fa, yt = + ty. 
 
LECTURE VII, 635 
 
 And because these suppositions give, 
 
 (a'^y + aHy= (a^yy+ a'^ja'H + a'^ta'^y + (a'^0'^ 
 = -a""7^+ ta'^ (ya'^ + a'^y) + t"a'^ = a'~ {f- - y") = c, 
 a (a'^y + a'^i) - {a'^y + a'^t) a = {aa^ + a'^a) y = 2^, 
 
 we see that the substitution succeeds, without restriction on the 
 sign of ^ : so that we have both 
 
 ^1^ =qia+ 6, and q^^ = q2(i + b, 
 
 if qi, q^ have the values assigned in the foregoing article. And 
 it is important to observe that, in the preceding verification, we 
 have made no use of any supposition respecting the reality of 
 the scalar t, but only of its commutativeness with other factors, 
 as regards arrangement in a product {ta = at, ty = 7^). 
 
 640. If we now suppose that t is real, and different from 
 zero, so that 
 
 t^ = Ca- + 7- > 0, - (T . a" ^ 7)^ c < - ( T . a' ^ 7)^^, 
 
 then c and c-\- a? are negative scalars ; and the quadratic factor 
 
 (see 636, 637, 638), 
 
 a;^-(c+ of) a; + ^^ = 0, 
 
 of the cubic equation in x, has two real and negative roots (one 
 algebraically greater and the other less than the negative scalar 
 a^), giving ybz^r imaginary values for the scalar iv, ox four ima- 
 ginary roots of the biquadratic equation, 
 
 i^* - (c + a-) w~ + i!^ =:= 0, 
 
 which is here the remaining factor of the equation of the sixth 
 degree. Let the two roots of the quadratic in x be denoted by 
 
 X2 — — U , 3/3 = — V", 
 
 where u and v are reals, and may be supposed to be positive 
 
 scalars, such that 
 
 ?/2 ^ y2 = _ (c + a^), uv = t; 
 
 then the four roots of the biquadratic in w may be thus denoted : 
 
 Wz = + u -\/-l, Wi = -u s/-\, w^ =^ + v^/-l, W6 = -v .^-l ; 
 
 where it is very necessary to observe that the symbol ^/ -\ de- 
 notes the old AND ORDINARY IMAGINARY OF COMMON ALGEBRA, 
 
636 ON QUATERNIONS. 
 
 and NOT ANY ONE of those square roots of negative unity which 
 have HITHERTO occurred in these Lectures, and have been con- 
 structed by vector units, or by directed unit-lines in space. The 
 symbol V - 1 , as here employed, in these last expressions for the 
 four new values of w, denotes an imaginary scalar, instead of 
 denoting a real vector : and it admits, as in algebra, of being 
 COMMUTED with all other factors, as regards arrangement in a 
 product; which our peculiar roots of negative unity do not. 
 641. The linear equation of article 636, 
 
 \. {w + a) p = y, 
 may have its solution thus expressed (compare 514, 559) : 
 V. -ya w'^y — a^.a'y 
 
 In general, therefore, the six roots of the equation q'^ = qa + b, 
 which were spoken of in art. 637, are the six values of the ex- 
 pression, 
 
 a Y.ya '^ n y -W^aS . ay. 
 
 where w is some one of the six roots of the equation f (jv~) =0, 
 in article 636. When we suppose S . a7 = 0, as in 638, then (by 
 that article) two of the six values of w vanish, and the recent 
 expression for q becomes, for each, illusory; but the same article 
 assigns the two values ^'i, q2, of q, which answer to that case. 
 Under the same supposition (S.ay^O), if the recently consi- 
 dered scalar t be real, the four other values oi w give, by 640, 
 these four other and imaginary values of q : 
 
 ^3 = ^'3 + /- 1 /a ; ^4 = q'z - V- 1 q'% ; 
 ^5 = g's + V- 1 (/a ; ^6 = q^ - V- 1 q\ ; 
 
 where c[z, q"z, ^'5, q'o are four real quaternions, namely : 
 
 a' -^ + —n.— - a" --a ^V 
 
 a' - '' , "^ ■ a" -"La y \ 
 
 2 2(u-+a-)'^^ 2' v- + a^ 
 
LECTURE VII. 637 
 
 642. It may be interesting and useful to prove, d posteriori, 
 that these ybwr imaginary quaternions, just assigned, are in fact 
 symbolical roots of the proposed quadratic equation. And this is 
 
 easy. For since, by 640, the symbol V - 1 is here commutative 
 as a factor, and is distinct from all those square roots of negative 
 unity which enter into the expressions of 7'eal quaternions, such 
 as a and b are at present supposed to be, the equation 
 
 (^ + V^ qy = (q' + V~\ q") a + b 
 
 breaks up into the two following real equations, or equations be- 
 tween realSi which it is necessary and sufficient to verify : 
 
 qq+<^'q=q'a. 
 
 And there is no difficulty in proving that these two equations are 
 satisfied, when, retaining the recent significations of the other 
 symbols, we suppose 
 
 ^ 2^2{y + a^y^ 2 ^^ 2/ + «^> 
 
 and treat w y as a new scalar, or commutative symbol, such that 
 = 2/2 + (c + a^) ?/ + if^ = (?/ + a-) (2/ + c) + 7^ : 
 
 the reality of this scalar Vy being here again unimportant. 
 
 643. If we now choose to consider the following supposition, 
 
 t" = c'^a^ + 7' < 0, 
 instead of that opposite supposition of inequality, which was con- 
 sidered in 640, t becomes an imaginary scalar of the form^'v^-1 
 where ^'is real; and the two expressions of 638 for qi and q.^ be- 
 come imaginary quaternions, but are still, by 639, symbolical 
 solutions oi the quadratic equation proposed in 636. At the same 
 time the ordinary quadratic equation referred to in 640, namely, 
 
 x~ -{c + a^) X + Ca^ + y^ = 0, 
 
 has one of its two real roots positive, the other root being still 
 negative; thus one of the two roots of the lately mentioned equa- 
 dratic in y, namely, 
 
638 ON QUATERNIONS. 
 
 2/2 + (c + a^) y+Ca^^y^' = 0, 
 
 remains still positive, as before, but the other becomes now 7ie- 
 gative; one value of?/ has therefore still a real square o^oot, as 
 when t was real, but the other value of y'?/ becomes imaginary: 
 and finally, in 641, we may still suppose that the scalar u is real^ 
 but must then treat v as an imaginary scalar of the form 
 vV-lj ^' being supposed real. Thus, with the present suppo- 
 sitions, the six roots of the quadratic equation q^ = qa+b may be 
 collected into the following table: 
 
 qi = q'l + \^- 1 q"i, q^^cfi- V- 1 ^"i, 
 qz = q's + ^/- 1 q'3, qi = q'3 - '^ - 1 ^"sj 
 
 q-o = 9 5 + q'si q& = q^ - q& ; 
 
 where q\^ q"i, q'z, q^, q^, q\ are six real quaternions, expressed 
 as follows : 
 
 q\ = \{a + a'^-{); q\ = \aH'', 
 
 , a ay / ^'/i 7 N 
 
 f, V2/5 ^"^ v' being three real scalars, namely, 
 
 i'= V(-ca^-r), 
 
 where the quantity under the radical sign is now a positive sca- 
 lar; u= V^/i' ^^2/1 ^^ ^^^^ positive root of the lately written qua- 
 dratic equation in y; and v ^V-y^, if ?/2 be the negative root of 
 that quadratic. 
 
 644. We see, however, that the imaginary solutions of the 
 proposed equation in quaternions still present themselves under 
 the general form, 
 
 q = q'+\/-\q\ 
 
 where q and g"are real quaternions, while v - 1 is still, as in 627, 
 the old and ordinary imaginary of algebra, and is distinguished 
 from all those other roots of negative unity which ox e peculiar to 
 the present calculus, P*, by its not denoting any real line, on the 
 plan of interpretation which we adopt ; and 1 1"^, by its being, as 
 
LECTURE vri. 639 
 
 a factor, commutative ivith every other. An expression of this 
 general form is called by me Biquaternion. The theory of such 
 biquaternions is as necessary and important a complement to the 
 theory of single or real quaternions, as in algebra the theory of 
 couples, or of expressions of the form 
 
 re' + V^- 1 A'", 
 where a'anda?" denote some two positive or negative or null num- 
 bers, is to the theory oi single or real numbers or quantities. It 
 is admitted that the doctrine of algebraic equations would be en- 
 tirely incomplete, if their imaginary roots, or solutions of the 
 above written and well known couple form {x + 'V -ly), were to 
 be neglected, or kept out of view. And in like manner we may al- 
 ready clearly see, from the foregoing remarks and examples, that 
 no theory of equations in quaternions can be considered as com- 
 plete, which refuses or neglects to take into account the biquater- 
 nion solutions that may exist, of the form above assigned, in any 
 particular or general inquiry. The subject indeed is one of vast 
 extent, and of no little difficulty: but it appears to me to be one 
 which will amply repay the labour of future research. 
 
 645. To give a numerical example, or at least an example 
 with numerical coefficients, let us take the quadratic equation, 
 
 q^ = 5qi + 1 Oj. 
 
 Here (see 636), Vv'e have the values, a = 5i, b= \Qj, and there- 
 fore a = 5i, c = - 25, y = 20/. These values give (compare 638), 
 
 ay = lOOA; S . ay = ; a" = - 25 ; 7^ == _ 400 ; a'y=-4ij=-4k; 
 t~ = ca^ + j' = 625~40() = 225; t^\5; aH^-^ii 
 ^1 == ^ (pi -Ak + Si) = 4i - 2k ; 
 q.^ = ^{5i-4k-Si)^i-2k. 
 
 Such then are, in this example, the two real roots of the qua- 
 dratic. Accordingly we have, by the values of the squares and 
 products of ijk, 
 
 {U - 2ky = - 20 = 5 {Ai - 2k)i + \Qj, 
 (i-2ky = -5 = 5{i~2k)i+l0j; 
 
 and therefore, with the recent expressions for qi, 5-2, 
 qi^ = 5qii + 1 Oy ; q^^ = 5q2i + 1 0/. 
 
640 ON QUATERNIONS. 
 
 646. Proceeding to investigate ihefour imaginary roots of 
 the same quadratic, orthej^wr different biquaternions which sa- 
 tisfy it, we are (by 640, 641, 642) to seek the two real and posi- 
 tive numbers, U", v^^ which are the values of y in the ordirt^'y 
 quadratic equation, 
 
 = 2/2 + (c + a?)y + ca~ + 7^ 
 that is, here, 
 
 = 2/2 _ ^Qy + 225 ; giving u^ = 5, v^ = 4 1 . 
 
 Hence 
 
 mM a2 = -20; i;^ + „2 ^ + 20 ; and by 641, 
 
 , 5.. ,. ,, 3-v/5 
 
 and finally the four biquaternion solutions of the equation q~ = 
 5qi + lOj may be thus written : 
 
 5,. ,. ^^,. ■. 
 ^4 = 2^*~^~~2~^ •^^' 
 
 9^6=2 (^+^) 2~^ •^^' 
 
 where v - 5 is to be treated as an ordinary or scalar imaginary. 
 
 647. To verify that each of these biquaternion expressions 
 does in fact satisfy the proposed quadratic equation, it is suffi- 
 *ient to shew, on the plan of 642, that the^o^^r real or single 
 quaternions, q\, q"z, q\, q'^i satisfy the^oi^r following equations : 
 
 qz' - q"z~ = 5q3 i + 1 Oj ; q'^q^ + q'^q's = 5q"., i ; 
 S'V - q'\" = 5q',i+l Oj ; ^'5 /s + /s q'o = ^q"o i- 
 
 And accordingly it will be found that the common value of each 
 
 5 
 member of the first of these equations is - - (5 +j) ; of the se- 
 
LECTURE VII. 641 
 
 5 i/5 —5 
 
 cond, — — — (i-k); of the third, -^ (5 - 9;') ; and of the fourth, 
 
 — - — {i + K). We find, therefore, a posteriori, that 
 
 ^a' = 5q^ i + 1 Oj ; q:^ = 5qJ+l0j; 
 q,^ = 5qJ+ lOj; q^- = 5q,i+l Oj. 
 
 648. To exemplify the case of 643, let us consider this other 
 quadratic equation, 
 
 q- = qi +j. 
 
 Here a= i, b =j, and therefore a = i, c = ~l, y = 2j, a^ = -l, y^ = 
 - 4, ay = 2k, a^ = -i, a-^y = -2k, Ca" + y~ = I - 4 = - d == P =- t'^ ; 
 SO that t becomes imaginary, and = y'- 3, but t' real, and = ^3. 
 At the same time, c + a- = - 2, and the quadratic in y becomes 
 0=y^-2y-3 = (?/-3)('i/+l); we have thus u= V^, v=\/^, 
 v'= 1, M^ + a^ == 2, v"'--a^=2. Thus the six real quaternions, ^'i, 
 &c., of the article above cited, become, in this example, 
 
 g''i = - - ^ ; q\ = - iz -v/ 3 ; 
 , i h „ ^3 
 
 ?3 = 2 + ^;?3 = -Y-(l-j); 
 
 The two real roots of the proposed quadratic are, therefore, 
 
 and the four imaginary roots, or the four biquaternion solutions, 
 are given by the expressions : 
 
 q = ii(l+V-^)-k; q = i(i + k)±i(\-j)V^; 
 
 where V - 3 is the old imaginary so denoted, and is 7iot here to 
 be interpreted as any real line. It is easy to verify the fact of 
 calculation, that each of these six values of q gives q"^ = qi+j. 
 
 649. More generally let 
 
 q- = qa+[5, 
 
 where a and j3 shall be supposed to denote any two rectangular 
 
 2 T 
 
642 ON QUATERNIONS. 
 
 vectors. Then a = a, b = (5, c = a\y = 2(5, t^ = a^+ 4j3% {y + a=)- 
 + 4j3^ = 0, m2 = Ta^ + 2Tj3, V' = Ta^ - 2Tj3, and the six values of 
 q are included in the three expressions following : 
 
 I. J + a-^j3±K'(«'+4j3^)*5 
 
 II. i(l + U/3){a±(a^ + 2T/3)*}; 
 III. l(l-Ui3){a±(a^-2Tj3)*}. 
 
 Of these expressions, the third gives always two imaginary qua- 
 ternions, because a~-2Tj3 is always negative; and according as 
 Ta^ is < or > 2Tj3, and therefore a*+ 4j3^< or > 0, we shall have 
 two real quaternions from the second expression, and two imagi- 
 nary vectors from the first ; or else two real vectors from the first 
 expression, and two imaginary quaternions from the second. It 
 may be noted that when a^ + 4j3- < 0, the two real quaternion roots 
 of the quadratic equation have a common tensor, = V Tj3 ; 
 whereas, when a* + 4/3- > 0, the two real vector roots have unequal 
 tensors, or lengths, one tensor being greater and the other being 
 less than ^Tj3; which is, however, still the geometrical mean 
 between them. And it is easy to see that the distinction between 
 these two cases corresponds to the imaginariness or reality of the 
 intersections of the sphere and i^ight line, whose equations are, 
 respectively, 
 
 p"" = S. ap, and V. ap = j3. 
 
 650. It may also be worth while to observe, that since 
 
 q^- qa = - q {a- q) = {r - a) r, if r = a-q, 
 
 the method given in the foregoing articles (636, &c.), for resolv- 
 ing a quadratic equation in quaternions of the form g^ = qa + b, 
 serves also to resolve a quadratic of this other form, r~ -ar + b ; 
 and that if a and b be the sanie given quaternions in these tivo 
 equations, each of the six roots, q, of one, will be connected with 
 a root, r, of the other, by the relations, 
 
 q+r = a; qr = -b. 
 
 Conversely, this last system of two equations between two qua- 
 ternions, q and r, in which their sum and product are given, may 
 be resolved by the foregoing methods. And we see tliat there 
 
LECTURE VII. 643 
 
 will be, in general, two real systems^ and^wr imaginary systems, 
 or pairs, of quaternions satisfying the conditions. 
 
 651. One way in which such a quadratic equation may pre- 
 sent itself in a research is the following. Let it be required to 
 estimate the value, or to change the form, of the following con- 
 tinued FRACTION, 
 
 h 
 
 \a + 
 the notation implying that 
 
 b h „ 
 Ux = , Ut = , &c. ; 
 
 a + Uo a + Ui 
 
 and a, b, Uq being here any three given quaternions, but x being 
 a positive whole number. Assume at pleasure any two quater- 
 nions, qi, qi\ then because, by supposition, 
 
 we shall have 
 
 Uoc^^■^q^ = {h + q^a■\■ q-^u^ {a + u^- \ 
 Ux^x + q2={b + q^a + q^u^) {a + Ux)'\ 
 
 and therefore, 
 
 Ux+x + qz b + q2a+qzUx_ q^'^b + a + Ux .^ 
 Ux+i + qi b + qxa + qiUx qi~^b + a + Ux 
 
 If, then, we suppose that qx and q2 are any tivo roots {real or 
 imaginary^ of the quadratic equation in quaternions, 
 
 q'^ = qa + b, or q = a + q'^b, 
 
 so that 
 
 qx'^b + a-=qx, q2'^b + a = q2, 
 
 and if we make, for abridgment, 
 
 Ux + q2 ,, . Uo + q2 
 
 Vx = —, SO that Vo = —, 
 
 Ux + qi Uo + qx 
 
 we shall have 
 
 Va+i = 5'2^':i<7rS and therefore Vx= q2^'^oq\^', 
 
 which is the transformation that we desired to effect, and from 
 
 2 t2 
 
644 ON QUATERNIONS, 
 
 which the continued fraction u^ can easily be deduced, by the 
 formula, 
 
 652. A less elementary mode of accomplishing the same 
 transformation, but one which it is instructive to notice, is the 
 following. Assuming 
 
 ai + a2+ ' ' ' ar; + c D.r D' x {cix + c) + D" ^bj 
 
 and changing c to b^^i («r+i + c)"S and Ux to 2<a+ij we obtain the 
 following equations in finite differences, with quaternion coeffi- 
 cients and variables : 
 
 D'x + 1 == D'xttx + D",-cbx, D"x^ 1 = Bx ; 
 together with the initial conditions, 
 
 N\=^Q,N'\^l,D\=\,D\ = Q, 
 
 which allow us to suppose 
 
 iV'o = l, Z)'o=0. 
 Making next 
 
 aa: = a, bx = b, c= Uo, 
 we have 
 
 Nx = N'x {a + Uo) + N'x.i b, Dx = D'x {a + u^ + Z)',,- 1 b, 
 N'x. 1 = N'xa + N'x^.b, D'x., = D'xtt + D'x.,b ; 
 
 and may thus be led to assume 
 
 N'x = Iqi'^ + mqf, D'x = I'qi" + mq-f", 
 qi = a^-qi^b, qo = a + qo~^b, 
 l+m= 1, ^5-1 + mq2 = 0, l'+ wz'= 0, I'q, + m'qn = 1 ; 
 
 whence are obtained the values, 
 
 l-{qf^-q,-^y^qf^ = -q,{q,-q,y\ 
 m = - (qr-^ - q,-')- 'qi' = + qi {q, - q,)-\ 
 r = - m' = {qi - qo)-\ 
 
 Hence we are conducted to express the continued fraction Ux as 
 the quotient of the two following expressions, 
 
 Nx = Iqi""' (qi 4 Uo) + mq/ {q. 4 Uo), 
 Dx = Iqi""' {qi+ Uo) 4 w'^-.-^' (qo 4 M„) ; 
 
LECTURE VII. 645 
 
 and this may suggest the consideration of another and auxiliary 
 quotient, Vj^, which in this process is defined by the formula 
 (which in the foregoing article was deduced), 
 
 qi" {qi + Uo) qi+ ih 
 
 for thus we deduce, by the present process, a relation between Ux 
 and Vx (which in the former article was defined to exist), since 
 we find that 
 
 _ /_^Y =-^= ^ ^ ^^^-^ = - ^2 (gi - q^)'^ + qx (qi - q2)-'Vx 
 ''~\a+ J ° Dx l' + m% (qi - q^y^ (1 - v^) 
 
 = - 5-1 + ( 1 - Vx)- ' {qi - go) = (1 - Vx)-' (Vxqi - qz), 
 
 as before. 
 
 653. As an example, let a = i, b=j, Uo= 0, so that the con- 
 tinued fraction becomes 
 
 «.= (fJo. 
 
 Here the quadratic equation becomes q^ = qi +J, as in article 
 648 ; and by that article, its two real roofs are the following : 
 
 q, = ^{l+i+J-k); 5'2 = i (- 1 + i -j-k) ; 
 
 whence, by 651, 
 
 Vx = {-\ + i -j-ky^' (1 + ^ +j-k)-''-K 
 
 To transform these powers, or the corresponding powers of the 
 two quaternion roots of the quadratic, 1 observe that those two 
 roots are versors, the tensor of each being unity, Tqi = Tq2 = l; 
 which agrees with a remark made in 649, the j3 of that article 
 being here a vector-unit, namely, j. We have also, 
 
 Z?, = 3 = Z(-g.); UVy.= ^; UV(-J.) = _^; 
 
 and, therefore, 
 
 , -. / xir k-i-^j . ccttx 
 j/ = (-l)'(cos-+-^s,n-), 
 
 xtt k -i-j , xtt 
 
 XTT k -i-j , Xtt 
 qr^' = COS — + — —— Sin 
 
646 ON QUATERNIONS. 
 
 Vx-i-?2^^r^ = (-l)-'|(^cos-^j +— sin — 
 
 2i +2/^-1 
 
 
 " 3 — V'"' Y) 
 
 Thus 
 
 VQ = -k, Vi = i, V2=-] ; Vs = + k, Vi = -i, z?5 = + 1 ; 
 
 Ve = - kf y, = i, ^8 = - 1 ; &c. ; 
 and generally, 
 
 Hence, as a verification, by the last formula of 651, 
 
 Wo = (1 - Vo)-^ (VoQi - ^a) = - (1 + A)-i (kg^ + g's) = ; 
 
 and by continuing to apply that formula, we find 
 
 Ml = (1 - i)-^ {iq, - S's) = i (1 + {j+k)=k; 
 2^2 = (1 + l)-i (- ^1 - ^2) = - i (^1 + ^2) = i (^ - ; 
 
 W3 = (1 - A;)-i (Ag-i - ^2) = - (1 + k) q^^k-i] 
 
 Ui = {l + i)-' (- iqi - ^2) = - i (1 - (2 - I) = - e ; 
 
 M5-(l-l)-^(^i-g2)-0-i(l+j) = Qo; 
 
 after which the values of the continued fraction recur, in the pe- 
 riod, 
 
 0, k,^{h~ i), k-i, - i, 00, 
 
 because we have here 
 
 Accordingly, division gives, directly, 
 
 W5=--00; Me = -i^ = 0; ?/7 = ^=^, &c. 
 
 u 00 z 
 
LECTURE VII. 647 
 
 654. To exemplify now the use of the imaginary roots of 
 the same quadratic equation, 
 
 q-^qi^j, 
 
 let us suppose, as by 648 we are allowed to do, that qi and qz 
 are the two following imaginary vectors : 
 
 qi = 2i-k, q2 = z-^i-k; 
 
 where 2 = ^(1 + \/-3) =(- l)^= (cos + aZ-I sin) -; 
 
 o 
 
 the old imaginary of algebra being here the one employed, so 
 that z is commutative in multiplication (compare 640, 644). As 
 a preliminary verification, we have, 
 
 (zi -ky ^ - z"" - \ = - z- {zi - k) i +y, 
 {z-H-ky = -z-^~-l=-z-' = {z-H- k) i +j, 
 
 so that the recent expressions qi, q-i do in fact satisfy the quadra- 
 tic. They give 
 
 q^ = -z''-\ = -z, qi = -z-^ = z~, Vi = q^^ q-c'^^-z -, 
 Vzn.i = qz^"" ^i"'" = (- zY = (cos + \/~l sin) -^ ; 
 
 ^-'^ , /— . X 2>Z7r 
 
 9-;;z (cos - y 
 
 2 sni 
 
 (1 - V.n-lY' = 2U7r (^^^ ~ ^~ ^ ^^") ~/' 
 
 3 
 
 (1 - Vzn-i)'^ Vzn-i= 2^^ (cos + )/- 1 Sin) — » 
 
 2 sin -^ 
 
 and therefore by the last formula of 651, with the present values 
 of qii q%i we have 
 
 1 / \ ^-^^ N . 2w7r 
 
 Utn-x = - i (<?i + q'l) + —2" ((/i - ^2) cot -^ 
 
 . (w-l)7r 
 _ sin ^ — ~ — 
 
 = A;+2(--| + v| cot — - ) = ^ - i ™— — — J 
 
 sin- 
 
 an expression from which the imaginary symbol has disappeared. 
 
648 ON QUATERNIONS. 
 
 and which gives the following real values of the continued frac- 
 tion, Uxi for odd values of x : 
 
 Ui = k, M3 = A - i, U5 = 00, 
 u^ = k, Ug = k- i, Un - CO, &e. ; 
 
 agreeing perfectly with the results of the foregoing article, 
 
 although here deduced by the help of the two imaginary vectors 
 
 (zi-k, z'H-k)^ which have been taken as the two values of q^ 
 
 and which may be said to be the vectors of the two imaginary 
 
 POINTS OF INTERSECTION of the Sphere p^ = S .if), and the right 
 
 line V. ?p =J, which line is situated wholly exteiHor to the sphere 
 
 (compare 649). 
 
