CORNELL 
 
 UNIVERSITY 
 
 LIBRARY 
 
 GIFT OF 
 
 John Henry Tanner 
 
 MATHEMATICS 
 
Cornell University Library 
 QA 196.T13 1890 
 
 An elementary treatise on quaternions. 
 
 3 1924 001 571 094 
 
Cornell University 
 Library 
 
 The original of tiiis book is in 
 tine Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924001571094 
 
AN ELEMENTAKY TREATISE 
 
 ON 
 
 QUATEENI0N8 
 
aonDon : C. J. CLAY and SONS, 
 
 CAMBEIDGE UNIVERSITY PEESS WAREHOUSE,. 
 
 AVE MARIA LANE. 
 
 CAMBRIDGE ; DEIGHTON, BELL, AND CO. 
 LEIPZIG : F. A. BEOCKHAUS. 
 
AN ELEMENTARY TREATISE 
 
 ON 
 
 QUATEENIONS 
 
 P. G. TAIT, M.A., Sec. R.S.E., 
 
 HONOKAKT FELLOW OF ST PETEB'S COLLEGE, OAMBEIDGE 
 PEOFESSOB OF NATCEAL PHILOSOPHY IN THE UNIVEESITI OF EDINBUKGH 
 
 TerpaKTVV, 
 
 nayav dcvaov <j>ii(T€as pi^djiaT ^xovcrav. 
 
 THIRD EDITION, MUCH ENLARGED 
 
 CAMBRIDGE 
 
 AT THE UNIVERSITY PRESS 
 
 1890 
 
 [All Rights reserved.] 
 
PBINIED BY C. 3. ChAY, M.A. AND SONS, 
 AI THE nNIVEBSITY PBESS. 
 
PREFACE. 
 
 TN the present edition this work has been very greatly en- 
 -■- larged ; to the extent, in fact, of more than one-third. Had I 
 not determined to keep the book in moderate compass it might 
 easily have been doubled in size. A good deal of re-arrangement 
 
 4,. 
 
 ERRATA. 
 
 P. xxiv, 1. 18, for Vpp = read Vpp = 0. 
 
 — — for Vpp = y read Vpp = y. 
 
 P. 232, 1. 15, for To = l read Tp' = l. 
 
 to whatever branch of science he apphes such a method, my chief 
 contributions are still those contained in the fifth and the two 
 last Chapters. When, twenty years ago, I published my paper 
 on the application of V to Green's and other Allied Theorems, I 
 was under the impression that something similar must have been 
 contemplated, perhaps even mentally worked out, by Hamilton as 
 the subject matter of the (unwritten but promised) concluding 
 section of his Elements. It now appears from his Life (Vol. III. 
 p. 194) that such was not the case, and thus that I was not in any 
 way anticipated in this application (from my point of view by far 
 the most important yet made) of the Calculus. But a bias in 
 such a special direction of course led to an incomplete because one- 
 sided presentation of the subject. Hence the peculiar importance 
 of the contribution from an Analyst like Prof Cayley. 
 
 T. Q. I. h 
 
PREFACE. 
 
 TN the present edition this work has been very greatly en- 
 -L larged ; to the extent, in fact, of more than one-third. Had I 
 not determined to keep the book in moderate compass it might 
 easily have been doubled in size. A good deal of re-arrangement 
 also has been thought advisable, especially with reference to the 
 elementary uses of g'( )q~\ and of V. Prominent among the 
 additions is an entire Chapter, on the Analytical Aspect of Quater- 
 nions, which I owe to the unsolicited kindness of Prof Cayley. 
 
 As will be seen by the reader of the former Preface (reprinted 
 below) the point of view which I have, from the first, adopted 
 presents Quaternions as a Calculus uniquely adapted to Euclidian 
 space, and therefore specially useful in several of the most im- 
 portant branches of Physical Science. After giving the necessary 
 geometrical and other preliminaries, I have endeavoured to de- 
 velope it entirely from this point of view ; and, though one can 
 scarcely avoid meeting with elegant and often valuable novelties 
 to whatever branch of science he applies such a method, my chief 
 contributions are still those contained in the fifth and the two 
 last Chapters. When, twenty years ago, I published my paper 
 on the application of V to Green's and other Allied Theorems, I 
 was under the impression that something similar must have been 
 contemplated, perhaps even mentally worked out, by Hamilton as 
 the subject matter of the (unwritten but promised) concluding 
 section of his Elements. It now appears from his Life (Vol. III. 
 p. 194) that such was not the case, and thus that I was not in any 
 way anticipated in this application (from my point of view by far 
 the most important yet made) of the Calculus. But a bias in 
 such a special direction of course led to an incomplete because one- 
 sided presentation of the subject. Hence the peculiar importance 
 of the contribution from an Analyst like Prof Cayley. 
 
 T. Q. T. ^ 
 
Vi PREFACE. 
 
 It is disappointing to find how little progress has recently been 
 made with the development of Quaternions. One cause, which 
 has been specially active in France, is that workers at the subject 
 have been more intent on modifying the notation, or the mode of 
 presentation of the fundamental principles, than on extending the 
 applications of the Calculus. The earliest offender of this class 
 was the late M. Houel who, while availing himself of my permis- 
 sion to reproduce, in his Theorie des Quantites Complexes, large parts 
 of this volume, made his pages absolutely repulsive by introducing 
 fancied improvements in the notation. I should not now have 
 referred to this matter (about which I had remonstrated with M. 
 Hoiiel) but for a remark made by his friend, M. Laisant, which 
 peremptorily calls for an answer. He says: — "M. Tait...trouve 
 que M. Hoiiel a alt^rd I'oeuvre du maitre. Perfectionner n'est pas 
 ddtruire." This appears to be a parody of the saying attributed 
 to Louis XIV.: — "Pardonner n'est pas oublier": — but M. Laisant 
 should have recollected the more important maxim " Le mieux est 
 I'ennemi du bien." A line of Shakespeare might help him: — 
 
 "...with taper-light 
 To seeli: the beauteous eye of heaven to garnish, 
 Is wasteful and ridiculous excess." 
 
 Even Prof Willard Gibbs must be ranked as one of the retarders 
 of Quaternion progress, in virtue of his pamphlet on Vector 
 Analysis; a sort of hermaphrodite monster, compounded of the 
 notations of Hamilton and of Grassmann. 
 
 Apropos of Grassmann, I may advert for a moment to some 
 comparatively recent German statements as to his anticipations 
 &c. of Quaternions. I have given in the last edition of the Encyc. 
 Brit. (Art. Quaternions, to which I refer the reader) all that is 
 necessary to shew the absolute baselessness of these statements. 
 The essential points are as follows. Hamilton not only published 
 his theory complete, the year before the first (and extremely 
 imperfect) sketch of the Ausdehnungslehre appeared; but had 
 given ten years before, in his protracted study of Sets, the very 
 processes of external and internal multiplication (corresponding to 
 the Vector and Scalar parts of a product of two vectors) which 
 have been put forward as specially the property of Grassmann. 
 The scrupulous care with which Hamilton dreAV up his account of 
 
PREFACE. VU 
 
 the work of previous writers {Lectures, Preface) is minutely detailed 
 in his correspondence with De Morgan (Hamilton's Life, Vol. ill.). 
 
 Another cause of the slow head-way recently made by Qua- 
 ternions is undoubtedly to be ascribed to failure in catching the 
 "spirit" of the method: — especially as regards the utter absence 
 of artifice, and the perfect naturalness of every step. To try to 
 patch up a quaternion investigation by having recourse to quasi- 
 Cartesian processes is fatal to progress. A quaternion student loses 
 his self-respect, so to speak, when he thus violates the principles 
 of his Order. Tannhauser has his representatives in Science as 
 well as in Chivalry ! One most insidious and dangerous form of 
 temptation to this dabbling in the unclean thing is pointed out in 
 § 500 below. All who work at the subject should keep before 
 them Hamilton's warning words (Lectures, § 513): — 
 fii " J regard it as an inelegance and imperfection in this calculus, 
 or rather in the state to which it has hitherto [1853] been un- 
 folded, whenever it becomes, or seems to become, necessary to have 
 
 recourse to the resources of ordinary algebra, for the solution 
 
 OF EQUATIONS IN QUATERNIONS." 
 
 As soon as my occupation with teaching and with experimental 
 work perforce ceases to engross the greater part of my time, I 
 hope to attempt, at least, the full quaternion development of 
 several of the theories briefly sketched in the last chapter of 
 this book ; provided, of course, that no one have done it in the 
 meantime. From occasional glimpses, hitherto undeveloped, I feel 
 myself warranted in asserting that, immense as are the simplifi- 
 cations introduced by the use of quaternions in the elementary 
 parts of such subjects as Hydrokinetics and Electrodynamics, they 
 are absolutely as nothing compared with those which are to be 
 effected in the higher and (from the ordinary point of view) vastly 
 more complex branches of these fascinating subjects. Complexity 
 is no feature of quaternions themselves, and in presence of their 
 attack (when properly directed) it vanishes from the subject 
 also: — provided, of course, that what we now call complexity 
 depends only upon those space-relations (really simple if rightly 
 approached) which we are in the habit of making all but incom- 
 prehensible, by surrounding them with our elaborate scaffolding of 
 non-natural coordinates. 
 
Vlll PREFACE. 
 
 Mr Wilkinson has again kindly assisted me in the revision of 
 the proofs; and they have also been read and annotated by Dr 
 Plarr, the able French Translator of the second edition. Thanks 
 to their valuable aid, I may confidently predict that the present 
 edition will be found comparatively accurate. 
 
 With regard to the future of Quaternions, I will merely quote 
 a few words of a letter I received long ago from Hamilton : — 
 
 " Could anything be simpler or more satisfactory ? Don't you 
 feel, as well as think, that we are on a right track, and shall be 
 thanked hereafter ? Never mind when." 
 
 The special form of thanks which would have been most 
 grateful to a man like Hamilton is to be shewn by practical 
 developments of his magnificent Idea. The award of this form of 
 thanks will, I hope, not be long delayed. 
 
 P. G. TAIT. 
 
 ADDITIONS, CHANGES, ETC. IN THE PRESENT EDITION. 
 
 (Only the more important are noticed, and they are indicated by the 
 sectional numbers.) 
 
 Chap. I. 31 (k), (m), 40, 43. 
 
 II. 51, 89. 
 
 III. 105, 108, 116, 119—122, 133—4. 
 
 IV. 140(8)— (12), 145—149. 
 
 V. 174, 187, 193—4, 196, 199. 
 
 VI. The whole. 
 
 VIII. 247, 250, 250*. 
 
 IX. 285, 286, 287. 
 
 X. 326, 336. 
 
 XI. 357—8, 382, 384—6. 
 
 Xn. 390, 393—403, 407, 458, 473 (a)—{l], 480, 489, 493, 499, 500, 
 503, 508—511, 512—13. 
 
 There are large additions to the number of Examples, some in fact 
 to nearly every Chapter. Several of these are of considerable importance; 
 as they contain, or suggest, processes and results not given in the text. 
 
PREFACE TO THE SECOND EDITION. 
 
 To the first edition of this work, published in 1867, the 
 following was prefixed : — 
 
 'The present work was commenced in 1859, while I was a 
 Professor of Mathematics, and far more ready at Quaternion 
 analysis than I can now pretend to be. Had it been then 
 completed I should have had means of testing its teaching 
 capabilities, and of improving it, before publication, where found 
 deficient in that respect. 
 
 ' The duties of another Chair, and Sir W. Hamilton's wish that 
 my volume should not appear till after the publication of his 
 Elements, interrupted my already extensive preparations. I had 
 worked out nearly all the examples of Analytical Geometry in 
 Todhunter's Collection, and I had made various physical applica- 
 tions of the Calculus, especially to Crystallography, to Geometrical 
 Optics, and to the Induction of Currents, in addition to those on 
 Kinematics, Electrodynamics, Fresnel's Wave Surface, &c., which 
 are reprinted in the present work from the Quarterly Mathematical 
 Journal and the Proceedings of the Royal Society of Edinburgh. 
 
 ' Sir W. Hamilton, when I saw him but a few days before his 
 death, urged me to prepare my work as soon as possible, his being 
 almost ready for publication. He then expressed, more strongly 
 perhaps than he had ever done before, his profound conviction of 
 the importance of Quaternions to the progress of physical science ; 
 and his desire that a really elementary treatise on the subject 
 should soon be published. 
 
 ' I regret that I have so imperfectly fulfilled this last request 
 
X PREFACE TO THE SECOND EDITION. 
 
 of my revered friend. When it was made I was already engaged, 
 along with Sir W. Thomson, in the laborious work of preparing 
 a large Treatise on Natural Philosophy. The present volume has 
 thus been written under very disadvantageous circumstances, 
 especially as I have not found time to work up the mass of 
 materials which I had originally collected for it, but which I 
 had not put into a fit state for publication. I hope, however, 
 that I have to some extent succeeded in producing a thoroughly 
 elementary work, intelligible to any ordinary student ; and that 
 the numerous examples I have given, though not specially 
 chosen so as to display the full merits of Quaternions, will 
 yet sufficiently shew their admirable simplicity and naturalness 
 to induce the reader to attack the Lectures and the Elements; 
 where he will find, in profusion, stores of valuable results, and 
 of elegant yet powerful analytical investigations, such as are 
 contained in the writings of but a very few of the greatest 
 mathematicians. For a succinct account of the steps by which 
 Hamilton was led to the invention of Quaternions, and for 
 other interesting information regarding that remarkable genius, 
 I may refer to a slight sketch of his life and works in the 
 North British Review for September 1866. 
 
 ' It will be found that I have not servilely followed even so 
 great a master, although dealing with a subject which is entirely 
 his own. I cannot, of course, tell in every case what I have 
 gathered from his published papers, or from his voluminous 
 correspondence, and what I may have made out for myself 
 Some theorems and processes which I have given, though wholly 
 my own, in the sense of having been made out for myself before 
 the publication of the Elements, I have since found there. Others 
 also may be, for I have not yet read that tremendous volume 
 completely, since much of it bears on developments unconnected 
 with Physics. But I have endeavoured throughout to point out 
 to the reader all the more important parts of the work which I 
 know to be wholly due to Hamilton. A great part, indeed, may 
 be said to be obvious to any one who has mastered the pre- 
 liminaries ; still I think that, in the two last Chapters especially, 
 a good deal of original matter will be found. 
 
PREFACE TO THE SECOND EDITION. XI 
 
 ' The volume is essentially a working one, and, particularly in 
 the later Chapters, is rather a collection of examples than a 
 detailed treatise on a mathematical method. I have constantly 
 aimed at avoiding too great extension ; and in pursuance of 
 this object have omitted many valuable elementary portions 
 of the subject. One of these, the treatment of Quaternion 
 logarithms and exponentials, I greatly regret not having given. 
 But if I had printed all that seemed to me of use or interest to 
 the student, I might easily have rivalled the bulk of one of 
 Hamilton's volumes. The beginner is recommended merely to 
 read the first five Chapters, then to work at Chapters VI, 
 VII, VIII* (to which numerous easy Examples are appended). 
 After this he may work at the first five, with their (more 
 difficult) Examples ; and the remainder of the book should 
 then present no difficulty. 
 
 'Keeping always in view, as the great end of every mathe- 
 matical method, the physical applications, I have endeavoured 
 to treat the subject as much as possible from a geometrical 
 instead of an analytical point of view. Of course, if we premise 
 the properties of i, j, k merely, it is possible to construct from 
 them the whole system-I"; just as we deal with the imaginary 
 of Algebra, or, to take a closer analogy, just as Hamilton 
 himself dealt with Couples, Triads, and Sets. This may be 
 interesting to the pure analyst, but it is repulsive to the 
 physical student, who should be led to look upon i, j, k, from 
 the very first as geometric realities, not as algebraic imagi- 
 naries. 
 
 ' The most striking peculiarity of the Calculus is that mul- 
 tiplication is not generally commutative, i.e. that qr is in general 
 different from rq, r and q being quaternions. Still it is to 
 be remarked that something similar is true, in the ordinary 
 coordinate methods, of operators and functions : and therefore 
 
 * [In this edition these Chapters are numbered VII, Vni, IX, respectively — 
 Aug. 1889.] 
 
 + This has been done by Hamilton himself, as one among many methods he has 
 employed; and it is also the foundation of a memoir by M. AU^gret, entitled Essai 
 sur le Galcul des Quaternions (Paris, 1862). 
 
xii PREFACE TO THE SECOND EDITION. 
 
 the student is not wholly unprepared to meet it. No one is 
 puzzled by the fact that log. cos. a; is not equal to cos. log. a?, 
 
 or that J^ is not equal to ^-V^/- Sometimes, indeed, this 
 
 rule is most absurdly violated, for it is usual to take cos^a; as, 
 equal to (cos a;)", while cos"'* is not equal to (cosa;)"\ No such 
 incongruities appear in Quaternions ; but what is true of operators 
 and functions in other methods, that they are not generally com- 
 mutative, is in Quaternions true in the multiplication of (vector) 
 coordinates. 
 