 665. Again, to calculate the values of the same continued 
 
 fraction, Ux, for even values of ;r, by the help of the same two 
 
 imaginary vectors, t^i, q^, we may proceed as follows. Since, by 
 
 651, 
 
 {ux + ^72) (^^x + ^1)" ' = Vx = qi" Vo ^T^ ; 
 and 
 
 Vo = {uo + 5'o) (uo + ^1)"^ = q2qi~\ because Wo = ; 
 we have therefore 
 
 ^3"^ (^2' ' + Ux-^) = qi-"" {qi ' + Ux-^), 
 
 and finally 
 
 Ux = (qi-""-' - qi^'-')-' iqz''^ - q,-% 
 
 as a general expression for the value of the continued fraction 
 
 b V- 
 0, 
 
 a + 
 
 qi and qi being still any two roots of the quadratic equation 
 
 q'= qa + h. 
 In the present example, 
 
 ST^^-s^'S qn:" = -z, q{^ = kz-^-i, q.{^ = kz-i, 
 
 and the formula gives, 
 
 u'^l = i - (2;" - 5-")" ' (e""' - ^""'0 ^ 
 
 . - . (W + 1 ) TT 
 
 I -k ^\n - — „ 
 
 cosec -— , 
 o 
 
LECTURE VII. 649 
 
 the imaginary symbol disappearing here again. And accord- 
 ingly, this last expression gives the values, 
 
 Uo'^ = GO, U2'^ = I - k, Ui^ = 2, Ue'^ = GC, &C., 
 
 or, 
 
 Wo = 0, U2=^ (k- i), Ui = - i, Ue = 0, &c., 
 
 as found in article 653. The method of the present article may 
 also be applied to the case of odd values of x, and gives, for such 
 values, the expression, 
 
 . . (w-Htt 
 « sin-^^ — ^ 
 
 sin- 
 
 as in 654. And the other pair oj" imaginary roots of the quadra- 
 tic, which was determined in 648, would be found to give still 
 the same real results. 
 
 656. It may be considered as still more remarkable that we 
 are even at liberty to employ one real and one imaginary root of 
 the quadratic, in order to calculate the real values of the conti- 
 nued fraction : the imaginary symbol still disappearing, when the 
 prescribed operations are performed. For example, if we sup- 
 pose, with the recent signification of z, but with a new selection 
 oi the pair of roots employed, 
 
 ^1 ^zi-h, ^2 = i (- 1 + i -j-k), 
 we shall have, 
 
 q-^n ^ (_ £yn ^ (cog ^ ^ri sin) — ^ ; 
 
 yf 2«-i = (_ z)-"^ {kz-' -i) = k (cos + \/^ sin) ^^^ ~ -^^ "^ 
 
 o 
 
 -I (cos+ v-lsin) -^— ; 
 o 
 
 4.'C7r i-j-k . AXTT 
 <?2-^ = C0S -^+ ^g sin — ; 
 
 2w7T i-j-k . 2mr 
 
 g,-^n ^ cos -^ + V3~ ^^" ~3~' 
 
 .„ J 2(w-l)7r i-j-k . 2(«-l)7r 
 ^, 2«-i^cos ^-^ + ^g sin ' g ^ ; 
 
650 ON QUATERNIONS. 
 
 ?i 
 
 -2n 
 
 , /—- k-i + i . 2mr 
 
 2ra-l 
 
 But by 655, we have the formula, 
 
 comparing then the coefficients of i/ZY we find 
 
 , . , . (2n-l)Tr 2mr 
 
 Uin =t ~k sm — ^ cosec — — - 
 
 o o 
 
 . J . (n+l) TT 
 = 1 - ksm - — —^ — 
 
 nir 
 3 "^''"-3' 
 
 as in the article just cited. Or we might have compared the real 
 parts (those independent of the ordinary y/ -\),\x\ the same ge- 
 neral formula, and so have obtained the same result, under the 
 form, 
 
 k-i + j . 2mr k-i+j,. . 2mr , . 2(w-2)7r, 
 — ^.«»->sm-^=— ^{»sin — + Asm-A-3 ); 
 
 because this last product would easily be found to be 
 = qz'^'"''^ - (real part of) yf ^'^-^ 
 
 Or we may write, at once, 
 
 and the imaginary symbol will still be found to disappear, and 
 the same real result as before be obtained, when the proper re- 
 ductions are made, in the manner indicated above. 
 
 657. It must, however, be confessed that such calculations as 
 these with biguaternions, or with mixed expressions involving 
 ijk and \/-l, are sometimes very delicate, and require great cau- 
 tion, from the following circumstance, to which nothing analo- 
 gous occurs in the theory of pure or single or real quaternions. 
 This circumstance is that the product of two biquaternions 
 may vanish, without either factor separately vanishing. To give 
 a very simple example, the product 
 
 {k+ y/-\){k- v'-l)-^-+l=0. 
 
 While k+ v'- 1 and k- ^ -\ must each be considered as different 
 
LECTURE VII. 651 
 
 from zero^ if k be still one of the peculiar symbols of this calcu- 
 lus, while v' - 1 is the old imaginary. We might therefore write 
 
 {k + V^^)-i = (^ - V^) q, 
 
 where g' is an arbitrary quaternion, not necessarily equal to zero. 
 In the recent question, we might in like manner have written, 
 
 q being an arbitrary quaternion, reducible to the real kind : be- 
 cause, by the rules of this calculus, we have 
 
 'k -i+jy _ 
 
 V3 J 
 
 And thus it might appear that an arbitrary addition would be 
 made to the value lately found for u^n'^- Such arbitrary addition 
 might indeed present itself, in some other investigation with bi- 
 quaternions. But in the example of the foregoing article, we 
 knew, by the nature of the question, that the final and reduced 
 expression for the continued fraction, Ux, could contain no ima- 
 ginary term. We were therefore, in this case, justified in adopt- 
 ing those reductions, which caused the symbol -vZ-l to disappear, 
 and which we found to be consistent among themselves. Still 
 the remark of the present article may shew, how cautiously it 
 might become needful to proceed in other cases, where no such 
 check was previously known to exist, on the results of operations 
 with biquaternions, in which anything like division is involved. 
 658. In the example of art. 653, it was supposed that Uq = 0; 
 But if we had considered, more generally, the continued fraction, 
 
 where c = Mq = any real and given quaternion, while ^i and q^ 
 shall still be supposed to denote, as in 653, the two real roots 
 of the quadratic equation q'^=qi-\-j^ we might then calculate the 
 value of Ux by the two last formulae of 651, combined with the 
 following initial value of Vx : 
 
 I'o = (9-2 + c) (q, -f cy\ 
 
652 ON QUATERNIONS. 
 
 And because the quadratic gives, 
 
 ^3 = q2i + qj ^ (^qi + J) i + ^j^ q (J -l)-k, 
 
 and in like manner, 
 
 q^ = -q^i-q{\ +J) = -qj+k, 
 q^ = -q^j+qk = -j^=l, 
 
 we see that the common value of the sixth powers of all the six 
 roots q is unity, a result which may easily be otherwise proved, 
 from the expressions assigned in former articles, for each of those 
 roots in particular. Thus, 
 
 qi — qi = -t} 'Vx+e — '^xi ^x+6 — Ux', 
 
 and the values of the continued fraction form still a pen'ot^ of six 
 terms. Indeed if it happen that the quaternion c is a real root 
 of this other quadratic equation, 
 
 c^ + ci =j\ 
 so that either 
 
 c=^-qi = -l(l+i+j-k), 
 or 
 
 c = -q^=-l(-l+i-j-k), 
 
 we shall then have 
 
 = c,Ux=l4-]c=^c; 
 
 and the value of the continued fraction will become, in this case, 
 constant. But for every other real value of c, the fraction circu- 
 lates, as above. 
 
 659. The following is an example of a continued fraction of 
 the foregoing form, which converges generally to a limit, instead 
 of circulating in a pei'iod. Let there be now, 
 
 c still denoting some real and given quaternion, as the initial va- 
 lue of the fraction. The quadratic in q becomes now 
 
 q'^ = Qqi + 1 0/, 
 
LECTURE VII. 653 
 
 of which the two real and the four imaginary roots have been al- 
 ready assigned. Attending only to the former, we have by 
 645, 651, 
 
 qy = 4z - 2k, q2=i- Ik, 
 Vo={c + i- 2k) (c + 41 - 2ky\ 
 v^={i-2kyvo{4i-2k)-^, 
 
 Mx ^ (1 - V^)- 1 {V,r: ^i - 5-2) • 
 
 Here 
 
 T{Ai-2k)=2^J5^, T(i-2k)=V5; 
 and therefore 
 
 Tq,= 2Tq,; T?;., = 2 - T?;o. 
 
 If we suppose that c is a real root of this new quadratic, 
 
 c^ + 5ci = lOj, 
 so that either 
 
 c = -qi = 2k — 41, or c = - qi=2k- i, 
 
 then in the first case we shall have 
 
 Vo = GO , V3; = cc , Uj. = -qi=2k- 4i, 
 
 and in the second case, 
 
 Vo =0, Vx = 0, Ux = -q2 = 2k- L 
 
 In these two cases, then, the value of the continued fraction re- 
 mains constant (as in the example at the end of 658) ; in fact 
 these two real values of the initial quaternion c give 
 
 In fact if we assume UQ = 2k- 4i, we find 
 
 Ml = lOj {5i + Mo)-' = lOj {2k + 0-1 = - 2j (2k + i) ^2k- 4i, 
 
 and similarly for all subsequent values of m.^ ; or if, on the other 
 hand, we assume the initial value, Uo = 2k- i, we find 
 
 Ml = lO;' (2^ + 4«)-i = 5j {k + 2i)-^ = -j {k + 2%) = 2k-i, 
 
 and the fraction will still be constant. In every other case, 
 that is, for every other assumed and real quaternion value of c, 
 
654 ON QUATERNIONS. 
 
 the value of the fraction will vary^ v^^i being always different 
 fvovciUx', ^M^ this value \\\\\ converge to a definite quaternion, 
 namely, to 2h - i, as its limit : for we shall have, 
 
 It might then, perhaps, seem not too fanciful to say, that these 
 two values, 
 
 2k - i. and 2h - 4i, 
 
 correspond respectively to positions of stable and unstable equi- 
 libf'ium^ for the continued fraction u^ which has been the subject 
 of the present article. If we set out with assuming either, we 
 shall never leave that assumed position, or value : but if we begin 
 with any other Uq, the fraction will tend indefinitely to become 
 equal to the stable value, 2h - i, and will not tend to equality with 
 the unstable value, Ih - Ai. 
 
 660. If the initial value c, of the fraction considered in the 
 foregoing article, be assumed equal to a vector po perpendicular 
 toy, so that 
 
 U(j= C = pa= IXq + fiZQ, 
 
 where Xq and Zq may be regarded as the rectangular co-ordinates 
 of a point Pq in the plane oi xz ; then 
 
 ,„.,,_, . . ,, . lO{(5 + a?o)A;-0o«} 
 «.= iq;{(5 + ^.). + ^rf) . = _i_^Z--^; 
 
 SO that we may write, 
 
 Ml = pi = ixi + kzi = the vector of Pi, 
 
 the new or derived point Pi being, like the assumed point Pq, in 
 the plane of xz or of ik, but having its coordinates therein deter- 
 mined by the two expressions, 
 
 - 10 -^0 _ 10(5 + a;o) 
 
 {5 + Xof + Zo''' ' {5 + x^y^z,^' 
 
 In like manner, from this^rs^ derived point Pi, we may pass to 
 a second derived point Pj, of which the vector and the co-ordinates 
 are, respectively, 
 
 U2 = pz= 1X2 + hz^, 
 
X2 
 
 LECTURE VII. 655 
 
 - lO^Ti 10(5 + a;i) 
 
 (5 + XiY + Zi" ' (5 + A'i)2 + z-^ ' 
 
 so that, by substitution of the recent values for a?i, z^-, we have 
 these other values : 
 
 . -4(^0 + 5) .^ 9^ 4(00-2) 
 
 (aro + 5)^+(0o-2)^' (:ro+5)2 + (2ro-2)^ 
 
 If we assume a?o = -4, ^0 = 2, we shall have, by these formulae, 
 a;i = -4, 01 = 2, a;2 = -4, 2^3 = 2, &e.; or if we assume Xq = -\^ 
 00 = 2, then a:;i = - 1, 2^1 = 2, a?2 = -l, ^2 = 2, &c. ; but if we begin 
 with any other initial values of x and z, the results of the suc- 
 cessive substitutions will give a series of varying values for those 
 co-ordinates : for the equations 
 
 -lOz \Q(6 + x) 
 
 (5 + xy + 2^' (5 + xy + 0^' 
 
 give 
 
 (5 + a?) a; +02 = 0, (5 + a;)2 + 22^5(5 + a;), 
 
 and therefore 
 
 = 2, a?2 + 5ar + 4=0, a;=-l, or = -4. 
 
 We may however prove, even without quaternions, what the 
 analysis of the foregoing article enables us at once to foresee, 
 namely, that if Fi and Fg be the. two fixed points whose co-ordi- 
 nates are respectively (-4, 2) and (- 1, 2), then any other as- 
 sumed initial point Po will have its ultimate derivative at the lat- 
 ter of the two fixed points, as a limiting position : or in symbols 
 that 
 
 Poo = ^2- 
 
 In fact we have 
 
 P^2 ,^ (^^ + 4)2 + (2-^ _ 2)2, ^^z = (x^ + 1)2 + (^Zo - 2)2, 
 
 and similarly, 
 
 p^i' = (^1 + 4)2 + (01 - 2)2, 7^,-^ ={x, + \y+ (01 - 2y. 
 
 But 
 
 a:i2+0i2=lOO{(5 + a;o)2 + ^o')"M 
 
 and hence, after a few other easy reductions, we find that 
 
656 ON QUATERNIONS. 
 
 {x, + 4y+(^^-2y 
 
 {Xo + 5)- + 2:0^ 
 
 (., + l)'+(z.-2)= = ^f(^"^'>'^(^"-^)' 
 
 (0^0 + 5)^+2:0' 
 
 and therefore that 
 
 PiF2 -j-PiFi = iPoF2-;- PoFi. 
 
 Hence 
 
 P«F2 ^ P„Fi = 2-«Po F3 -f- PoFi ; 
 
 and therefore, unless it happen that the assumed initial point co- 
 incides with the fixed point Fi, the derived point p„ must tend to 
 coincide with the other fixed point Fo; or in symbols, at the limit, 
 
 p„F3 = 0, and p^ = F2, as above. 
 
 And the law of this approach, of the point p^ to its limiting po- 
 sition, is at the same time seen to be the contimial bisection of 
 the quotient, of its distances from the two fixed points. 
 
 661. The recent calculations with co-ordinates, by which 
 this law and limit have been established, are no doubt suflaciently 
 easy : yet 1 think that they cannot compete in simplicity with 
 the quaternion method, which expresses both (and indeed also 
 other and more general results, depending on other suppositions 
 respecting the initial value c), by the formula of 659, 
 
 where the quaternion Vo is the initial quotient, and Vj^ is the va- 
 riable quotient, of the two vectors drawn from the fixed points 
 to the point p. The formulse of the article just cited give also 
 easily, 
 
 and therefore 
 
 V2n = 2-^"Vo', V.n^i = 2-~"Vr, 
 
 Uy3n = Uro, Uz;2„,+i = Ut'i. 
 
 An interesting geometrical interpretation may be assigned to 
 these last results. For, from the geometrical significations just 
 now stated, of the quaternions Vo, v^:, combined with the princi- 
 ples of art. 321, &c., it may be easily inferred that the alternate 
 
LECTURE VII. 657 
 
 points, Po, P2, P4, . . F2«j • . are all situated on one common circle 
 passing through the two fixed points ; and that in like manner, 
 the other series of alternate points, Pj, P3, P5, &c., are all situated 
 on another circular circumference^ which contains also the two 
 fixed points Fi and F2. Accordingly, we may confirm this result 
 by the method of co-ordinates, by shewing that the values found 
 in 660 for x-i and 2^2 give, 
 
 5/2 ~r ^2 "• *^^2 ^0 ' ^0 ' *^Xq 
 
 Zi — J!i Zq — Z 
 
 As a numerical example, if we place the initial point Pq at the 
 origin of vectors, we shall have the following co-ordinates, for 
 points of the two alternate series: 
 
 ,^ ^x /-20 50\ /-500 1050\ 
 
 P,= (0,2); P3=(^:^^2); Pa = (=^, 2 
 
 so that Po, P2, and P4, are situated on the circle of which the 
 equation is 
 
 a;2 + z'- + 5x = 0, 
 
 and which evidently passes through the fixed points (- 4, 2) and 
 (-1,2); while Pi, P3, and P5 are on the straight line 
 
 z=2, 
 
 which passes through the same pair of fixed points, and must be 
 regarded as the limit of a circle. 
 
 662. As regards the general relation between the two circu- 
 lar loci, considered in the preceding article, it may suffice to 
 observe that if o be the origin of vectors, and if we introduce the 
 symbols ki and jc2 to denote the vectors of the two fixed points, 
 making 
 
 Ki = 0Fi = 2k-ii, K2=OFz = 2k-i, 
 
 we shall have, by 659, 660, 
 
 ^0 = (po - Ks) (po - Kl) 'S Vi = (CsVoKf ^ = Kf ^ . K2V0K1, 
 
 and therefore, 
 
 2 u 
 
658 ON QUATERNIONS. 
 
 UVi = -Uk2 U. VoKi =U. K2Ao"S 
 
 where Xo=?^oK:i=a certain vector oLq in the plane oi ik, namely 
 (see the Fourth Lecture) the fourth proportional to the three 
 vectors p^- ki, po-K^, and ki, or to ki - po, Ko-poj and ki, that is, 
 to PoFi, P0F2, and OFi, which are lines in the same given plane. 
 But we have also (compare 651, 661) in the present question, 
 
 Vi = (pi - K2) (pi - Ki)-' = (ico - /oi) ((ci - pi)-' = PjFs -h PiFi ; 
 
 thus, equating the angles of the two quaternions Vj and k2Xo"S 
 which have been proved to have equal versors, we find that the 
 angle FiPjFa in the second circular segment, or the angle sub- 
 tended at the derived point Pj by the fixed line F1F2, or the rota- 
 tion from PjFi to PiFo, is equal to the rotation from Ao to K2, or 
 from oLo to 0F2 ; while the rotation from ki to Xo, or from ofi to 
 OLo, is equal (by the above-mentioned proportionality) to the ro- 
 tation from Ki -po to Ko -po> or from PqFi to P0F2, or to the angle 
 FiPoFj in the first circular segment, which the same fixed line 
 FiFo subtends at the assumed point Pq. But the sum of the two 
 rotations, from ki to Xo and from Xo to 1C2, is equal to the rotation 
 from Ki to K2, or from oFi to gFj, or to the fixed angle F1OF2 which 
 the same fixed line subtends at the origin o. The following is 
 therefore the required relation between the two circular loci, or 
 between the angles subtended therein, by the common chord 
 F1F2 : " the sum of these two angles, in the two circles, or in those 
 segments of them which contain alternately the successive and 
 derived points p, is equal to the fixed angle at the origin ;" or in 
 symbols, 
 
 F1P0F3 + F1P1F2 = F1OF2. 
 
 If this formula should give a negative value for an angle, the 
 fixed angle F1OF2 being considered as positive, it would imply 
 that the derived point which is the vertex of that angle lies in a 
 segment situated at the opposite side of the fixed line F1F2. 
 
 663. The following is a shorter mode of obtaining the same 
 result. In general, let k, k be any two vectors, and v any qua- 
 ternion coplanar with k, so that 
 
 S . i?K = 0, vk = - K . vk = kKv. 
 Then 
 
LECTURE VII. 659 
 
 and therefore, if ic'be also a line in the plane (or perpendicular to 
 the axis) of v, so that S . W = 0, we shall have the formula, 
 
 Z . kVk'^ + /. V = A . k'»c"S 
 
 where the angles are to be interpreted as rotations, and added 
 with their proper signs, as such. Applying this result to the 
 expressions for Vq, Vi, assigned in the foregoing article, we might 
 infer at once, that (with this interpretation of the angles, as ro- 
 tations, which will not always coincide with that adopted in the 
 Fourth Lecture) the following relation holds good : 
 
 zvo+ z Vi = z.K2»cr^; 
 
 which agrees with that recently found. As an example, when 
 we suppose that p© is at o, or that po = 0, then Vq = kzKi''^, and the 
 last formula gives zz;i = 0,- and accordingly we saw in 661 that 
 in this particular case the alternate derived points Pi, P3, P55 are 
 situated on the straight line F1F2, prolonged through Fo, since 
 we had, for the co-ordinates of each of them, x>-\, 2 = 2. But 
 I cannot say that such conjirmations by co-ordinates add any- 
 thing to my own conviction of the truth of a conclusion obtained 
 by calculation with quaternions. 
 
 664. It may b€ satisfactory, however, to generalize the con- 
 struction of art. 660, for deriving the point Pi from Pq, or Pg from 
 Pi, &c., and at the same time to state it, and its results, under a 
 more purely geometrical form, and one which shall be indepen- 
 dent, as to its expression, of both co-ordinates and quaternions. 
 And you will (I think) have little difficulty in now perceiving how 
 the consideration of the continued fraction 
 
 ^. = (— )po, 
 
 where a, j3, po, px are real vectors, /3 being perpendicular to the 
 other three, and the condition a* + 4j3^ > being satisfied (see 
 art. 649), conducts to the following results, under the form of a 
 geometrical theorem, or rather series of theorems, which seem 
 to be somewhat new in their kind. 
 
 QQ5. Let c and d be two given points, and p an assumed 
 
 2 u 2 
 
660 ON QUATERNIONS. 
 
 point. Join dp, and draw cq perpendicular thereto, and towards 
 a given hand, in the assumed plane cdp, so that the rectangle 
 CQ . DP may be equal to a given area. From the derived point 
 Q, as from a new assumed point, derive a new point r, by the 
 same rule of construction. Again conceive that s is derived from 
 R, and T from s, &c., by an indefinite repetition of the process. 
 Then, if the given area he less than half the square of the given 
 line CD, and if a semicircle (towards the proper hand) be con- 
 structed on that line as diameter, it will be possible to inscribe a 
 parallel chord ab, such that the given area shall be represented 
 by the product of the diameter cd, and the distance of this chord 
 therefrom. We may also conceive that b is nearer than a to c, 
 so that ABCD is an uncrossed trapezium inscribed in a circle, and 
 the angle abc is obtuse. This construction being clearly under- 
 stood, it becomes obvious, P', that because the given area is equal 
 to each of the two rectangles, ca . da and cb . db, while the an- 
 gles in the semicircle are right, then, whether we begin by assum- 
 ing the position of the point p to be at the corner a, or at the 
 corner b, of the trapezium, every one of the derived points, q, r, 
 s, T, &c., will coincide with the position so assumed for p, how- 
 ever far the process of derivation may be continued. But I also 
 say, 11"*^, that \i any other point in the plane, except these two 
 Jixed points, a, b, be assumed for p, then not only will its suc- 
 cessive derivatives, q, r, s, t, . . be all distinct from it, and from 
 each other, but they will tend successively and indefinitely to 
 coincide with that one of the two fixed points which has been 
 above named b. I add, 11 P'"^, that if, from any point t, distinct 
 from A and from b, we go hack, by an inverse process of deriva- 
 tion, to the next preceding point s of the recently considered se- 
 ries, and thence, by the same inverse law, to r, q, p, &c., this 
 process will produce an indefinite tendency to, and an ultimate 
 coincidence with, the other of the two fixed points, namely, a. 
 IV"\ The common law of these two tendencies, direct and in- 
 verse, is contained in the formula, 
 
 QB . PA cb 
 
 = — • = constant ; 
 
 QA . PB CA 
 
 which may be variously transformed, and in which the constant 
 
 J 
 
, LECTURE VII. 661 
 
 is independent of the position of p. V"". The alternate points, p, 
 R, T, &c., are all contained on one common circular segment ape ; 
 and the other system of alternate points, q, s, &c., has for its 
 locus another circular segment, aqb, on the same fixed base, ab. 
 VI*"^. The relation between these two segments is expressed by 
 this other formula, connecting the angles in them, 
 
 APB + AQB = ACB J 
 
 the angles being here supposed to change signs, when their ver- 
 tices cross the fixed line ab. The symbols a, b, c, p, q, r, s, 
 T, of the present article correspond evidently to the less general 
 Fi, F2, o, Po, Pi, P2, P3, P4, P5, of 660, &c. It has not been thought 
 necessary, at this stage, to draw any illustrative diagram. 
 
 QQQ. If the given area under dp and CQ were greater than the 
 half square of the given line cd, there would then be no tendency 
 of the derivative points to converge to any limiting position; the 
 points A, b, of the recent construction becoming then imaginary : 
 or the right line ab no longer intersecting the semicircle on cd 
 (compare 649). This answers to the case where a*+4j3^<0, 
 Ta^<2Tj3, for which we saw (in 649) that the two vector roots 
 ofthe quadratic equation q^ = qa + j3 became imaginary ; and it may 
 be exemplified by the continued fraction of art. 658, for which it 
 was shewn that there is circulation instead of convergence. Geo- 
 metrically, if the rectangle cq . dp be equal to the square on cd, 
 instead of being less than its half, the construction of the forego- 
 ing article gives a. period of six points (of which one may go off 
 to infinity), instead of giving a series of points, tending to a li- 
 mit. In the case of transition from real to what may be called 
 imaginary convergence, namely, in the case when a* + 4/3- = 0, or 
 when the rectangle is just equal to the half square, so that the 
 line ab touches the semicircle, some diflBculties of a peculiar kind 
 present themselves, on which I cannot enter now. 
 
 667. But in connexion with them, and with the whole sub- 
 ject recently discussed, I may remark that the quadratic equa- 
 tion q^ = qa+ (5 of 649, where a and j3 denote two real and rect- 
 angular vectors, will be found to conduct (compare 658) to the 
 following biquadratic equation, 
 
 q' = q''a'+(5\ 
 
662 ON QUATERNIONS. 
 
 which is satisfied by the imaginary as well as by the real quater- 
 nion roots q of the former quadratic equation. In fact, the qua- 
 dratic gives, 
 
 q^=q^a + q^ = {qa + ^)a + q^ = q {a? + /3) + /3a ; 
 g* = q^ (a^ + j3) + gjSa = ^ (a^ + a/3 + j3a) + jS {a" + /3) 
 = qa" + jSa^ + ^^ = a^ {qa + j3) + /3^ = a^^^ + /3^ 
 
 This new and biquadratic equation in g is only oi quadratic form, 
 relatively to q^ \ and on account of the scalar character of its co- 
 efficients a? and /3% it gives, as in algebra, 
 
 (2g2_a0' = a^ + 4/3^. 
 