 'It will be observed by those who are acquainted with the 
 Calculus that I have, in many cases, not given the shortest or 
 simplest proof of an important proposition. This has been done 
 with the view of including, in moderate compass, as great a 
 variety of methods as possible. With the same object I have 
 endeavoured to supply, by means of the Examples appended 
 to each Chapter, hints (which will not be lost to the intelli- 
 gent student) of farther developments of the Calculus. Many 
 of these are due to Hamilton, who, in spite of his great origi- 
 nality, was one of the most excellent examiners any University 
 can boast of. 
 
 ' It must always be remembered that Cartesian methods are 
 mere particular cases of Quaternions, where most of the distinctive 
 features have disappeared ; and that when, in the treatment of 
 any particular question, scalars have to be adopted, the Quaternion 
 solution becomes identical with the Cartesian one. Nothing there- 
 fore is ever lost, though much is generally gained, by employing 
 Quaternions in preference to ordinary methods. In fact, even 
 when Quaternions degrade to scalars, they give the solution 
 of the most general statement of the problem they are applied 
 to, quite independent of any limitations as to choice of particular 
 coordinate axes. 
 
 ' There is one very desirable object which such a work as this 
 may possibly fulfil. The University of Cambridge, while seeking 
 to supply a real want (the deficiency of subjects of examination for 
 mathematical honours, and the consequent frequent introduction 
 of the wildest extravagance in the shape of data for " Problems "), 
 
PREFACE TO THE SECOISB EDITIOX. XIU 
 
 is in danger of making too much of such elegant trifles as Trilinear 
 Coordinates, while gigantic systems like Invariants (which, by the 
 way, are as easily introduced into Quaternions as into Cartesian 
 methods) are quite beyond the amount of mathematics which 
 even the best students can master in three years' reading. 
 One grand step to the supply of this want is, of course, the 
 introduction into the scheme of examination of such branches 
 of mathematical physics as the Theories of Heat and Electricity. 
 But it appears to me that the sttidy of a mathematical method 
 like Quaternions, which, while of immense power and compre- 
 hensiveness, is of extraordinary simplicity, and yet requires 
 constant thought in its applications, would also be of great 
 benefit. M'^ith it there can be no " shut your eyes, and write 
 down your equations," for mere mechanical dextexity of analysis 
 is certain to lead at once to error on account of the novelty of 
 the processes employed. 
 
 ' The Table of Contents has been drawn up so as to give the 
 student a short and simple summary of the chief fundamental 
 formulae of the Calculus itself, and is therefore confined to an 
 analysis of the first five [and the two last] chapters. 
 
 ' In conclusion, I have only to say that I shall be much obliged 
 to any one, student or teacher, who will point out portions of the 
 work where a difficulty has been found; along with any inaccuracies 
 which may be detected. As I have had no assistance in the revision 
 of the proof-sheets, and have composed the work at irregular in- 
 tervals, and while otherwise laboriously occupied, I fear it may 
 contain many slips and even errors. Should it reach another 
 edition there is no doubt that it will be improved in many 
 important particulars.' 
 
 To this I have now to add that I have been equally surprised 
 and delighted by so speedy a demand for a second edition — and the 
 more especially as I have had many pleasing proofs that the 
 work has had considerable circulation in America. There seems 
 now at last to be a reasonable hope that Hamilton's grand 
 invention will soon find its way into the working world of science, 
 to which it is certain to render enormous services, and not be laid 
 T. Q. I. ^ 
 
XIV PREFACE TO THE SECOKD EDITION. 
 
 aside to be unearthed some centuries hence by some grubbing 
 antiquary. 
 
 It can hardly be expected that one whose time is mainly en- 
 grossed by physical science, should devote much attention to the 
 purely analytical and geometrical applications of a subject like this; 
 and I am conscious that in many parts of the earlier chapters I 
 have not fully exhibited the simplicity of Quaternions. I hope, 
 however, that the corrections and extensions now made, especially 
 in the later chapters, will render the work more useful for my chief 
 object, the Physical Applications of Quaternions, than it could 
 have been in its first crude form. 
 
 I have to thank various correspondents, some anonymous, for 
 suggestions as well as for the detection of misprints and slips of 
 the pen. The only absolute error which has been pointed out to 
 me is a comparatively slight one which had escaped my own notice : 
 a very grave blunder, which I have now corrected, seems not to 
 have been detected by any of my correspondents, so that I cannot 
 be quite confident that others may not exist. 
 
 I regret that I have not been able to spare time enough to 
 rewrite the work ; and that, in consequence of this, and of the 
 large additions which have been made (especially to the later 
 chapters), the whole will now present even a more miscellaneously 
 jumbled appearance than at first. 
 
 It is well to remember, however, that it is quite possible to 
 make a book too easy reading, in the sense that the student may 
 read it through several times without feeling those difficulties 
 which (except perhaps in the case of some rare genius) must 
 attend the acquisition of really useful knowledge. It is better to 
 have a rough climb (even cutting one's own steps here and there) 
 than to ascend the dreary monotony of a marble staircase or a 
 well-itiade ladder. Royal roads to knowledge reach only the 
 particular locality aimed at — and there are no views by the way. 
 It is not on them that pioneers are trained for the exploration of 
 unknown regions. 
 
 But I am happy to say that the possible repulsiveness of my 
 early chapters cannot long be advanced as a reason for not attack- 
 ing this fascinating subject. A still more elementary work than 
 
PREFACE TO THE SECOND EDITION. XV 
 
 the present will soon appear, mainly from the pen of my colleague 
 Professor Kelland. In it I give an investigation of the properties 
 of the linear and vector fiinction, based directly upon the Kine- 
 matics of Homogeneous Strain, and therefore so different in method 
 from that employed in this work that it may prove of interest to 
 even the advanced student. 
 
 Since the appearance of the first edition I have managed (at 
 least partially) to effect the application of Quaternions to line, 
 surface, and volume integrals, such as occur in Hydrokinetics, 
 Electricity, and Potentials generally. I was first attracted to 
 the study of Quaternions by their promise of usefulness in 
 such applications, and, though I have not yet advanced far in 
 this new track, I have got far enough to see that it is certain 
 in time to be of incalculable value to physical science. I have 
 given towards the end of the work all that is necessary to put 
 the student on this track, which will, I hope, soon be followed to 
 some purpose. 
 
 One remark more is necessary. I have employed, as the 
 positive direction of rotation, that of the earth about its axis, or 
 about the sun, as seen in our northern latitudes, i.e. that opposite 
 to the direction of motion of the hands of a watch. In Sir W. 
 Hamilton's great works the opposite is employed. The student 
 will find no difficulty in passing from the one to the other ; but, 
 without previous warning, he is liable to be much perplexed. 
 
 With regard to notation, I have retained as nearly as possible 
 that of Hamilton, and where new notation was necessary I 
 have tried to make it as simple and as little incongruous with 
 Hamilton's as possible. This is a part of the work in which great 
 care is absolutely necessary; for, as the subject gains development, 
 fresh notation is .inevitably required ; and our object must be to 
 make each step such as to defer as long as possible the revolution 
 which must ultimately come. 
 
 Many abbreviations are possible, and sometimes very useful in 
 private work ; but, as a rule, they are unsuited for print. Every 
 analyst, like every short-hand writer, has his own special con- 
 tractions ; but, when he comes to publish his results, he ought 
 invariably to put such devices aside. If all did not use a common 
 
 c2 
 
XVI PREFACE TO THE SECOND EDITION. 
 
 mode of public expression, but eacPi were to print as he is in the 
 habit of writing for his own use, the confusion would be utterly 
 intolerable. 
 
 Finally, I must express my great obligations to my friend 
 M. M. U. Wilkinson of Trinity College, Cambridge, for the care 
 with which he has read my proofs, and for many valuable sug- 
 gestions. 
 
 P. G. TAIT. 
 
 College, Edinbckgh, 
 October 1873. 
 
CONTENTS. 
 
 Chapter I. — Vectors and their Composition . pp. 1 — 28 
 
 Sketoli of the attempts made to represent geometrically the imaginary of 
 
 algebra. §§ 1—13. 
 De Moivre's Theorem interpreted in plane rotation. §§ 7, 8. 
 Curious speculation of Servois. § 11. 
 
 Elementary geometrical ideas connected with relative position. § 14. 
 Definition of a Veotoe. It may be employed to denote translation. Definition 
 
 of currency. § 16. 
 Expression of a vector by one symbol, containing implicitly three distinct 
 
 numbers. Extension of the signification of the symbol = . § 18. 
 The sign + defined in accordance with the interpretation of a vector as 
 
 representing translation. § 19. 
 Definition of - . It simply reverses the currency of a vector. § 20. 
 Triangles and polygons of vectors, analogous to those of forces and of simul- 
 taneous velocities. § 21. 
 When two vectors are parallel we have 
 
 a = x^. § 22. 
 Any vector whatever may be expressed in terms of three distinct vectors, which 
 
 are not coplanar, by the formula 
 
 p = xa + y^ + zy, 
 
 which exhibits the three numbers on which the vector depends. § 23. 
 Any vector in the same plane with a and ^ may be written 
 p = xa + yp. § 24. 
 
 The equation rar=p, 
 
 between two vectors, is equivalent to three distinct equations among 
 
 numbers. § 25. 
 The Commutative and Associative Laws hold in the combination of vectors by 
 
 the signs + and - . § 27. 
 
XVm CONTENTS. 
 
 The equation p=xp, 
 
 where p is a variable, and ^ a fixed, rector, represents a line drawn through 
 the origin parallel to ;8. 
 
 p=a+x^ 
 is the equation of a line drawn throiigh the extremity of a and parallel 
 to /3. § 28. 
 
 pz=ya + x^ 
 represents the plane through the origin parallel to a and /3, while 
 
 p=y + ya + xp 
 denotes a parallel plane through the point 7. § 29. 
 The condition that p, u,, |8 may terminate in the same line is 
 pp + qa + r^=0, 
 subject to the identical relation 
 
 p + q + r=zO. 
 Similarly pp + qa + rp + sy = 0, 
 
 with p + q + r + s = 0, 
 
 is the condition that the extremities of four vectors lie in one plane. § 30. 
 Examples with solutions. Conditions that a vector equation may represent a 
 line, or a surface. 
 
 The equation p = <pt 
 
 represents a curve in space; while 
 
 p = U(pt 
 
 is a cone, and p = (pt + ua 
 
 is a cylinder, both passing through the curve. § 31. 
 Differentiation of a vector, when given as a function of one number. §§ 32 — 37. 
 If the equation of a curve be 
 
 p=<t>{s) 
 
 where s is the length of the arc, dp is a vector-tangent to the curve, and its 
 
 length is ds. §§ 38, 39. 
 
 Examples with solutions. §§ 40 — 43. 
 
 Examples to Chapter I. 28 30 
 
 Chapter II. — Products and Quotients of Vectors . 31 — 57 
 
 Here we begin to see what a quaternion is. When two vectors are parallel 
 their quotient is a number. §§ 45, 46. 
 
 When they are perpendicular to one another, their quotient is a vector perpen- 
 dicular to their plane. §§ 47, 64, 72. 
 
 When they are neither parallel nor perpendicular the quotient in general 
 involves /our distinct numbers — and is thus a Quaternion. § 47. 
 
 A quaternion q regarded as the operator which turns one vector into another. 
 It is thus decomposable into two factors, whose order is indifferent, the 
 stretching factor or Tensor, and the turning factor or Versor. These are 
 denoted by Tq, and Uq. § 48. 
 
CONTENTS. XIX 
 
 The equation ^ = qa 
 
 gives - = 2. ov /3a~i = o, hut not in general 
 
 a 
 
 a-^^=q. §49. 
 
 q or (3a~i depends only on the relative lengths, and directions, of ^ and a. 
 
 §50. 
 
 Beciprocal of a quaternion defined, 
 
 ^ . 1 , o 
 
 3=^ gives -or 3-1 = -, 
 
 r.s-i = ^, U.q'^ = {Uq)-K §51. 
 
 Definition of the Conjugate of a quaternion, 
 Kq = (Tqyq-\ 
 
 and qKq=Kq.q = (TqY. §52. 
 
 Representation of versors by arcs on the unit-sphere. § 53. 
 Versor multiplication illustrated by the composition of ares. The process 
 
 proved to be not generally commutative. § 54. 
 Proof that K(qr) = Kr.Kq. §55. 
 
 Proof of the Associative Law of Multiplication 
 
 p.qr=pq .r. §§57 — 60. 
 [Digression on Spherical Conies. § 59*.] 
 Quaternion addition and subtraction are commutative. § 61. 
 Quaternion multiplication and division are distributive. § 62. 
 Integral powers of a versor are subject to the Index Law. § 63. 
 Composition of quadrantal versors in planes at right angles to each other. 
 
 Calling them i, j, k, we have 
 i'=p=k^=-l, ij=-ji = ]c, jk=-kj = i, ki=-ik=j, 
 ijk=-l. §§64—71. 
 A unit-vector, when employed as a factor, may be considered as a quadrantal 
 
 versor whose plane is perpendicular to the vector. Hence the equations 
 
 just written are true of any set of rectangular uuit-'y«ctors i, j, k. § 72. 
 The product, and also the quotient, of two vectors at right angles to each other 
 
 is a third perpendicular to both. Hence 
 Ka = — a, 
 
 and {Ta.)^=aEa=-a\ §73. 
 
 Every versor may be expressed as a power of some unit-vector. § 74. 
 Every quaternion may be expressed as a power of a vector. § 75. 
 The Index Law is true of quaternion multiplication and division. § 76. 
 Quaternion considered as the sum of a Scaiab and a Veotok. 
 
 q=^=x + y = Sq + Vq. §§77,78. 
 a 
 
 Proof that SKq = Sq, VKq=-Vq, 2,Kq = K^q. §79. 
 
 Quadrinomial expression for a quaternion 
 
 q=w + ix+jy + kz. 
 
 An equation between quaternions is equivalent to four equations between 
 numbers (or soalars). § 80. 
 
XX CONTENTS. 
 
 Second proof of the distributive law of multiplication. §§ 81—83. 
 
 Algebraic determination of the constituents of the product and quotient of two 
 
 vectors. §§ 83, 84. 
 Second proof of the associative law of multiplication. § 85. 
 Proof of the formulae Sa^ = S^a, 
 
 Fa/3= - V8a, 
 
 2Fa/3 = a/3-^a, 
 S .qr=S.rq, 
 S . qrs = S .rsq = S . sqr, 
 S . a^y = S . pya = S .yap= - S .ayp = &c., 
 
 25. a/3.. .0x1 
 
 the upper sign belonging to the scalar if the number of factors is even. 
 §§ 86-89. 
 Proof of the formulae 
 
 V. a.V^y = ySa^ - ^Sya, 
 V. a/37 = "S^i^T - PSya + ySa^, 
 V.a^y=V.y^a, 
 V . Fa/3 F7S = aS . /SyS - /3S . ayd, 
 = SS .a§y-yS. a^S, 
 SS .a^y = aS.^yS + ^S.yad + yS .a^d, 
 
 = Va^SyS + V^ySaS + VyaS^d. §§ 90—92. 
 Hamilton's proof that the product of two parallel vectors must be a scalar, and 
 that of perpendicular vectors, a vector; if quaternions are to deal with 
 space indifferently in all directions. § 93. 
 
 Examples to Chaptee II. ... . . 57— 58 
 
 Chapter III. — Interpretations and Transforma- 
 tions OF Quaternion Expressions . . . 59 — 88 
 
 If 8 be the angle between two vectors, a and /3, we have 
 
 S?=^cos9, Sa8=-Ta.T^oose, 
 
 a la 
 
 TV^='^sme, TVaB= TaTBsine. §§94—96. 
 
 a la 
 
 Applications to plane trigonometry. § 97. 
 
 The condition Sa^ = 
 
 shews that a is perpendicular to ;8, while 
 Fa/3 = 0, 
 
 shews that a and ^ are parallel. 
 
 The expression S . a^y 
 
 is the volume of the parallelepiped three of whose conterminous edges are 
 
 u, /3, y. Hence 
 
 S.a/37 = 
 
 shews that a, /3, 7 are coplanar. 
 
CONTENTS. XXI 
 
 Expression of S . a,37 as a determinant. §§ 98—102. 
 Proofthat {Tqf={Sq)^ + (TVq)\ 
 
 and T(qr) = TqTr. §103. 
 
 Simple propositions in plane trigonometry. § 104. 
 
 Proof that - a^a~^ is the vector reflected ray, when /3 is the incident ray and a 
 
 normal to the reflecting surface. § 105. 
 Interpretation of 0187 when it is a vector. § 106. 
 Examples of variety in simple transformations. § 107. 
 
 The relation among the distances, two and two, of five points in space. § 108. 
 De Moivre's Theorem, and Plane Trigonometry. §§ 109 — 111. 
 Introduction to spherical trigonometry. §§ 112 — 116. 
 
 Representation, graphic, and by quaternions, of the spherical excess. §§ 117, 118. 
 Interpretation of the Operator 
 
 9 ( ) 3-^ 
 in connection with rotation. Astronomical examples. §§ 119 — 122. 
 
 Loci represented by different equations — points, lines, surfaces, and volumes. 
 
 §§ 123-126. 
 Proof that r-i (r^q^ q-i= U (rq + KrKq). § 127. 
 
 Proof of the transformation 
 
 BlQUATEKNIONS. §§ 180—132. 
 
 Convenient abbreviations of notation. §§ 133, 134. 
 
 Examples to Chaptek III 89 — 93 
 
 Chapter IV. — Differentiation of Quaternions . 94—105 
 
 Definition of a differential, 
 
 dr=dFq = ^^n \f L + ^\- Fq\ , 
 
 where dq is any quaternion whatever. 
 We may write dFq =/ {q, dq) , 
 
 where / is linear and homogeneous in dq ; but we cannot generally write 
 dFq = t{q)dq. §§185-188. 
 Definition of the differential of a function of more quaternions than one. 
 
 d (qr) = qdr + dq.r, but not generally d (qr) = qdr + rdq. § 139. 
 Proofs of fundamental differential expressions : — 
 dTp_ dp 
 
 i^=F^-e,&c. §140. 
 Up p 
 
 Successive differentiation ; Taylor's theorem. §§ 141, 142. 
 
XXll CONTENTS. 
 
 If the equation of a surface be 
 
 F(p) = C, 
 the differential may be written 
 
 Spdp = 0, 
 where » is a vector normal to the surface. § 144. 
 Definition of Hamilton's Operator 
 
 „ . d . d , d 
 dx dy dz 
 Its effects on simple scalar and vector functions of position. Its square 
 the negative of Laplace's Operator. Expressions for the condensation and 
 rotation due to a displacement. Application to fluxes, and to normals to 
 surfaces. Precautions necessary in its use. §§ 145 — 149. 
 
 Examples to Chapteb IV 106, 107 
 
 Chapter V. — The Solution of Equations of the 
 
 First Degree 108 — 141 
 
 The most general equation of the first degree in an unknown quaternion q, 
 may be written 
 
 SF. aqb + S .cq = d, 
 where a, 6, c, d are given quaternions. Elimination of Sq, and reduction 
 to the vector equation 
 
 0p = S . aS^p=y. §§ 150, 151. 
 General proof that (j>^p is expressible as a linear function of p, ipp, and 0V- 
 
 § 152. 
 Value of (p for an ellipsoid, employed to illustrate the general theory. 
 
 §§ 153—155. 
 Hamilton's solution of c)>p=y. 
 
 If we write S<rcjip=Sp<j>'a, 
 
 the functions and 0' are said to be conjugate, and 
 
 m0-iFXM = F0'X0'/i- 
 Proof that m, whose value may be written as 
 
 S . 0'X0'/X0V 
 
 S.\ij.v ' 
 is the same for all values of X, /<,, v. §§ 156 — 158. 
 Proof that if mg=m- mig + m^g^ - g^, 
 
 where ^ S (X0'^0'i' + 0'X/^0V + 0'X0V»-) 
 
 and m _ ^ (^^0'" + 0'^" + M't^") 
 
 (which, like m, are Invariants,) 
 
 then m, (0 - gy^VX/j. = (m0-i - gx + g^) V\/i.. 
 
 Also that X=»"a-0. 
 
 whence the final form of solution 
 
 m0-i = mi - m20 + <pK §§ 159, 160. 
 
CONTENTS. XXIU 
 
 Examples. §§ 161—173. 
 The fundamental cubic 
 
 Wien is its own conjugate, the roots of the cubic are real; and the 
 equation 
 
 Vp<pp=0, 
 or {4,-g)p=0, 
 
 is satisfied by a set of three real and mutually perpendicular vectors. 
 Geometrical interpretation of these results. §§ 174 — 178. 
 Proof of the transformation of the self-conjugate linear and vector function 
 (f>p=fp + hV. {i + ek)p{i-ek) 
 where {<p-gj)i — 0, 
 
 (<p-gs)lc=0, 
 
 91-92 
 f=i{9i+9s), 
 
 Another transformation is 
 
 tj>p = aaVap + b^S^p. §§ 179—181. 
 Other properties of (/>. Proof that 
 
 Sp{<p-g)-^p = 0, and Sp(<t>-h)-^p=0 
 represent the same surface if 
 
 mSp(j>~'-p=ghp^. 
 Proof that when tj> is not self-conjugate 
 
 (t>p=<t>'p+Vep. §§182—184. 
 Proof that, if g = att>a + ptp^ + y<f>y, 
 
 where o, /S, y are any rectangular unit- vectors whatever, we have 
 
 Sq=-m^, Vq = e, 
 where Vep = i{ij)-(p')p. 
 
 This quaternion can be expressed in the important form 
 g = V0p. §185. 
 A non-conjugate linear and vector function of a vector differs from a self- 
 conjugate one solely by a term of the form 
 Vep. §186. 
 Graphic determination of the conditions that there may be three real vector 
 solutions of 
 
 Vp^p = Q. § 187. 
 Degrees of indeterminateness of the solution of a quaternion equation — 
 
 Examples. §§ 188—191. 
 The linear function of a quaternion is given by a symbolical biquadratic. 
 
 §192. 
 Particular forms of linear equations. Differential of the nth root of a quaternion. 
 
 §§ 193—196. 
 A quaternion equation of the with degree in general involves a scalar equation 
 
 of degree m*. § 197. 
 Solution of the equation q^^qa + l. § 198. 
 
 Examples to Chapter V 142 — 145 
 
xxiv CONTENTS. 
 
 Chapter VI.— Sketch of the Analytical Theory 
 OF Quaternions 
 
 146—159 
 
 Chapter VIL— Geometry of the Straight Line 
 
 AND Plane. . . ^ 160-174 
 
 Examples to Chapter VII. ... ... 175—177 
 
 Chapter VIII.— The Sphere and Cyclic Cone . 178—198 
 
 Examples to Chapter VIII. ■'■"" ^"■'■ 
 
 Chapter IX.— Surfaces of the Second Degree . 202—224 
 
 Examples to Chapter IX 224 229 
 
 Chapter X.— Geometry of Curves and Surfaces 230—269 
 
 Examples to Chapter X 270 278 
 
 Chapter XL— Kinematics 279—304 
 
 A. Kinematics of a Point. §§ -354 — 366. 
 
 If p=(t>t be the vector of a moving point in terms of the time, p is the vector 
 velocity, and p the vector acceleration. 
 
 (7 = p = (j>' (t) is the equation of the Hodograph. 
 'p=vp' + v^p" gives the normal and tangential accelerations. 
 Vpp=0 if acceleration directed to a point, whence Vpp = y. 
 Examples. — Planetary acceleration. Here the equation is 
 .. P'Up 
 
 giving Vpp = -y; whence the hodograph is 
 
 p = £y-^ — fj,Up .y~^, 
 and the orbit is the section of 
 
 IMTp = Se{yh-^-p) 
 by the plane Syp = 0. 
 
 Cotes' Spirals, Epitrochoids, &c. §§ 354—366. 
 
 B. Kinematics of a Rigid System,. §§ 367 — ST."!. 
 
 Rotation of a rigid system. Composition of rotations. If the position of a 
 system at time t is derived from the initial position by q{ )g~', the 
 instantaneous axis is 
 
 e = 2Vqq-K 
 Eodrigues' coordinates. §§ 367 — 375. 
 
CONTENTS. XXV 
 
 C. Kinematics of a Deformable System. §§ 376 — 386. 
 
 Homogeneous strain. Criterion of pure strain. Separation of the rotational 
 from the pure part. Extraction of the square root of a strain. A strain 
 is equivalent to a pure strain ij(j>'(j> followed by the rotation (pj ij<f>'<p. 
 Simple Shear. §§ 376—383. 
 
 Displacements of systems of points. Consequent condensation and rotation. 
 Preliminary about the use of V in physical questions. For displacement 
 a, the strain function is 
 
 </>t = t-StV.u. §§384—386. 
 
 J). Axes of Inertia. § 387. 
 Moment of inertia. Biuet's Theorem. § 387. 
 
 Examples to Chapter XI 304 — 308 
 
 Chapter XII. — Physical Applications . . 309—409 
 
 A. Statics of a Rigid System. §§ 389 — 403. 
 
 Condition of equilibrium of a rigid system is XS . ^Sa = Q, where |8 is a vector 
 force, a. its point of application. Hence the usual six equations in the 
 form S/3 = 0, SFa;8 = 0. Central axis. Minding's Theorem, &c. §§389 
 —403. 
 
 B. Kinetics of a Rigid System. §§ 404 — 425. 
 
 For the motion of a rigid system 
 
 SS(ma-/3)5a = 0, 
 whence the usual forms. Theorems of Poinsot and Sylvester. The 
 
 equation 
 
 2g = 70-1(5-172), 
 where y is given in terms of t and q if forces act, but is otherwise constant, 
 contains the whole theory of the motion of a rigid body with one point 
 fixed. Eeduction to the ordinary form 
 
 dt _dw _dx _ dy _ dz 
 
 '2~W ~T~ Y~2" 
 Here, if no forces act, W, X, Y, Z are homogeneous functions of the third 
 degreein jc, X, 2/, z. §§404 — 425. 
 
 C Special Kinetic Problems. §§ 426 — -430. 
 
 Precession and Nutation. General equation of motion of simple pendulum. 
 Foucault's pendulum. §§ 426—430. 
 
 D. Geometrical and Physical Optics. §§ 431 — 452. 
 
 Problem on reflecting surfaces. § 431. 
 
 Fresnel's Theory of Double Refraction. Various forms of the equation of 
 Fresnel's Wave-surface; 
 
 The conical cusps and circles of contact. Lines of vibration, &c. 
 §§ 432—452. 
 
Xxvi CONTENTS. 
 
 E. Mectrodynamics. §§ 453 — 472. 
 Electrodynamios. The vector action of a closed circuit on an element of 
 current a-^ is proportional to Va^fi where 
 
 the integration extending round the circuit. It can also be expressed 
 as - Vfi, where O is the spherical angle subtended by the circuit. This 
 is a many-valued function. Special case of a circular current. Mutual 
 action of two closed circuits, and of solenoids. Mutual action of magnets. 
 Potential of a closed circuit. Magnetic curves. §§ 453—472. 
 
 F. General Expressions for the Action between Linear Elements. § 473. 
 
 Assuming AmpSre's data I, 11, III, what is the most general expression for 
 the mutual action between two elements? Particular cases, determined 
 by a fourth assumption. Solution of the problem when I and II, alone, 
 are assumed. Special cases, including v. Hehnholtz' form. § 473 (a). ..(I). 
 
 G. Application of V to certain Physical Analogies. §§ 474 — 478. 
 
 The effect of a current-element on a magnetic particle is analogous to dis- 
 placement produced by external forces in an elastic solid, while that 
 of a smaU closed circuit (or magnet) is analogous to the corresponding 
 vector rotation. 
 
 H. Elementan-y Properties of '^ . §§ 479 — 481. 
 SdpV= -d = Sd(rV^ 
 
 1 J v7 ■ ^ . d , d 
 where <r = i|-Hj7;-l-*f and V,r=» j^-l- J ^-I-k ^; 
 
 so that, if dff=tpdp, 
 
 Vcr=0'-lV. 
 
 J. Applications of V to Line, Surface and Volume Integrals. 
 §§ 482—501. 
 
 Proof of the fundamental theorem for comparing an integral over a closed 
 surface with one through its content 
 
 fJJSVadt=ffS<!Vvds. 
 Hence Green's Theorem. Examples in potentials and in conduction of 
 heat. Limitations and ambiguities. Determination of the displacement 
 in a fluid when the consequent rotation is given. §§ 482 — 494. 
 Similar theorem for double and single integrals 
 
 fS(rdp=ffS . VvWads. §§ 495—497. 
 The fundamental forms, of which all the others are simple consequences, are 
 Sdpu=SSdiV(UvV)u, 
 f//Vuds=f/Ui'uds. §§498,499. 
 The first is a particular case of the second. § 500. 
 
 Expression for a surface in terms of an integral through the enclosed volume. 
 §499. 
 
CONTENTS. XXVll 
 
 K. Application of the V Integrals to Magnetic (be. Problems. 
 §§ 502—506. 
 
 Volume and surface distributions due to a given magnetic field. Solenoidal 
 and Lamellar distributions. § 502. Magnetic Induction and Vector 
 potential. § 503. Ampere's Directrice. § 504. Gravitation potential of 
 homogeneous solid in terms of a surface-integral. § 505. 
 
 L. Application of V to the Stress- Function. §§ 507 — 511. 
 
 When there are no molecular couples the stress-function is self-conjugate. 
 
 § 507. Properties of this function which depend upon the equilibrium 
 
 of any definite portion of the solid, as a whole. § 508. Expression for 
 
 the stress-function, in terms of displacement, when the solid is isotropic : — • 
 
 <pa= -n (SuV . 0- -|- VSuir) - (c - §n) wSVcr. 
 
 Examples. § 509. Work due to displacement in any elastic solid. Green's 
 21 elastic coefficients. § 511. 
 
 M. The Hydrokinetic Equations. §§ 512, 513. 
 
 Equation of continuity, for displacement a : — 
 
 de de „ _ __ 
 
 at dt 
 Equation for rate of change of momentum in unit volume: — 
 
 9T=_vfp+f^V-V«. §512. 
 
 3« 
 Term introduced by Viscosity: — 
 
 ocVV-H^VSViT. (Miscellaneous Examples, 36i) 
 Helmholtz's Transformation for Vortex-motion: — 
 
 ^FV(r=F.VF.(rFVi<ri. 
 
 Thomson's Transformation for Circulation: — 
 
 . 1= [q - k]^ . where /= - i^adp. 
 
 N. Use of V in connection with Taylor's Theorem. §§ 514 — 517. 
 
 Proof that 
 
 e-«'^/(p)=/(p+<r). 
 Apphcations and consequences. Separation of symbols of operation and their 
 treatment as quantities. 
 
 0. Applications of V in connection with the Calculus of Variations. 
 
 §§ 518—527. 
 
 If A=fQTdp 
 
 we have dA= - [QSUdpdp] +/{ dQTdp + S.dp{Q U5p) } , 
 whence, if ^ is a maximum or minimum, 
 
 ^(«P')-VQ = 0. 
 
 Applications to Varying Action, Brachistochrones, Catenaries, &o. §§ 518 — 526. 
 Thomson's Theorem that there is one, and but one, solution of the equation 
 S.V{e-Vu) = iTn: §527. 
 
 Miscellaneous Examples 409 — 421. 
 
.(.(.U, 
 
 QUATEENIONS. 
 
 CHAPTEE I. 
 
 VECTORS, AND THEIR COMPOSITION. 
 
 1. For at least two centuries the geometrical representation 
 of the negative and imaginary algebraic quantities, —1 and J — 1 
 has been a favourite subject of speculation with mathematicians. 
 The essence of almost all of the proposed processes consists in 
 employing such expressions to indicate the Direction, not the 
 length, of lines. 
 
 2. Thus it was long ago seen that if positive quantities were 
 measured off in one direction along a fixed line, a useful and lawful 
 convention enabled us to express negative quantities of the same 
 kind by simply lajang them off on the same line in the opposite 
 direction. This convention is an essential part of the Cartesian 
 method, and is constantly employed in Analytical Geometry and 
 Applied Mathematics. 
 
 3. Wallis, towards the end of the seventeenth century, proposed 
 to represent the impossible roots of a quadratic equation by going 
 out of the line on which, if real, they would have been laid off. 
 This construction is equivalent to the consideration of 7 — 1 as a 
 directed unit-line perpendicular to that on which real quantities 
 are measured. 
 
 4. In the usual notation of Analytical Geometry of two 
 dimensions, when rectangular axes are employed, this amounts 
 to reckoning each unit of length along Oy as +^-1, and on 
 Oy' as - J^l ; while on Ox each unit is + 1, and on Ox it is - 1. 
 
 T. Q. I. 1 
 
QUATERNIONS. 
 
 [5- 
 
 If we look at these four lines in circular order, i.e. in the order of 
 positive rotation (that of the northern hemisphere of the earth 
 about its axis, or opposite to that of the hands of a watch), they 
 give 
 
 1, J^i, -1, -J^T. 
 
 In this series each expression is derived from that which precedes 
 it by multiplication by the factor J^l. Hence we may consider 
 n/ - 1 as an operator, analogous to a handle perpendicular to the 
 plane of xy, whose effect on any line in that plane is to make it 
 rotate (positively) about the origin through an angle of 90°. 
 
 5. In such a system, (which seems to have been first developed, 
 in 1805, by Bude) a point in the plane of reference is defined by a 
 single imaginary expression. Thus a + hJ — 1 may be considered 
 as a single quantity, denoting the point, P, whose coordinates are 
 a and b. Or, it may be used as an expression for the line OP 
 joining that point with the origin. In the latter sense, the ex- 
 pression a + h J — 1 implicitly contains the direction, as well as the 
 length, of this line; since, as we see at once, the direction is 
 inclined at an angle tan"' hja to the axis of x, and the length is 
 Ja^ 4- V. Thus, say we have 
 
 the line OP considered as that by which we pass from one 
 extremity, 0, to the other, P. In this sense it is called a Vector. 
 Considering, in the plane, any other vector, 
 
 OQ = a' + h'J^^; 
 the addition of these two lines obviously gives 
 
 OR = a + a' + (b + h')J~^; 
 and we see that the sum is the diagonal of the parallelogram on 
 OP, OQ. This is the law of the composition of simultaneous 
 velocities ; and it contains, of course, the law of subtraction of one 
 directed line from another. 
 