 But in the critical case just mentioned, where 
 
 a* + 4/32 = 0, or Ta^ = 2T/3, a^ = - 2T/3, 
 
 we are ?20^ to infer that 
 
 2^2_a2^0, 
 
 except for the rea/ roots of the original quadratic, which roots may 
 in this case be said to he four real and equal vectors ; namely, 
 by the formulae I. or II. of the lately cited article 649, 
 
 ^=ia+a-i/3=i(l + U/3)a, 
 
 these two last expressions becoming equal here, because 
 
 a-^/3=-/3a-i=-a--T/3U/3a = iU/3.a. 
 
 For besides these real and equal roots, the formula III. of 649 
 affords also in this case the two imaginary or hiquaternion solu- 
 tions included in the expressions, 
 
 5 = (l-U/3){ia±/^/T3"} = S^ + Vg; 
 
 S^- being a pure imaginary scalar (compare 637, 640), namely, 
 
 Sg = ± /~1 \/T^, giving S^2 = - T/3 = la^' ; 
 
 and Yq a mixed imaginary vector, of the form 
 
 while p'and jo'are two rea/and rectangular and equally long vec- 
 tors, namely, 
 
LECTURE VII. 663 
 
 so that 
 
 Hence, for these two biquaternion values of q, we have 
 
 = Yq^ = (p'±V^py; 
 2q^-a^=4Sqyq; 
 and finally 
 
 (2^'-a0^=0, as above, 
 
 without 2q^ - a^ itself here vanishing. These results, so far as 
 they relate to biquaternions, will soon be stated more generally. 
 668. The analysis of articles 651, 659, &c., enables us easily 
 to prove the following general theorem : if a and b denote any 
 two real quaternions, and if c be any other real quaternion, which 
 is not a root of the quadratic equation 
 
 c^ + ca= b, 
 then 
 
 b 
 
 c= c. 
 
 c being that real root of the last-mentioned quadratic, which has 
 the lesser tensor. In the case of the continued fractions consi- 
 dered in 653, 658, the two real roots of the quadratic in c had 
 equal tensors (each = 1) ; and the recent theorem of convergence 
 was therefore in that case inapplicable, being replaced (as we 
 have seen) by a certain circulating property. In the more ge- 
 neral case, when such equality of tensors does not exist, if we 
 change a, b, c, respectively, to 
 
 a + ia' -vja + hd'\ b + ib' +jb" + kb'\ c + ic +jc" + kc'", 
 
 where the twelve new symbols aa'a"d"bb'b"b"'cc'c'c"' are supposed 
 to denote so many real scalars, whereof a . .b . . may be supposed 
 to be given, and c . . to be assumed; if we also make, for abridg- 
 ment, 
 
 e2 = (a + cy + (a' + cj + («" + c"y + {a!" + c'y, 
 
 and then derive four new scalars Ci . . . from c ... by the formulae, 
 
664 ON QUATERNIONS. 
 
 Ci = e-2 { 5 (a + c) + Z.' {a + c') + b" {a + c") + b'" {a" + c") ] , 
 Ci' = e-2 { 6' (a + c) - 6 (a + c') + 6'" (a" + c") - b" {a" + c'") ] , 
 Ci" =e-^ b" (a + c)- b'" (a' + c)-b (a + c") + b' \d" + c') ] , 
 c"' =€-'{ h" (« + c) + y {d + c) - b' {a +c")-b {a'" + d") ] ; 
 
 and so proceeding, derive a new system of four scalars, c^ . . . 
 from a . . 6 . . Ci . . , as Ci . . have been derived from a . .b . .c . . , 
 and anotlier new system from this, 8ic.,ad infinitum, we have the 
 following Theorem: ^^ the ultimate result of the process thus de- 
 fined will generally be one fixed and limiting system of four 
 values^ 
 
 namely, that one of the two real systems of values of these last 
 symbols, satisfying the system of the four equations 
 
 C=^-^{/^(« + C) + 6'(a'+C') + 6"(«"+C'") + ^"'(«"'+C"")) 
 
 C' = &c., C" = &c., C'" = &c., 
 
 where jB ^ = (a + C)^ + (a' + CJ + (a" + C")' + {a" + C'")', 
 
 which gives the lesser of two real values to the following other 
 sum of four squares: 
 
 669. We may here dismiss the consideration of that class of 
 continued fractions which has been the subject of several recent 
 articles: but a few more words must be said o"n the theory of the 
 biquaternions. In general (see again 637, 640, 644) a biquater- 
 nion, such as the following, 
 
 Q = q+^~lq', 
 
 may be decomposed into a scalar part, of the form 
 
 SQ=w + V- 1 w, 
 
 and a vector part, of the form (compare 667), 
 
 where 
 
 w = Sq, w = Sq, p = V^, p = Yq' ; 
 
 w and w denoting here two real scalars, p and p two real vec- 
 
LECTURE VII. 665 
 
 tors, and q, q two real quaternions. And by the same analogy 
 of nomenclature, we may agree to call an expression of the form 
 w+V^-l 2^'aBiscALAR; and an expression of the form p +\/- 1 p' 
 a BivECTOR ; so that we shall have this general formula of 
 decomposition : 
 
 BiQUATERNION = BiSCALAR + BlVECTOR ; 
 
 the grand distinction, in calculation, between these two compo- 
 nent parts of a biquaternion being, that a hiscalar, although 
 imaginary as a number, is yet commutative in multiplication with 
 every other factor, so far as regards arrangement in a product 
 (like the -vZ-l of 644, or the z of 654); whereas a bivector, al- 
 though it may be said to denote an imaginary line in space (an- 
 swering, for instance, as in 649, 654, to geometrically unreal in- 
 tersections of loci), is yet (like the real vectors of the present 
 calculus) in general non-commutative as a factor. We may also 
 write, by analogy to a formula of 408, 
 
 KQ=SQ-VQ; 
 
 and may say that the conjugate, or, more fully, that the Biconju- 
 gate of a biquaternion is equal to the biscalar, minus the bivector. 
 With these enlarged meanings of the symbols S, V, K, it is easy 
 to extend to biquaternions a great variety of formulae, already es- 
 tablished for quaternions ; for instance, those of art. 499, all of 
 which are frequently useful; and the following (compare 190, 
 519), which we shall shortly have occasion to employ : 
 
 K.i2Q = KQ. Ki2; Kn = n'K. 
 
 670. Pursuing the same train of notation and nomenclature, 
 1 propose to write, by analogy to a formula of article 409 (or 
 432), 
 
 and to call the TQ thus found the tensor, or more fully the Bi- 
 TENSOR, of the biquaternion Q; so that we shall have the gene- 
 ral relation, 
 
 Bitensor squared = Biscalar squared - Bivector squared. 
 It is to be observed that the square of a bivector, like that of a 
 
666 ON QUATERNIONS. 
 
 biscalar, is generally a biscalar ; the square of a bitensor is there- 
 fore also in general a biscalar, or of the mixed imaginary but or- 
 dinary form, 
 
 where u and u are reals^ of the ordinary algebraic kind; it is 
 therefore always possible, by the usual rules of algebra, to ex- 
 press the bitensor itself under the analogous form, 
 
 where t and t' are reals, satisfying the two conditions, 
 
 And because these two conditions admit generally two solutions, 
 or leave the signs of t and t' ambiguous, although related, I pro- 
 pose to remove this ambiguity, for the purposes of our calculus, 
 by defining that the real part of a bitensor is never to be nega- 
 tive. Indeed it may happen that this real part vanishes, by the 
 square of the bitensor becoming equal to a real and negative 
 scalar ; to meet which case, I propose to define that the coeffi- 
 cient of \/- 1 in the imaginary part of a bitensor is to be taken 
 positively, when the real part of the bitensor vanishes* For in- 
 stance, the biquaternion expressions of article 646 give, 
 
 5 /-25 25 5\ ,^ 
 
 an 
 
 4 V 4 4 4^ 
 T^4^ = 10, T^5^=T^6^ = -10; 
 
 d therefore (v 10 being regarded as positive), 
 
 T^3 = T^, = \/T0, Tq, = Tq, = \/~\ V To". 
 
 In general the notations of the present and preceding articles 
 give, 
 
 T Q2 = («^; + \f~\ w'y - (p + /^ pj ={t+ \/^ t'Y 
 
 = w''-p'- w"" + p'=^ + 2\/n^ {WW - S . f>p) ; 
 that is (compare 538), 
 
LECTURE VII. 667 
 
 { T (^ + V^g') }' = Tq'- Tq'^ + 2 \/~l S . qKq\ 
 because 
 
 q = w + py q' = w + p, Kq' =w' - p. 
 We may then write, generally, 
 
 and shall have, to determine this real and positive scalar ^, the 
 formula, 
 
 2^2 = T^' - Tq' + { (Tg2 _ Tq'y + 4(S . q^qj] K 
 
 We have also, generally, this other and simpler equation, 
 
 QKQ = (TQ)^ 
 
 so that the product of two conjugate hiquaternions is equal to 
 the square of their common hitensor : vehich may be compared 
 with a result of the lately quoted article 409, or of the earlier 
 article 163. We may also agree to write (compare 90) the ge- 
 neral formula, 
 
 Q = TQ.UQ=UQ.TQ; 
 
 and to say that the quotient of a biquaternioUf divided by its hi- 
 tensor, is generally the versor, or, more fully, the Biversor, of 
 that biquaternion. 
 
 671. A large number of other general formulae may be ex- 
 tended in like manner to biquaternions ; especially all those which 
 depend only on the symbolic rules for calculating with scalars 
 and vectors (v - 1 being still treated as a scalar), including the 
 commutative and associative principles of addition, and the dis- 
 tributive and associative principles of multiplication; which prin- 
 ciples have been so fully illustrated, and indeed proved (as theo- 
 rems) in earlier articles, in connexion with their geometrical 
 significations, while only real (or geometrically interpretable) 
 quaternions were involved: whereas they are now defined to hold 
 good also, for certain new or extended forms, considered as crea- 
 tures and subjects of calculation. Among these extended results, 
 or generalized ^rmMte, it seems worth while to notice here the 
 following : 
 
668 ON QUATERNIONS. 
 
 (T.RQy = {TRy{TQy', 
 
 where Q and R may denote any two biquaternions. When a 
 corresponding formula was proved in article 189, for any two 
 real quaternions, it was done, at least partly, by an appeal (as 
 just now hinted) to the geometrical meanings of the acts of ten- 
 sion^ which were to be compounded and compared. But because 
 the acts ofhitension^ to be now combined, are geometrically ima- 
 ginary (or at least hitherto uninterpreted), we must employ some 
 symbolical processy such as the following, which depends upon 
 the final formulae of the two foregoing articles, 
 
 (T.i2Q)^=i2Q.K.i2Q=i2.Q.KQ.Ki2 
 = R{T€iyKR = R}LR.{TQ)~={TRy{TQ)\ 
 
 Or we might observe that 
 
 (T.i2Q)2=(S.22Q)2-(V.i2Q) 
 
 and that 
 
 S.i2Q = Si2SQ + i(Vi2VQ+VQVi2), 
 
 V./JQ = Si2VQ+Vi2SQ + i(Vi2VQ-VQVi2); 
 
 whence 
 
 (S.i2Qy=Si2^SQH2S22SQS.Vi2VQ 
 
 + 1 (Vi2 V qy + i (V orvRf + i vi?-- v €t ; 
 
 (V.RQy=SR'YCt + YR'SQ' + 2SRSQS.YRYQ 
 + i (YRY qy + i ( V QYRy - \YR'Y q\ 
 
 and therefore, 
 
 {T .RQy = (SR'-YR')(SQ'-YQ') = {TRy(TQy,asahove. 
 
 Hence, taking on both sides the square-roots, but prefixing now 
 an ambiguous sign, which it was unnecessary to do when we 
 were dealing only with real and positive tensors, we have, for 
 any two biquaternions, the formula : 
 
 T.RQ = ±TR.Ta; 
 
 and more generally, for any number of such factors, we may 
 write (compare 208), 
 
 TnQ = ±nTQ. 
 
LECTURE VII. 669 
 
 For instance, the bitensor of a power of a biquaternion can only 
 differ in sign (at most), from the corresponding poicer of the bi- 
 tensor. But such diiFerences of sign may arise, in the applica- 
 tions of the definition given in article 670, which will occasion- 
 ally require us to take the negative of a product of bitensors, in 
 order to obtain a new bitensor, with a real and positive part. 
 
 672. We saw in 667 that the square of a certain bivector va- 
 nished, without that bivector vanishing itself It must then be 
 possible (as in the case of that bivector for example), to have a 
 null bitensor of a biquaternion which is not itself equal to zero. 
 And it is easy to assign the conditions under which such a result 
 will take place. For by 670, if the biquaternion be Q = ^+ a/-1 
 q\ where q and q' are real quaternions, its bitensor will vanish 
 when, and only when, the two following equations are satisfied : 
 
 Tq = Tq'; S.qKq'=0. 
 
 But q'Kq'=Tq'^; thus, if we still suppose that Q itself does not 
 vanish, we are to make 
 
 qq'-' = S-'0 = T-n=i, q=iq, 
 
 and the expression for the biquaternion becomes, 
 
 I here denoting some real unit-vector. We may, however, trans- 
 form this expression, by writing 
 
 K = q'-^iq\ iq = ^'k, Q = ^' (k + /- 1) ; 
 
 where k, by 286, will denote another real unit-line. It is easy 
 to infer, as a corollary from this general theorem, or to prove by 
 a process more direct, that a bivector p + a/- 1 p' will have a 
 null bitensor, when the two real vectors p and p on which it de- 
 pends represent lines whose lengths are equal, and whose direc- 
 tions are rectangular ; or that 
 
 T(p+ /^ p') = 0, if T^ = Tp, and S.pp= 0. 
 
 Accordingly these conditions were satisfied in the case of article 
 667. 
 
 673. The following appears to be a remarkable example of 
 
670 ON QUATERNIONS. 
 
 the occurrence of biquaternions whose tensors are null. Sub- 
 tracting the expression in 641 for a root q of the quadratic equa- 
 tion q^ = qa-\^ h, from the analogous expression for another root 
 ^, which answers to another value w of w, supposed to corres- 
 pond to a different root of the cubic equation (636) in w"", and 
 dividing the remainder by \ {w -w), we find, after some easy re- 
 ductions, the following biquaternion value, 
 
 {w - w) {w~ - a^) {w- - a") '^ 
 
 where X is an imaginary vector (or bivector), namely, 
 
 \={w + w') V. a7 - 7 {lUW 4 a^) 
 
 + W'^ w' ^ {w"^ + WW +w'''- a^) aS .ay, 
 
 and fi is another bivector, on account of one only of the scalar 
 values of w, w' being real. Squaring and reducing, we obtain 
 the equation, 
 
 w" W- X" {w' - a')- 1 (w^^ -ay = w' w'^ f - {w" + w"" - aO ( S . ay)'. 
 
 But if we denote by vJ'- the third root of the equation =f{yf) 
 of article 636, regarded as a cubic, we have 
 
 W~ + w'"- + w'"" = c + a^ (w2 + w'^) w"^ + 10"" w'^ = ca^ + y^ ; 
 w'^ W^ w"^ = (S . ajy. 
 
 Eliminating therefore w"'^ and c, we are conducted to the rela- 
 tion, 
 
 w^ w"^ {w^ - cf) (w"^ - aF) = w"" w'"^ 7^ - (w;2 + iv"^ -a^) (S . 07)^. 
 
 Comparing, we perceive that 
 
 X^ = (w' - a'Y (w' - a^y ; or, ^^ = 1 . 
 
 Thus, 
 
 and finally 
 
 TQ:=sa'-Ya'=l-fi'=o 
 
 TQ=^0;T(q-q) = 0. 
 
 If, then, q and q be (as above) two different roots of a quadra- 
 tic equation in quaternions, of the form q^=qa + b, which corre- 
 spond to two different roots of the auxiliary and cubic equation 
 
LECTURE VII. 671 
 
 (636, 637), their difference, q - q, is a biquaternion with an 
 evanescent tensor. For example, if we take the six roots as- 
 signed in 645, 646, of the particular quadratic q^ = 5qi + 10/, we 
 shall easily find that the twelve following differences, 
 
 qz - qi, qz - q^, qi - qi, qi - qz, 
 qs - qi, q^ - q-z, qs - qu qe - qz, 
 qs - qs, qi - qi, qe - qs, q^ - qi, 
 
 are biquaternions of this particular kind ; thus 
 
 ^3 - 2'i = - I «■ - i^ + i (1 +j) \^~5, 
 
 -'•(-|J-(-lJ^(TXT)-»'Tto-..) = o. 
 
 But the tensors of the three following differences of pairs of 
 roots of the quadratic (each pair answering to only 07ie root of 
 the auxiliary cubic), 
 
 qz - qi, qi - qs, qe - q^, 
 
 will be found to be different from zero. A more general verifi- 
 cation may be had from the formulae of 649. 
 
 674. We saw, in 657, that the product of two biquaternions 
 might vanish, without either ^c^or vanishing separately. If we 
 now propose to inquire into the general conditions under which 
 such a result may occur, we may proceed as follows. Breaking 
 up the imaginary (or biquaternion) equation, 
 
 (r + /^ r') (q + \/-[ q) = 0, 
 
 into the two real equations, 
 
 rq - r'q' = 0, rq' + r'q = 0, 
 
 and making for a moment r'q = s= a. real quaternion, which in 
 the present question is different from zero, we find, 
 
 q = r''^s, q=-r~^s, {rr'''^ + r'r~^)s = 0, 
 
 y+rV-1 =(1 +^^-1)^' g' + g'V-l = -r-^{L+ y/ -\)s; 
 
 so that the evanescence of the product may be said to depend on 
 the identity, 
 
672 ON QUATERNIONS. 
 
 (1 + t^-\)(t+ V-1) = (1 + t^) V -1 + 1 (1 + /^O = 0, 
 
 where \/-\ is still the ordinary symbol of that form, and i is a 
 real unit vector, of which, by the principles of the present cal- 
 culus, the square is negative unity. We may, however, also 
 write (compare 672), ir = rK, where k denotes another real unit 
 vector; and therefore, with equal generality, under the conditions 
 of the present investigation, 
 
 r+r'^/-l = r{\ +k v'-l)> 
 
 g' + 5'V-l = (K+\/-l)^'; 
 and we see that when two biquaternion factors thus give a null 
 product (of the form + V-l)? ^^'''^°^* either separately va- 
 nishing, the tensor of each is zero. Conversely, it is obvious now 
 (see again 672), that wheii the tensor of a biquaternion vanishes, 
 that biquaternion may always be associated as a factor y whether 
 as multiplier or as multiplicand, with another, in such a way that 
 their product may be zero; and indeed that this may be done in 
 indefinitely many ways, because an arbitrary but finite biquater- 
 nion factor may be introduced at pleasure. It seems convenient, 
 therefore, to call biquaternions of this class nullific, or to say 
 that they are nullifiers; and it is worth observing, that the 
 reciprocal of such a nullifier is infinite. For in general we may 
 write, as a formula for the reciprocal of a biquaternion, the fol- 
 lowing : 
 
 {q + r ^J-l)~^ = {q-^ rq-^r)-^ -{r + qr^qy^^s/ -\ ; 
 where, by 672, we have now, 
 
 qr~^ = L, rq'^ - I, qr~^q = -r, rq'^r=- q; 
 and therefore, 
 
 (q + ryZ-iy = 00 + 00 V-1, if T(^ + rV-l) = 0. 
 We may also write this other general expression, 
 
 where, when the tensor of q + r\/ -I is zero, the denominator of 
 the fraction vanishes, without the numerator vanishing generally. 
 
LECTURE VII. 673 
 
 It is scarcely necessary to add, after what has been shewn above, 
 that whenever (as in 667) the square of a biquaternion vanishes, 
 the biquaternion itself must belong to the nullific class. But it 
 may be noted here that the equation 
 
 where g- is a given and real quaternion, admits generally of the 
 following imaginary or biquaternion pair of solutions, 
 
 in addition to the obvious and real pair, 
 
 Q = ±q. 
 
 675. To give now, although very briefly, for the subject is 
 of great extent, some notion of the manner in which biquater- 
 nions may be use/ill in geometry, let us resume the equation of 
 the unit sphere (168), p^ + 1 = 0, and change the vector ^o to a bi- 
 vector form, such as o-+tv/-1. The equation of the sphere then 
 breaks up into the system of the two following, 
 
 <t2-t2+1= 0, S. (77 = 0; 
 
 and suggests our considering a and r as two real and rectangular 
 vectors, such that Tt = (Ta^ - l)i Hence it is easy to infer that 
 if we assume o- 1| X, where X is a vector given in position, the new 
 real vector a- + r will terminate on the surface of a double-sheeted 
 an^ equilateral hyperboloid ; and that if, on the other hand, we 
 assume r || X, then the locus of the extremity of the real vector 
 er + r will be an equilateral but single-sheeted hyperboloid. The 
 study of these two hyperboloids is, therefore, in this way con- 
 nected very simply, through biquaternions, with the study of the 
 sphere rand thus it may be understood that the eminently simple 
 equation, |o'^=-l, of the latter surface, may be made to furnish 
 the solutions of many difficult problems, respecting other surfaces 
 of the second degree. I intend to reprint, as an Appendix to 
 this Course of Lectures, the abstract of a communication made 
 by me to the Royal Irish Academy in May, 1850, on the sub- 
 ject of the inscription of a gauche polygon in an ellipsoid, or in 
 a hyperboloid, when the n successive sides of the polygon are re- 
 quired to pass through the same number of given points of space, 
 
 2 X 
 
674 ON QUATERNIONS. 
 
 distinguishing between the two great cases^ where the number of 
 the sides is odd^ and where it is even. The Abstract referred to 
 has been drawn up in a geometrical form, but it is altogether a 
 translation mio'gQom^txicdX language of investigations conducted 
 with quaternions, and extended by the aid of biquaternions on 
 principles already indicated. I may just remark here, that cer- 
 tain formulae of the Sixth Lecture (in particular those of articles 
 335, 336) played an important part in the quaternion analysis 
 employed. Other geometrical uses of biquaternions will suggest 
 themselves to any one who will take the trouble to compare (for 
 example) the equations of 436 and 438, for the ellipsoid and dou- 
 ble-sheeted hyperboloid, namely, 
 
 (S.pa-)^-(V.pi3-0^ = l, 
 
 (S.pa-0^+(V.pi3-^)^=l, 
 
 and to see how the one passes into the other, by merely changing 
 /3 to /3 V-l 5 or to compare on the same plan either of the two 
 equations just cited, with the equation of the single-sheeted hy- 
 perboloid in 439, namely, with the following. 
 
 In general all such investigations as those of Poncelet, respect- 
 ing ideal secants in geometry, admit of being conducted by bi- 
 quaternions. 
 
 676. Without longer dwelling at present on the general 
 theory of biquaternions, it may be proper to give here some ra- 
 pid sketch of the manner in which the present calculus applies to 
 the inscription of a gauche polygon in the unit sphere, under con- 
 ditions of the sort alluded to in the foregoing article. I observe, 
 then, P*, that when the number of the sides of the polygon is 
 even, n= 2m, the equation of closure in article 336 becomes, 
 
 pq^vi = Qimpi or = V. p V^-om ; 
 
 but, II""^, that when the number n is odd, =2m + 1, the equation 
 of closure in the same article becomes, 
 
 P^2«n-i = -g2jn+i/o, giving 0=S^2m+ij and = S .qzm+ip. 
 
 IIP% that from 335, we easily infer that it is allowed to write 
 generally, whether n be even or odd, 
 
LECTURE VII. 675 
 
 where q^ and q'n are two real quaternions independent of p, and 
 satisfying the two equations in finite differences, 
 
 which may be collected into the single formula, 
 
 qn ±V-\ q'n = (on ± V-1) (S''n-I + V^-1 /n-i), 
 
 and are to be combined with the initial conditions, 
 
 qo = 1, /o = 0, or q\ = ai, q'l = 1. 
 
 IV*, that these equations give, by a species of finite integration, 
 the two following among other relations, 
 
 T^V - T^V = (-1)" {an' + 1) (a\.i + 1) . . . (a^ + 1), 
 and = S . qrJ^q'n = ab-S . aj3, 
 
 if a = Sq'n, b = Sq"n, a ^Vq'n, /3 = V/„. 
 
 V*\ that if w be odd, n = 2m + l, the equations of closure in II. 
 take thus the forms, 
 
 = a-S./3p, 0=6+S.a|o; 
 
 which are both included in the single equation, 
 
 V. py = aa + b[5i where y = V. )3a, 
 
 VP^, that this equation determines the position of a certain real 
 right line, or chord of solution, which cuts the unit sphere p^ +1 
 = in two points (real or imaginary), whose vectors are given by 
 the formula, 
 
 p = {aa + 6/3) 7"^ ± (a^ + (5')^ (b' + a'Y y-\ 
 
 and which are adapted, and are alone adapted, to be the positions 
 of the initial point f of the inscribed and odd-sided polygon. 
 VII*^ that if w be even, n = 2m, the equation of closure in I. 
 assumes then a form essentially different from the forms in V., 
 namely, the following, 
 
 V. pa = pY. jOj3, 
 
 which, when combined with p^ = -l, conducts to one or other of 
 
 2x 2 
 
676 ON QUATERNIONS. 
 
 the two following systems of scalar equations of the first degree 
 in p, 
 
 (Vll.y. . .S.7p = a^-cc-iS.j3a, S . (jS -^'a) p = 0, 
 (VII.)". ..S.7|0 = a^-+a;S.i3a, S . (/3 + a^'^a) p = 0, 
 
 where y still denotes V. j3a, and a; is a real scalar satisfying the 
 condition, 
 
 VIII*^, that these two systems of equations represent two real 
 right lines, which relatively to the sphere are reciprocal polars 
 of each other, because 
 
 (a^ + ^S.j3a)(a2-a;-iS./3a) = -7Sand S.(/3-a?a) ((i + x'^ a) = ; 
 
 and these two lines may be said to be chords of real and imagi- 
 nary/ solution, of the problem of inscribing the sought even-sided 
 polygon, one of them giving two real positions of the initial point 
 r, and, consequently, two real inscribed polygons, while the other 
 line, which is wholly external to the sphere, may yet be said to 
 give two imaginary positions of that point, and therefore two 
 imaginary polygons: which latter may, however, become real 
 when we pass, by imaginary deformation, from the sphere to a 
 single-sheeted hyperboloid. IX"', that, for example, we can ge- 
 nerally, by VIII., inscribe (or conceive inscribed) in a given 
 sphere two real and two imaginary gauche quadrilaterals, whose 
 sides shall pass successively through any four given points of 
 space; but X*^, that we can on the other hand, by VI., inscribe 
 generally in the given sphere two real or two imaginary gauche 
 pentagons, but not two of one kind, and also two of the other, 
 whose sides shall pass through ^'ue such points. No account is 
 taken here of any exceptional or limiting cases, such as might 
 arise, for instance, from the supposition that the given points, or 
 some of them, were situated on the given spheric surface. 
 