 6. Operating on the first of these symbols by the factor 7^. 
 it becomes -h + aJ-1; and now, of course, denotes the point 
 whose X and y coordinates are - 6 and a ; or the line joining this 
 point with the origin. The length is still Ja' + b\ but the angle 
 the line makes with the axis of x is tan~'(-a/6); which is 
 evidently greater by 7r/2 than before the operation. 
 
lO.] VECTORS, AND THEIR COMPOSITION. 3 
 
 7. De Moivre's Theorem tends to lead us still further in the 
 same direction. In fact, it is easy to see that if we use, instead 
 of J — 1, the more general factor cos a + ^ — 1 sin a, its effect on 
 any line is to turn it through the (positive) angle a in the plane 
 of X, y. [Of course the former factor, J — 1, is merely the par- 
 ticular case of this, when « = 7r/2.1 
 
 Thus (cos a + J — 1 sin a) (a + bj — 1) 
 
 = a cos a — bsina+J — 1 (a sin a + 6 cos a), 
 
 by direct multiplication. The reader will at once see that the new 
 form indicates that a rotation through an angle a. has taken place, 
 if he compares it with the common formulae for turning the co- 
 ordinate axes through a given angle. Or, in a less simple manner, 
 thus — 
 
 Length = J {a cos a — 6 sin txf + (a sin a + b cos a)^ 
 = V a" + h^ as before. 
 
 Inclination to axis of x 
 
 b 
 
 , tan a -t- - 
 
 _, a sm a -I- cos a ^ _, a 
 
 = tan i — -. = tan , 
 
 a cos a — sin a ■, b , 
 
 1 tana 
 
 a 
 
 = a + tan"' bja. 
 
 8. We see now, as it were, why it happens that 
 
 (cos a + V^l sin o)" = cos ma + 7 - 1 sin ma. 
 
 In fact, the first operator produces m successive rotations in the 
 same direction, each through the angle a ; the second, a single 
 rotation through the angle ma. 
 
 9. It may be interesting, at this stage, to anticipate so far as to 
 remark that in the theory of Quaternions the analogue of 
 
 cos 4- 7^ sine 
 is cos + OT sin Q, 
 
 where w^= — \. 
 
 Here, however, vs is not the algebraic J^l, but is any directed 
 vmt-Une whatever in space. 
 
 10. In the present century Argand, Warren, Mourey, and 
 others, extended the results of Wallis and Bude. They attempted 
 
 1—2 
 
QUATERNIONS. 
 
 [II. 
 
 to express as a line the product of two lines each represented by a 
 symbol such as a+hj^l. To a certain extent they succeeded, 
 but all their results remained confined to two dimensions. 
 
 The product, IT, of two such lines was defined as the fourth 
 proportional to unity and the two lines, thus 
 
 1 -.a + hJ^^-.-.a+Vj^l:^, 
 or n = (aa' - W) + (ah + h'a) J^^. 
 
 The length of U is obviously the product of the lengths of the 
 factor lines ; and its direction makes an angle with the axis of x 
 which is the sum of those made by the factor lines. From this 
 result the quotient of two such lines follows immediately. 
 
 11. A very curious speculation, due to Servois and published 
 in 1813 in Gergonne's Annates, is one of the very few, so far as has 
 been discovered, in which a well-founded guess at a possible mode 
 of extension to three dimensions is contained. Endeavouring to 
 extend to space the form a + 6 ^ - 1 for the plane, he is guided by 
 analogy to write for a directed unit-line in space the form 
 p cos a -I- 2 cos /3 + r cos 7, 
 
 where a, /3, 7 are its inclinations to the three axes. He perceives 
 easily that p, q, r must be non-reals : but, he asks, " seraient-elles 
 imaginaires r^ductibles h. la forme gdndrale A+BJ—W The 
 i,j, h of the Quaternion Calculus furnish an answer to this question. 
 (See Chap. II.) But it may be remarked that, in applying the 
 idea to lines in a plane, a vector OP will no longer be represented 
 (as in § 5) by 
 
 OP = a + hJ^^, 
 but by OP=pa + qb. 
 
 And if, similarly, OQ = pa' + qb', 
 
 the addition of these two lines gives for OR (which retains its 
 previous signification) 
 
 OB=p(a + a') + q(b + b'). 
 
 12. Beyond this, few attempts were made, or at least recorded, 
 in earlier times, to extend the principle to space of three dimen- 
 sions; and, though many such had been made before 1843, none, 
 with the single exception of Hamilton's, have resulted in simple, 
 practical methods ; all, however ingenious, seeming to lead almost 
 at once to processes and results of fearful complexity. 
 
1 4-] VECTORS, AND THEIR OOMPOSITION. 5 
 
 For a lucid, complete, and most impartial statement of the 
 claims of his predecessors in this field we refer to the Preface to 
 Hamilton's Lectures on Quaternions. He there shews how his long 
 protracted investigations of Sets culminated in this unique system 
 of tridimensional-space geometry. 
 
 13. It was reserved for Hamilton to discover the use and 
 properties of a class of symbols which, though all in a certain sense 
 square roots of — 1, may be considered as real unit lines, tied down 
 to no particular direction in space ; the expression for a vector is, 
 or may be taken to be, 
 
 p = ix+jy + kz; 
 
 but such vector is considered in connection with an extraspatial 
 magnitude w, and we have thus the notion of a Quaternion 
 
 w + p. 
 
 This is the fundamental notion in the singularly elegant, and 
 enormously powerful, Calculus of Quaternions. 
 
 While the schemes for using the algebraic J — 1 to indicate 
 direction make one direction in space expressible by real numbers, 
 the remainder being imaginaries of some kind, and thus lead to 
 expressions which are heterogeneous ; Hamilton's system makes all 
 directions in space equally imaginary, or rather equally real, there- 
 by ensuring to his Calculus the power of dealing with space 
 indifferently in all directions. 
 
 In fact, as we shall see, the Quaternion method is independent 
 of axes or any supposed directions in space, and takes its reference 
 lines solely from the problem it is applied to. 
 
 14. But, for the purpose of elementary exposition, it is best 
 to begin by assimilating it as closely as we can to the ordinary 
 Cartesian methods of Geometry of Three Dimensions, with which 
 the student is supposed to be, to some extent at least, acquainted. 
 Such assistance, it will be found, can (as a rule) soon be dispensed 
 with; and Hamilton regarded any apparent necessity for an oc- 
 casional recurrence to it, in higher applications, as an indication 
 of imperfect development in the proper methods of the new 
 Calculus. 
 
 We commence, therefore, with some very elementary geometrical 
 ideas, relating to the theory of vectors in space. It will subsequently 
 appear how we are thus led to the notion of a Quaternion. 
 
6 QUATERNIONS. [l5- 
 
 15. Suppose we have two points A and B in space, and sup- 
 pose A given, on how many numbers does jB's relative position 
 depend ? 
 
 If we refer to Cartesian coordinates (rectangular or not) we find 
 that the data required are the excesses of B's three coordinates 
 over those of A. Hence three numbers are required. 
 
 Or we may take polar coordinates. To define the moon's 
 position with respect to the earth we must have its Geocentric 
 Latitude and Longitude, or its Eight Ascension and Declination, 
 and, in addition, its distance or radius-vector. Three again. 
 
 16. Here it is to be carefully noticed that nothing has been 
 said of the actual coordinates of either A or B, or of the earth 
 and moon, in space ; it is only the relative coordinates that are 
 contemplated. 
 
 Hence any expression, as AB, denoting a line considered with 
 reference to direction and currency as well as length, (whatever 
 may be its actual position in space) contains implicitly three 
 numbers, and all lines parallel and equal to AB, and concurrent 
 with it, depend in the same way upon the same three. Hence, all 
 lines which are equal, parallel, and concurrent, may he represented 
 by a common symbol, and that symbol contains three distinct numbers. 
 In this sense a line is called a vectoe, since by it we pass from 
 the one extremity. A, to the other, B; and it may thus be 
 considered as an instrument which carries A to £: so that a 
 vector may be employed to indicate a definite translation in space. 
 
 [The term " currency " has been suggested by Cayley for use 
 instead of the somewhat vague suggestion sometimes taken to 
 be involved in the word "direction." Thus parallel lines have 
 the same direction, though they may have similar or opposite 
 currencies. The definition of a vector essentially includes its 
 currency.] 
 
 17. We may here remark, once for all, that in establishing a 
 new Calculus, we are at liberty to give any definitions whatever 
 of our symbols, provided that no two of these interfere with, or 
 contradict, each other, and in doing so in Quaternions simplicity 
 and (so to speak) naturalness were the inventor's aim. 
 
 18. Let AB be represented by a, we know that a involves 
 three separate numbers, and that these depend solely upon the 
 position of B relatively to A. Now if CD be equal in length to AB 
 
20.] VECTORS, AND THEIR COMPOSITION. 7 
 
 and if these lines be parallel, and have the same currency, we may 
 evidently write 
 
 where it will be seen that the sign of equality between vectors 
 contains implicitly equality in length, parallelism in direction, 
 and concurrency. So far we have extended the meaning of an 
 algebraical symbol. And it is to be noticed that an equation 
 between vectors, as 
 
 a = /3, 
 
 contains three distinct equations between mere numbers. 
 
 19. We must now define + (and the meaning of — will follow) 
 in the new Calculus. Let A,B, be any three points, and (with 
 the above meaning of = ) let 
 
 If we define + (in accordance with the idea (§ 16) that a vector 
 represents a translation) by the equation 
 
 a + /3 = 7, 
 or AB + BG=AG, 
 
 we contradict nothing that precedes, but we at once introduce the 
 idea that vectors are to he compounded, in direction and magnitude, 
 like simultaneous velocities. A reason for this may be seen in 
 another way if we remember that by adding the (algebraic) differ- 
 ences of the Cartesian coordinates of B and A, to those of the 
 coordinates of G and B, we get those of the coordinates of G and 
 A. Hence these coordinates enter linearly into the expression for 
 a vector. (See, again, § 5.) 
 
 20. But we also see that if G and A coincide (and G may be 
 any point) 
 
 AG = 0, 
 
 for no vector is then required to carry A to G. Hence the above 
 relation may be written, in this case, 
 
 AB + BA = 0, 
 or, introducing, and by the same act defining, the symbol — , 
 
 BA = -AB. 
 Hence, the symbol — , applied to a vector, simply shews that its 
 currency is to he reversed. 
 
8 QUATERNIONS. [2 1 • 
 
 And this is consistent with all that precedes ; for instance, 
 
 and AB = AC-BG, 
 
 or =AG + GB, 
 
 are evidently but diffefenT'expressions of the same truth. 
 
 21. In any triangle, ABG, we have, of course, 
 
 lB + BG+CA = 0; 
 and, in any closed polygon, whether plane or gauche, 
 
 AB + BG+ + YZ+ZA = 0. 
 
 In the case of the polygon we have also 
 
 AB + BC+ + YZ = AZ. 
 
 These are the well-known propositions regarding composition 
 of velocities, which, by Newton's second law of motion, give us 
 the geometrical laws of composition of forces acting at one point. 
 
 22. If we compound any number oi parallel vectors, the result 
 is obviously a numerical multiple of any one of them. 
 
 Thus, if A, B, G are in one straight line, 
 
 W = xAB; 
 where a; is a number, positive when B lies between A and G, other- 
 wise negative : but such that its numerical value, independent 
 of sign, is the ratio of the length of BG to that of AB. This is 
 at once evident if AB and BG be commensurable ; and is easily 
 extended to incommensurables by the usual reductio ad absurdum. 
 
 23. An important, but almost obvious, proposition is that any 
 vector may be resolved, and in one way only, into three components 
 parallel respectively to any three given vectors, no two of which are 
 parallel, and which are not parallel to one plane. 
 
 Let OA, OB, OG be the three fixed c" 
 vectors, OP any other vector. JFrom P draw 
 PQ parallel to CO, meeting the plane BOA 
 in Q. [There must be a definite point Q, 
 else PQ, and therefore CO, would be parallel 
 to BOA, a case specially excepted.] From Q 
 draw QR parallel to BO, meeting OA in B. 
 Then we have OP^OR + RQ + QP (§ 21), 
 and these components are respectively parallel to the three given 
 
26.] ' VECTORS, AND THEIR COMPOSITION. 9 
 
 vectors. By § 22 we may express OR as a numerical multiple 
 of OA, RQ of OB, and QP of OC. Hence we have, generally, for 
 any vector in terms of three fixed non-coplanar vectors, a, jS, 7, 
 
 OP = p = xa + y^ + zy, 
 
 which exhibits, in one form, the three numbers on which a vector 
 depends (§ 16). Here a;, y, z are perfectly definite, and can have 
 but single values. 
 
 24. Similarly any vector, as OQ, in the same plane with OA 
 and OB, can be resolved (in one way only) into components OR, 
 RQ, parallel respectively to OA and OB ; so long, at least, as these 
 two vectors are not parallel to each other. 
 
 25. There is particular advantage, in certain cases, in em- 
 ploying a series of three mutually perpendicular unit-vectors as 
 lines of reference. This system Hamilton denotes by i, j, k. 
 
 Any other vector is then expressible as 
 p = oci + yj + zk. 
 Since i, j, k are unit-vectors, x, y, z are here the lengths of 
 conterminous edges of a rectangular parallelepiped of which p 
 is the vector-diagonal ; so that the length of p is, in this case, 
 
 J^f+f + I. 
 Let ■07 = ^ + 7]j + ^k 
 
 be any other vector, then (by the proposition of § 23) the vector 
 equation p = ct 
 
 obviously involves the following three equations among numbers, 
 
 «=?. y = v, « = ?• 
 Suppose i to be drawn eastwards, j northwards, and k upwards, 
 this is equivalent merely to saying that if two points coincide, they 
 are equally to the east (or west) of any third point, equally to the 
 north (or south) of it, and equally elevated above (or depressed below) 
 its level. 
 
 26. It is to be carefully noticed that it is only when a, /3, 7 
 are not coplanar that a vector equation such as 
 
 p = ■57, 
 
 or xcc + y0 + zy = ^ci + 7)^ + ^y, 
 
 necessitates the three numerical equations 
 
 « = ?, y = v, ^ = ?- 
 
10 QUATERNIONS. [2 7- 
 
 For, if a, ^, 7 be coplanar (§ 24), a condition of the following form 
 must hold 
 
 7 = aa + 6/3. 
 
 Hence p = {x + za)a + {y + zh) ^, 
 
 ^ = (| + ra)a+(77 + ?6)A 
 and the equation /a = •ar 
 
 now requires only the two numerical conditions 
 
 27. The Commutative and Associative Laws hold in the com- 
 bination of vectors by the signs + and — . It is obvious that, if we 
 prove this for the sign + , it will be equally proved for — , because 
 — before a vector (§ 20) merely indicates that it is to be reversed 
 before being considered positive. 
 
 Let A, B, C, D be, in order, the corners of a parallelogram ; we 
 have, obviously, 
 
 AB = DG, AD = BG. 
 And AB + BG = AG=AI) + DG = BG + AB. 
 
 Hence the commutative law is true for the addition of any two 
 vectors, and is therefore generally true. 
 
 Again, whatever four points are represented hy A, B, C, D, we 
 have 
 
 AI) = AB + BI>==AG+CI), 
 
 or substituting their values for AD, BJD, AC respectively, in these 
 three expressions, 
 
 AB + BG + CD = AB + {BC + GD) = (AB + BC) + CD. 
 And thus the truth of the associative law is evident. 
 
 28, The equation 
 
 p = xl3, 
 where p is the vector connecting a variable point with the origin, 
 /3 a definite vector, and x an indefinite number, represents the 
 straight line drawn from the origin parallel to /3 (§ 22). 
 
 The straight line drawn from A, where 0-4= a, and parallel 
 to /3, has the equation 
 
 P = a + xl3 (1). 
 
 In words, we may pass directly from to P by the vector OP or 
 p; or we may pass first to -4, by means of OA or a, and then to P 
 along a vector parallel to /S (§ 16). 
 
30.] VECTORS, AND THEIR COMPOSITION. H 
 
 Equation (1) is one of the many useful forms into which 
 Quaternions enable us to throw the general equation of a straight 
 line in space. As we have seen (§ 25) it is equivalent to three 
 numerical equations ; but, as these involve the indefinite quantity 
 X, they are virtually equivalent to but two, as in ordinary Geometry 
 of Three Dimensions. 
 
 29. A good illustration of this remark is furnished by the fact 
 that the equation 
 
 which contains two indefinite quantities, is virtually equivalent to 
 only one numerical equation. And it is easy to see that it re- 
 presents the plane in which the lines a and yS lie ; or the surface 
 which is formed by drawing, through every point of OA, a line 
 parallel to OB. In fact, the equation, as written, is simply § 24 
 in symbols. 
 
 And it is evident that the equation 
 
 /> = 7 + 2/a + a;/3 
 
 is the equation of the plane passing through the extremity of 7, 
 and parallel to a and /8. 
 
 It will now be obvious to the reader that the equation 
 
 where a^, a^, &c. are given vectors, and j3j, p^, &c. numerical quan- 
 tities, represents a straight line if p^, p^, &c. be linear functions of 
 one indeterminate number; and a. plane, if they be linear expres- 
 sions containing two indeterminate numbers. Later (§ 31 (I)), 
 this theorem will be much extended. 
 Again, the equation 
 
 p = xa + y^ -f- ^7 
 
 refers to any point whatever in space, provided a, /3, 7 are not 
 coplanar. {Ante, § 23.) 
 
 30. The equation of the line joining any two points A and B, 
 where OA = a. and OB = /8, is obviously 
 
 p = a + a; (/3 - a), 
 or p = ^ + y(a-^). 
 
 These equations are of course identical, as may be seen by putting 
 1 — 2/ for «!. 
 
12 QUATEENIONS. 
 
 [31- 
 
 The first may be written 
 
 or pp + qa + r^ = 0, 
 
 subject to the condition ^ + g- + r = identically. That is — A 
 homogeneous linear function of three vectors, equated to zero, 
 expresses that the extremities of these vectors are in one straight 
 line, if the sum of the coefficients be identically zero. 
 
 Similarly, the equation of the plane containing the extremities 
 A, B, G oi the three non-coplanar vectors a, /3, 7 is 
 
 p = a + a;(^- a) + 2/(7-/3), 
 ■where x and y are each indeterminate. 
 This may be written 
 
 pp + qa + r^ + sy = 0, 
 with the identical relation 
 
 p + q + r + s=0, 
 
 which is one form of the condition that four points may lie in one 
 plane. 
 
 .31. We have already the means of proving, in a very simple 
 manner, numerous classes of propositions in plane and solid 
 geometry. A very few examples, however, must suffice at this 
 stage ; since we have hardly, as yet, crossed the threshold of the 
 subject, and are dealing with mere linear equations connecting two 
 or more vectors, and even with them we are restricted as yet to 
 operations of mere addition. "We will give these examples with a 
 painful minuteness of detail, which the reader will soon find to be 
 necessary only for a short time, if at all. 
 
 (a) The diagonals of a parallelogram bisect each other. 
 
 Let ABCD be the parallelogram, the point of intersection of 
 its diagonals. Then 
 
 which gives AO- 00 = DO -OB. 
 
 The two vectors here equated are parallel to the diagonals respect- 
 ively. Such an equation is, of course, absurd unless 
 
 (1) The diagonals are parallel, in which case the figure 
 is not a parallelogram ; 
 
 (2) A0 = 0C,3,ndD0 = OB, the proposition. 
 
31 (&').] VECTORS, AND THEIR COMPOSITION. 13 
 
 (6) To shew that a triangle can he constructed, whose sides 
 are parallel, and equal, to the bisectors of the sides of any 
 triangle. 
 
 Let ABG be any triangle, Aa, Bh, Cc the bisectors of the 
 sides. 
 
 Then A'a = AB + B'a = AB + \BG, 
 
 Eb - - =BG + \GA, 
 
 Gc - - - =GA+\AB. 
 
 Hence Z^ + m + Cc = f (ZB + 50+ Cl) = 0; 
 
 which (§ 21) proves the proposition. 
 
 Also A'a = AB + \BG 
 
 = AB-l{GA+AB) 
 
 = \{AB-GA) = \{AB + TG), 
 
 results -which are sometimes useful. They may be easily verified 
 by producing ^a to twice its length and joining the extremity 
 with B. 
 
 (b') The bisectors of the sides of a triangle meet in a point, 
 which trisects each of them. 
 
 Taking A as origin, and putting a, j8, 7 for vectors parallel, and 
 equal, to the sides taken in order BG, GA, AB ; the equation of 
 Bb is (§ 28 (1)) 
 
 That of Gc is, in the same way, 
 
 p = -(H-2/)/3-|7. 
 At the point 0, where Bb and Gc intersect, 
 
 p = (l + ^)7 + |i8 = -(H-2/)/3-|7. 
 Since 7 and /3 are not parallel, this equation gives 
 l+a; = -|, and| = -(l+2/). 
 
 From these oo = y = -^. 
 
 Hence l0 = H7-^) = l^- (See Ex. (6).) 
 
 This equation shews, being a vector one, that Aa passes 
 
 through 0, and that AOiQa :: 2:1. 
 
14 
 
 QUATERNIONS. 
 
 [31 (4 
 
 (c) If 6a = a, 
 
 6B = ^, 
 0G = la + m/3, 
 be three given co-planar vectors, c the intersection of AB, OC, and 
 if the lines indicated in the figure he drawn, the points a,, 6,, c^ lie 
 in a straight line. 
 
 We see at once, by the process indicated in § 30, that 
 
 jr- la + m^ 
 l + m 
 
 l—m 
 
 7^ '^^ 
 
 Hence we easily find 
 
 
 
 0^ "^^ 
 
 m ^" 
 
 7=r- - Za + m/3 
 
 Oc. = j — 
 
 ' m-l 
 
 ^«i- l-l-2m' 
 
 ^"^ l-2Z-m-- 
 
 These give 
 
 - (1 - i - 2m) Oa, + ( 1 - 2 i - m) O&i - (m - Oc, = . 
 
 But -{\-l- 2m) + (1 - 2? - m) - (m - = identically. 
 This, by § 30, proves the proposition. 
 
 {d) Let OA = a, OB = /S, be any two vectors. If MP he a 
 given line parallel to OB ; and OQ, BQ, be drawn parallel to AP, 
 OP respectively ; the locus of Q is a straight line parallel to OA. 
 
 
 
 f. 
 
 ^^^^' 
 
 
 0^ 
 
 ^^ 
 
 X / 
 
 
 
 B M 
 
 Let 
 
 
 
 OM=ea. 
 
 Then 
 
 
 
 AP = e-la + x^. 
 
31 (e)-] VECTORS, AND THEIR COMPOSITION. 15 
 
 Hence the equation of OQ is 
 
 and that oi BQ is p = ^j^z{ea + x0). 
 
 At Q we have, therefore, 
 
 xy = l+zx, ) 
 
 y(e-l)==ze. ] 
 These give xy = e, and the equation of the locus of Q is 
 
 p = e/3 + y'a, 
 i.e. a straight line parallel to OA, drawn through N in OB pro- 
 duced, so that 
 
 ON : OB : : OM : OA. 
 
 _C0E. If SQ meet MP in q,F^ = /3; and if J.P meet NQ in 
 p, Qp = a. 
 
 Also, for the point R we have pE = AP, QB = Bq. 
 
 Further, the locus of iJ is a hyperbola, of which MP and NQ 
 are the asymptotes. See, in this connection, § 31 (k) below. 
 
 Hence, if from any two points, A and B, lines be drawn inter- 
 cepting a given length Pq on a given line Mq; and if, from R their 
 point of intersection, Rp be laid off= PA, and RQ = qB; Q and p 
 lie on a fixed straight line, and the length of Qp is constant. 
 
 (e) To find the centre of inertia of any system of masses. 
 If OA=a, 05 = 3^, be the vector sides of any triangle, the 
 vector from the vertex dividing the base AB in G so that 
 
 BG: GA :: TO:m, 
 • ma. + m,g^ 
 
 m + m^ 
 
 For .45 is a, - a, and therefore AG is 
 
 TO, 
 
 TO + TO, 
 
 Hence OG=02 + A0 
 
 TO, 
 TO + TO, ' 
 
 _ ma + m^a^ 
 m + m^ 
 This expression shews how to find the centre of inertia of two 
 masses ; to at the extremity of a, to, at that of a,. Introduce to^ 
 
16 
 
 QUATERNIONS. 
 
 [31 (/)• 
 
 at the extremity of a^, then the vector of the centre of inertia of the 
 three is, by a second application of the formula, 
 
 (m + wij) f 
 
 /ma + m.aA 
 
 m + m 
 
 ma + m^a^ + m^a^ 
 m + m^ + m^ 
 
 (m + m^) + m^ 
 
 From this it is clear that, for any number of masses, expressed 
 generally by m at the extremity of the vector a, the vector of the 
 centre of inertia is 
 
 o S (ma) 
 '^~ S(m)- 
 This may be written 2m (a — /3) = 0. 
 
 Now Wj - /3 is the vector of m, with respect to the centre of inertia. 
 Hence the theorem, If the vector of each element of a mass, drawn 
 from the centre of inertia, be increased in length in proportion to the 
 mass of the element, the sum of all these vectors is zero. 
 
 (/) We see at once that 
 the equation 
 
 p = at + 
 
 ^f 
 
 T/fc 
 
 where t is an indeterminate 
 number, and a, given vec- 
 tors, represents a parabola. 
 The origin, 0, is a point on 
 the curve, /3 is parallel to 
 the axis, i.e. is the diameter 
 OB drawn from the origin, 
 and a is OA the tangent at the origin. In the figure 
 
 QP=oit, oq=^. 
 
 The secant joining the points where t has the values t and t' 
 represented by the equation 
 
 IS 
 
 (§30) 
 
 = od-{-~-\-x 
 
 {1f-t)\a+^t+lY 
 
 Write X for x (f - t) [which may have any value], then put H = t, 
 and the equation of the tangent at the point {t) is 
 
 P = oct + ~ +x{a+^t). 
 
31 (i^)-] VECTORS, AND THEIR COMPOSITION. 17 
 
 In this put x = — t, and we have 
 
 or the intercept of the tangent on the diameter is equal in length 
 to the abscissa of the point of contact, but has the opposite 
 currency. 
 
 Otherwise : the tangent is parallel to the vector a + j3t or 
 
 at + ^f or ^ + a< + '^- or OQ + OP. But TP = TO + OP, 
 
 hence TO =^0Q. 
 
 (g) Since the equation of any tangent to the parabola is 
 
 Of 
 
 p = at + ^+x{(x + ^t), 
 
 let us find the tangents which can be drawn from a given point. 
 Let the vector of the point be 
 
 p=pa + q^ (§24). 
 
 Since the tangent is to pass through this point, we have, as con- 
 ditions to determine t and x, 
 
 t + x=p, 
 
 f 
 
 ^+xt = q; 
 
 by equating respectively the coefficients of a and y3. 
 
 Hence t=p±Jp''-2q. 
 
 Thus, in general, two tangents can be drawn from a given point. 
 
 These coincide if p" = 2q; 
 
 that is, if the vector of the point from which they are to be xirawn 
 
 p' 
 is p=poi + q^ = pi^ + Y^' 
 
 i.e. if the point lies m the parabola. They are imaginary if 
 2q >p^, that is, if the point be 
 
 r being positive. Such a point is evidently within the curve, as at 
 R, where 0Q = ^ 13, QP =pa, PR = r/3. 
 
 T, Q. I. ^ 
 
18 QUATERNIONS. [3I {k). 
 
 (h) Calling the values of t for the two tangents found in (g) 
 «, and t^ respectively, it is obvious that the vector joining the 
 points of contact is 
 
 a^i + —^ — <x.% — 2' ' 
 
 which is parallel to « + yS ^"^ ' 
 
 or, by the values of f, and t^ in {g), 
 
 a + p/3. 
 
 Its direction, therefore, does not depend on q. In words. If pairs oj 
 tangents he drawn to a parabola from points of a diameter produced, 
 the chords of contact are parallel to the tangent at the vertex of the 
 diameter. This is also proved by a former result, for we must have 
 OT for each tangent equal to QO. 
 
 (i) The equation of the chord of contact, for the point whose 
 vector is 
 
 p=pa + q^, 
 
 Bt^ 
 is thus p = OLt^ + -^ + y{a + p^). 
 
 Suppose this to pass always through the point whose vector is 
 
 p — aa. + 6/3. 
 Then we must have t^ + y = a,' 
 
 t^ 
 
 or tj=p±J p^-2pa + 2b. 
 
 Comparing this with the expression in (g), we have 
 
 q = pa — b; 
 that is, the point from which the tangents are drawn has the vector 
 
 p=pa + ipa-b)^ 
 = -b^+p(a + a/3), a straight line (§ 28 (1)). 
 
 The mere form of this expression contains the proof of the usual 
 properties of the pole and polar in the parabola ; but, for the sake 
 of the beginner, we adopt a simpler, though equally general, 
 process. 
 
 Suppose a = 0. This merely restricts the pole to the particular 
 
31 {^yi VECTOES, AND THEIE COMPOSITION. 19 
 
 diameter to which we have referred the parabola. Then the pole 
 is Q, where p = b^; 
 
 and the polar is the line TU, for which 
 
 p = -hp +pa. 
 Hence the polar of any point is parallel to the tangent at the 
 extremity of the diameter on which the point lies, and its inter- 
 section with that diameter is as far beyond the vertex as the pole 
 is within, and vice versd. 
 
 (j) As another example let us prove the following theorem. 
 If a triangle be inscribed in a parabola, the three points in which 
 the sides are met by tangents at the angles lie in a straight line. 
 
 Since is any point of the curve, we may take it as one corner 
 of the triangle. Let t and t^ determine the others. Then, if 
 ■nr , CTjj, ro-g represent the vectors of the points of intersection of the 
 tangents with the sides, we easily find 
 
 2t^~t\ 2 
 
 2t-t^\ 2 
 
 ■■ — '—a. 
 
 t, + t 
 
 These values give 
 
 2t-t 2t-t, t'-f 
 
 " tt, ' 
 
 Also ^^ - ?^ - ^-^-^ = identically. 
 
 t t tt^ 
 
 Hence, by § 30, the proposition is proved. 
 
 (k) Other interesting examples of this method of treating 
 curves will, of course, suggest themselves to the student. Thus 
 
 p =acos t + 13 sin t 
 or p = ax + ^Jl-oif 
 
 represents an ellipse, of which the given vectors a and ^ are semi- 
 conjugate diameters. If t represent time, the radius-vector of this 
 ellipse traces out equal areas in equal times. [We may anticipate 
 so far as to write the following : — 
 
 2 Area = TJVpdp = TFa/S . fdt ; 
 
 which will be easily understood later.] 
 
 2—2 
 
20 QUATERNIONS. [3I (/)■ 
 
 Again, p = at + ^ or /a = a tana; + ^ cot a; 
 
 V 
 
 evidently represents a hyperbola referred to its asymptotes. [If 
 t represent time, the sectorial area traced out is proportional to 
 log t, taken between proper limits.] 
 Thus, also, the equation 
 
 p = a(t + sin t)+^ cos t 
 in which a and /S are of equal lengths, and at right angles to one 
 another, represents a cycloid. The origin is at the middle point of 
 the axis (2/8) of the curve. [It may be added that, if t represent 
 time, this equation shews the motion of the tracing point, provided 
 the generating circle rolls uniformly, revolving at the rate of a 
 radian per second.] 
 
 When the lengths of a, /3 are not equal, this equation gives the 
 cycloid distorted by elongation of its ordinates or abscissae : — not a 
 trochoid. The equation of a trochoid may be written 
 
 p = a(et + sin t) + /3 cos t, 
 e being greater or less than 1 as the curve is prolate or curtate. 
 The lengths of a and /B are still taken as equal. 
 
 But, so far as we have yet gone with the explanation of the 
 calculus, as we are not prepared to determine the lengths or in- 
 clinations of vectors, we can investigate only a very limited class of 
 the properties of curves, represented by such equations as those 
 above written. 
 
 (1) We may now, in extension of the statement in § 29, make 
 the obvious remark that 
 
 p = Xpct 
 
 (where, as in § 23, the number of vectors, a, can always be reduced 
 to three, at most) is the equation of a curve in space, if the 
 numbers p^, p^, &c. are functions of one indeterminate. In such 
 a case the equation is sometimes written 
 
 p = <P(t). 
 
 But, if Pj, p^, &c. be functions of two indeterminates, the locus of 
 the extremity of p is a surface; whose equation is sometimes written 
 
 p = 4>(t,u). 
 
 [It may not be superfluous to call the reader's attention to the 
 fact that, in these equations, ^ (t) or (t, u) is necessarily a vector 
 expression, since it is equated to a vector, p.] 
 
32.] VECTORS, AND THEIR COMPOSITION. 21 
 
 (m) Thus the equation 
 
 p = acost + ^sint + iyt (1) 
 
 belongs to a helix, while 
 
 p = a cos t + ^ sint + yu (2) 
 
 represents a cylinder whose generating lines are parallel to 7, and 
 whose base is the ellipse 
 
 p = a cos f + /9 sin t. 
 The helix above lies wholly on this cylinder. 
 Contrast with (2) the equation 
 
 p = u(a.cost + ^sint+j) (3) 
 
 which represents a cone of the second degree : — made up, in fact, 
 of all lines drawn from the origin to the ellipse 
 p = a cos t + ^ sint + j. 
 If, however, we write 
 
 p = u(acost + ^sint + yt), 
 
 we form the equation of the transcendental cone whose vertex is 
 at the origin, and on which lies the helix (1). 
 In general 
 
 p = u^ (t) 
 is the cone whose vertex is the origin, and on which lies the curve 
 
 while p = (f){t) + ua. 
 
 is a cylinder, with generating lines parallel to a, standing on the 
 
 same curve as base. 
 
 Again, p=pa+q^ + ry 
 
 with a condition of the form 
 
 ap" + hif + cr" = 1 
 belongs to a central surface of the second order, of which a, ^, 7 
 are the directions of conjugate diameters. If a, b, c be all positive, 
 the surface is an ellipsoid. 
 