 677. If instead of conceiving, as above, a polygon PP1P2 . • 
 Pn-iPj whose n successive sides pPi, &c., are required to pass 
 through n given points, Ai, &c., we now conceive a polygon 
 pPi . . p„ of ?^ + 1 sides, whereof only the n first are obliged to 
 pass through those n points, while the last side p,iP \%free, then 
 it is clear that the initial point p of this new polygon is also free, 
 
LECTURE VII. 
 
 6n 
 
 or may be taken at pleasure anywhere upon the spheric surface : 
 but that when this initial point p is once assumed^ the^wa/ point 
 p„, and the closing side p„p, become entirely determined. There 
 will thus be a determined system of such closing chords in the 
 sphere, namely, one for each point of its surface assumed as the 
 initial corner of the polyg^on : and a variety of interesting ques- 
 tions may be proposed, respecting the arrangement of those 
 chords, considered as lines hdi\\\\^ position in space. For some 
 results respecting such arrangement, with extensions to other 
 surfaces of the second order, I may refer to the Numbers of the 
 Philosophical Magazine for September, 1849, and April, 1850, in 
 which Magazine a number of other papers on Quaternions, and 
 on connected subjects, by myself and others, have within the 
 last few years appeared ; also to the Abstract printed in the Pro- 
 ceedings of the Royal Irish Academy, of the communication made 
 by me in June, 1849, which, together with that already mentioned 
 of May, 1850, will perhaps appear in a fuller form, after no long 
 time, in the Transactions of that Academy. Meanwhile, 1 may 
 remark, XI*, that a very useful formula, for the case of the unit 
 sphere, is the following, which assigns the vector pn of the final 
 point p„ as a function of the assumed vector p of the initial 
 point p, and is easily deduced from the principles of 335 and 676 : 
 
 -q"n^{r\Yqnp , 
 
 qn+{-i-Tq np 
 
 but XIP'^, that, even without employing this expression XI. for 
 Pm the formula VI. of 676 enables us to infer that when the num- 
 ber of the given points Aj . . or of the given vectors ai . . is even, 
 = 1m, so that the number of sides of the variable polygon is oddy 
 the final or closing side touches two distinct surfaces of the second 
 order, represented by the two separate equations, 
 
 a2 + j3^ = 0, 6Ha^ = 0, 
 
 in which «, b, a, /3 are regarded as linear functions of the vector 
 «2»«+i5 and which will be found to represent an inscribed ellipsoid, 
 and an exscribed and double- sheeted hyperboloid, having double 
 contact with the sphere and with each other, at two real points 
 which on them are umhilics, and being also otherwise remarkably 
 
678 ON QUATERNIONS. 
 
 related; whereas, XIIP'*, if the number of the given points be 
 odd, = 2m - 1, or of the sides even, = 2m, then, by making the roots 
 equal in the quadratic equation VII. for x, or by other processes 
 unnecessary here to be described, we are conducted to an equa- 
 tion of the fourth degree in azmi which breaks up (for the case of 
 the sphere) into two imaginary and quadratic factors, of the 
 forms, 
 
 /32-a=^ = + 2/^S./3a, or(j3 + aV-l? = 0, 
 
 representing two imaginary cones, which jointly compose the en- 
 velope of the closing side, or are the surfaces which are both 
 touched by it in all its varying positions; XIV*, that these ima- 
 ginary cones may become real, namely, by changing the sphere 
 to a single-sheeted hyperboloid, in which case the bases of the 
 developable surfaces, composed by mutually intersecting chords, 
 which bases are analogous to lines of curvature, are real right 
 lines (the generatrices), although for the sphere they are imagi- 
 nary lines, represented in the present analysis by the equation 
 
 which admits of being solved (compare 667, 672, 675) by biqua- 
 ternions, without our supposing dp itself toyoxn^h; XV*'', that 
 for the case XII. the two analogous curves through any point p 
 have their tangents parallel to two conjugate semidiameters of 
 the surface, in which the variable and odd-sided polygon is to be 
 inscribed ; so that these curves everywhere cross each other at 
 right angles when that given surface is a sphere. Finally it may 
 be noticed, XVI*, that in the case XIII. the two imaginary 
 cones touch the given sphere along two imaginary circles, the 
 equations of whose planes are, 
 
 a + hy/-l = 0,a-b^-\^0, 
 
 and which may become two real and plane conies, by that imagi- 
 nary deformation which was referred to in XIV.; their planes 
 being, in all cases, harmonic conjugates with respect to the pair 
 of planes represented by the equations « = 0, Z> = 0, which latter 
 planes are also otherwise important in these investigations. 
 
 678. Reserving for another occasion (as has been hinted) the 
 fuller developement and elucidation of this whole theory of the 
 
LECTURE VII. 679 
 
 inscription of polygons in surfaces, with the corresponding theory 
 of the circumscription of polyhedra, and some comparisons of the 
 results so obtained with other and better known ones, which have 
 been discovered by geometers for plane polygons^ inscribed in or 
 circumscribed about plane conies, 1 wish to offer here a few re- 
 marks on the geometrical signification of the equation 
 
 V. pa = p V. |0j3, 
 which occurred in 676, VII., and might give occasion for a 
 longer discussion than we can at present afford to bestow. Sup- 
 posing still, as in the recent investigations respecting inscriptions 
 of polygons in a sphere, that a and j3 denote two ;'ea^and known 
 vectors, while p denotes a sought vector (real or imaginary), we 
 may endeavour to find this last vector by resolving the last-cited 
 equation, without any reference now to any other equation in- 
 volving p, such as the equation p^ = - 1, of the unit sphere. And 
 it might at first sight appear that, even without any such em- 
 ployment of any additio?ial equation, the problem was more than 
 determinate. For if we should choose to substitute, in both mem- 
 bers of the equation, for the sought vector p a trinomial expreS' 
 sion of the form ix +jy + kz (as in 507, &c.), with analogous re- 
 presentations for the given vectors a and j3, and then equate the 
 two resulting expressions of the standard quadrinomial form, 
 namely, w + ix+Jx + kz (arts. 450, &c.), it might seem that we 
 should have to satisfy four equations, of the ordinary algebraical 
 kind, with only three disposable quantities, real or imaginary. 
 And even after perceiving, as we should soon do, from inspection 
 of the formula itself, that neither member contributes any scalar 
 term, and therefore that only three ordinary equations (at most) 
 are to be satisfied by the three sought co-ordinates, x, y, z, on 
 which the vector p depends, it might still seem that (as in 513, 
 &c.) these three equations should suffice to determine those three 
 co-ordinates. But because a closer inspection of the formula 
 would shew that each member represents not only some vector, 
 but a vector perpendicular to p, we might thence perceive that 
 after expanding the equation into the trinomial form, 
 
 iX-vjY+kZ = 0, 
 the coefficients X, Y, Z, which would be certain scalar functions 
 
680 ON QUATERN10$fS. 
 
 of the second degree of the sought co-ordinates x, y, 2, must be 
 connected by the relation^ 
 
 xX+yY-^zZ = 0', 
 
 and therefore that the three scalar equations^ 
 
 X=0, Y = 0, Z = 0, 
 
 are not independent of each other. Accordingly, without resort- 
 ing to co-ordinates (compare again 513), we may perceive, 
 merely from the principles of the present calculus, that the equa- 
 tion in question may be thus written : 
 
 V.p(V./3p + a) = 0; 
 or thus 
 
 Y.qp = - a, where q = ff + (i, 
 
 g being here an arbitrary scalar. Hence, by 514 (or by 559), 
 we satisfy the equation by making 
 
 p = -(g^(5y'ia+g-'^.(5a); 
 
 or, as it may be also written, 
 
 g(g'-(5^p = (5S.(5a + gY.i3a-g'a. 
 
 To each assumed value of the scalar g corresponds a certain de- 
 rived value of the vector p ; and the locus of the termination of 
 this variable vector, p, is a curve of double curvature, which is of 
 the THIRD ORDER, in the sense that it is cut by an arbitrary plane 
 in three points, real or imaginary ; because if the equation of the 
 assumed plane be thus written, 
 
 S .fip = tn, 
 
 the condition for determining its points of intersection with the 
 locus is the following : 
 
 ^9{9'^-(^^) = S ./ijSS . (5a+gS . julia - g^S . fxa ', 
 
 which is an ordinary cubic in g. The curve just mentioned has 
 some interesting properties, respecting which it may suffice to 
 mention here that it is the common intersection of all the surfaces 
 of the second order, which are jointly represented by the equa- 
 tion, 
 
LECTURE VII. 681 
 
 S . aXp = (0^ S . j3X - S . /3joS . \p, 
 
 obtained by operating on the proposed equation with the symbol 
 S . A, where X is an arbitrary/ vector ; and that by making suc- 
 cessively, and separately, \ = a, X = j3j and X = 7, where y = V. j3a, 
 we obtain, in particular, the three following surfaces of the second 
 order, whereof the curve is the common intersection : 
 
 p'^ S . aj3 = S . a/oS . (dp ; 
 
 S . yap = S . /3|0 S .yp; 
 
 of which three surfaces the first is a cone, the second a cylinder, 
 and the third an hyperbolic paraboloid ; while the cone and cy- 
 linder are connected as having a common rectilinear generatrix, 
 represented by the equation 
 
 which right line is contained in one of the two asymptotic planes, 
 
 of the paraboloid, namely, in the second of them, but is not a 
 part of the sought locus, or of the curve of the third order, here 
 considered (compare the Paper by the Rev. George Salmon, on 
 the classification of curves of double curvature, published in the 
 Cambridge and Dublin Mathematical Journal for February, 
 1850). As to the intersections of this curve with the unit sphere, 
 I obtained the formulae (VII.)', (VII.)", of art. 676, by seeing 
 that when p^ = - 1 the equation gives, 
 
 S . 7^ = (V. M' = (V. apf = (S . ^pY + /3- (S . apf + a\ 
 and 
 
 - S . /3a = S . a/)S ./3p = a; (S . apy = x-^ (S . )3/o)^ 
 if we make for abridgment a? = S . j3jo -r- S . ojo ; whence, 
 (a^-.t-0S./3a = (S.ap)^-(S.i3^)^ = /3^--«% 
 as in 676, (VII.); and 
 
 S . yp = a^ - a;"' S . |3a, S . (j3 - Xa) p = 0, 
 as in the equations (VII.)'; from which those marked (VII.)" 
 
682 ON QUATERNIONS. 
 
 were derived, by simply changing x to -x''^. But conditions es- 
 sentially equivalent, for determining the intersections of the 
 sphere and curve, might be deduced in quite another way, namely, 
 by squaring the expression of the present article for p in terms of 
 g ; which process, after suppression of a common factor, namely, 
 g"^ - /3^, would give (compare 636), 
 
 p-={g'-^^yna--g-^{^.^aY]', 
 
 and therefore would lead, for p'^ = -\, to the following biquadra- 
 tic equation in g, which is, however, only of quadratic form rela- 
 tively to ^^ : 
 
 0^g'-i5' + a'-g-'(S.(3aY;or,g^-g^{l^'-a-^) = {S.(5a)\ 
 
 In fact, the positive value of g^ would give the two real values of 
 |0, answering to the two real intersections of the sphere with the 
 curve, or with the chord of real solution in 676, VIII.; while 
 the negative value of ^^ would give the two imaginary values ofp, 
 answering to the two imaginary intersections of the sphere with 
 the same curve, or with the chord of imaginary solution, men- 
 tioned in the same paragraph 676, VIIL, which was there shewn 
 to be the reciprocal polar of the former chord, and to lie wholly 
 outside the sphere. It must be remarked that the common fac^ 
 tor g"^ - /3S which was suppressed in the recent process, and 
 which cannot vanish except when g takes one of the two imagi- 
 nary values, 
 
 ^ = ±T/3v/-l, 
 
 appears to indicate two imaginary and infinite values for p, or 
 two imaginary points at infinity, as two other intersections oi the 
 sphere with the curve of the third order (compare the remark 
 made at the end of 553) : but I do not at present see of what geo- 
 metrical utility these two new points can be, even when we pass 
 by imaginary deformation from the sphere to the single-sheeted 
 hyperboloid. 
 
 679. Without introducing the consideration of any but real 
 quaternions, a variety of new forms might be assigned, in this 
 calculus, for the representation of real loci, in addition to those 
 which have been already pointed out, and of which some appear 
 to be remarkable. Thus if we assume any fixed vector oa = o. 
 
LECTURE VII. 683 
 
 and denote (as usual) by p another and generally variable vector 
 OP, drawn from the same fixed origin o to a point p of which the 
 locus is required, introducing also for abridgment the following 
 symbol of a certain quaternion which depends on the position 
 of p, 
 
 then the equation 
 
 [i]..g=o, 
 
 as giving p = 0, expresses that p coincides with o ; but the equa- 
 tion 
 
 [2]..g=l, 
 
 which gives p = ±a, expresses that p is situated either at a, or at 
 another fixed point a', such that o bisects aa'; while this other 
 equation, of almost the same apparent form, 
 
 [3]..^=-l, 
 
 gives, as the locus of p, a circular circumference (compare 170), 
 namely, a great circle with a for pole, on the spheric surface, 
 with o for centre: and this spheric surface itself is represented 
 by the equation, 
 
 [4]..T^=1. 
 
 The indefinite right line through o and a is denoted by writing 
 
 [5]..U^ = 1; 
 
 and the indefinite plane through o, perpendicular to this line, is 
 represented (see 172) by this other formula, 
 
 [6]..U^ = -1; 
 
 while the system of this line and plane may be expressed by the 
 equation 
 
 [7]..V^ = 0, 
 
 since this requires (compare 504) that we should have either 
 
 VV^ = 0, or SV^ = 0. 
 To write on the other hand, 
 
 [8]..S^ = 0, 
 is to express (see again 504) that 
 
684 ON QUATERNIONS. 
 
 and therefore (by 438), this locus [8] is an equilateral right cone, 
 containing all the indefinite lines op which are inclined at 45° to 
 the fixed line oa. The equations 
 
 [9] . .S^=l, and [10] . .89 = -!, 
 
 represent respectively (by 438, 439) a double- sheeted and equi- 
 lateral hyperholoid of revolution^ and the conjugate and single- 
 sheeted hyperboloid ; their common axis of revolution being the 
 indefinite line oa, and the finite line oa itself being the real semi- 
 axis of the former. Any other assumed and constant scalar va- 
 lues of 85- would give other, concentric, similar, and similarly 
 placed hyperboloids ; and if, on the contrary, we assign a con- 
 stant vector value j3 to V^, where j3 = ob = a fixed line perpen- 
 dicular to a, writing thus, 
 
 [ll]..V^ = /3,/3 J_«, 
 
 the locus of p will be found to be no surface, but a curve, namely, 
 an equilateral hyperbola, in a plane perpendicular to ob, with o 
 for centre, and oa for one of its asymptotes. Another mode of re- 
 presenting an hyperbola by a single equation in this calculus oc- 
 curred in 505, and will be more fully discussed in the next arti- 
 cle. Meanwhilcj I observe that an ellipse may in like manner 
 be represented in various ways by a single equation in real qua- 
 ternions, for instance, by the following, 
 
 [12]..(yV.ap)^ + (7V.i3p)^ = l, 
 
 in which a, j3, 7 denote any three real and rectangular vectors ; 
 because on developing the squares of the two quaternions, 
 
 7 V. OjO = S . yap - aS . y/j, y V. j3jO = 8 . yjS/O - j3S . -yp, 
 
 it will be found that the only way of making the sum of those 
 squares equal to unity, by any real vector p, is to suppose that 
 tKis vector satisfies the system of the two scalar equations, 
 
 [13]..(8.yapr-+(S.7j3py=l, S.jp = 0, 
 
 whereof the latter represents a plane, and the former an elliptic 
 cylinder: the locus of the termination of p is therefore (as just 
 
LECTURE VII. 685 
 
 now asserted) an ellipse, which has its centre at the origin, and 
 its axes in the directions of the two lines a and j3. For example, 
 the equation 
 
 [14] . . (a-i kY.Jpy + (6-1 kY. ipf = 1, 
 
 where p = ix -^-jy + kz^ can only be satisfied, for real co-ordinates 
 xyz, by supposing that those co-ordinates satisfy the two equa- 
 tions, 
 
 [15] ..a-2a;2 + 5-2 2/2 = 1,2 = 0. 
 
 On the other hand the equation, 
 
 [16]..(S.ap)2+(y V.ap)^=l, 
 
 where 7 is still supposed X a, admits of an alternative of two so- 
 lutionsj and conducts to the following system, of two real curves: 
 
 [17] . . S . 7p = 0, (S . apy + (S . yapY = 1, 
 [18] . . S . 7a,o = 0, (S . apy - Ta' (S . jpy = 1, 
 
 whereof the former represents generally an ellipse, and the latter 
 an hyperbola, these two curves having one common axis, and one 
 common pair of summits, but being situated in two rectangular 
 planes. For example, the circle and equilateral hyperbola, which 
 have their equations in co-ordinates as follows, 
 
 x^ + y^= I, z = 0, andx^ - z'^=l,y = 0, 
 
 and of which the consideration has presented itself to some for- 
 mer writers, in connexion with modes of interpreting certain re- 
 sults respecting the ordinary V-lj are jointly represented in 
 this calculus by the one equation, 
 
 [19] ..(S.^»2+(^V.^»2=1. 
 
 Again, the equation, 
 
 , [20] ..p'- + b'- + {ekY.jpY = 0, where e^ < 1, 
 
 represents a system, of two ellipses, in two rectangular planes, but 
 having in like manner two common summits; namely, the two 
 principal sections through the mean axis of the ellipsoid, of which 
 the equation in co-ordinates is, 
 
 [21] . . (1 - e^) a;- + 2/^ + (1 + e2) ^2 ^ i2. 
 
686 ON QUATERNIONS. 
 
 Again, if t and k denote any two fixed vectors from the origin, 
 the equation 
 
 [22] . . ipicp = pKpi, or = V. ipKpt 
 may easily be shewn to represent a system of two rectangular 
 right lines, bisecting the angles between i and k ; whereas this 
 other equation, of nearly similar form, 
 
 [23] . . ipKp = pipK, or V. jO V. ipK = 0, 
 which may also be thus written (compare 520), 
 
 [24] . . V. ipS .Kp + Y. KpS. ip = 0, 
 or thus, 
 
 l25-]..(tpy^(pK)MU^ = KS 
 represents a system of three rectangular right lines, 
 namely, the two bisecting lines }ust mentioned, in the directions 
 of Ui ± Uk, and also a third line, perpendicular to the given plane 
 of the two given lines f, k, and having therefore the direction of 
 V. LK. Accordingly, if we seek the directions of the three axes of 
 an ellipsoid, by inquiring where the diameters are normals, or by 
 making, in 474, 
 
 [26].. V.v/)=0, 
 
 we are conducted precisely to the recent equation [24]. Or we 
 might, on the same principle [26], have deduced the equation 
 [23] from the last formula of 593 or of 596. This seems to be 
 a natural occasion for remarking, that the general equation of 
 surfaces of the second order may in this calculus be written thus 
 (compare 476, 552), 
 
 [27]..l=/(p)=^,o^+2SS.a/>S./3|0+S.yp, 
 
 giving for the vector of proximity (compare 474, 475, 481, 575) 
 the expression, 
 
 [28] ..v=0(jo) = ^|O + S(aS.i3/>+ j3S.o/o) + 7; 
 
 and that when, by suitable reductions, the sign of summation is 
 removed, the two cyclic normals of the surface, or the normals to 
 what have been called by MacCullagh the two directive planes, 
 have the directions of the two constant vectors a and j3, in the 
 one remaining term of the form 2S . ap S . (5p (compare 469, 
 593). As regards curves and surfaces of higher orders, it may 
 
LECTURE VII. 687 
 
 suffice for the present to observe, in addition to what is sug- 
 gested by the remarks in 552, that any proposed equation in x, 
 y, z, may he transformed from co-ordinates into quaternions^ by 
 simply making the substitutions , 
 
 [29] . . a? = i" ^ S .ip,y=j'^ S .ip,z = k'^ S . kp^ 
 or 
 
 [30] . .x = -iS .ip,y=: -jS .jp, z = -kS .kp; 
 
 for instance, one form of the quaternion equation of Fresnel's 
 Wave, obtained on this plan, is the following : 
 
 (s.apy (s.jdpy (s.ypy_^ 
 
 •- -^ ' ' |0--a2 jo'-iS^ /o'-7' 
 
 But it is usually possible, in interesting questions, to obtain ex- 
 pressions more elegant, or at least better adapted to be treated by 
 the peculiar methods of this calculus, than the forms which result 
 immediately from the foregoing very general substitution : and 
 accordingly I have been able to obtain other expressions by qua- 
 ternions for the lately mentioned wave surface, which put in 
 evidence those conical cusps, and those circles of contact there- 
 upon, on which appear to depend the optical phenomena of co- 
 nical refraction in crystals with two axes, that were ex- 
 perimentally observed by the Rev. Humphrey Lloyd about the 
 end of the year 1832, with a carefully cut specimen of arrago- 
 nite. Finally, as additional illustrations oi the flexibility , combined 
 with distinctness, of the symbolical language of the present cal- 
 culus, it may be noticed that by subjecting a variable quaternion, 
 q, instead of merely a variable vector, p, to satisfy a given equa- 
 tion, and allowing the scalar part to vai-y, new s^owxcq^ oi expres- 
 sion arise. For example, if we write (as we have often done) 
 q = w + p, and regard the part w as arbitrary, and p as variable, 
 but both as real, while a and j3 are any two given and constant 
 and real vectors from the origin, the equation, 
 
 will be found to represent a full circle, inasmuch as the va- 
 riable vector p will now be free to terminate at any one of all 
 those points of space which are contained upon, or included 
 
688 ON QUATERNIONS. 
 
 within^ that circular circumference of which the vector of the 
 centre is a, while j3 is perpendicular to its plane, and its radius is 
 = T/3 : because the quaternion analysis shews that we have here, 
 
 [33] . .S.(p-a)/3 = 0, Tip-ay^^T^-'-ioK 
 
 The equation 
 
 [34].. (^y=., 
 
 would represent, on the same plan, the system of a full circle 
 and of two points, related to each other as the equator and poles 
 of a sphere. And the very simple equation, 
 
 [35] ..T^=l, or T{iv+p)=\, 
 represents in like manner a full sphere, namely, the unit- 
 sphere, regarded now as no mere surface, but as a solid locus, 
 whereof all the internal points are here to be taken into account, 
 as beUig all included in the formula. Results of the sorts as- 
 signed in the present article might be almost indefinitely multi- 
 plied : and if the subject shall be hereafter pursued, the difficulty 
 will much less be to interpret than to class the expressions. 
 
 680. After these general remarks on equations in the present 
 calculus, let us resume the particular equation of art. 505, 
 
 N.-np.Y.pB^iy.nBf, 
 
 and treat it as if it had now for the first time presented itself, in 
 some geometrical investigation. One general and always per- 
 mitted process of transformation, of any equation in quaternions, 
 has been seen to be the taking separately the scalar and the vec- 
 tor parts of the two members, and then equating them respec- 
 tively. Taking therefore the vector parts, the first member of 
 the equation gives, 
 
 Y{V.np-y'pO) = pS.nep; 
 
 but also by the scalar character of the square of a vector, 
 
 {y.nOy=y-'o, y.(v.-ney=o; 
 
 and the proposed equation forbids us to suppose p = 0, it being 
 understood that jj and are not parallel ; we are therefore con- 
 ducted to this other equation, 
 
 J 
 
LECTURE VII. 689 
 
 Thus, 
 
 p\\\ri,9; p = xri + ye; 
 V.r,p = yY.ne', Y.pe = xY.r,e; 
 
 and finally the equation of condition, which the two variable 
 scalar coefficients x and y are obliged to satisfy, is found to be 
 the following : 
 
 xi/=l. 
 
 It is therefore necessary and sufficient to admit that the variable 
 vector p has sotne one of the values included in the expression, 
 
 p = xi] + x~^Q^ 
 
 where x is an arbitrary scalar. The locus of the extremity of p 
 is consequently a (plane) hyperbola, having its centre at the origin 
 of vectors, with tj and Q for portions of its two asymptotes, and 
 with rj + for one of the values of p, or for the vector of one point 
 of the curve. But rj and B have been seen in earlier articles 
 (compare 497, 503), to be portions of the axes of the two cylin- 
 ders of revolution, within which the two spheres slide, in one of 
 our modes o^ generating the ellipsoid (art. 496), and within each 
 of which two cylinders the ellipsoid itself h inscribed. We saw 
 also (in 502) that rj + is an umbilicar vector of the ellipsoid. 
 No uncertainty therefore can now remain, respecting the fitness 
 and adequacy of the equation assigned in art. 505, to represent, 
 in this calculus, that known curve which has been named the 
 focal hyperbola, of a certain ellipsoid, and of its confocals. In- 
 deed, that the equation expressed, among other things, the co- 
 planarity of rj, 9, p, might have been more rapidly inferred 
 from the consideration that because the vectors V. tjp and V. p9 
 are asserted to have a scalar product, they must be supposed to 
 be parallel to some one line ; to which one line therefore the th?'ee 
 lines Tj, 9, p must be perpendicular, and consequently must be 
 coplanar with each other. 
 