 32. In Example (/) above we performed an operation equi- 
 valent to the differentiation of a vector with reference to a single 
 numerical variable of which it was given as an explicit function. 
 As this process is of very great use, especially in quaternion investi- 
 gations connected with the motion of a particle or point ; and as it 
 will afford us an opportunity of making a preliminary step towards 
 
22 QUATERNIONS. [33. 
 
 overcoming the novel difificulties which arise in quaternion dififeren- 
 tiation ; we will devote a few sections to a more careful, though 
 very elementary, exposition of it. 
 
 33. It is a striking circumstance, when we consider the way 
 in which Newton's original methods in the Differential Calculus 
 have been decried, to find that Hamilton was obliged to employ 
 them, and not the more modern forms, in order to overcome the 
 characteristic difficulties of quaternion differentiation. Such a thing 
 as a differential coefficient has absolutely no meaning in quaternions, 
 except in those special cases in which we are dealing with degraded 
 quaternions, such as numbers, Cartesian coordinates, &c. But a 
 quaternion expression has always a differential, which is, simply, 
 what Newton called a fluxion. 
 
 As with the Laws of Motion, the basis of Dynamics, so with the 
 foundations of the Differential Calculus ; we are gradually coming 
 to the conclusion that Newton's system is the best after all. 
 
 34. Suppose p to be the vector of a curve in space. Then, 
 generally, p may be expressed as the sum of a number of terms, 
 each of which is a multiple of a constant vector by a function of some 
 one indeterminate ; or, as in § 31 {I), if P be a point on the curve, 
 
 6P=p = <i>{t). 
 
 And, similarly, if Q be any other point on the curve, 
 
 OQ = p. = p + 8/J=<^(g = </)(«+S0, 
 where ht is any number whatever. 
 
 The vector-chord PQ is therefore, rigorously, 
 Sp = pj^ — p = (f)(t + St) - 4>t. 
 
 35. It is obvious that, in the present case, because the vectors 
 involved in (f> are constant, and their numerical multipliers alone vary, 
 the expression ^ (< + U) is, by Taylor's Theorem, equivalent to 
 
 ,,,,^),,^)»V 
 
 And we are thus entitled to write, when ht has been made inde- 
 finitely small. 
 
36.] VECTORS, AND THEIR COMPOSITION. 
 
 23 
 
 In such a case as this, then, we are permitted to differentiate, 
 or to form the differential coefficient of, a vector, according to the 
 ordinary rules of the Differential Calculus. But great additional 
 insight into the process is gained by applying Newton's method. 
 
 36. Let OP be 
 
 _ P = ^{t\ 
 
 and OQ, 
 
 p, = ^{t + dt), 
 where dt is any number whatever. 
 
 The number t may here be taken 
 as representing time, i.e. we may 
 suppose a point to move along the 
 curve in such a way that the value 
 of t for the vector of the point P of 
 the curve denotes the interval which 
 has elapsed (since a fixed epoch) when the moving point has 
 reached the extremity of that vector. If, then, dt represent any 
 interval, finite or not, we see that 
 
 will be the vector of the point after the additional interval dt. 
 
 But this, in general, gives us little or no information as to the 
 velocity of the point at P. We shall get a better approximation 
 by halving the interval dt, and finding Q^, where OQ^ = (j}(t + ^dt), 
 as the position of the moving point at that time. Here the vector 
 virtually described in ^dt is PQ^. To find, on this supposition, 
 the vector described in dt, we must double PQ^, and we find, as a 
 second approximation to the vector which the moving point would 
 have described in time dt, if it had moved for that period in the 
 direction and with the velocity it had at P, 
 
 Pq, = 2PQ, = 2(OQ,-OP) 
 
 = 2[4>(t + idi)-(l>{t)}. 
 The next approximation gives 
 
 P^, = 3PQ, = 3{OQ,-OP) 
 
 = S{cf>{t + m-'f>ii)}- 
 And so on, each step evidently leading us nearer the sought truth. 
 Hence, to find the vector which would have been described in time 
 dt had the circumstances of the motion at P remained undisturbed, 
 we must find the value of 
 
 dp = P^ = <.=. a; I (^ (« + ^ c^«) - </. (0 
 
24 QUATERNIONS. [37- 
 
 We have seen that in this particular case we may use Taylor's 
 Theorem. We have, therefore, 
 
 = 4>' (t) dt. 
 And, if we choose, we may now write 
 
 37. But it is to be most particularly remarked that in the 
 whole of this investigation no regard whatever has been paid to 
 the magnitude of dt. The question which we have now answered 
 may be put in the form — A point describes a given curve in a given 
 manner. At any point of its path its motion suddenly ceases to he 
 accelerated. What space will it describe in a definite interval ? As 
 Hamilton well observes, this is, for a planet or comet, the case 
 of a ' celestial Atwood's machine.' 
 
 38. If we suppose the variable, in terms of which p is expressed, 
 to be the arc, s, of the curve measured from some fixed point, we 
 find as before 
 
 dp = <f)'(s) ds. 
 
 From the very nature of the question it is obvious that the length 
 of dp must in this case be ds, so that <j}'{s) is necessarily a unit- 
 vector. This remark is of importance, as we shall see later ; and 
 it may therefore be useful to obtain afresh the above result without 
 any reference to time or velocity. 
 
 39. Following strictly the process of Newton's Vllth Lemma, 
 let us describe on Pq^ an arc similar to PQ^, and so on. Then 
 obviously, as the subdivision of ds is carried farther, the new arc 
 (whose length is always ds) more and more nearly (and without 
 limit) coincides with the line which expresses the corresponding 
 approximation to dp. 
 
 40. As additional examples let us take some well-known 
 plane curves ; and first the hyperbola (§ 81 {k)) 
 
 p = at + ^. 
 Here dp = (a—^]dt. 
 
 d, = («-^^ 
 
41.] VECTORS, AND THEIR COMPOSITION. 25 
 
 This shews that the tangent is parallel to the vector 
 
 In words, if the vector {from the centre) of a point in a hyperbola 
 he one diagonal of a parallelogram, two of whose sides coincide with 
 the asymptotes, the other diagonal is parallel to the tangent at the 
 point, and cuts off a constant area from the space between the 
 asymptotes. (For the sides of this triangular area are t times the 
 length of a, and Ijt times the length of /8, respectively ; the angle 
 between them being constant.) 
 
 Next, take the cycloid, as in § 31 (k), 
 
 p = a(t+sm <) + /3 cos t. 
 We have 
 
 dp = {«(!+ cos t) — ^ sin t] dt. 
 At the vertex 
 
 ^ = 0, cos i = 1, sin t=0, and dp = 2<xdt. 
 At a cusp 
 
 t = TT, cos t = —l, sin f = 0, and dp = 0. 
 This indicates that, at the cusp, the tracing point is (instan- 
 taneously) at rest. To find the direction of the tangent, and the 
 form of the curve in the vicinity of the cusp, put t=7r + T, where 
 powers of t above the second are omitted. We have 
 
 dp = ^rdt + -„- dt, 
 
 so that, at the cusp, the tangent is parallel to /3. By making the 
 same substitution in the expression for p, we find that the part of 
 the curve near the cusp is a semicubical parabola, 
 
 p = a(7r-|-T76)-^(l-T72); 
 
 or, if the origin be shifted to the cusp (p = Tra- yS), 
 
 p = aTV6+y8T72. 
 
 41. Let us reverse the first of these questions, and seek the 
 envelop of a line which cuts off from two fixed axes a triangle of 
 constant area. 
 
 If the axes be in the directions of a and /3, the intercepts may 
 
 evidently be written at and ^. Hence the equation of the line is 
 
 (§30) 
 
 ■■oct + r '^ 
 
26 QUATERNIONS. [42. 
 
 The condition of envelopment is, obviously, (see Chap. IX.) 
 
 dp = 0. 
 
 This gives = L-x(^ + oSidt+(^- at) dx*. 
 
 Hence {1 -x)dt—tdx = 0, 
 
 1 X ,, dx . 
 
 and _ £^^ 4- = 0. 
 
 t t 
 
 From these, at once, a; = |, since dx and dt are indeterminate. 
 Thus the equation of the envelop is 
 
 the hyperbola as before ; a, j3 being portions of its asymptotes. 
 
 42. It may assist the student to a thorough comprehension 
 of the above process, if we put it in a slightly different form. 
 Thus the equation of the enveloping line may be written 
 
 p = at{l-x)+/3^, 
 
 fx 
 which gives c?/3 = = ad{<(l — ;»)} +ySd f- 
 
 Hence, as a is not parallel to /3, we must have 
 
 d{t{\-x)\ = ^, d{^ = 0; 
 
 and these are, when expanded, the equations we obtained in the 
 preceding section. 
 
 43. For farther illustration we give a solution not directly 
 employing the differential calculus. The equations of any two of 
 the enveloping lines are 
 
 * Here we have opportunity for a remark (very simple indeed, but) of the 
 utmost importance. We are iiot to equate separately to zero the coefficients of dt 
 and dx ; for we must remember that this equation is of the form 
 
 0=pa + qp, 
 
 where p and q are numbers ; and that, so long as o and /3 are actual and non-parallel 
 vectors, the existence of such an equation requires (§ 24) 
 
 p^O 3 = 0. 
 
43-1 VECTORS, AND THBlE, COMPOSITION. 27 
 
 p= at + x(j-at\, 
 
 p = a.t, + x^(j-at}j, 
 
 t and t^ being given, while x and x^ are indeterminate. 
 
 At the point of intersection of these lines we have (§ 2(j), 
 t(l-x) = t,il-x,)/ 
 X _x^ 
 
 t-J, 
 
 These give, by eliminating a;,, 
 
 t{i-x)=t,(i-*-^xy 
 t 
 
 or x = ^ . 
 
 t^ + t 
 
 Hence the vector of the point of intersection is 
 
 att, + /3 
 
 and thus, for the ultimate intersections, where <^-^ = 1, 
 
 V 
 
 iM) 
 
 T^l as before. 
 
 OoR. If, instead of the ultimate intersections, we consider 
 the intersections of pairs of these lines related by some law, we 
 obtain useful results. Thus let 
 
 tt, = 1, 
 
 OL + P 
 
 or the intersection lies in the diagonal of the parallelogram on a, /3. 
 If t^ = mt, where m is constant, 
 
 mta + - 
 
 V 
 
 But we have also x = 
 
 ^ m + \ 
 1 
 
 m+ 1 
 
 Hence the locus of a point which divides in a given ratio a line 
 cutting off a given area from two fixed axes, is a hyperbola of which 
 these axes are the asymptotes. 
 
28 QUATERNIONS. [44- 
 
 If we take either 
 
 U^(t + 1^) = constant, or —S- = constant, 
 
 the locus is a parabola ; and so on. 
 
 It will be excellent practice for the student, at this stage, to 
 work out in detail a number of similar questions relating to the 
 envelop of, or the locus of the intersection of selected pairs from, a 
 series of lines drawn according to a given law. And the process 
 may easily be extended to planes. Thus, for instance, we may 
 form the general equation of planes wbich cut off constant tetra- 
 hedra from the axes of coordinates. Their envelop is a surface of 
 the third degree whose equation may be written 
 
 where xyz = a^ 
 
 Again, find the locus of the point of intersection of three of 
 this group of planes, such that the first intercepts on /8 and 7, the 
 second on 7 and a, tbe third on a and /3, lengths all equal to one 
 another, &c. But we must not loiter with such simple matters as 
 these. 
 
 44. The reader who is fond of Anharmonic Ratios and Trans- 
 versals will find in the early chapters of Hamilton's Elements of 
 Quaternions an admirable application of the composition of vectors 
 to these subjects. The Theory of Geometrical Nets, in a plane, 
 and in space, is there very fully developed; and the method is 
 shewn to include, as particular cases, the corresponding processes of 
 Grassmann's Ausdehnungslehre and Mobius' Barycentrische Galcul. 
 Some very curious investigations connected with curves and surfaces 
 of the second and third degrees are also there founded upon the 
 composition of vectors. 
 
 EXAMPLES TO CHAPTER I. 
 
 ■^ 1. The lines which join, towards the same parts, the extremities 
 of two equal and parallel lines are themselves equal and parallel. 
 {Euclid, I. xxxiii.) 
 
 '' 2. Find the vector of the middle point of the line which joins 
 the middle points of the diagonals of any quadrilateral, plane or 
 
VECTORS, ANT) THEIR COMPOSITION. 29 
 
 gauche, the vectors of the comers being given ; and so prove that 
 this point is the mean point of the quadrilateral. 
 J If two opposite sides be divided proportionally, and two new 
 quadrilaterals be formed by joining the points of division, the mean 
 points of the three quadrilaterals lie in a straight line. 
 •^ Shew that the mean point may also be found by bisecting the 
 line joining the middle points of a pair of opposite sides. 
 
 '' 3. Verify that the property of the coefficients of three vectors 
 whose extremities are in a line (§ 30) is not interfered with by 
 altering the origin. 
 
 v' 4. If two triangles ABC, abc, be so situated in space that Aa, 
 Bh, Cc meet in a point, the intersections of AB, ah, of BG, he, and 
 of GA, ca, lie in a straight line. 
 
 3. Prove the converse of 4, i. e. if lines be drawn, one in each 
 of two planes, from any three points in the straight line in which 
 these planes meet, the two triangles thus formed are sections of a 
 common pyramid. 
 
 6. If five quadrilaterals be formed by omitting in succession 
 each of the sides of any pentagon, the lines bisecting the diagonals 
 of these quadrilaterals meet in a point. (H. Fox Talbot.) 
 
 •^ 7. Assuming, as in § 7, that the operator^^^^^^ i i^ifi+o'i 'Afi =^(A1 
 
 COS y + J — 1 sm y ^-.^ ,^^ , 
 
 turns any radius of a given circle through an angle d in the 
 positive direction of rotation, without altering its length, deduce 
 the ordinary formulae for cos {A + B), cos {A — B), sin {A + B), and 
 sin (A — B), in terms of sines and cosines of A and B. 
 
 "^ 8. If two tangents be drawn to a hyperbola, the line joining 
 the centre with their point of intersection bisects the lines join- 
 ing the points where the tangents meet the asymptotes : and the 
 secant through the points of contact bisects the intercepts on 
 the asymptotes. 
 
 '^ 9. Any two tangents, limited by the asymptotes, divide each 
 other proportionally. 
 
 </ 10. If a chord of a hyperbola be one diagonal of a parallelogram 
 whose sides are parallel to the asymptotes, the other diagonal passes 
 through the centre. 
 
30 QUATERNIONS. 
 
 11. Given two points A and B, and a plane, G. Find the 
 locus of P, such that if AP cut G in Q, and BP cut C in R,QR 
 may be a given vector. 
 
 J ^ 12. Shew that p = x^a + y^^ +(x + yy'y 
 
 is the equation of a cone of the second degree, and that its section 
 by the plane 
 
 P + q + i' 
 is an ellipse which touches, at their middle points, the sides of the 
 triangle of whose corners a, ^, y are the vectors. (Hamilton, 
 Elements, p. 96.) 
 
 V 13. The lines which divide, proportionally, the pairs of opposite 
 
 sides of a gauche quadrilateral, are the generating lines of a hyper- 
 bolic paraboloid. (Ibid. p. 97.) 
 
 ■' 14. Shew that p = x^a + y'/S + z^y, 
 
 where ^a; + y + z = 0, 
 
 represents a cone of the third order, and that its section by the 
 plane 
 
 _ pa + q/3 + ry 
 p+q + r 
 is a cubic curve, of which the lines 
 
 P^±q^ ^^ 
 '^ p + q 
 Sire the asymptotes and the three (real) tangents of inflection. Also 
 that the mean point of the triangle formed by these lines is a 
 conjugate point of the curve. Hence that the vector a -|-/8 + 7 is 
 a conjugate ray of the cone. {Ibid. p. 96.) 
 
 £^ f ^ €<<^^^--^'' ^ 
 
 . (^.^^v^^ V-h/- ^7^.-- x::Oj ^■^./l^,,^,,:^. Tt^,: '^■-'^ 
 
CHAPTER II. 
 
 PRODUCTS AND QUOTIENTS OF VECTORS. 
 
 45. We now come to the consideration of questions in which 
 the Calculus of Quaternions differs entirely from any previous 
 mathematical method; and here we shall get an idea of what a 
 Quaternion is, and whence it derives its name. These questions 
 are fundamentally involved in the novel use of the symbols of 
 multiplication and division. And the simplest introduction to 
 the subject seems to be the consideration of the quotient, or ratio, 
 of two vectors. 
 
 46. If the given vectors be parallel to each other, we have 
 already seen (§ 22) that either may be expressed as a numerical 
 multiple of the other; the multiplier being simply the ratio of 
 their lengths, taken positively if they have similar currency, 
 negatively if they run opposite ways. 
 
 47. If they be not parallel, let OA and OB be drawn parallel 
 and equal to them from any point ; and the question is reduced 
 to finding the value of the ratio of two vectors drawn from the 
 same point. Let us first find upon how many distinct numbers this 
 ratio depends. 
 
 "We may suppose OA to be changed into OB by the following 
 successive processes. 
 
 1st. Increase or diminish the length of OA till it becomes 
 equal to that of OB. For this only one number is required, viz. 
 the ratio of the lengths of the two vectors. As Hamilton remarks, 
 this is a positive, or rather a signless, number. 
 