 681. Let p and p, expressed as follows, 
 
 P = xy} + x''^9, p' = xrj + x''^9, 
 
 be any two vectors, ap, ap', of the focal hyperbola; their diffe- 
 rence is evidently, 
 
 2y 
 
690 ON QUATERNIONS. 
 
 Tv' = p- p = {x -x)r\ + {x''^ -x'^)0 ; 
 
 and if this difference, or the cAorc? joining the extremities of the 
 two vectors, is to be parallel to tj - 0, we must have 
 
 x'+ x'''^ = x + x''^, 
 and therefore generally 
 
 XX'= 1, p=X'^r] + x9, 
 
 the scalar diiference a?'- a; being supposed not generally to vanish. 
 The same chord pp' meets the asymptotes »), 9, in two points q, 
 q', of which the vectors are, 
 
 - 1 ' 
 
 AQ=-^ --^ix+x'^) n: Aq'=(x+ X'^)0; 
 
 X-X'^ 
 
 whence, 
 
 FQ = X-^(ri-0)', TQ'=-x{ri-B); PQ.PQ'=T(i7-0)2; 
 
 and, as is known, 
 
 p'q = q'p, p'q'= qp- 
 
 But as X approaches to 1, or as the variable vector p approaches 
 to the particular value ri+ 0, or lo (art. 502), the chord p- p tends 
 to vanish in length, and to become in direction tangential to the 
 curve ; and the portion of the tangent intercepted between the 
 asymptotes is seen, by the recent analysis, to be (as is well 
 known) bisected at the point of contact. Thus, at the umbilic 
 of the ellipsoid, which is (by 502) the termination of the vector 
 w, the tangetit to the focal hyperbola has the direction of rj - 0, or 
 of L (art. 498) ; that is (as is known), of the umbilicar normal 
 (compare 501) to the ellipsoid. Or we might haiVQ differentiated 
 the scalar variable x in the expression for p, and then made x = \; 
 which would have given dp -^ do^* = ij - 0, when p = ri + 0, and 
 would have conducted to the same conclusion respecting the di- 
 rection of the tangent to the hyperbola, at the same umbilic of 
 the surface. And hence we may prove, by quaternions, the 
 known theorem already alluded to (505), that the focal hyper- 
 bola cuts the ellipsoid perpendicularly, at each umbilicar point. 
 Combining the recent results with others somewhat earlier ar- 
 rived at, we are conducted without difficulty to the following con- 
 struction. At an umbilic u, draw a tangent tuv to the focal hy~ 
 
LECTURE VII. 
 
 691 
 
 perbola, meeting the asymptotes in t and v, as in the annexed 
 
 figure 102. Then the sides of 
 
 the triangle tav are, as res- ^^' 
 
 peets their lengths, av = 2Tr) ; " ^^ 
 
 A? = 2T0; TV = 2T {ri-B); 
 that is, by 501, 
 
 AV = a + c ; AT = « - c ; TV = 26. 
 
 And the r} and 9 of this Lec- 
 ture are precisely the halves 
 of the sides Av and at of this 
 triangle ; or they are the two 
 oblique co-ordinates ay, ax of 
 the umbilic u, referred to the asymptotes of the hyperbola, when 
 directions as well as lengths are attended to. 
 
 682. It has been so much my wish, in the present Course of Lec- 
 tures, now drawing rapidly to its close, to lay a sound and strong 
 geometrical foundation for future applications of this Calculus ; 
 and 1 so well foresee that through necessary future extensions of 
 the theory, such as the introduction, already sketched, of what I 
 have called Biquaternions, many difficulties as yet unapproached 
 will arise : that I have anxiously sought to provide a large 
 amount of what might become, through the united exertions of 
 myself and others, a settled, established, and common ground, 
 respecting the validity of which no diversity of opinion could 
 ever afterwards occur. And, in this spirit, I ask you now to allow 
 me to state a few geometrical reasonings, of a very simple kind, 
 by which the recent results, and some earlier geometrical conclu- 
 sions, of this new mode of calculation may be confirmed. 
 
 The sum of the squares of any three conjugate semi-diameters 
 of a given ellipsoid being known to be a constant quantity (= a"^ 
 + b^+ c^), while the umbilicar vector au (= u), and any two rect- 
 angular radii (each = b), of the circular and diametral section 
 made by a plane parallel to the umbilicar tangent plane, compose 
 a conjugate system, we are to subtract 2b^ from a'^ + b-+ c^, and 
 shall thus obtain the value u'^=a^-b'^+ c^ as in art. 502. Again, 
 the parallelepipedon under any three conjugate semi-diameters 
 
 2y 2 
 
692 ON QUATERNIONS. 
 
 being known to be constant, and = abc, we are to divide this by 
 6^, and so obtain ab''^c (compare 501), as an expression for the 
 perpendicular let fall from the centre A on the umbilicar tangent 
 plane; or for the projection su, of the umbilicar vector Au (in 
 fig. 102), on the umbilicar normal tuv to the ellipsoid, which 
 normal is known to coincide with the tangent to the focal hyper- 
 bola (as proved by quaternions in the foregoing article). Thus 
 ^(a'-Z>^ + c2) jg t}^e hypotenuse AU, and Z>~^ac is one side su 
 about the right angle, in the triangle asu; so that the other side, 
 AS, must be =b''^{d^-lr)^{¥-c^^^. Such, then, is the altitude of 
 the triangle tav, if the centre a of the ellipsoid, or of the hyper- 
 bola, be considered as the vertex. But, by the properties of the 
 curve, this area does not vary when we change the point of con- 
 tact u ; it is therefore equal to the rectangle under the semiaxes 
 of the focal hyperbola, or to the product {a"^- i^) ^ (i^ - c-)2 ; and 
 it is known that the tangent tv is bisected at the point of con- 
 tact; the semibase, tu, or uv, of the triangle tav, must therefore 
 be = 6 : which would be a geometrical confirmation, if such were 
 needed, of the proof previously given by quaternions (see 498, 
 499), that T(j] -B) = b. To find the lengths of the sides, av, at, 
 of the last-mentioned triangle, we have, as before, the altitude as 
 = b'^{a~ - b"^)^ (p^ - c^)^, and the segments, 
 
 sv = su + uv = 6"iac+ 5 = Z»'^ («c + Zi^), 
 ST = su- VY = b'^ac -b = b~^ {ac - 6^); 
 whence by two right-angled triangles, 
 
 AV = (a- + c- + 2ac)i = a + c, 
 AT = (a'^ + c- - 2ac)^ = a - c; 
 
 these sides are therefore the sum and difference of the two ex- 
 treme semi-axes of the ellipsoid : a result which agrees with the 
 values found otherwise in article 501, namely, Ti} =^(a + c), TO 
 = ^{a-c). It maybe remarked that the triangle bcg of figure 98 
 would admit of being superposed on the triangle yax of fig. 102, 
 if both triangles were constructed for one common ellipsoid. 
 
 683. Resuming (partly as an exercise) the calculations with 
 quaternions, it is easy to see that 
 
 I 
 
LECTURE VII. 693 
 
 S . (pr,-Op) (,, - 0) = S {pr-pri9 - Opri + epB) = -2S . nOp, 
 because 
 
 = S . pr,^ = S . SpO, and S.pne=S.epv = S. nOp. 
 
 Hence generally, for any three vectors, rj, 6, p, we have the 
 transformations, 
 
 T.(pr^-ep)U(n-e)^T(pn-ep)', 
 S.(pn-ep)V(n-e) = -2T(r,-e)"S.nep; 
 TV . (pn- dp) U(., -0) = V{ T^ipv - Opf - 4T{r, - Oy^S . tiOpY} 
 = V { (pj? - Op) inp - pO) + (r, - 0)-2 {rjOp - pSnY} ; 
 
 also for any two conjugate quaternions, q, q', and any vector a, 
 we have the identity, 
 
 TY.qa = TY.(/a=V{{TY.aYqy + {TaSqy}l 
 
 and therefore, 
 
 TV. (np - pB) U (t, - 6) = TV. (pr, - Op) U (r, - 0). 
 
 For the ellipsoid, by 499, we have the equation, 
 
 TV.(„p-pO) U {r,-6) = e^^-r,'^', 
 
 and hence, by squaring, we obtain this new form of the equation 
 of that surface : 
 
 {9- - ri~y = {pn - dp) {np - pB) + (?} - 0)-^ {r\Bp - pBnY- 
 
 Or, by a partial re-introduction of the signs S and T, we find 
 this somewhat shorter form : 
 
 T{pn - BTpy + 4(r, - oy (s .-nOpy = (O^ - n^y ; 
 
 of which we shall presently assign the interpretation, and in 
 which, instead of the square of the tensor of the quaternion 
 prt - Bp, we may write any one of several general expressions for 
 that square, of which the proofs will easily suggest themselves 
 to those who have studied with attention the transformations 
 already given, and the principles of the present calculus; for in- 
 stance, any of the following : 
 
 T(pr,-Bpy=T(r,p-pBy 
 = (pn - Bp) (rip - pB) = {np - pB) (pri - Bp) 
 
 . = (rj^ + 0^) p^ - prfpB - Bprjp = (?j^ + B~) p~ - ripBp - pBpr\ 
 
694 ON QUATERNIONS. 
 
 = (r, + ey p' - {np + pn) {Op + pO) 
 = {ri^+e')p^-2S.np9p 
 
 = (», - 0)V' + 4S ( V. 7,|0 . V. p0). 
 
 All these transformations, it must be remarked, hold good, inde- 
 pendently of any relation between the three vectors 17, 6, p. 
 
 684. To interpret that form of the equation of the ellipsoid, 
 which was assigned at the beginning of article 500, we may ob- 
 serve that 
 
 V — — ^ = pi + p2 ; 
 if for conciseness we write, 
 
 p,=(ri-ey's.in-e)p', p,=^y.iv-0)-'y^p(ri+e). 
 
 But jOi is the perpendicular from the centre a of the ellipsoid on 
 the plane of a circular section, passing through the extremity of 
 the vector or semidiameter p, and perpendicular to the cyclic 
 normal tj-0; and p^ may be easily shewn (compare 441) to be 
 a radius of the same circular section, multiplied by a scalar co- 
 efficient, namely, by 
 
 i} + 0_ v^-O^ Tyi'^-TO^ ac 
 
 n-e~{r,-ey~'T{r,-ey''b'-' 
 
 If then, from the foot of the perpendicular, let fall (as above) on 
 
 the plane of a circular section, we draw a right line in that plane, 
 
 which bears to the radius of that section the constant ratio of the 
 
 rectangle ac under the two extreme semi-axes to the square b~ of 
 
 the mean semi-axis of the ellipsoid, the equation for that surface, 
 
 which was given at the beginning of article 500, expresses that 
 
 the line so drawn will terminate on a spheric surface, which has 
 
 ac 
 its centre at the centre of the ellipsoid, and has its radius = —, 
 
 It was thus, in fact, that I happened to perceive this property of 
 the surface, by interpreting as above one of the quaternion forms 
 of its equation ; but it is not difficult to "^xo^q geometrically that 
 the described construction conducts to the last-mentioned spheric 
 locus ; namely, to the sphere concentric with the ellipsoid, which 
 touches at once the four umbilicar tangent planes. 
 
LECTURE VII. 695 
 
 685. Proceeding to the interpretation of the equation of the 
 ellipsoid, which was arrived at in 683, we may remark that since 
 
 pr)-ep = S.p{tj-9)+V.piTi + 0), 
 
 the quaternion pri - Op gives a pure vector as a product, or as a 
 quotient, if it be multiplied or divided by the vector r} + d (com- 
 pare 500) ; we may therefore write 
 
 pn-Op^Xiiri + B), 
 
 Ai being a new vector symbol, of which the value may be thus 
 expressed : 
 
 Xi=p-2{r,+ey's.ep. 
 
 This vector Ai is evidently such as to give, 
 
 T (pr, - Op) = TX,. Tin + 9); 
 
 T(pn-epy=Xi^{r, + ey. 
 
 We have also the identity, 
 
 (02 _ „2)2 = (^ _ ey (^ + oy + (nO - Ony ; 
 which may be shewn to be such, by observing that 
 
 (r, - ey in + ey = ('n'+e'-2S. nS) {r + Q'~ + 2S . 7J0) 
 
 = {rf + Q-'y - 4 (s . nBy = (rj^ - e-'y + 4 (t . r^ey - 4 (s . ^ey 
 =(r-e'y-4{Y.riey=:{e'-v''y-{ri9-er,yi 
 
 or by remarking that (compare 454), 
 
 n'-e'=^S.{r,-e) (rj+0), r,0-en=y.(v-0) (ri + O), 
 
 and {n-ey {ri + ey = {T . {n-O) {v + 0)}'; 
 
 or in several other ways. Introducing then a new vector c, such 
 that 
 
 rie-erj = eT{ri + e), or f = 2 V. r,0 . T (», + 6>)-' ; 
 
 and that therefore 
 
 (ri9-er,y = -a'iri + 0y> 
 and 
 
 2S.'n9p = S.ep.T{ri + 9), 4iS .nBpy= - (S . epy . (ri + 9y; 
 while, by 498, or 499, 
 
 T(n-9) = b, (r)~9y = -b'; 
 
69(i ON QUATERNIONS. 
 
 we find that the equation of the ellipsoid above referred to, 
 namely, 
 
 T (pn - OpY + 4 („ - ey (S . nOpf = {&" - r,% 
 
 after being divided by (ij + 9)\ assumes the following form : 
 
 But also, by the recent values of Xi and c, 
 
 S . eXi = S . £p ; 
 the equation just found may therefore be also written thus : 
 
 = (X^-ey+{b + b-^S.epy; 
 
 and the scalar b + b'^ S. ep is positive, even at an extremity of the 
 mean axis of the ellipsoid, because 
 
 (02 _ ^2)2 ^ _ (^2 + ,2) (^ + Qy = (52 _ T£^-) T (r, + ey, 
 
 and therefore 
 
 Te<b. 
 
 We have then this new form of the equation of the ellipsoid, de- 
 duced by transposition and extraction of square roots, according 
 to the rules of the present calculus : 
 
 T{X^-E) = b + b-^S.£p. 
 
 By a process exactly similar to the foregoing, we find also the 
 fojm 
 
 T{X,+ e) = b-b-'S.ep; 
 
 which differs from the equation last found, only by a change of 
 sign of the auxiliary and constant vector £ ; and hence, by addi- 
 tion of the two last equations, we find still another form, namely, 
 
 T(Ai-£) + T(Ai + £) = 26; 
 
 or substituting for Ai, e, and b their values, in terms of »j, 6, and 
 p, and multiplying into T (r/ + 6), 
 
 = 2T.O,-6l)(„ + 0). 
 G8G. The locus of the termination Lj of the auxiliary and 
 
LECTURE VII. 697 
 
 variable vector Xi, which is derived from the vector p of the ori- 
 ginal ellipsoid by the linear formula of the last article, namely, 
 
 \i^p-2(rj + e)-'S.ep, 
 
 being thus represented by the equation of the same article, 
 
 T(Xi + £) + T(Xi-e) = 26, 
 
 is evidently a certain new ellipsoid; namely, an ellipsoid of 
 revolution, which has the mean axis 2b of the old or given ellip- 
 soid for its major axis, or for its axis oj^ revolution, while the 
 vectors of its two Jbci are denoted by the symbols + e and -s. 
 In fact if we still place the origin of vectors at the centre a of the 
 ellipsoid of arts. 466, &c., and make 
 
 Xi = ALi, £ = AFi = F2A, 
 
 we shall have, for the locus of the point Li, the following equa- 
 tion of a very simple and well-known form : 
 
 F2L1 + FiLi = 26. 
 We have also, by the foregoing article, combined with. 501, 502, 
 
 Te~ = b' + (02 - ify {ri + ey = 6^ - a^c'u-^ ; 
 or 
 
 « = & - ~i — T- — = •'^ -r- \ if « = Te. 
 
 Such then is the expression for the square of the distance (e) of 
 either focus (Fi or F3) of the new or derived ellipsoid, which has 
 Xi for its varying vector, from the common centre a of the new 
 and old ellipsoids, which centre is also the common origin of the 
 vectors Xi and p : while these two foci of the new ellipsoid are 
 situated upon the mean axis of the old one. There exist also 
 other remarkable relations, between the original ellipsoid with 
 three unequal semi-axes «, b, c, and the new ellipsoid of revolu- 
 tion, of which some will be brought into view, by pursuing the 
 quaternion analysis in a way which we shall proceed to point 
 out. 
 
 687. Combining the recent expression for Xi with three other 
 analogous expressions, as follows : 
 
 . _py]-Op . pO-yp 
 rj + y Tj + l> 
 
698 ON QUATERNIONS. 
 
 . _ pQ-^ -r\-^p _pr)-^-0-^p 
 
 it is easy to prove (compare 494) that 
 
 and that 
 
 S . r,0Xi = S . nOXo = S . r,ex, = S . r,eXi = S . r,6lp ; 
 whence it follows that the four vectors Xi, X2, X3, X4, being sup- 
 posed to be all drawn from the centre a of the original ellipsoid, 
 terminate in four points, Li, Lo, L3, L4, which are the^m- cor- 
 ners of a quadrilateral ifisc?'ibed in a circle of the lately derived 
 ellipsoid of revolution ; the plane of this circle being parallel to 
 the plane of the greatest and least axes of the original ellipsoid 
 {abc)i and passing through the point e of that ellipsoid, which is 
 the termination of the vector p. We shall have also the equa- 
 tions, 
 
 Xi — jO S . dp Xi- p S.O'^p 
 
 which shew that the two opposite sides L1L2, L3L4, of this in- 
 scribed quadrilateral, being prolonged if necessary, intersect in 
 the lately mentioned point e of the original ellipsoid. And be- 
 cause the recent expressions give also 
 
 y Xa-Xl X4-X3 Q 
 
 these opposite sides LjLo, L3L4, of the plane quadrilateral thus in- 
 scribed in a circle of the derived ellipsoid, are parallel respectively 
 to the vectors 17 + 0, t}"^+ 0"S or (by 502, 503) to the two umbi- 
 licar vectors a>, to', of the original ellipsoid, constructed with the 
 semi-axes abc. At the same time, the equations 
 
 1] if 
 
 hold good, and shew that the two other and mutually opposite 
 sides of the same inscribed quadrilateral, namely, the sides L2L3, 
 L4L1, are respectively parallel to the two vectors 1], 9, or to the 
 axes of the two cylinders of revolution which can be circum- 
 scribed about the same original ellipsoid. 
 
LECTURE VII. 699 
 
 688. Hence it is easy to infer the following Theorem, else- 
 where already published by me as a result of the Calculus of 
 Quaternions : '•'•If on the mean axis^ 2b, of a given ellipsoid, abc, 
 as the major axis, and with two foci Fi, Fj, of which the common 
 distance from the centre a is 
 
 V (a^ - o^ + c^) 
 we construct an ellipsoid of revolution ; and if in any circular 
 section of this new ellipsoid, we inscribe a quadrilateral, L1L2L3L4, 
 of which the two opposite sides L1L2, L3L4 are respectively paral- 
 lel to the two umhilicar diameters of the given ellipsoid ; while 
 the two other and mutually opposite sides LnLg, l^Lj, of the same 
 inscribed quadrilateral, are respectively parallel to the axes of 
 the two cylinders of revolution which can he circumscribed about 
 the same given ellipsoid; then the point of intersection e of the 
 first pair of opposite sides (namely, of those parallel to the um- 
 bilicar diameters) will be a point upon that given ellipsoid." It 
 seems to me that, in consequence of this remarkable relation 
 between these two ellipsoids, the two foci Fi, F3 of the above-de- 
 scribed ellipsoid oi revolution, which have been seen to be situated 
 upon the mean axis of the original ellipsoid, may not inconve- 
 niently be called the two medial foci of that original ellipsoid 
 (a5c) ; and that the new or c^en'vec? ellipsoid of revolution itself may 
 be called the mean ellipsoid ; but 1 gladly submit the question of 
 the propriety of these designations, to the judgment of other and 
 better geometers. Meanwhile it may be noticed, that the two 
 ellipsoids intersect each other in a system of two ellipses, of 
 which the planes are perpendicular to the axes of the two cylin- 
 ders of revolution above mentioned ; and that those four common 
 tangent planes of the two ellipsoids, which are parallel to their 
 common axis, that is to the mean axis of the original ellipsoid 
 abc, are parallel also to its two umbilicar diameters. It may be 
 added that if U denote the minor semi-axis (= {b"^- e'^)^=acu-'^) of 
 the above-mentioned tnean ellipsoid, and if we construct another 
 concentric ellipsoid, ab'c, which will thus not be of revolution, the 
 equation of this third ellipsoid may in our symbols be written thus : 
 
 T (rjjO - /O0) = 0^ - rf ; 
 
700 ON QUATERNIONS. 
 
 and that its cyclic normals have the same directions as those of 
 thdit fourth ellipsoid dbc\ for which ac =h~ = cd, and which is, in 
 a well-known sense, reciprocal to the Jirst or given ellipsoid, 
 abc, having also the same mean axis, but having its major axis 
 in the same direction as the minor axis of the other. As to the 
 intersection of the other pair of sides L2L3, L4L1, of the inscribed 
 quadrilateral, it is easy to see (compare again 494) that if we 
 call this point s, and denote its vector as by <t, we shall have the 
 expression, 
 
 o- = (rj + Oymn' + 0-) P - 2 V. rtpO}; 
 
 so that (compare 597) the locus of the point s is a certain Jiff h 
 ellipsoid, on the properties of which I cannot enter here. 
 
 689. The same general methods of calculation (compare the 
 remarks made at the end of 624) admit of a vast variety of other 
 geometrical applications. For instance, if we combine the for- 
 mula S.vdvd|O = 05 of article 609, with the last expression for 
 V in 593, we find, for the lines of curvature on an ellipsoid, the 
 differential equations, 
 
 = S . vdp, = S . vdpidpK, or = S . vnr/c, = S . it, 
 
 if T be a vector parallel to the tangent to such a line ; and then, 
 by combining these two last equations, we find that r may be 
 expressed as follows, r = UV. vi+UY. vk; which reproduces the 
 theorem, discovered (I believe) by M. Chasles, that the lines of 
 curvature on an ellipsoid (or other surface of the second order) 
 bisect at each point the angles between the two circular sections 
 of the surface. Again, if the last formula of 604, or of 605, be 
 suitably combined with quaternion forms of the equation of a co?ie 
 of the second degree, such as those assigned in 438, where /3 is 
 afocal line, and in 678, where a, j3 are cyclic normals, those theo- 
 rems may be deduced, respecting the curvature of a spherical 
 conic, which have been published by me in the Cambridge and 
 Dublin MathematicalJournal, as part of a Paper entitled " Sym- 
 bolical Geometry." But it is manifestly impossible, in any sin- 
 gle Course of Lectures such as the present, to include all such 
 applications : and with thanks to those persons who have favoured 
 me so far by their attention, I now heartily bid them farewell. 
 
 END OF THE LECTURES. 
 
APPENDIX. 
 
 [The following is the Abstract of a Communication by the Author to the 
 Royal Irish Academy, which was referred to in article 675, page 673, of the 
 foregoing Lecture, and is reprinted here from the published Proceedings of 
 the Academy.] 
 
 Royal Irish Academy, May 13, 1850, 
 
 Sir William Rowan Hamilton gave an account of some 
 geometrical reasonings, tending to explain and confirm certain 
 results to which he had been previously conducted by the 
 method of quaternions, respecting the inscription of gauche 
 polygons in central surfaces of the second order. 
 
 1. It is a very well known property of the conic sections, 
 that if three of the four sides of a plane quadrilateral inscribed 
 in a given plane conic be cut by a rectilinear transversal in 
 three given points, the fourth side of the same variable qua- 
 drilateral is cut by the same fixed right line in a fourth point 
 likewise fixed. And whether we refer to the relation of invo- 
 lution discovered by Desargues, or employ other principles, 
 it is easy to extend this property to surfaces of the second 
 order, so far as the inscription in them of plane quadrilaterals 
 is concerned. If then we merely wish to pass from one point 
 p to another point r of such a surface, under the condition 
 that some other point q of the same surface shall exist, such 
 that the two successive and rectilinear chords, pq and qr, 
 shall pass respectively through some two given guide-points, 
 A and B, internal or external to the surface ; we are allowed 
 to substitute, for this pair of guide-points, another pair, 
 such as b' and a', situated on the same straight line ab ; and 
 may choose one of these two new points anywhere upon that 
 line, provided that the other be then suitably chosen. In fact. 
 
702 APPENDIX. 
 
 if c and c' be the two (real or imaginary) points in which the 
 surface is crossed by the given transversal ab, we have only 
 to take care that the three pairs of points aa', bb', cc', shall 
 be in involution. And it is important to observe, that in 
 order to determine one of the new guide-points, b' or a', when 
 the other is given, it is by no means necessary to employ the 
 points c, c', of intersection of the transversal with the surface, 
 which may be as often imaginary as real. We have only to 
 assume at pleasure a point p upon the given surface ; to draw 
 from it the chords paq, qbr; and then if a' be given, and b' 
 sought, to draw the two new chords ra's, sb'p ; or else if a' is 
 to be found from b', to draw the chords pb's, sa'r. For ex- 
 ample, if we choose to throw off the new guide-point b' to in- 
 finity, or to make it a guide-stai', in the direction of the given 
 line AB, we have only to draw, from the assumed initial and 
 superficial point p, a rectilinear chord ps of the surface, which 
 shall be parallel to ab, and then to join sr, and examine in 
 what point a' this joining line crosses the given line ab. The 
 point a' thus found will be entirely independent of the assumed 
 initial point p, and will satisfy the condition required : in 
 such a manner that if, from any other assumed superficial 
 point p', we draw the chords p'aq', qbr', and the parallel 
 p's' to ab, the chord rV shall pass through the same point a'. 
 All this follows easily from principles perfectly well known. 
 
 2. Since then for two given guide-points we may thus 
 substitute the system of a guide-star and a guide-point, it 
 follows that for three given guide-points we may substitute a 
 guide-star and two guide-points; and, therefore, by a repeti- 
 tion of the same process, may substitute anew a system of two 
 stars and one point. And so proceeding, for a system of n 
 given guide-points, through which n successive and rectilinear 
 chords of the surface are to pass, we may substitute a system 
 oin-\ guide-stars, and of a single guide-point. The pro- 
 blem of inscribing, in a given surface of the second order, a 
 gauche polygon of n sides, which are required to pass succes- 
 sively through w given points, is, therefore, in general, redu- 
 cible, by operations with straight lines alone, to the problem 
 
APPENDIX. 703 
 
 of inscribing in the same surface another gauche polygon, of 
 which the last side shall pass through a new fixed point, while 
 all its other (n - 1) sides shall be parallel to so many fixed 
 straight lines. And if the^r*^ n sides of an inscribed poly- 
 gon of n+l sides, ppi Pg . . . p„, be obliged to pass, in order, 
 through n given points, Ai As . . . a„ , namely, the side or chord 
 ppi through Ai, &c., it will then be possible, in general, to 
 incribe also another polygon, pqi Qs . . , Pn, having the same 
 first and wth points, p and p„ , and therefore the same final or 
 closing side p„p, but having the other n sides different^ and 
 such that the n-\ first of these sides, pqi, Qi Qs, • • • Qn-3 
 Q„ _ 1, shall be respectively parallel to w - 1 given right lines, 
 while the nth side Qts . i p^ shall pass through a fixed point b„. 
 The analogous reductions for polygons in conic sections have 
 long been familiar to geometers. 
 
 3. Let us now consider the inscribed gauche quadrilateral 
 PQi Q2 Q3, of which the four corners coincide with the four 
 first points of the last-mentioned polygon. In the plane 
 Qi Q2 Q3 of the second and third sides of this gauche quadri- 
 lateral, draw a new chord Qi R3 , which shall have its direction 
 conjugate to the direction of pqi , with respect to the given 
 surface. This new direction will itself be fixed, as being pa- 
 rallel to a fixed plane, and conjugate to a fixed direction, not 
 generally conjugate to that plane ; and hence in the plane in- 
 scribed quadrilateral R2Q1Q3Q3J the three first sides having 
 fixed directions, the fourth side Q3 R2 will also have its direction 
 fixed : which may be proved, either as a limiting form of the 
 theorem referred to in (1), respecting four points in one line, 
 or from principles still more elementary. And there is no diffi- 
 culty in seeing that because pqi and Qi R2 have fixed and con- 
 jugate directions, the chord PR2 is bisected by a fixed diameter 
 of the surface, whose direction is conjugate to both of their's ; or 
 in other words, that if o be the centre of the surface, and if we 
 draw thevariable diameter pon, the variable chord NRg will then 
 be parallel to the fixed diameter just mentioned. So far, then, 
 as we only concern ourselves to construct the fourth or closing 
 
704 APPENDIX. 
 
 side Qs p of the gauche quadrilateral pQi Q2 Q3, whose three first 
 sides have given or fixed directions, we may substitute it for ano- 
 ther gauche quadrilateral PNR2 Q3, inscribed in the same surface, 
 and such that while its first side pn passes through the centre 
 o, its second and third sides, nro and Rg Q3 , are parallel to 
 two fixed right lines. In other words, we may substitute, for 
 a system of three guide-stars^ a system of the centre and two 
 stars, as guides for the three first sides ; or, if we choose, in- 
 stead of drawing successively three chords, pqi, QiQa? Q2Q3» 
 parallel to three given lines, we may draw a first chord PRg, 
 so as to be bisected by a given diameter, and then a second 
 chord Rg Q3 , parallel to a given right line. 
 
 4. Since, for a system of three stars, we may substitute a 
 system of the centre and two stars, it follows that for a system 
 oifour stars we may substitute a system of the centre and 
 three stars ; or, by a repetition of the same process, may sub- 
 stitute a system of the centre, the same centre again, and two 
 stars ; that is, ultimately, a system of two stars may be sub- 
 stituted for a system oifour stars, the two employments of the 
 centre as a guide having simply neutralized each other, as 
 amounting merely to a return from n to p, after having gone 
 from p to the diametrically opposite point n. For five stars 
 we may therefore substitute three ; and for six stars we may 
 substitute four, or two. And so proceeding we perceive that 
 for any proposed system of guide-stars, we may substitute tico 
 stars, if the proposed number be even ; or three, if that num- 
 ber be odd. And by combining this result with what was 
 found in (2), we see that for any given system of ?^ guide-points 
 we may substitute a system of two stars a?id a point, if n be 
 odd ; or \in be even, then in that case we may substitute a sys- 
 tem of three stars and a point : which may again be changed, 
 by (3), to a system of the centre, two stars, and one point. 
 
 5. Let us now consider more closely the system of two 
 guide-stars, and one guide-point; and for this purpose let us 
 conceive that the two first sides pQi and Q1Q2 of an inscribed 
 gauche quadrilateral FQ1Q0P3 are parallel to two given right 
 
APPENDIX. 705 
 
 lines, while the third side Q2P3 is obliged to pass through a 
 fixed point B3 ; the first point p, and therefore also the qua- 
 drilateral itself, being in other respects variable. In the plane 
 PQi Q2 of the two first sides* which is evidently parallel to a 
 fixed plane, inscribe a chord q^s, whose direction shall be 
 *onjugate to that of the fixed line 0B3, and therefore shall 
 itself also be fixed, o being still the centre of the surface; and 
 draw the chord PS. Then, in the plane inscribed quadrilateral 
 PQi Q2S, the three first sides have fixed directions, and there- 
 fore, by (3), the direction of the fourth side sp is also fixed. 
 In the plane SQ2P3, which contains the given point B3, draw 
 through that point an indefinite right line B3C3, parallel to 
 SQ2; the line so drawn will have a given position, and will be 
 intersected, at some finite or infinite distance from B3, by the 
 chord SP3, which is situated in the same plane with it, namely, 
 in the plane SQ2 P3. But if we consider the section of the sur- 
 face, which is made by this last plane, and observe that the 
 two first sides of the triangle sq.j P3 pass, by the construction, 
 through a star or point at infinity conjugate to B3, and through 
 the point B3 itself, we shall see that, in virtue of a well-known 
 and elementary principle respecting triangles in conies, the 
 third side PgS must pass through the point D3, if D3 be the pole 
 of the right line B3C3, which contains upon it the two conju- 
 gate points ; this pole being taken with respect to the plane 
 section lately mentioned. If then we denote by D3E3 the in- 
 definite right line which is, with respect to the surface, the 
 polar of the fixed line B3C3, we see that the chord SP3 must in- 
 tersect this reciprocal polar also, besides intersecting the line 
 B3C3 itself. Conversely this condition, of intersecting these 
 two fixed polars, is sufficient to enable us to draw the chord 
 SP3 when the point s has been determined, by drawing from 
 the assumed point p the chord ps parallel to a fixed right line. 
 We may then substitute, for a system of two guide-stars and 
 one guide-point, the system of one guide-star and two guide- 
 lines; these lines being (as has been seen) a pair o{ reciprocal 
 polars, with respect to the given surface. 
 
 2z 
 
706 APPENDIX. 
 
 6. If, then, it be required to inscribe a polygon ppi Pg .. Pgn 
 with any odd number 2n+ I of sides, which shall pass suc- 
 cessively through the same number of given points, ai Ag . . 
 A2n + i> we may begin by assuming a point p upon the given 
 surface, and drawing through the given points 2n+l successive 
 chords, which will in general conduct to a final point P2n + 1#^ 
 distinct from the assumed initial point p. And then, by pro- 
 cesses of which the nature has been already explained, we can 
 find a point s such that the chord ps shall be parallel to a fixed 
 right line, or shall have a direction independent of the assumed 
 and variable position of p; and that the chord SP2n + i shall at 
 the same time cross two other fixed right lines, which are reci- 
 procal polars of each other. In order then to find a new point 
 p, which shall satisfy the conditions of the proposed problem, 
 or shall be such as to coincide with the point Psb + i, deduced 
 from it as above, we see that it is necessary and sufficient to 
 oblige this sought point p to be situated at one or other ex- 
 tremity of a certain chord ps, which shall at once be parallel 
 to a fixed line, and shall also cross two fixed polars. It is 
 clear then that we need only draw two planes, containing re- 
 spectively these two polars, and parallel to the fixed direction ; 
 for the right line of intersection of these two planes will be the 
 chord of solution required ; or in other words, it will cut the 
 surface in the two (real or imaginary) points, p and s, which 
 are adapted, and are alone adapted, to be positions of the first 
 corner of the polygon to be inscribed. 
 
 7. But if it be demanded to inscribe in the same surface a 
 polygon pPiPg .. P2n- ij with an even number 2w of sides, pass- 
 ing successively through the same even number of given points, 
 Ai Ag . . AgTC, the problem then acquires a character totally dis- 
 tinct. For if, after assuming an initial point p upon the sur- 
 face, we pass, by 2?i successive chords, drawn through the 
 given points Ai, &c., to a final point Pgn upon the surface, 
 which will thus be in general distinct from p ; it will indeed be 
 possible to assign generally two fixed polars, across which, as 
 two given guide-lines, a certain variable chord sp-3„ is to be 
 
APPENDIX. 707 
 
 drawn, like the chord sP3„ + i of (6); but the chord PS will not, 
 in this question, be parallel to a given line, or directed to a 
 given star; it will, on the contrary, by (3) (4) (5), be bisected 
 by a given diameter^ which we may call ab ; or, if we prefer to 
 state the result so, it will be now the supplementary chord ns 
 of the same diametral section of the surface (n being still the 
 point of that surface opposite to p), which will have a given 
 direction, and not the chord PS itself. In fact, at the end of 
 (4), we reduced the system of 2n guide-points to a system of 
 the centre, two stars, and one point; and in (5) we reduced 
 the system of two stars and a point to the system of a star 
 and two polars. In order then to find a point p which shall 
 coincide with the point v^n deduced from it as above, or which 
 shall be adapted to be the first corner of an inscribed polygon 
 of In sides passing respectively through the 2n given points, 
 Ai . . A2„, we must endeavour to find a chord ps which shall be 
 at once bisected by the fixed diameter ab, and shall also inter- 
 sect the two fixed polars above mentioned. And conversely, 
 if we can find any such chord ps, it will necessarily be at least 
 one chord of solution of the problem ; understanding hereby, 
 that if we set out with either extremity, p or s, of this chord, 
 and draw from it 2w successive chords pPi, &c., or sSi, &c., 
 through the 2n given points Ai, &c., we shall be brought back 
 hereby (as the question requires) to the point with which we 
 started. For, in a process which we have proved to admit of 
 being substituted for the process of drawing the 2w chords, we 
 shall be brought first from p to s, and then back from s to p ; 
 or else first from s to p, and then back from p to s : provided 
 that the chord of solution ps has been selected so as to satisfy 
 the conditions above assigned. 
 
 8. To inscribe then, for example, a gauche chiliagon in an 
 ellipsoid, ppi .. P999, or ssi .. S999, under the condition that its 
 thousand successive sides shall pass successively through a 
 thousand given points X\ .. Aioooj we are conducted to seek to 
 inscribe, in the same given ellipsoid, a chord ps, which shall 
 be at once bisected by a given diameter ab, and also crossed by 
 
 2z2 
 
708 APPENDIX. 
 
 a given chord cd, and by the polar of that given chord. Now 
 in general when any two proposed right lines intersect each 
 other, their respective polars also intersect, namely, in the 
 pole of the plane of the two lines proposed. Since then the 
 sought chord PS intersects the polar of the given chord cd, it 
 follows that the polar of the same sought chord PS must in- 
 tersect the given chord cd itself. We may therefore reduce 
 the problem to this form : to find a chord ps of the ellipsoid 
 which shall be bisected by a given diameter ab, and shall also 
 be such that while it intersects a given chord cd in some point 
 E, its polar intersects the prolongation of that given chord, in 
 some other point f. 
 
 9. The two sought points e, f, as being situated upon two 
 polars, are of course conjugate relatively to the surface ; they 
 are therefore also conjugate relatively to the chord cd, or, in 
 other words, they cut that given chord harmonically. The 
 four diametral planes abc, abe, abd, abf, compose therefore 
 an harmonic pencil ; the second being, in this pencil, har- 
 monically conjugate to the fourth ; and being at the same 
 time, on account of the polars, conjugate to it also with re- 
 spect to the surface, as one diametral plane to another. When 
 the ellipsoid becomes a sphere, the conjugate planes abe, abf 
 become rectangular ; and consequently the sought plane abe 
 bisects the angle between the two given planes abc and abd. 
 This solves at once the problem for the sphere ; for if, con- 
 versely, we thus bisect the given dihedral angle cabd by a 
 plane abe, cutting the chord cd in e, and if we take the har- 
 monic conjugate f on the same given chord prolonged, and 
 draw from e and f lines meeting ordinately the given diame- 
 ter AB, these two right lines will be situated in two rectangu- 
 lar or conjugate diametral planes, and will satisfy all the other 
 conditions requisite for their being polars of each other ; but 
 each intersects the given chord cd, or that chord prolonged, 
 and therefore each intersects also, by (8), the polar of that 
 chord ; each therefore satisfies all the transformed conditions of 
 the problem, and gives a chord of solution, real or imaginary. 
 
APPENDIX. 709 
 
 More fully, the ordinate ee' to the diameter ab, drawn from 
 the internal point of harmonic section e of the chord cd, 
 gives, when prolonged both ways to meet the surface, the 
 chord of real solution, PS ; and the other ordinate rr' to the 
 same diameter ab, which is drawn from the external point of 
 section f of the same chord cd, and which is itself wholly ex- 
 ternal to the surface, is the chord of imaginary solution. But 
 because when we return from the sphere to the ellipsoid, or 
 other surface of the second order, the condition of bisection of 
 the given dihedral angle cabd is no longer fulfilled by the 
 sought plane abe, a slight generalization of the foregoing 
 process becomes necessary, and can easily be accomplished as 
 follows. 
 
 10. Conceive, as before, that on the diameter ab the or- 
 dinate ee' is let fall from the internal point of section e, and 
 likewise the ordinates cc' and dd' from c and d ; and draw also, 
 parallel to that diameter, the right lines cc", dd", ee", from 
 the same three points c, d, e, so as to terminate on the dia- 
 metral plane through o which is conjugate to the same dia- 
 meter; in such a manner that oc", od", oe" shall be parallel 
 and equal to the ordinates c'c, d'd, e'e ; and that the segments 
 CE, E D of the chord cd shall be proportional to the segments 
 c"e", e"d" of the base c"d" of the triangle c"od", which is 
 situated in the diametral plane, and has the centre o for its 
 vertex. For the case of the sphere, the vertical angle c^'od" of 
 this triangle is, by (9), bisected by the line oe"; wherefore 
 the'sides oc", o d", or their equals, the ordinates c'c, d'd, are, 
 jn this case, proportional to the segments g"e", E"D"ofthe 
 base, or to the segments ce, ed of the chord : while the 
 squares of the ordinates are, for the same case of the sphere, 
 equal to the rectangles ac'b, ad'b, under the segments of the 
 diameter ab. Hence, ^or the sphere, the squares of the seg- 
 ments of the given chord are proportional to the rectangles 
 under the segments of the given diameter, these latter seg- 
 ments being found by letting fall ordinates from the ends of 
 the chord ; or, in symbols, we have the proportion, 
 
710 APPENDIX. 
 
 CF^ : DF^ : : ce^ : ed^ : : ac'b : ad'b. 
 
 But, by the general principles of geometrical deformation , the 
 property, thus stated, cannot he peculiar to the sphere. It 
 must extend, without any further modification, to the ellipsoid; 
 and it gives at once, for that surface, the two points of har- 
 monic section, e and f, of the given chord cd, through which 
 points the two sought chords of real and imaginary solution 
 are to pass ; these chords of solution are therefore completely 
 determined, since they are to be also ordinates, as before, to 
 the given diameter ab. The problem of inscription for the 
 ellipsoid is therefore fully resolved ; not only when, as in (6), 
 the number of sides of the polygon is odd, but also in the 
 more difficult case (7), when the number of sides is even. 
 
 11. If the given surface be a hyperboloid of two sheets, 
 one of the two fixed polars will still intersect that surface, and 
 the fixed chord cd may still be considered as real. If the 
 given diameter ab be also real, the proportion in (10) still 
 holds good, without any modification from imaginaries, and 
 determines still a real point e, with its harmonic conjugate f, 
 through one or other of which two points still passes a chord 
 of real solution, while through the other point of section still 
 is drawn a chord of imaginary solution, reciprocally polar to 
 the former. But if the diameter ab be imaginary, or in other 
 words if it fail to meet the proposed hyperboloid at all, we 
 are then led to consider, instead of it, an ideal diameter a'b', 
 having the same real direction, but terminating, in a well- 
 known way, on a certain supplementary surface; in such a 
 manner that while a and b are now imaginary points, the 
 points a' and b' are real, although not really situated on the 
 given surface ; and that 
 
 OA^ = ob^ = - oa'^ = - ob'2. 
 
 The points c' and d' are still real, and so are the rectangles 
 ac'b and ad'b, although a and b are imaginary ; for we may 
 write, 
 
 ac'b = OA^ - OC'2, ad'b = OA- - OD'^, 
 
APPENDIX. 711 
 
 and the proportion in (10) becomes now, 
 
 cf2 : DF^ : : ce^ : ed^ : : oc'^ + oa'^ : od'^ + oa'^. 
 It gives therefore still a real point of section e, and a. real con- 
 jugate point F ; and through these two points of section of CD 
 we can still draw two real right lines, which shall still ordi- 
 nately cross the real direction of ab, and shall still be two re- 
 ciprocal polars, satisfying all the transformed conditions of the 
 question, and coinciding still with two chords of real and 
 imaginary solution. For the double-sheeted hyperboloid, there- 
 fore, as well as for the ellipsoid, the problem of inscribing a 
 gauche chiliagon, or other even-sided polygon, whose sides 
 shall pass successively, and in order, through the same given 
 number of points, is solved by a system of tivo polar chords, 
 which we have assigned geometrical processes to determine; 
 and the solutions are still, in general, four in number; two of 
 them being still real, and two imaginary. 
 
 12. If the given surface be a hyperboloid of one sheet, then 
 not only may the diameter ab be real or imaginary, but also 
 the chord cd may or may not cease to be real ; for the two 
 fixed polars will now either both meet the surface, or else both 
 fail to meet it in any two real points. When ab and cd are 
 both real, the proportion in (10), being put under the form 
 
 CF^ : DF^ : : ce^ : ed^ : : oa^ - oc'^ : oa^ - od'^, 
 shews that the point of section e and its conjugate f will be 
 real, if the points c' and n' fall both on the diameter ab itself, 
 or both on that diameter prolonged ; that is, if the extremities 
 c and D lie both within or both without the interval between 
 the two parallel tangent planes to the surface which are drawn 
 at the points a and b : under these conditions therefore there 
 will still be ttvo real right lines, which may still be called the 
 two chords of solution ; but because these lines will still be 
 two reciprocal polars, they will now (like the two fixed polars 
 above mentioned) either both meet the hyperboloid, or else 
 both fail to meet it; and consequently there will now be either 
 four real, or else four imaginary solutions. If ab and cd be 
 still both real, but if the chord en have one extremity within 
 
712 
 
 APPENDIX. 
 
 and the other extremity without the interval between the two 
 parallel tangent planes, the proportion above written will 
 assign a negative ratio for the squares of the segments of cd ; 
 the points of section e and f, and the two polar chords of so- 
 lution, become therefore, in this case, themselves imaginary ; 
 and of course, by still stronger reason, the four solutions of 
 the problem become then imaginary likewise. If cd be real, 
 but AB imaginary, the proportion in (11) conducts to two real 
 points of section, and consequently to two real chords, which 
 may, however, correspond, as above, either to four real or to 
 four imaginary solutions of the problem. And, finally, it will 
 be found that the same conclusion holds good also in the re- 
 maining case, namely, when the chord cd becomes imaginary, 
 whether the diameter ab be real or not ; that is, when the two 
 fixed polars do not meet, in any real points, the single-sheeted 
 hyperboloid. 
 
 13. Although the case last mentioned may still be treated 
 by a modification of the proportion assigned in (10), which 
 was deduced from considerations relative to the sphere, yet in 
 order to put the subject in a clearer (or at least in another) 
 point of view, we may now resume the problem for the ellip- 
 soid as follows, without making any use of the spherical de- 
 formation. It was required to find two lines, reciprocally 
 polar to each other, and ordinately crossing a given diameter 
 AB of the ellipsoid, which should also cut a given chord cd of 
 the same surface, internally in some point e, and externally 
 in some other point f. Bisect cd in g, and conceive ef to be 
 bisected in h ; and besides the four old ordinates to the dia- 
 meter AB, namely cc', dd', ee', and ff', let there be now sup- 
 posed to be drawn, as two new ordinates to the same diameter, 
 the lines gg' and hh'. Then g will bisect c'd', and h' will 
 bisect eV; while the centre o of the ellipsoid will still bisect 
 AB. And because the points e' and f' are harmonic conju- 
 gates, not only with respect to the points a and b, but also 
 with respect to the points c' and d', we shall have the follow- 
 ing equalities : 
 
Hence, 
 that is, 
 
 appendix. 713 
 
 h'f'2 = h'e'2 = h'a . h'b = h'c' . h'd', 
 = h'o^ - oa2 = h'g'2 - g'c'2. 
 
 OH'2 - g'h'2 = OA^ - c'g'2, 
 OA^ + OG'2 - c'g'2 Oa2 + OC' . OD' 
 
 OH = 
 
 2og' oc' + od' 
 
 Now each of these two last expressions for oh' remains 
 real, and assigns a real and determinate position for the point 
 h', even when the points c', d', or the points a, b, or when 
 both these pairs of points at once become imaginary ; for the 
 points o and G'are still in all cases real, and so are the squares 
 of OA and c'g', the rectangle under oc' and od', and the sum 
 oc'+ od'. Thus h' can always be found, as a real point, and 
 hence we have a real value for the square of h'e', or h'f', which 
 will enable us to assign the points E'and r' themselves, or else 
 to pronounce that they are imaginary. 
 
 14. We see at the same time, from the values h'o^- oa^ 
 and h'g'2- c'g'2 above assigned for h'e'^ or h'f'^, that these two 
 sought points e' and f' must both be real, unless the two fixed 
 points A and c' are themselves both real, since o, g', h', are, all 
 three, real points. But for the ellipsoid, and for the double 
 sheeted hyperboloid, we can in general oblige the points c, d, 
 and their projections c', d', to become imaginary, by selecting 
 that one of the two fixed polars which does not actually meet 
 the surface ; for these two sorts of surfaces, the two polar chords 
 of solution of the problem of inscription of a gauche polygon 
 with an even number of sides passing through the same num- 
 ber of given points, are therefore found anew to be two real 
 lines, although only one of them will actually intersect the 
 surface, and only two of the four polygons will (as before) be 
 real. And even for the single sheeted hyperboloid, in order 
 to render the two chords of solution imaginary lines, it is ne- 
 cessary that the two given polars should actually meet the 
 surface ; for otherwise the polar lines deduced will still be 
 real. It is necessary also, for the imaginariness of the two 
 
714 APPENDIX. 
 
 lines deduced, that the given diameter ab should be itself a 
 real diameter, or in other words that it should actually inter- 
 sect the hyperboloid. But even when the given chord CD 
 and the given diameter ab are thus both real, and when the 
 surface is a single sheeted hyperboloid, it does not follow that 
 the two chords of solution may not be real lines. We shall 
 only have failed to prove their reality by the expressions re- 
 cently referred to. We must resume, for this case, the reason- 
 ings of (12), or some others equivalent to them ; and we find, 
 as in that section of this Abstract, for the imaginariness of the 
 two sought polar lines, the condition that one of the two ex- 
 tremities of the given and real chord cd shall fall within, and 
 that the other extremity of that chord shall fall without the 
 interval between the two real and parallel tangent planes to 
 the single sheeted hyperboloid, which are drawn at the extre- 
 mities of the real diameter ab. Sir W. R. Hamilton confesses 
 that the case where all these particular conditions are com- 
 bined, so as to render imaginary the two polar lines of solu- 
 tion, had not occurred to him when he made to the Royal 
 Irish Academy his communication of June, 1849. 
 
 15. It seems to him worth while to notice here that instead 
 of the foregoing metric processes for finding (when they exist) 
 the two lines of solution of the problem, the foWowing graphic 
 process of construction of those lines may always, at least in 
 theory, be substituted, although in practice it will sometimes 
 require modification for imaginaries. In the diametral plane 
 ABC, draw a chord kd'l, which shall be bisected at the known 
 point d' by the given diameter ab ; and join ck, cl. These 
 joining lines will cut that diameter in the two sought points 
 e', f'; which being in this manner found, the two sought 
 lines of solution ee', ff', are constructed without any diflfi- 
 culty. For the sphere, the ellipsoid, and the hyperboloid of 
 two sheets, although not always for the single sheeted hyper- 
 boloid, this simple and graphic process can actually be applied, 
 without any such modification from imaginaries as was above 
 alluded to. The consideration of non-central surfaces does 
 
APPENDIX. 715 
 
 not enter into the object of the present communication; nor 
 has it been thought necessary to consider in it any limiting 
 or exceptional cases, such as those where certain positions or 
 directions become indeterminate, by some peculiar combina- 
 tions of the data, while yet they are m general definitely as- 
 signable, by the processes already explained. 
 