 2nd. Turn OA about 0, in the common plane of the two 
 vectors, until its direction coincides with that of OB, and (remem- 
 
32 QUATERNIONS. [Ad- 
 
 hering the effect of the first operation) we see that the t>^o vectors 
 now coincide or become identical. To specify this operation three 
 numbers are required, viz. two angles (such as node and inclination 
 in the case of a planet's orbit) to fix the plane in which the rotation 
 takes place, and one angle for the amount of this rotation. 
 
 Thus it appears that the ratio of two vectors, or the multiplier 
 required to change one vector into another, in general depends upon 
 four distinct numbers, whence the name QUATERNION. 
 
 A quaternion q is thus defined as expressing a relation 
 
 between two vectors a, /3. By what precedes, the vectors a, 0, 
 which serve for the definition of a given quaternion, must be in a 
 given plane, at a given inclination to each other, and with their 
 lengths in a given ratio ; but it is to be noticed that they may be 
 any two such vectors. [Inclination is understood to include sense, 
 or currency, of rotation from a to /3.] 
 
 The particular case of perpendicularity of the two vectors, where 
 their quotient is a vector perpendicular to their plane, is fully 
 considered below ; §§ 64, 65, 72, &c. 
 
 48. It is obvious that the operations just described may be 
 performed, with the same result, in the opposite order, being per- 
 fectly independent of each other. Thus it appears that a quaternion, 
 considered as the factor or agent which changes one definite vector 
 into another, may itself be decomposed into two factors of which 
 the order is immaterial. 
 
 The stretching factor, or that which performs the first operation 
 in § 47, is called the Tensor, and is denoted by prefixing T to the 
 quaternion considered. 
 
 The turning factor, or that corresponding to the second operation 
 in § 47, is called the Versor, and is denoted by the letter U pre- 
 fixed to the quaternion. 
 
 49. Thus, if OA = a, OB = 0, and if q be the quaternion 
 which changes a to /S, we have 
 
 which we may write in the form 
 
 B 
 
 -=q, or 0a-' = q, 
 
 if we agree to define that 
 
 B 
 
 ~.a=B(x-\aL = B. 
 
50.J 
 
 PRODUCTS AND QUOTIENTS OP VECTORS. 
 
 33 
 
 Here it is to be particularly noticed that we write q before a to 
 signify that a is multiplied by (or operated on by) q, not q 
 multiplied by a. 
 
 This remark is of extreme importance in quaternions, for, as we 
 shall soon see, the Commutative Law does not generally apply to 
 the factors of a product. 
 
 We have also, by §§ 47, 48, 
 
 q=Tq.Uq=Uq.Tq, 
 
 where, as before, Tq depends merely on the relative lengths of 
 a and /3, and Uq depends solely on their directions. 
 
 Thus, if a, and ;S, be vectors of unit length parallel to a and /3 
 respectively, 
 
 T ^ = m/Ta, = 1, U^=-UpJUa=U-. 
 
 As will soon be shewn, when a is perpendicular to /3, i.e. when the 
 vefsor of the quotient is quadrantal, it is a unit-vector. 
 
 50. We must now carefully notice that the quaternion which 
 is the quotient when /3 is divided by a in no way depends upon 
 the absolute lengths, or directions, of these vectors. Its value 
 will remain unchanged if we substitute for them any other pair 
 of vectors which 
 
 (1) have their lengths in the same ratio, 
 
 (2) have their common plane the same or parallel, 
 and (3) make the same angle with each other. 
 
 Thus in the annexed figure 
 OB 
 
 if, and only if, 
 0,B, 
 
 (1) 
 
 0£, 
 
 OB 
 
 OA 
 
 0,A^ OA ' 
 
 (2) plane AOB parallel to plane 4,0,5,, 
 
 (3) ZAOB= AAfi,B,. 
 
 [Equality of angles is understood to include 
 concurrency of rotation. Thus in the annexed 
 figure the rotation about an axis drawn upwards 
 from the plane is negative (or clock- wise) from 
 OA to OB, and also from 0^^ to 0,B^.] 
 T. Q. I. 
 
34 QUATERNIONS. [SI- 
 
 It thus appears that if 
 
 the vectors a, yS, 7, 8 are parallel to one plane, and may be repre- 
 sented (in a highly extended sense) as proportional to one another, 
 
 thus : — 
 
 /3 : a = 8 : 7. 
 
 And it is clear from the previous part of this section that this 
 may be written not only in the form 
 
 a : /3 = 7 : 8 
 but also in either of the following forms : — 
 
 7 : a = 8 ; /3. 
 
 a : 7 = /3 : 8. 
 
 While these proportions are true as equalities of ratios, they 
 do not usually imply equalities of products. 
 
 Thus, as the first of these was equivalent to the equation 
 
 f = ^ = ^, or^a- = 87- = ?; 
 the following three imply separately, (see next section) 
 
 ^"6 ^ ' a~/S~'^' 78*"' 
 or, if we please, 
 
 «^-> = 78-' = q-\ yoT' = S/3-' = r, 07"' = /SS"' = r" ; 
 where r is a new quaternion, which has not necessarily anything 
 (except its plane), in common with q. 
 
 But here great caution is requisite, for we are not entitled to 
 conclude from these that 
 
 a8 = ,87, &c. 
 
 This point will be fully discussed at a later stage. Meanwhile 
 we may merely state that from 
 
 a 7 /3 8 
 
 — = -- or — ^ — 
 /3 8' a ry- 
 
 we are entitled to deduce a number of equivalents such as 
 
 a/3~'8 = 7, or a = yB~^0, or /3"'8 = a~'7, &c. 
 
 51. The Reciprocal of a quaternion q is defined by the 
 equation 
 
 ^q = q q=l = q- = qq . 
 
5I-] PRODUCTS 
 
 AND QUOTIENTS OF VECTORS, 
 
 Hence if 
 
 /8 
 
 - = g, or 
 
 
 /Q = ?«, 
 
 we must have 
 
 « 1 _, 
 
 For this gives 
 
 ^,0 = q-\qa, 
 
 35 
 
 and each member of the equation is evidently equal to a. 
 Or thus : — 
 
 Operate by q~^, 
 
 g-'/8 = a. 
 Operate on ^'\ 
 
 Or, we may reason thus : — since q changes OA to OB, q^ must 
 change OB to OA, and is therefore expressed by 3 (§ 49). 
 
 The tensor of the reciprocal of a quaternion is therefore the 
 reciprocal of the tensor; and the versor differs merely by the 
 reversal of its representative angle. The versor, it must be 
 remembered, gives the plane and angle of the turning — it has 
 nothing to do with the extension. 
 
 [Remark. In §§' 49 — 51, above, we had such expressions as 
 
 - =ySa"\ We have also met with a"^/3. Cayley suggests that this 
 
 also may be written in the ordinary fractional form by employing 
 the following distinctive notation : — 
 
 a |a a\ 
 
 (It might, perhaps, be even simpler to use the solidus as 
 recommended by Stokes, along with an obviously correlative 
 type : — thus, 
 
 ^=l3cr' = ^/a, a-^^ = a\/3.) 
 a 
 
 I have found such notations occasionally convenient for private 
 work, but I hesitate to introduce changes unless they are abso- 
 lutely required. See remarks on this point towards the end of the 
 Preface to- the Second Edition — reprinted above.] 
 
 3—2 
 
36 QUATERNIONS. [S-- 
 
 52. The Conjugate of a quaternion q, written Kq, has the 
 same tensor, plane, and angle, only the angle is taken the reverse 
 way; or the versor of the conjugate is the reciprocal of the versor 
 of the quaternion, or (what comes to the same thing) the versor of 
 the reciprocal. 
 
 Thus, if OA, OB, OA', lie in one plane, and if 
 OA' = OA, and ZA'0B = Z BOA, we have 
 
 OB ,6b • ^ t v„ 
 
 -= = a, and = coniugate oi q = Kq. 
 
 OA ^ OA' ^ 
 
 By last section we see that 
 
 Kq = {Tqfq-\ j-,^^ ■ 
 
 Hence qKq = Kq.q=^{Tq)\ 
 
 This pi'oposition is obvious, if we recollect that 
 the tensors of q and Kq are equal, and that the 
 versors are such that either annuls the effect of the other ; while 
 the order of their application is indifferent. The joint effect of 
 these factors is therefore merely to multiply twice over by the 
 common tensor. 
 
 53. It is evident from the results of § 50 that, if a and /3 be 
 of equal length, they may be treated as of unit-length so far as 
 their quaternion quotient is concerned. This quotient is therefore 
 a versor (the tensor being unity) and may be represented indif- 
 ferently by any one of an infinite number of concurrent arcs of 
 given length lying on the circumference of a circle, of which the 
 two vectors are radii. This is of considerable importance in the 
 proofs which follow. 
 
 OB 
 Thus the versor ^= may be represented 
 
 in magnitude, plane, and currency of rota- 
 tion (§ 50) by the arc AB, which may in 
 this extended sense be written AB. 
 
 (Tb 
 
 And, similarly, the versor ^^ is repre- 
 sented by J. ,5,; which is equal to (and concurrent with)-4B if 
 
 ^AfiB^ = ^AOB, 
 
 i.e. if the versors are equal, in the quaternion meaning of the 
 word. 
 
55-1 PRODUCTS AND QUOTIENTS OF VECTORS. 37 
 
 54. By the aid of this process, when a versor is represented as 
 
 an arc of a great circle on the unit-sphere, we can easily prove 
 
 that quaternion multiplication is not generally commutative. 
 
 ^-^ CYR 
 
 Thus let q be the versor AB or , 
 
 OA 
 
 where is the centre of the sphere. 
 
 Take BC = AB, (which, it must be re- 
 membered, makes the points A, B, G, lie 
 in one great circle), then q may also be 
 
 00 
 represented by =^ . 
 
 In the same way any other versor r may be represented by 
 
 DB or BE and by -^ or = . 
 •' OD OB 
 
 [The line OB in the figure is definite, and is given by the 
 
 intersection of the planes of the two versors.] 
 
 Now rOI) = OB, and qOB=OC. 
 
 Hence qrOD = OC, 
 
 00 -"^ 
 
 or qr = ■= , and may therefore be represented by the arc BC of 
 
 a great circle. 
 
 But rq is easily seen to be represented by the arc AE. 
 
 For qOA=OB, and rOB=OE, 
 
 OE 
 
 whence rqOA = OE, and rq = ■= ■ 
 
 ^ ^ OA 
 
 Thus the versors rq and qr, though represented by arcs of equal 
 
 length, are not generally in the same plane and are therefore 
 
 unequal: unless the planes of q and r coincide. 
 
 Remark. We see that we have assumed, or defined, in the 
 
 above proof, that q .ra = qr.a and r .qoc = rq.a in the special case 
 
 when qa, ra, q . ra, and r . qa. are all vectors. 
 
 55. Obviously CB is Kq, BD is Kr, and CD is K (qr). But 
 GD = BI). CB, as we see by applying both to 00. This gives us 
 the very important theorem 
 
 K{qr) = Kr.Kq, 
 i.e. the conjagnte of the produot of two versors is the product of their 
 
38 QUATERNIONS. [S^- 
 
 conjugates in inverted order. This will, of course, be extended to 
 any number of factors as soon as we have proved the associative 
 property of multiplication. (§ 58 below.) 
 
 56. The propositions just proved are, of course, true of quater- 
 nions as well as of versors; for the former involve only an additional 
 numerical factor which has reference to the length merely, and not 
 the direction, of a vector (§ 48), and is therefore commutative with 
 all other factors. 
 
 57. Seeing thus that the commutative law does not in general 
 hold in the multiplication of quaternions, let us enquire whether 
 the Associative Law holds generally. That is if p, q, r be three 
 quaternions, have we 
 
 p . qr=pq. rl 
 
 This is, of course, obviously true if^, q, r be numerical quantities, 
 or even any of the imaginaries of algebra. But it cannot be con- 
 sidered as a truism for symbols which do not in general give 
 
 pq = qp. 
 We have assumed it, in definition, for the special case when r, 
 qr, and pqr are all vectors. (§ 54.) But we are not entitled to 
 assume any more than is absolutely required to make our 
 definitions complete. 
 
 58. In the first place we remark that p, q, and r may be 
 considered as versors only, and therefore represented by arcs of 
 great circles on the unit sphere, for their tensors may obviously 
 (| 48) be divided out from both sides, being commutative with the 
 versors. 
 
 Let AB=p, m) = 61 = q, And FE = r. 
 
 Join BG and produce the great circle till it meets JEF in H, and 
 
 make KH= FE = r, and HG = CB =pq (§ 54). 
 
 Join GK. Then jfo = HG . KH = pq . r. 
 
59*-J PRODUCTS AND QUOTIJBNTS OP VECTORS. 39 
 
 Join FD and produce it to meet AB in M. Make 
 LM=^FD, and MN = AB, 
 and join iVX. Then LN=MN . LM^p . qr. 
 
 Hence to shew that p .qr = pq.r 
 
 all that is requisite is to prove that LN, and KG, described as 
 above, are equal arcs of the same great circle, since, by the figure, 
 they have evidently similar currency. This is perhaps most easily 
 effected by the help of the fundamental properties of the curves 
 known as Spherical Conies. As they are not usually familiar to 
 students, we make a slight digression for the purpose of proving 
 these fundamental properties ; after Chasles, by whom and Magnus 
 they were discovered. An independent proof of the associative 
 principle will presently be indicated, and in Chapter VIII. we shall 
 employ quaternions to give an independent proof of the theorems 
 now to be established. 
 
 59.* Def. a spherical conic is the curve of intersection of a 
 cone of the second degree with a sphere, the vertex of the cone being 
 the centre of the sphere. 
 
 Lemma. If a cone have one series of circular sections, it has 
 another series, and .any two circles belonging to different series lie 
 on a sphere. This is easily proved as follows. 
 
 Describe a sphere. A, cutting the cone in one circular section, 
 G, and in any other point whatever, and let the side OpP of the 
 cone meet A m p, P; P being a point in G. Then PO . Op is 
 constant, and, therefore, since P lies in a plane, p lies on a sphere, 
 a, passing through 0. Hence the locus, c, of p is a circle, being 
 the intersection of the two spheres A and a. 
 
 Let OqQ be any other side of the cone, q and Q being points in 
 c, G respectively. Then the quadrilateral qQPp is inscribed in a 
 circle (that in which its plane cuts the sphere A) and the exterior 
 
40 QUATERNIONS. [6o. 
 
 angle at p is equal to the interior angle at Q. If OL, OM be the 
 lines in which the plane POQ cuts the cyclic planes (planes through 
 parallel to the two series of circular sections) they are obviously- 
 parallel to pq, QP, respectively ; and therefore 
 
 Z LOp = Z Opq = Z OQP = Z MOQ. 
 
 Let any third side, OrR, of the cone be drawn, and let the 
 plane OPE cut the cyclic planes in 01, Om respectively. Then, 
 evidently, 
 
 Z lOL = Z qpr, 
 zMOm= ZQPR, 
 
 and these angles are independent of the position of the points p 
 and P, if Q and R be fixed points. 
 
 In the annexed section of the above space-diagram by a sphere 
 whose centre is 0, IL, Mm are the great circles which represent 
 the cyclic planes, PQR is the spherical conic which represents the 
 cone. The point P represents the line OpP, and so with the 
 others. The propositions above may now be stated thus, 
 
 Arc PL = arc MQ ; 
 
 and, if Q and R be fixed. Mm and IL are constant arcs whatever be 
 the position of P. 
 
 60. The application to § 68 is now obvious. In the figure of 
 that article we have 
 
 FE = KH, ED = CA, HG = CB, LM=FD. 
 
 Hence L, G, G, D are points of a spherical conic whose cyclic 
 planes are those of AB, FE. Hence also KO passes through L, 
 
 and with LM intercepts on AB an arc equal to AB. That is, it 
 passes through iV, or KG and LN are arcs of the same great circle : 
 and they are equal, for G and L are points in the spherical 
 conic. 
 
62.] 
 
 PRODUCTS AND QUOTIENTS OF VECTORS. 
 
 41 
 
 Also, the associative principle holds for any number of 
 quaternion factors. For, obviously, 
 
 qr .st = qrs.t = &c., &c., 
 since we may consider qr as a single quaternion, and the above 
 proof applies directly. 
 
 61. That quaternion addition, and therefore also subtraction, 
 is commutative, it is easy to shew. 
 
 For if the planes of two quaternions, 
 q and r, intersect in the line OA, we 
 may take any vector OA in that line, 
 and at once find two others, OB and 
 OC, such that 
 
 6B = qOA, 
 and Cd = rOA. 
 
 And (q + r)OA = OB+OC=OG+OB=(r + q) OA, 
 
 since vector addition is commutative (§ 27). 
 
 Here it is obvious that (q + r) OA, being the diagonal of the 
 parallelogram on OB, OG, divides the angle between OB and DC 
 in a ratio depending solely on the ratio of the lengths of these 
 lines, i.e. on the ratio of the tensors of q and r. This will be useful 
 to us in the proof of the distributive law, to which we proceed. 
 
 62. Quaternion multiplication, and therefore division, is 
 distributive. One simple proof of this depends on the possibility, 
 shortly to be proved, of representing any quaternion as a linear 
 function of three given rectangular unit-vectors. And when the 
 proposition is thus established, the associative principle may readily 
 be deduced from it. 
 