 16. Sir William Rowan Hamilton is unwilling to add to 
 the length of this communication by any historical references; 
 in regard to which, indeed, he does not consider himself pre- 
 pared to furnish anything important, as supplementary to what 
 seems to be pretty generally known, by those who feel an in- 
 terest in such matters. He has however taken some pains to 
 inquire, from a few geometrical friends, whether it is likelj/ 
 that he has been anticipated in his results respecting the in- 
 scription of gauche polygons in surfaces of the second order; 
 and he has not hitherto been able to learn that any such an- 
 ticipation is thought to exist. Of course he knows that he 
 must, consciously and unconsciously, be in many ways in- 
 debted to his scientific contemporaries, for their instructions and 
 suggestions on these and on other subjects ; and also to his 
 acquaintance, imperfect as it may be, with what has been done 
 in earlier times. But he conceives that he only does justice 
 to the yet infant Method of Quaternions (communicated to the 
 Royal Irish Academy for the first time in 1843), when he 
 states that he considers himself to owe, to that new method 
 of geometrical research, not merely the results stated to the 
 Academy in the summer of 1849, respecting these inscriptions 
 of gauche polygons, and several other connected although 
 hitherto unpublished results, which to him appear remarkable, 
 but also the suggestion of the mode of geometrical investiga- 
 tion which has been employed in the present Abstract. No 
 doubt the principles used in it have all been very elementary, 
 and perhaps their combination would have cost no serious 
 trouble to any experienced geometer who had chosen to attack 
 the problem. But to his own mind the whole foregoing in- 
 vestigation presents itself as being (what in fact in his case it 
 
716 APPENDIX. 
 
 was) a mere translation of the quaternion analysis into ordi- 
 nary geometrical language, on this particular subject. And 
 he will not complicate the present Abstract by giving, on this 
 occasion, any account of those other theorems respecting po- 
 lygons in surfaces, to which the Calculus of Quaternions has 
 conducted him, but of which he has not yet seen how to 
 translate the proofs (for it is easy to translate the results) into 
 the usual language oi geometry.* 
 
 * It will not have escaped the notice of geometrical readers of the fore- 
 going Abstract of May, 1850, that, instead of the centre and guide-s?ars, we 
 may as easily conceive any fixed point o, with points in its polar (or conju- 
 gate) plane Q ; and that then, by using the two principles : P', that for any 
 two guide-points two others on the same right line may be substituted, 
 whereof one may be assumed at pleasure ; and, II"'', that a system of two 
 conjugate guide-points is equivalent to a system of two conjugate guide-lines, 
 namely, the line of the two given points, and its reciprocal polar, and there- 
 fore also to a system of two other conjugate points, on this latter polar line ; 
 we may first transform any proposed system of n guide-points into another 
 system of which all but the last shall be contained in the assumed plane Q ; 
 and may then substitute for any three points in that plane the system of the 
 assumed pole o, and of two points in Q. In this way, by an easy extension 
 of the process employed in the Abstract, we may transform any proposed 
 odd system of n guide-points into a system of thkee such points, which will 
 then give easily (as in the plane problem) one right line, as the unique chord 
 of real or imaginary solution, for the problem of the inscription of an odd-sided 
 polygon, whose sides shall pass in order through the n given guide-points. 
 But in the contrary case, namely, when n is even, the same general process 
 conducts to a transformed system of fouk guide-points, conjugate two by 
 two ; namely, the assumed pole o, a point in the plane Q, and a second pair 
 of mutually conjugate points, which may all be replaced by two polar pairs 
 of guide-lines ; across which four lines there may generally be drawn (as in 
 the Abstract) two polar chords of solution (real or imaginary), for the prob- 
 lem of the inscription of an even-sided polygon : this latter problem being thus 
 again reduced (by a slight modification of the process in art. 13) to the well- 
 known one of finding two points on a given line, which shall be at once har- 
 monically conjugate with respect to two given pairs of points thereon. The 
 writer is still unable to say whether these general reductions, of the problem 
 of inscribing a gauche polygon in a surface of the second order (or even in a 
 sphere"), involving as they do a proof of the essential distinction (in results, 
 and not merely in methods') betiaeen the odd and even cases, have hitherto oc- 
 curred to geometers. (April, 1853.) 
 
APPENDIX B. 717 
 
 APPENDIX B.* 
 
 [Reprinted (with Notes) from the Proceedings of the Academy.] 
 
 Royal Irish Academy, June 25, 1849. 
 
 Sir William Rowan Hamilton communicated to the Aca- 
 demy some results, obtained by the quaternion analysis, re- 
 specting the inscription of gauche polygons in surfaces of the 
 second order. 
 
 If it be required to inscribe a rectilinear polygon p, Pj, 
 P2 . . . Pk_i in such a surface, under the conditions that its n 
 successive sides, pPi, Pi P2, . . . p^-iP, shall pass respectively 
 through n given points, Aj, A2, . . . a„, the analysis of Sir W. 
 R. H. conducts to one, or to two real^ right lines, as contain- 
 ing the first corner p, according as the number n of sides is 
 odd or even : while, in the latter of these two cases, the two 
 real right lines thus found are reciprocal polars of each 
 other, with reference to the surface in which the polygon is 
 to be inscribed. Thus, for the inscription of a plane triangle, 
 
 • It had been designed that with the foregoing Appendix, which has been 
 reprinted without any alteration from the Proceedings of the Royal Irish Aca- 
 demy, of the date already mentioned (May 13th, 1850), the present Volume 
 should conclude. But it has since been thought that those persons who may 
 have done the author the honour to read so far, might like to have at hand a 
 copy of the published Abstract of an earlier communication to the Academy, 
 made at the Meeting of June 23th, 1849, which is intimately connected with 
 the subject of the foregoing Appendix, and is indeed referred to in it (at 
 page 714), and also in Lecture VII. (at page 677). It is therefore now 
 thought useful to reprint that earlier Abstract, with a few notes annexed, 
 as a second Appendix to this work : and indeed to follow it up by another 
 short and appended paper. 
 
 f For a case in which the two lines become imaginary, see the foi*egoing 
 Appendix, Art. 14 (page 714). 
 
718 APPENDIX B. 
 
 or of a gauche pentagon, heptagon, &c., in a surface of the 
 second order, where three, five, seven, &c. points are given 
 upon its sides, a single right line is found, which may or may 
 not intersect the surface ; and the problem of inscription ad- 
 mits generally of two real or of two imaginary solutions. 
 But for the inscription of a gauche quadrilateral, hexagon, 
 octagon, &c., when four, six, eight, &c. points are given on 
 its successive sides, two real right lines are found, which (as 
 above stated) are polars of each other; and therefore, if the 
 surface be an ellipsoid, or a hyperboloid of two sheets, the 
 problem admits generally of two real and of two imaginary 
 solutions : while if the surface be a hyperboloid of one sheet, 
 the four solutions are then, in general, together real, or toge- 
 ther imaginary. 
 
 When a gauche pentagon, or polygon with 2m + 1 sides, 
 is to be inscribed in an ellipsoid or in a double-sheeted hyper- 
 boloid, and when the single straight line, found as above, lies 
 wholly outside the surface, so as to give two imaginary solu- 
 tions of the problem as at first proposed, this line is still not 
 useless geometrically; for its reciprocal polar intersects the 
 surface in two real points, of which each is the first corner of 
 an inscribed decagon, or polygon with Am + 2 sides, whose 
 2m + 1 pairs of opposite sides intersect each other respectively 
 in the 2m + 1 given points, Ai, A2, . . . Aom+i. Thus when, in 
 the well-known problem of inscribing a triangle in a plane 
 conic, whose sides shall pass through three given points, the 
 known rectilinear locus of the first corner is found to have no 
 real intersection with the conic, so that the problem, as usually 
 viewed, admits of no real solution, and that the inscription 
 of the triangle becomes geometrically impossible ; we have 
 only to conceive an ellipsoid, or a double-sheeted hyperboloid, 
 to be so constructed as to contain the given conic upon its 
 surface; and then to take, with respect to this surface, the 
 polar of this known right line, in order to obtain two real or 
 geometrically possible solutions of another problem, not less 
 interesting : since this rectilinear polar will cut the surface in 
 
APPENDIX B. 719 
 
 two real points, of which each is the first corner of an inscribed 
 gauche hexagon whose opposite sides intersect each other in the 
 three points proposed. (It may be noticed that the three 
 diagonals of this gauche hexagon, or the three right lines 
 joining each corner to the opposite one, intersect each other in 
 one common point * namely, in the pole of the given plane.) 
 
 If we seek to inscribe a polygon of 4m sides in a surface of 
 the second order, under the condition that its opposite sides 
 shall intersect respectively in 2m given points, the quaternion 
 analysis conducts generally to two polar right lines, as loci of 
 the first corner, which lines are the same with those that would 
 be otherwise found as loci of the first corner of an inscribed 
 polygon of 2m sides, passing respectively through the 2/« 
 given points. Thus, iii general, the polygon of 4m sides, 
 found as above, is merely the polygon of 2m sides, with each 
 side twice traversed by the motion of a point along its peri- 
 meter. But if a certain condition be satisfied, by a certain 
 arrangement of the 2m give7i points in space ; namely, if the 
 last point K^m be on that real right line which is the locus of 
 the first corner of a real or imaginary inscribed polygon of 
 2m- 1 sides, which pass respectively through the first 2m- 1 
 given points Ai, . . . Aa^.i; then the inscribed polygon of 4m 
 distinct sides becomes not only possible but indeterminate, 
 its first corner being in this case allowed to take any posi- 
 tion on the surface. For example, if two triangles p' p'l p'j, 
 p" p"i p"2 be inscribed in a conic, so that the corresponding 
 sides p'p'i and p" p"i intersect each other in Ai; p'i p'a and 
 p"i p"2 in A2 ; and p'3 p', p"2 p", in A3 ; and if we take a 
 fourth point A4 on the right line p' p", and conceive any sur- 
 face of the second order constructed so as to contain the given 
 conic ; then any point P, on this surface, is fit to be the first 
 corner of a plane or gauche octagon, p Pi . . . P7, inscribed in 
 the surface, so that the first and fifth sides p Pi, P4 P5 shall 
 
 * More generally, if the opposite sides of an inscribed gauche polygon of 
 4m + 2 sides intersect upon one common plane, the lines connecting opposite 
 corners intersect in the pole of that plane. 
 
720 APPENDIX B. 
 
 intersect in Ai ; the second and sixth sides in A2 ; the third and 
 seventh sides in A3; and the fourth and eighth in a^. And 
 generally if 2m given points be points of intersection of oppo- 
 site sides of any one inscribed polygon of 4m sides, the same 
 2m points are then fit to be intersections of opposite sides of 
 infinitely many other inscribed polygons, plane or gauche, of 
 Am sides. A very elementary example is furnished by an in- 
 scribed plane quadrilateral, of which the two points of meet- 
 ing of opposite sides are well known to be conjugate, relatively 
 to the conic or to the surface, and are adapted to be the points 
 of meeting of opposite sides of infinitely many other inscribed 
 quadrilaterals. 
 
 When all the sides but one, of an inscribed gauche poly- 
 gon, pass through given points, the remaining side may be 
 said generally to be doubly tangent to a real or imaginary sur- 
 face of the fourth order, which separates itself into two real 
 or imaginary surfaces of the second order, hsLving real or ima- 
 ginary double* contact with the original surface of the second 
 order, and with each other. If the original surface be an 
 ellipsoid (e), and if the number of sides of the inscribed po- 
 lygon, pPi . . , P2OT, be odd, =2m+l, so that the number of 
 fixed points Ai, . . . A2,„ is even, = 2m, then the two surfaces 
 enveloped by the last side Psm p are a real inso'ibed ellipsoid 
 (e'), and a real exscribed hyperboloid of two sheets (e") ; and 
 these three surfaces (e) (e') (e") touch each other at the two 
 real't points b, b', which are the first corners of two inscribed 
 polygons BBi . . . Bsm-i and b'b'i . . . b'sm-u whose 2m sides pass 
 
 * It will be seen below that this contact may become quadruple, namely, 
 for the case of an even-sided polygon, in accordance with an acute remark 
 which was made in 1849 by Arthur Cayley, Esq., in a letter to the Rev. 
 George Salmon, F. T. C. D. Perhaps I may be permitted to add, that be- 
 fore I saw Mr. Cayley's letter, I had been conducted to the same result in 
 my own unpublished researches. 
 
 t The three surfaces must be considered to touch each other also at the 
 two imaginary points which are situated on the polar of the chord bb' : and the 
 four points of contact become all real, or all imaginary, when the original sur- 
 face becomes a single-sheeted hyperboloid. 
 
APPENDIX B. 721 
 
 respectively through the 2m given points (a). If these three 
 surfaces of the second order be cut by any three planes pa- 
 rallel to either of the two common tangent planes at b and b', 
 the sections are three siinilar and similarly placed ellipses; 
 thus B and b' are two of the four umhilics of the ellipsoid (e'), 
 and also of the hyperboloid (e") when the original surface e 
 is a sphere. The closing chords Po„^ p touch a series of real 
 curves (jc') on (e'), and also another series of real curves (c") 
 on (e"), which curves are the aretes de rebrousseinent of two 
 series of developable* surfaces, (d') and (d"j, into which latter 
 surfaces the closing chords arrange themselves ; but these two 
 sets of developable surfaces are not generally rectangular to 
 each other, and consequently the closing chords themselves 
 are not generally perpendicular to any one common surface. 
 However, when (e) is a sphere, the developable surfaces cut 
 it in two series of curves, (f'), (f"), which everywhere cross 
 each other at right angles ; and generally at any point P on 
 (e), the tangents to the two curves (f') and (f") are parallel 
 to two conjugate semidiameters. 
 
 The centres^ of the three surfaces of the second order are 
 placed on one straight line; and every closing chord P2;» p is 
 cut harmonically at the points where it touches the two sur- 
 
 * Malus discovered that right lines proceeding from ayiy surface, accord- 
 ing to any law, arrange themselves into two series of developable surfaces, 
 and touch two series of curves (the aretes), which are contained upon two 
 other surfaces, or rather generally upon two sheets of one common surface. 
 What seemed to me remarkable in the present question, independently of the 
 non-rectangular ity of the developables, was chiefly the separability of the two 
 superficial envelopes, in both the odd and even cases, and their imaginariness 
 for the latter case ; at least if the original surface, in which the even-sided 
 gauche polygon is inscribed, be not a ruled one. 
 
 f Mr. Cayley observed, in that letter of his to Mr. Salmon which has 
 been mentioned in a former note, that this statement of mine, respecting the 
 coUinearity of the three centres, ought to be replaced by the more general 
 one, that the three poles of any arbitrary plane, with respect to the three 
 surfaces, are situated on one straight line. In general, as it was well re- 
 marked by Mr. Cayley, the relations between these three surfaces are merely 
 those between three which h^ve four generating lines in common. 
 
 3 A 
 
722 APPENDIX B. 
 
 faces* (e'), (e"), or the two curves (c'), (c"), which are the 
 aretes of the two developable surfaces {p), (d"), passing 
 through that chord v^m^- In a certain class oi cases the three 
 surfaces (e), (e'), (e") are all of revolution, round one common 
 axis ; and when this happens, the curves (c'), (c"), (f'), (f") 
 are certain spires^ upon these surfaces, having this common 
 character, that for any one such spire equal rotations round 
 the axis give equal anharmonic ratios; or that, more fully, if 
 on a spire (c'), for example, there be taken two pairs of points 
 c'l, c'2 and c'3, c'4, and if these be projected on the axis b b' 
 in points g'i, g'^ and g's, G'4, then the rectangle bg'j . g'jB' 
 will be to the rectangle bg'j . g'i b', as bg's . G'4 b' to BG'4 . G'3 b', 
 if the dihedral angle c'l BB'c'2 be equal to the dihedral angle 
 c'3 bb' c'4. In another extensive class of cases the hyperbo- 
 loid of two sheets (e") reduces itself to a pair of planes, touch- 
 ing the given ellipsoid (e) in the points b and b'; and then 
 the prolongations of the closing chords, P2mP, all meet the 
 right line of intersection of these two tangent planes : or the 
 inscribed ellipsoid (e') may reduce itself to the right line bb', 
 which is, in that case, crossed by all the closing chords. For 
 example, if the first four sides of an inscribed gauche penta- 
 gon pass respectively through four given points, which are 
 all in one common plane, then the fifth side of the pentagon 
 intersects a fixed right linej in that plane. 
 
 An example of imaginary envelopes is suggested by the 
 
 * In general, if any two points be conjugate relatively to any two of the 
 three surfaces, they are conjugate also relatively to the third; so that the 
 three polar planes of an arbitrary point, taken with respect to the three sur- 
 faces, intersect in one right line. 
 
 f In this case, if the surface (e) be a sphere, the spires (f) (f") may be 
 stereographically projected into two sets of logarithmic spirals, which cross 
 each other at right angles. 
 
 X This little theorem is perhaps well known ; it may, among other ways, 
 be obtained by projection from a property which is proved by quaternions 
 in Lecture VI., namely, that if the four first sides of a gauche pentagon in- 
 scribed in a sphere be respectively parallel to four given lines, the fifth side 
 will then be parallel to a given plane. 
 
APPENDIX B. 723 
 
 problem of inscribing a gauche quadrilateral,, hexagon, or po- 
 lygon of 2m sides in an ellipsoid, all the sides but the last 
 being obliged to pass through fixed points. In this problem 
 the last side may be said to touch two imaginary surfaces* of 
 the second order, which intersect each other in two real or 
 
 * Soon after this Abstract had been printed, I perceived, by continuing 
 the calculations with quaternions, that these two enveloped surfaces of the se- 
 cond order were two imaginary cones, which touched the original ellipsoid (e) 
 along two imaginary conies, and might be considered to have double contact 
 with it and with each other (in agreement with an earlier passage of the 
 Abstract) ; namely, at those two points where the two imaginary conies of 
 contact, just now mentioned, crossed each other, and which were also si- 
 tuated on the real line of intersection of the planes of the two conies of inter- 
 section (mentioned in the text) : the four (real and imaginary) planes through 
 that line com.'^osmg dM harmonic pencil ; and the line itself being the chord 
 of solution, of the problem of inscribing a polygon of 2m — \ sides, passing 
 through the 2?« — 1 given points. The developable surfaces were at the same 
 time found to become imaginary planes, touching the cones, and resting on 
 the imaginary generatrices of the original surface (e), as what might be called 
 their bases on that surface : so that the cones, planes, and lines became all 
 real, when the surface (e) became a single-sheeted hyperboloid. (Compare 
 art. 677, page 678, of the Lectures.) 
 
 These geometrical results, at least so far as related to the conical enve- 
 lopes, and to the generatrices of the original surface, were communicated by 
 me, without demonstration (in letters of October, 1849), to my friends 
 Mr. Townsend and Mr. Salmon. A short sketch of the analysis by which 
 those results were perceived will perhaps be given in a subsequent Appendix : 
 but in the meantime I may mention an easy geometrical confirmation of some of 
 them, which has only recently occurred to me, while reprinting the Abstract 
 as above. Let there be any four assumed points p, Q, R, s, on some one pri- 
 mary (generatrix') of a given and single-sheeted hyperboloid; that is on a line 
 belonging to one given system, which we may call the primary system, of ge- 
 neratrices of that surface : and let four chords ppi, qqi, bki, ssi, be drawn 
 from these four points, through some one given guide-point Ai. In like 
 manner, let the chords Pi Po, &c., be drawn through another given point Ao ; 
 P2 P3, &c., through A3 ; and so on for any odd number = 2?k + 1 of guide-points, 
 till a final set of four points on the surface is obtained. Then the four points 
 Pi Qi El Si will be situated on some one secondary {generatrix), and their an- 
 harmonic ratio will be the same as that of the points pqrs. Hence, on ac- 
 count of the supposed odd number of the guide-points Ai Aj A3 . . , the four 
 initial and four final points, pqrs and Pam + i Q2m + i R2m+i s-im + h are arranged 
 on two generatrices of opposite systems, which therefore meet in some point 
 
 3 A 2 
 
724 APPENDIX B. 
 
 imaginary conies, situated in two real planes ; and when these 
 two conies are real, they touch the original ellipsoid in two 
 real and common points, which are the two positions of the 
 first corner of an inscribed polygon, whose sides pass through 
 the 2m - 1 fixed points. Every rectilinear tangent to either 
 conic is a closing chord Pam-i?; but no position of that clos- 
 ing chord, which is not thus a tangent to one or other of these 
 conies, is intersected anywhere* by any infinitely near chord 
 
 T ; and they have the same anharmonic ratio : consequently (by a known 
 theorem) the four connecting lines (or closing sides of the inscribed and 
 even-sided polygon), namely, Pam+i p, Q2m+i Q, &c., envelope a conic (ci) in 
 their common plane; and this conic touches each of the two generating lines tp, 
 TP3m+i of the surface; one in some point u, and the other in some point v. 
 In like manner, if q' be an initial point taken on the secondary through p, 
 then the final point Q'2m + i will be on the primary through P2m+i ; and if t' be 
 the point of meeting of these two generating lines, then the new closing 
 chords Pom+i p, Q'sm+i Q') &c., envelope a new conic (co) in their own plane, 
 which conic touches also the generating lines tp, TPom+i, the 1*' in some 
 point u', and the 2"'^ in some other point V. Thus the original hyperboloid 
 being called (e), its generating lines pt, pt', may be called (fi) (Fj), by ana- 
 logy to a notation in the Abstract ; the developable surfaces (Di), (do), which 
 rest on these two lines, are seen to be the two planes ptv, ptV, touching the 
 hyperboloid (e) at T and t' ; while the two conies (ci) (C2) must be consi- 
 dered as their respective aretes; the first superficial envelope, (ei), is the 
 locus of the conic (Ci), and is at the same time the developable surface cir- 
 cumscribed about the hyperboloid (e), along that curve of contact which is 
 the locus of the point t' thereon ; and the second superficial envelope, (ej), 
 of the closing chords Pjm+i P, is at once the locus of the conies (co), and the 
 developable circumscribed about (e) along that other curve of contact which 
 is the locus of the point t. All these geometrical constructions agree per- 
 fectly with the results of calculation stated above : the two last developable 
 surfaces (ei) (E3), which thus contain each indefinitely many plane conies, 
 whereof each is touched by indefinitely many positions of the closing chord, 
 being evidently the two conical envelopes, which have been mentioned in the 
 present Note. We see, at the same time, that the reciprocal polar of the 
 closing chord Pjm+i p is always another chord drawn from some point T of 
 the one plane conic of contact, to some point t' of the other : this polar, and 
 these two conies of contact, as well as the enveloping cones, becoming thus 
 together imaginary, when the surface (e) becomes an ellipsoid or a double- 
 sheeted hyperboloid. (April, 1853.) 
 
 * That is to say, in any real point : for the analysis which was employed 
 did not fail to recognise the existence of two imagiyiary intersections. 
 
APPENDIX B. 725 
 
 of the system. These results were illustrated by an example,* 
 in which there were threef given points ; one conic was the 
 known envelope of the fourth side of a plane inscribed qua- 
 drilateral ; and this was found to be the ellipse de gorge of a 
 certain single-sheeted hyperboloid, a certain section of which 
 hyperboloid, by a plane perpendicular to the plane of the el- 
 Fipse, gave the hyperbola which was, in this example, the other 
 real conic, and was thus situated in a plane perpendicular to the 
 plane of the ellipse. And to illustrate the imaginary charac- 
 ter of the enveloped surfaces, or the general non-intersection 
 (in this example) of infinitely near positions of the closing 
 chords in space, one such chord was selected ; and it was 
 shewn that all the infinitely near chords, which made with this 
 chord equal and infinitesimal angles, were generatrices (of 
 one common system) of an infinitely thin and single-sheeted 
 hyperboloid. 
 
 Conceive that any rectilinear polygon of n sides, bBi . . . 
 B,i.i, has been inscribed in any surface of the second order, 
 and that n points Ai . . . a„ have been assumed on its n sides, 
 BBi, . . . B„.iB. Take then at pleasure any point p upon the 
 same surface, and draw the chords pAjPi, . . . Pw-iA„p„, passing 
 respectively through the n points (a). Again begin with p„, 
 
 * In the particular example which was thus used as an illustration, in the 
 communication of 1849, the polygons were quadrilaterals inscribed in a sphere; 
 and the particular closing chord, which was compared with infinitely many 
 others infinitely near to it, was a diameter : some degree of symmetry being 
 also introduced into the selection of the three fixed points, which rendered 
 the results slightly more simple than they would otherwise have been, with- 
 out essentially altering their cliaracter. 
 
 f Any odd number of guide-points may be reduced to three, as is shewn in 
 the Note to Appendix A (page 716) ; and then the system of these three 
 points may be indefinitely varied, according to fixed lav>'s, not only within 
 their own plane, but also (by the principles of the same Note) in a certain 
 other and co7ijugate plane, which passes through a certain chord of solution 
 determined by the given guide-points : and thus is furnished a geometrical 
 explanation of the existence of the second plane conic mentioned in the text, 
 as being enveloped by one set of closing chords, and as being real if the first 
 plane conic be so, even when the enveloped cones are imaginary. 
 
726 APPENDIX B. 
 
 and draw, through the same n points (a), n other successive 
 chords, p„AiP„+i, . . .P2M-iA„P2„. Again, draw the n chords, 
 Pan AiP2„+i, . . . P3n-iA„P3„. Draw tangent planes at P„ and P2„, 
 meeting the two new chords pPon and p^Psti in points r, r'; 
 and draw any rectilinear tangent bc at b. Then one or other 
 of the two following theorems will hold good, according as 
 n is an odd or an even number. When n is odd, the three 
 points brr' will be situated in one straight line.* When n is 
 even, the three pyramids which have bc for a common edge, 
 and have for their edges respectively opposite thereto the three 
 chords pPan, P2«Pkj p«P3M) being divided respectively by the 
 
 * It is clear (as was remarked in the Philosophical Magazine for April, 
 1850, page 306), that this collinearity enables us, by the help of two points R 
 and r' thus found, to determine the unique chord of solution bb', connecting 
 the two positions of the initial corner of an inscribed polygon, whose sides 
 are required to pass successively through the n given guide-points (a), n 
 being an odd number. More generally, if we pass, by means of chords drawn 
 through those points from q to q,j, as we have done from p to p,i, p and q 
 being both assumed at pleasure on the surface (provided that they be not 
 taken on one common generatrix) ; and if the transverse chords, f„ q, q,i p, 
 intersect in ani/ point B ; it will be found to follow, as a sort of converse of a 
 theorem of the present Appendix (see page 719), that this point of intersec- 
 tion B must be situated upon that sought chord of solution, bb'. The connexion 
 of this new theorem with the one above referred to is easily seen to consist 
 in this: that if we take r as a new guide-point, following the n = 2m— 1 given 
 ones, we shall be conducted, by the repeated employment of this system of 
 2m points, first from p to Q, and then back from q to p, describing thus a 
 closed and doubly even polygon (quadrilateral, or octagon, &c.) of 4m sides, 
 whereof the opposite sides intersect in the 2w — 1 given points (a), and in the 
 new point b. The case of exception to the converse of the theorem of page 
 719, or the case of possible inscription of a gauche polygon, whose opposite 
 sides shall intersect each other two by two in an even number of points, with- 
 out those points being obliged to satisfy the condition mentioned in that 
 page, namely, the case where opposite corners of the polygon are situated on 
 one common generatrix of the surface, at first escaped my notice, when inves- 
 tigating the theorem itself by means of my own analysis : which arose chiefly 
 from the circumstance that in representing by calculation with biquaternions 
 the passage from a ruled surface to a sphere, a.ny portion of a generatrix was 
 replaced by an imaginary vector, or bivector, of which the square was null. 
 (Compare the intei'pretation of the differential equation d|0- = 0, as repre- 
 
APPENDIX B. 727 
 
 squares of those three chords, and multiplied by the squares 
 of the three respectively parallel semidiameters of the surface, 
 and being also taken with algebraic signs which it is easy to 
 determine, have their sum equal to zero. Both theorems con- 
 senting the two systems of generatrices, in art.. 677 of the Lectures.) And 
 in fact the exception exists only in an imaginary sense, for polygons in a 
 sphere, ellipsoid, or c^ow^Ze-sheeted hyperboloid. But, for a sm^/e-sheeted 
 hyperboloid, the geometrical reasoning of a recent Note shews easily, 
 that if the two initial points p and Q be assumed upon one common ge- 
 neratrix T0 (the number w of the given guide-points being odd), the 
 transverse chords pq„, qp„ are then both situated in a certain common plane 
 UTV, and may cross each other anywhere on a certain chord uv, which is not 
 in general coincident with the unique chord of solution, of the problem of in- 
 scription of an odd-sided polygon. However, the theorem of the Appendix, 
 to which the present Note relates, and which may be thus stated, that "the 
 chord PP2» (if n be odd) intersects generally the chord of solution bb' in a point 
 R, which is situated on the tangent plane to the original surface at p„," receives 
 a satisfactory verification by the same geometrical reasoning. For if, in the 
 construction just referred to, and with the letters therein employed, we 
 place the point p at u, then p„ will be at t, and Po„ at v; and the chord uv, 
 or the polar of the point t wdth respect to the conic (ci), that is with respect 
 to the section of the cone (ei) made by the tangent plane UTV to the given 
 hyperboloid (e) at T, passes through the point x where that tangent plane in- 
 tersects the chord of solution bb'. In fact, by the theory sketched in this 
 Appendix, and in its Notes, this chord of solution (for an odd system of given 
 points) is the polar, relatively to the given surface (e), of the line connecting the 
 two (real or imaginary) vertices, of the two circumscribed cones (ei) (eo) ; and 
 therefore the point x of this chord, as being situated in the ^Zawe of contact 
 of (e) (ei), has the same polar plane with respect to those two surfaces: but 
 the point t is conjugate to it relatively to (what is here) the hyperboloid 
 (e), and therefore also relatively to the cone (ei), or to the conic (cj), so 
 that the three points u, v, x are collinear. The same polar relation of the 
 chord of solution to the line of vertices gives obviously a geometrical confir- 
 mation of an earlier theorem of the same Appendix (page 718), respecting 
 the inscription of a gauche polygon of 4m + 2 sides, which sides intersect their 
 respective opposites in 2m + 1 given points : of which polygon that line is 
 (in position) a diagonal. 
 
 It may be here remarked that, if we attend only to position in space, there 
 is in general only one such polygon, which however counts as tivo, in confor- 
 mity with the general theory, because either of two opposite corners may be 
 taken as the initial point upon the surface. Thus the two gauche hexagons 
 of page 719 are wholly superposed on each other. (April, 1853.) 
 
728 APPENDIX B. 
 
 duct to a form of Poncelet's construction* (the present writer's 
 knowledge of which is derived chiefly from the valuable work 
 on Conic Sections, by the Rev. George Salmon, F. T. C. D.), 
 when applied to the problem of inscribing a polygon in a plane 
 
 * My acquaintance with the great work of M. Poncelet (Traite des Pro- 
 prietSs Projectives, Paris, 1822) is very partial and imperfect: but I believe 
 that I am safe in stating, that after shewing (^Traite, p. 307) that the free 
 side of any polygon, inscribed in a plane conic, took in succession the same 
 positions as the free side of a triangle, and therefore (p. 243) that it enve- 
 loped a second conic having double contact with the given one, because it was 
 projectively equivalent to a chord of given length inscribed in a circle, and 
 touching another concentric therewith (pp. 65, 69), Poncelet inferred (p. 332) 
 that the lines (ah!, dk), joining opposite extremities of any two such posi- 
 tions (ak, dk'), intersected on the chord of contact, on account of the parallelism 
 of the lines oppositely joining the extremities of two equal chords in a circle 
 (pp. 248, 249) : and thence concluded that the chord of solution of the problem 
 of inscription of a polygon in a given conic, whose sides should pass succes- 
 sively and in an assigned order through the same number of given points, 
 was the PascaVs-line of a certain hexagon (ak'dkaUt), obtained by assuming 
 Qp. 332) any three points (o, a, a") on the conic, and thence deriving three other 
 points (h, k', k"), by drawing lines through the given guide-points. A sort of 
 extension of this beautiful construction to space, for the case of an odd system 
 of given points, has been given in a recent Note : the second and third trials 
 being supposed to begin where the first and second end, and tangent planes 
 being employed. It might at first sight seem that the rule thus stated should 
 apply, for space, as well as for the plane, not only for an odd, but also for an 
 even number of given points : but I have found that the locus of the point e, 
 in which the chord ppj,! intersects the tangent plane to the given surface at 
 p,„ is not a right line, but a surface of the second order (a double-sheeted hy- 
 perboloid, if the given surface be an ellipsoid), when the number n is even. 
 However, when the given points are all situated in one common plane, this 
 superficial locus of r is found to dwindle into a right line, namely, the one as- 
 signed by Poncelet's construction. A very elegant proof of that celebrated 
 construction was proposed some years ago by Mr. Townsend, who has 
 remarked that the same problem of inscription of a polygon in a conic may be 
 reduced to finding a point upon the latter, which shall have the same anhar- 
 monic ratio with three initial as with three final points thereon : or which 
 shall be, in the language of Chasles, one of the two double points oftwoAo- 
 mographic divisions on the curve. This has suggested to me some researches 
 respecting a new sort of syngraphy in geometry, and of syngraphical figures, 
 direct and inverse, on surfaces of the second order; with determinations of the 
 TWO POINTS (real or imaginary) on such a surface, of which each is its own 
 INVERSE SYNGRAPH, and of the FOUR POINTS of which each is its own direct 
 
APPENDIX B. 729 
 
 conic : and the second theorem may easily be stated generally 
 under a graphic* instead of a metric form. 
 
 The analysis! by which these results, and others connected 
 with them, have been obtained, appears to the author to be 
 sufficiently simple, at least if regard be had to the novelty and 
 difficulty of some of the questions to which it has been thus 
 applied ; but he conceives that it would occupy too large a 
 space in the Proceedings, if he were to give any account of 
 it in them : and he proposes, with the permission of the Coun- 
 cil, to publish his calculations as an appendage to his Second 
 Series of Researches respecting Quaternions, in the Transac- 
 
 STNGRAPH, relatively to THREE GIVEN PAIRS of points on the same surface : 
 respecting which researches I shall only at present say, that they confirm in 
 a new and satisfactory way some of the main results of this Appendix, 
 It may, however, be here added, that it is in general possible to pass, by three 
 or by four reflexions (through so many fixed points), from one of any two 
 given syngraphical figures to the other, according as the syngraphy is in- 
 verse or direct : but that the one or the other sort of syngraphy exists, with 
 the proposed signification of the words, when any odd or any even number of 
 reflecting points is thus employed. (April, 1853.) 
 
 * The graphic form thus referred to, of this second theorem, was ex- 
 pressed by me as follows, in the lately cited number of the Philosophical Ma- 
 gazine (for April, 1850), having been also previously communicated in an 
 unprinted paper, which was read in the Mathematical and Physical Section 
 of the British Association for the Advancement of Science, at Birmingham, 
 in September, 1849 : — " If n be even, and if we describe two pairs of plane co- 
 nies on the surface, each conic being determined by the condition of passing 
 through three points thereon, as follows : the first pair of conies passing 
 through BPPsn, and p,j P2k Psn ; and the second pair through bp«P3„ and pp,jP3„; 
 it will then be possible to trace, on the same surface, tiuo other plane conies, of 
 which the first shall touch the two conies of the first pair, at the two points B 
 and P« ; while the second new conic shall touch the two conies of the second pair, 
 at the two points b and Fn„." In other words, the tangent at b to the section 
 BPP'K intersects the tangent at p„ to the section P^PsiiPsn; and the tangent at 
 the same point b to the section BP,jP3,i intersects the tangent at Po,, to pp„P2„ : 
 the existence of both which intersections is proved by quaternions in the fol- 
 lowing Appendix C (with a slightly different notation), for the case of an 
 original sphere, and therefore generally. 
 
 t Some sketch (or at least some specimen) of this analysis, in addition to 
 what has been given in articles 676, 677 of the Lectures, will be found in the 
 following Appendix. 
 
730 APPENDIX B. 
 
 tions of the Academy. He would only further observe, on the 
 present occasion, that he has made, in these investigations, a 
 frequent use of expressions of the form q + -v/(-1)q', where 
 -v/(-l) is the ordinary imaginary of the older algebra, while 
 Q and q' are two different quaternions^ of the kind introduced 
 by him into analysis in 1843, involving the three new imagf- 
 naries, i,j, k, for which the fundamental formula, 
 
 i^=j'^k''=ijk = -l, 
 holds good. (See the Proceedings of November 13th, 1843). 
 And Sir W. R. Hamilton thinks that the name "Biqua- 
 TEUNioN," which he has been for a considerable time accus- 
 tomed to apply, in his own researches, to an expression of 
 this form Q + -v/(-l) Q'j is a designation more appropriate to 
 such expressions than to the entirely different (but very inte- 
 resting) octonomials of Messrs. J. T. Graves and Arthur Cay- 
 ley, to which Octaves* the Rev. Mr. Kirkman, in his paper 
 on Pluquaternions,] has suggested (though with all courtesy 
 towards the present author), that the name of hiquaternion 
 might be applied. 
 
 * Mr. Cayley was the first to publish (Phil. Mag., March, 1845, p. 210) 
 an octonomial expression of the form here referred to, namely, Xg-f Xi ti + . . . 
 Xt t-, where i\, ... i-, were seven imaginary square roots of—1, grouping ac- 
 cording to seven ternary types, or forming seven triads analogous to the triad 
 ijk : and he shewed that the product of two such octonomials was another of 
 the same form, having a certain modular relation to the factors. Results es- 
 sentially the same had been previously communicated to me (compare Lec- 
 tures, p. 539), by Mr. J. T. Graves, in letters of December 26th, 1843, and 
 January 4th, 1844 ; his octave being of the form 
 
 a + ib -\-jc -\-kd-\- le + mf+ ng + oh, 
 with the same modular property as Mr. Cayley's ; and the relations between 
 his seven imaginaries, ijklmno, admitting of being thus summed up (compare 
 a formula above) : 
 
 _ ] = ii =j2 = A2 = Z2 = m3 = k2 = o2 = 
 
 ijk = ilm = ion =jln =jmo = klo = knm. 
 (See Trans. R. I. A., Vol. XXI., Part ii., pp. 338, 339.) But in these octo- 
 nomial forms, no natural separation into tivo sets of four takes place, as it 
 does in what I call on that account a hiquaternion : namely (if h denote here 
 the ordinary imaginary of algebra), an expression of this other form, 
 (w -t- ix +jy + kz) +h {w + ix +jy' + kz). 
 t Phil. Mag. for December, 1848, p. 449. 
 
APPENDIX C. 731 
 
 APPENDIX C 
 
 I. If we suppose that p is an unit vector derived from a 
 proposed but variable unit vector p, by the process of drawing 
 n successive chords from an assumed point p of the unit sphere, 
 through a system of ^ given guide points, Ai, . . . a^, to a de- 
 rived point p', then, by principles already explained, in the 
 text of the present work, we shall have not only the equations, 
 
 p2^-l, p'3^-1, (1) 
 
 but also a relation of the form, 
 
 p={-yqpq-\ (2) 
 
 where g* is a quaternion, involving the variable vector p only 
 in the first degree, and including two constant quaternions in 
 its expression. Let Q be that biquaternio7i, which is formed 
 from q, by changing p to the ordinary square root of -1 ; and 
 let \ and p. be two constant and real vectors, entering into the 
 following expression of a certain derived bivector : 
 
 P + X^-1=1q. (3) 
 
 Then, instead of the relation (2), which involves (as has been 
 said) two constant quaternions, we shall have this other or 
 transformed relation, which is equally real with the former, 
 but is in some respects simpler, as involving only two constant 
 vpcfovs 
 
 p={-y{i+p+Xp)p(\+p + Xpy^; (4) 
 
 or, as by (1), it may also be written : 
 
 ^-^ l-,p + Xp ' (^) 
 
 the upper sign answering to the case where the number w of 
 
 * This third Appendix contains a rapid outline of the quaternion analysis 
 by which some of the foregoing results were obtained, and is designed as a 
 sort of supplement to articles 676, 677 (pages 674 to 678), of the Lectures. 
 
732 APPENDIX C. 
 
 the guide points is odd, and the lower sign to the case where 
 the number of those points is even. And for conciseness, we 
 shall sometimes call the former the case of an odd system., or 
 simply the odd case ; and the latter the case of an even st/s- 
 tem, or simply the even case. So far, these two great cases 
 appear to have much in common ; but the distinction of sign 
 (+) will be found to lead to an important difference oi proper- 
 ties. It may, however, be here noted that the formula (5) 
 conducts to this inverse iovmnXo,, in which the ambiguous sign 
 is retained, so as to comprehend both cases: 
 
 A+(l- /z)p-, 
 
 and which may be also thus written, 
 
 , A + (l-jtt)p 
 
 (T) 
 
 by changing p and p to p and p^ respectively, so that the unit 
 vector p" shall be derived from p, or the point p^ from p, by 
 drawing n chords backtvards, through the system of the fi guide 
 points reversed, or taken in the contrary order, as a„, . . . Ai. 
 II. Considering now specially the odd case, we find that 
 we may write, 
 
 , rj + Tj' , i^-Tt) 
 
 where 
 
 h'=2S.\np, r,' = 2V./x(X-io), (9) 
 
 but the scalar h and the vector rj are independent of the sign 
 
 of p ; so that 
 
 S . prj' = - h'=S . Xt]', S . pt]' = 0; (10) 
 
 and S.pK^-l = S.\K, S.pK = 0,iih'^=r,'. (11) 
 
 Now the equations, 
 
 S.Xp+l = S.pp = 0, (12) 
 
 are precisely those which belong to and determine that (real) 
 straight line, or chord of solution, which satisfies, for the odd 
 case here considered, the condition of closure, 
 
 p'=p, (13) 
 
APPENDIX C. 733 
 
 or the equation, 
 
 p{\+fx + \p) + {l+m)p-\ = 0. (14) 
 
 Hence it is easy to infer that this chord of solution (bb') is the 
 rectilinear locus of the terminal point e of the vector ^, which 
 point is, by (8) and (11), the intersection of the chord p'p' 
 with the tangent plane at p ; and thus is proved for the sphere, 
 and consequently (by obvious deformations) for other surfaces 
 of the second order, a theorem of Appendix B for the odd case, 
 or rather a theorem somewhat more general. 
 
 III. On the other hand, in the even case, by taking the 
 lower signs in (5) and (7), and attending to (1), we find that 
 
 Xp + ^ = (p^-p')-(^^+p'-2p); (15) 
 
 and therefore that 
 
 \p^fji={p-p"r{p+p"-2p'), (16) 
 
 if |o"be formed from p\ or p" from p, by going again forward 
 through the same even number of given guide points, asp' 
 was formed from p, or p' from p. Hence the two constant 
 vectors, X and ju, admit, in this even case, of being thus ex- 
 pressed, in terms of the four successive unit vectors, p" p p p": 
 
 2 2 2 ,_- 
 
 X = -^ ,+ ; + - ; (17) 
 
 p-p p-p p -p 
 
 f^ 
 
 p +p ^ p + p ^ p + p ^-|^g^ 
 
 p -p p-p p-p 
 If o- be the unit vector of a point b, which admits of being 
 taken as the first corner of an inscribed and even-sided poly- 
 gon, whose sides pass respectively and successively through 
 the given guide points, so that 
 
 (r'= (7, and (7^ = - 1, (19) 
 
 (T being formed from & as p from p in (5), where the lower 
 sign is to be taken ; or if, with cr" = - 1, we have also 
 
 (r(l + |U + Xa-) = (l+|U + X(7)(7: (20) 
 
 we find then that 
 
 0=^Y.<yfi-a\.<j\ = Y.cTY(fi-a\y, (21) 
 
 and therefore that 
 
 a\\Y{p-ty\), T ± V(^-a), ifr ± a; (22) 
 
734 APPENDIX C. 
 
 or that 
 
 = S.r(iU-(rX), if S.ffr = 0, (23) 
 
 that is, if T have the direction of any tangential vector bc, at 
 the point of solution b (real or imaginary). But if we make, 
 for abridgment, 
 
 so that ')^ . . ■^' are the four chords from b to p\ . p", we have, 
 by (17) (18), 
 
 u-(TA = ^^7^,+ ^^^ + ^-;r^; (25) 
 
 x-x x-x X -X 
 
 and consequently, by (23), 
 
 ix-xr ix-xr (x"-x?' ^ ^ 
 
 This result of calculation with quaternions gives, by an imme- 
 diate and easy interpretation, combined with a passage from 
 spheres to other surfaces of the second order, of which the 
 geometrical principles are obvious, that metric theorem for the 
 even case^ which was enunciated in Appendix B. And to de- 
 duce, from the same formula (26), that graphic theorem, for 
 the same even case, which has been stated in a Note (p. 729) 
 to the same Appendix, we have only to observe, that the for- 
 mula gives these two others : 
 
 = S . rx (x - X) (X - X") (X" - X). ^hen = S . rx^x' ; (27) 
 and 
 
 = S . rx (x' - X) (X - X )\x - X). when = S . rxx" : (28) 
 whereof the former (27) shews that the tangent at e to the 
 section ep'p' intersects the tangent at p to the section pp'p"; 
 and the latter (28) shews that the tangent at b to bpp" inter- 
 sects the tangent at p' to p'pp\ 
 
 IV. Let «, h, a, j3 retain the same significations as in 676, 
 IV. of the Lectures, n being now supposed even, and = 2m; 
 let the corresponding things, for n = 2m + 1, be denoted by 
 a', b', a j3'; and write for shortness, w instead of aam+i. We 
 shall then have, by 676, III., the values. 
 
APPENDIX C. 735 
 
 a'= b + S . ao); a' = j3 + «w - V. aw ; ) 
 
 which are to be substituted in the equations of the two enve- 
 loped surfaces of the second order, assigned in 677, XII., or 
 rather in the two following (obtained by accenting the letters), 
 
 a'2 + |3'3 = 0; b'' + a' = 0. (30) 
 
 Let o-i, (T2 be the two real, and o-'i, tr'a the two imaginary unit 
 vectors which satisfy the equation of closure in 676, VII.; 
 then, by the principles of that article and paragraph, and ge- 
 nerally of the present calculus, it will be found, after some 
 reductions, that if we make 
 
 29l = 1 + S . (TiW,/>3= 1 + S . (72W,p'l= 1 + S . (t'i(i),p'2= 1 
 + S . 0''2 tU, 
 
 7=V./3a, i: = a^ + /3^-2S.7(u + (S.atu)^ + (S.j3(u)% Y (31) 
 c + c =a^ + j3% cc' = -j", c> c', 
 U = b}^+l,U= a'^ + j3'^ v!' = 6'2 + a\ 
 
 then L = cu + c'pip2 = c'u + cp'ip'2, 1 {^2\ 
 
 u' = L+{¥- j3^) u, u"= L + (a"' - a-) u. ) 
 
 The original surface (e) being supposed to be the unit-sphere 
 M = 0, the two enveloped surfaces (e') (e") have for their equa- 
 tions u'= 0, u" = ; their three centres are seen to be collinear, 
 because they have for their respective vectors, 0, (6^- j3^)'^7, 
 (a^- a^y^y : and other geometrical relations, already mentioned, 
 may be deduced from the same equations. In particular, the 
 ^our imaginary right lines, for which pi .p^- 0, p\ . p'^ = 0, are 
 seen to be common to the three surfaces, because the equations 
 of these surfaces may be written thus : 
 
 cp'ip'z = c'piPi ; cp'ip'z = c'e'pipz ; cp\p2 = c'e"pip2 ; (33) 
 where 
 e(b^-(5' + c)=b'~-(5' + c; e" (a' - a: + c) = a' - a' + c; (34) * 
 
 and consequently, 
 
 b-^ e'(b^-^^ + cy = - «-^ e" {a^ - aH cy = 6^ - jS^ + „2 _ ^2, (35) 
 If this last constant be positive, then e > 0, e"<0; and the 
 surfaces (e') (e") are respectively an ellipsoid and a double- 
 
736 APPENDIX C. 
 
 sheeted hyperboloid, the surface (e) being still, for simplicity, 
 a sphere: but (e') and (e") interchange characters, when 6^ - j3^ 
 + a^- a^ changes sign. 
 
 V. The vectors A, /n of the present Appendix are con- 
 nected with a, b, a, j3, for an even system, by the relations, 
 
 a = afi-b\',^ = bfji + a\;{a^-b-')S.\fx = ab{l + X--ix^)\{^Q) 
 and for an odd system by these others, 
 
 a' = aV + b'X ; j3' = b'fji - a'X ; (6'^ - a") S . Xy = a'b' 
 
 (1+X'^-^'O: (37) 
 
 among the consequences of which it may suffice to mention 
 here, that when an even number of guide-points is given, the 
 equations of the two enveloped surfaces (b') (e") are jointly 
 included in the formula, /x^ = (V. A'ju')^ ; and that when the 
 number of given points is odd, the vectors of the summits of 
 the two imaginary cones, which are then touched by all the 
 closing chords, have for their joint expression, X'+ju' \/-l- 
 
 VI. Finally, as regards the conception of syngraphical 
 FIGURES ON A SURFACE of the second order, mentioned in a 
 note (pp.728, 729) to the preceding Appendix B, it may be 
 briefly remarked, in conclusion, that when the surface is the 
 unit-sphere, two constant vectors, X and f.i (or X'and fx) admit 
 in general of being definitely AetQvmmedL so as to satisfy three 
 conditions of the form (5), prepared so as to be equivalent to 
 six scalar equations, with one definite selection of the alge- 
 braical signs (+) ; three unit-vectors pi, p^, ps being assumed 
 or given as initial, and three others, p'l, p'^, p\, 2iS final; and 
 that then each neiv initial unit-vector p will give 07ie new 
 
 final unit- vector p ; or, in other words, each superficial point 
 p will give another such point p' as its syngraph : this syn- 
 graphy being inverse or direct, according as upper or lower 
 
 " signs are taken in the formula. 
 
 THE END. 
 
ERRATA.* 
 In Preface : — 
 
 Page (4), line 7 from foot, for than read as compared with 
 
 — (24), line 8, for not read nor 
 
 Ix Contents : — 
 
 Page ix., line 14 from foot, /or vector minus vehend read vectum minus vehend 
 
 — XV., line 4 of § xxiv., for = Tp, read = Tp", 
 
 — xvii., line 10 from foot, for bisects the supplement read is opposite to the 
 
 bisector 
 
 — xviii., line 11 of § xxxi., for q + 2lir read q + 2lir 
 
 — xxxii., line 7 from foot, for y'^jSya" read y'jS'Ja'' 
 
 — xxxviii., line 4 of § Lxiy.,for according as ap read according as ap 
 
 In Lectures : — 
 
 Page 76, line 7, dele " perpendicular thereto" 
 
 3. A 
 
 — 85, line 1, forja ^ readj^a 
 
 — 129, lines 5, 6, for quarter spire read quadrant at the pole 
 
 — 174, line 15, the exponent of — A should be - ^ 
 
 — 177, line 18, read {q ^'Kqy = TTJq, 
 
 — 208, line 8, for parallelipipedon read parallelepipedon 
 
 — 211, line 5, read U0 = (Uy -^ Ua) x Uj; ; 
 
 — 262, line 14 from foot, /or aba'qa read aqa'ba 
 
 — 321, line 19, for q,i-i read qn'^ 
 
 — 366, line 15, for c read a 
 
 — 377, line 7 from foot, /or 120° read 150" 
 
 — 379, line 15, /or so long read so long ago 
 
 — 408, lines 5 and 9 from foot, for a read a 
 
 — 460, line 10 from foot, /)r p" read /3'' 
 
 — 469, line 13 from foot, after ellipsoid insert if al = al' = Bk' 
 
 — 508, line 3 from foot, for beginning read middlf 
 
 — 645, line 9 from foot, /or f read Fm 
 
 — 546, line 10, for inequalities read formulae 
 
 — 560, line 5, for S\ff read S . Xcr 
 
 dj' 
 
 — 595, line 9 from foot, insert + before — 
 
 — 603, line 1, reads -^S^-sf-^T. 
 
 V V \ V J 
 
 — 612, line 10, for length read amomit 
 
 — 622, line 18, />r and Q read and q 
 
 — 629, line 7 from foot, /or qi read qi 
 
 — 638, line 18, for My read u 
 
 — 640, line 8, /or v^=Al read v^ =4:5 
 
 — 665, line 22, for 499 read 449 
 
 — 672, line 7 from foot, for rq- 1 — ( read rq- 1 = — t 
 
 — 687, line 5, forj-^S . ip readj-^S .jp 
 
 * A few other trlfiins typographical errors have hecn detected, which however (like most of those 
 in the present list) could not possibly embaiTass a reader. No pages have been printed, answering 
 to the numerals i. to viii. of the Contents. As regards the astronomical allusions in the First Lecture, 
 see a Note to page (63) of the Preface. 
 
 3b 
 
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