 [But Hamilton seems not to have noticed that we may employ 
 for its proof the properties of Spherical Conies already employed 
 
42 QUATERNIONS. [62. 
 
 in demonstrating the truth of the associative principle. For 
 continuity we give an outline of the proof by this process. 
 
 Let BA, CA represent the versors of q and r, and be the great 
 circle whose plane is that ot p. 
 
 Then, if we take as operand the vector OA, it is obvious that 
 U (q + r) will be represented by some such arc as I)A where 
 B, D, C are in one great circle; for (q + r) OA is in the same plane 
 as qOA and rOA, and the relative magnitude of the arcs BD and 
 DC depends solely on the tensors of q and r. Produce BA, DA, 
 CA to meet be in b, d, c respectively, and make 
 
 M=^BA, Fd = DA, 0^c = GA. 
 
 Also make b^ = dB = c'Y=p. Then JE, F, G, A lie on a spherical 
 conic of which BO and be are the cyclic arcs. And, because 
 b/3 = dB = C'y, $JE, SF, <yO, when produced, meet in a point H 
 which is also on the spherical conic (§ 59*). Let these arcs meet 
 BG in J, L, K respectively. Then we have 
 
 fk^m^pUq, 
 LH=n=p U{q + r), 
 KH=G^^ = 'pUr. 
 Also LJ = DB, 
 
 and KL = CD. 
 
 And, on comparing the portions of the figure bounded respectively 
 by HKJ and by AGB we see that (when considered with reference 
 to their etfects as factors multiplying OH and OA respectively) 
 
 p U{q + r) bears the same relation to p Uq and p Ur 
 that U{q + r) bears to Uq and Ur. 
 
 But T{q + r)U(q + r) = q+r = TqUq+TrUr. 
 
 Hence T {q + r) .pU(q + r) = Tq .pUq + Tr .pUr ; 
 
 or, since the tensors are mere numbers and commutative with all 
 other factors, 
 
 p{q + r)=pq+pr. 
 In a similar manner it may be proved that 
 (q + r)p = qp + rp. 
 And then it follows at once that 
 
 (p + q) (r + s) =pr +ps + qr+ qs, 
 where, by § 61 , the order of the partial products is immaterial] 
 
66. J PRODUCTS AND QUOTIENTS OF VECTORS. 43 
 
 63. By similar processes to those of § 53 we see that versors, 
 and therefore also quatemious, are subject to the index-law 
 
 at least so long as m and n are positive integers. 
 
 The extension of this property to negative and fractional 
 exponents must be deferred until we have defined a negative or 
 fractional power of a quaternion. 
 
 64. We now proceed to the special case of quadrantal versors, 
 from whose properties it is easy to deduce all the foregoing 
 results of this chapter. It was, in fact, these properties whose 
 invention by Hamilton in 1843 led almost intuitively to the 
 establishment of the Quaternion Calculus. We shall content 
 ourselves at present with an assumption, which will be shewn 
 to lead to consistent results; but at the end of the chapter we 
 shall shew that no other assumption is possible, following for this 
 purpose a very curious quasi-metaphysical speculation of Hamilton. 
 
 65. Suppose we have a system of three mutually perpendicular 
 unit-vectors, drawn from one point, which we may call for shortness 
 i, j, k. Suppose also that these are so situated that a positive 
 (i.e. left-handed) rotation through a right angle about i as an axis 
 brings j to coincide with k. Then it is obvious that positive 
 quadrantal rotation about j will make k coincide with i; and, about 
 k, willmake i coincide with j. 
 
 For definiteness we may suppose i to be drawn eastwards, j 
 northwards, and k upwards. Then it is obvious that a positive 
 (left-handed) rotation about the eastward line (i) brings the north- 
 ward line (j) into a vertically upward position (k) ; and so of the 
 others. 
 
 66. Now the operator which turns j into k is a quadrantal 
 versor (| 53) ; and, as its axis is the vector i, we may call it i. 
 
 Thus - = i, or k = ij' (1). 
 
 Similarly we may put ^=j, or i = jk (2), 
 
 and 4 =A;, or j = /,i (3). 
 
 [It may be here noticed, merely to shew the symmetry of the 
 system we are explaining, that if the three mutually perpendicular 
 
44 
 
 QUATERNIONS. 
 
 [67. 
 
 vectors i, j, k be made to revolve about a line equally inclined to 
 all, so that i fe brought to coincide with j, j will then coincide 
 with k, and k with i : and the above equations will still hold good, 
 only (1) will become (2), (2) will become (3), and (3) will become 
 (1)-] 
 
 67. By the results of § 50 we see that 
 
 k j' 
 
 i.e. a southwai^ unit- vector bears :^he same ratio to an upward 
 unit-vector that the latter does to a northward one ; and therefore 
 we have 
 
 Similarly 
 and 
 
 T-' 
 
 or 
 
 -j=t-k 
 
 -k . 
 
 i -3' 
 
 or 
 
 -k=ji 
 
 t'-' 
 
 or 
 
 -i=^j. 
 
 ..(4). 
 
 ■(•5); 
 .(6). 
 
 .(7). 
 
 68. By (4) and (1) we have 
 
 — j = rk = i(ij) (by the assumption in § 54) = rj. 
 
 Hence ij' — — \ 
 
 And in the same way, (5) and (2) give 
 
 / = -l (8), 
 
 and (6) and (3) &' = -! (9). 
 
 Thus, as the directions of i, j, k are perfectly arbitrary, we see 
 that the square of every quadrantal versor is negative unity. 
 
 [Though the following proof is in principle exactly the same as 
 the foregoing, it may perhaps be of use to the student, in shewing 
 him precisely the nature as well as the simplicity of the step we 
 have taken. 
 
 Let ABA' be a semicircle, whose centre 
 is 0, and let OB be perpendicular to AOA'. 
 
 m, 05 . , , 
 
 Inen -=- , = q suppose, is a quadrantal 
 
 versor, and is evidently equal to ^^ ; 
 i 50, 53. 
 Hence 
 
 A' -a 
 
 ^^_0A/ OB _^0A' _ 
 
 ^^ ~ OB'OA~ OA ■-' 
 
69.] 
 
 PRODUCTS AND QUOTIENTS OF VECTORS. 
 
 45 
 
 69. Having thus found that the squares of i, j, k are each 
 equal to negative unity ; it only remains that we find the values of 
 their products two and two. For, as we shall see, the result is such 
 as to shew that the value of any other combination whatever of 
 i,j, k (as factors of a product) may be deduced from the values of 
 these squares and products. 
 
 Now it is obvious that 
 
 - i ~ k ~-^ ' 
 
 (i.e. the versor which turns a westward unit-vector into an upward 
 
 one will turn the upward into an eastward unit) ; 
 
 or k=j(-i) = -ji* (10). 
 
 Now let us operate on the two equal vectors in (10) by the 
 same versor, i, and we have 
 
 ik. = i {— ji) = — iji. 
 But by (4) and (3) 
 
 ik = — j = — ki. 
 
 Comparing these equations, we have 
 -iji = -ki; 
 or, by § 54 (end), ij = k,\ 
 
 and symmetry gives jk = i,\ (11). 
 
 ki =j. J 
 
 The meaning of these important equations is very simple ; and 
 is, in fact, obvious from our construction in § 54 for the multiplica- 
 tion of versors ; as we see by the annexed figure, where we must 
 remember that i, j, k are quadrantal versors whose planes are at 
 right angles, so that the figure represents 
 a hemisphere divided into quadrantal 
 triangles. [The arrow-heads indicate the 
 direction of each vector arc] 
 
 Thus, to shew that ij = k, we have, 
 being the centre of the sphere, N, E, 
 8, W the north, east, south, and west, 
 and Zthe zenith (as in § 66) ; 
 
 j6W=0Z, S 
 
 whence y6W = iOZ=0S = kOW. 
 
 * The negative sign, being a mere numerical factor, is evidently commutative 
 with ./ ; indeed we may, if necessary, easily assure ourselves of the fact that to turn 
 the negative (or reverse) of a vector through a right (or indeed any) angle, is the 
 same thing as to turn the vector through that angle and then reverse it. 
 
 N 
 
 W 
 
 /k 
 
 i 
 
 A\ 
 
 \ 
 
 Z 
 
 i 
 
 j 
 
 ^ 
 
 
 J" 
 
40 QUATERNIONS. [/O. 
 
 70. But, by the same figure, 
 
 i6N=0Z, 
 whence jiON=jOZ=OE = -6W =-k6N. 
 
 71. From this it appears that 
 
 and similarly kj=—i, i- (12) 
 
 ik = -j, j 
 and thus, by comparing (11), 
 
 jk^-kj=d ((11), (12)). 
 
 ki = — ik=j,j 
 
 These equations, along with 
 
 {'=f=k' = -l ((7), (8), (9)), 
 
 contain essentially the whole of Quaternions. But it is easy to see 
 that, for the first group, we may substitute the single equation 
 
 ijk = -l, (13) 
 
 since from it, by the help of the values of the squares of i,j, k, all 
 the other expressions may be deduced. We may consider it proved 
 in this way, or deduce it afresh from the figure above, thus 
 
 kON= 6W, 
 
 jkON= J0W=6Z, 
 
 ijkON = ijOW = iOZ = 08 = - 6W. 
 
 72. One most important step remains to be made, to wit the 
 assumption referred to in | 64. We have treated i, j, k simply as 
 quadrantal versors ; and i, j, k as unit-vectors at right angles to 
 each other, and coinciding with the axes of rotation of these versors. 
 But if we collate and compare the equations just proved we have 
 
 (^^=-1. (7) 
 
 li^ = -l. (§9) 
 
 [y= !•; (11) 
 
 \ii= k (1) 
 
 \ji = -k (12) 
 
 b'i = -k, (5) 
 
 with the other similar groups symmetrically derived from them. 
 
75-J PEODUCTS AND QUOTIENTS OF VECTORS. 47 
 
 Now the meanings we have assigned to i, j, k are quite inde- 
 pendent of, and not inconsistent with, those assigned to i, j, k. 
 And it is superfluous to use two sets of characters when one will 
 suffice. Hence it appears that i, j, k may be substituted for i, j, k ; 
 in other words, a unit-vector when employed as a factor may be con- 
 sidered as a quadrantal versor whose plane is perpendicular to the 
 vector. (Of course it follows that every vector can be treated as the 
 product of a ■ number and a quadrantal versor.) This is one of the 
 main elements of the singular simplicity of the quaternion calculus. 
 
 73. Thus the product, and therefore the quotient, of two perpen- 
 dicular vectors is a third vector perpendicular to both. 
 
 Hence the reciprocal (§ 51) of a vector is a vector which has 
 the opposite direction to that of the vector, and its length is the 
 reciprocal of the length of the vector. 
 
 The conjugate (§ 52) of a vector is simply the vector reversed. 
 
 HencOj by § 52, if a be a vector 
 
 (Tay = aKoc = a (-«)=- a". 
 
 74. We may now see that every versor may be represented by 
 a power of a unit-vector. 
 
 For, if a be any vector perpendicular to i (which is any definite 
 unit-vector), 
 
 ia, = /3, is a vector equal in length to a, but perpendicular to 
 both i and a ; 
 
 i^a = — a, 
 
 i^a = — ia = — ^, 
 
 i*a= — i^ — — i'a = a. 
 Thus, by successive applications of i, a is turned round i as an axis 
 through successive right angles. Hence it is natural to define i" as 
 a versor which turns any vector perpendicular to i through m right 
 angles in the positive direction of rotation about i as an axis. Here 
 m may have any real value whatever, whole or fractional, for it is 
 easily seen that analogy leads us to interpret a negative value of m 
 as corresponding to rotation in the negative direction. 
 
 75. From this again it follows that any quaternion may be 
 expressed as a power of a vector. For the tensor and versor 
 elements of the vector may be so chosen that, when raised to the 
 same power, the one may be the tensor and the other the versor 
 of the given quaternion. The vector must be, of course, perpen- 
 dicular to the plane of the quaternion. 
 
48 
 
 QUATERNIONS. 
 
 [76. 
 
 76. And we now see, as an immediate result of the last two 
 sections, that the index-law holds with regard to powers of a 
 quaternion (§ 63). 
 
 77. So far as we have yet considered it, a quaternion has been 
 regarded as the product of a tensor and a versor : we are now to 
 consider it as a sum. The easiest method of so analysing it seems 
 to be the following. 
 
 Let ^= represent any quaternion. 
 
 Draw 
 
 Hence 
 
 = a; + 7. 
 
 BG perpendicular to OA, produced if neces- 
 sary. 
 
 Then, §19, OB=OG + GB. 
 
 But, §22, OG = xOA, 
 
 where a; is a number, whose sign is the same 
 as that of the cosine oi Z.A OB. 
 
 Also, § 73, since CB is perpendicular to OA , 
 
 06 = 702. 
 
 where 7 is a vector perpendicular to OA and GB, i.e. to the plane 
 of the quaternion; and, as the figure is drawn, directed towards the 
 reader. 
 
 OB_ xOA+r^ OA 
 
 OA" OA 
 
 Thus a quaternion, in general, may be decomposed into the sum 
 of two parts, one numerical, the other a vector. Hamilton calls 
 them the scalar, and the vectoe, and denotes them respectively 
 by the letters 8 and V prefixed to the expression for the 
 quaternion. 
 
 78. Hence q = Sq+ Vq, and if in the above example 
 
 OB 
 
 then 6B=0C+CB = Sq.0A + Vq.0A*. 
 
 The equation above gives 
 
 0G = 8q.0A, 
 CB=Vq.OA. 
 
 * The points are inserted to shew that S and F apply only to q, and not to qOA. 
 
8o.] PRODUCTS AND QUOTIENTS OF VECTORS. 49 
 
 79. If, in the last figure, we produce BG to D, so as to double 
 its length, and join OD, we have, by § 52, 
 
 ^ = Kq=^SKq+VKq; 
 
 so that 6D = 0C+Cl) = SKq.0A + VKq.0A. 
 
 Hence 0G = 8Kq.6A, 
 
 and CB=VKq.OA. 
 
 Comparing this value of OC with that in last section, we find 
 
 SKq = Sq, (1) 
 
 or the scalar of the conjugate of a quaternion is equal to the scalar of 
 the quaternion. 
 
 Again, GD = -GB by the figure, and the substitution of their 
 values gives 
 
 VKq = ~Vq, (2) 
 
 or the vector of the conjugate of a quaternion is the vector of the 
 quaternion reversed. 
 
 We may remark that the results of this section are simple con- 
 sequences of the fact that the symbols 8, V, K are commutative* 
 
 Thus SKq = KSq = Sq, 
 
 since the conjugate of a number is the number itself; and 
 
 VKq^KVq=-Vq{^ 73). 
 Again, it is obvious that, 
 
 tSq = Stq, 'SVq=Vlq, 
 and thence l,Kq = Klq. 
 
 80. . Since any vector whatever may be represented by 
 
 xi + yj + zk 
 
 where x, y, z are numbers (or Scalars), and i, j, k may be any three 
 non-coplanar vectors, §§ 23, 25 — though they are usually under- 
 stood as representing a rectangular system of unit-vectors — and 
 
 * It is curious to compare the properties of these quaternion symbols with those 
 of the Elective Symbols of Logic, as given in Boole's wonderful treatise on the 
 Laws of Thought; and to think that the same grand science of mathematical 
 analysis, by processes remarkably similar to each other, reveals to us truths in the 
 science of position far beyond the powers of the geometer, and truths of deductive 
 reasoning to which unaided thought could never have led the logician. 
 
 T. Q. t. 4 
 
50 QUATERNIONS. [8 1. 
 
 since any scalar may be denoted by w; we may write, for any 
 quaternion q, the expression 
 
 q^w + xi -\- yj + zk (^ 78). 
 
 Here we have the essential 'dependence on four distinct numbers, 
 from which the quaternion derives its name, exhibited in the most 
 simple form. 
 
 And now we see at once that an equation such as 
 
 where q' = 'w' + x'i + y'j + sfh, 
 
 involves, of course, the fovx equations 
 
 w' = w, x' = x, y' = y, z' = z. 
 
 81. We proceed to indicate another mode of proof of the dis- 
 tributive law of multiplication. 
 
 We have already defined, or assumed (§ 61), that 
 
 a ' a a ' 
 
 or ;8a"' + 7a"' = (/3 + 7) or\ 
 
 and have thus been able to understand what is meant by adding 
 two quaternions. 
 
 But, writing a for a~', we see that this involves the equality 
 
 (^ + 7) a = ;8a + 7a ; 
 
 from which, by taking the conjugates of both sides, we derive 
 
 a'('/S' + 7') = «'/S' + «V(§55)- 
 And a combination of these results (putting /3-f-7 for a' in the 
 latter, for instance) gives 
 
 (/S + 7) (^' + 7') = (/8 + 7) /3' + (/3 + 7) 7' 
 
 = /8/3' + 7/S' + /37' + 77' by the former. 
 
 Hence the distributive principle is true in the multiplication of 
 vectors. 
 
 It only remains to shew that it is true as to the scalar and 
 vector parts of a quaternion, and then we shall easily attain the 
 general proof. 
 
 Now, if a be any scalar, a any vector, and q any quaternion, 
 
 {a + a)q = aq + aq. 
 For, if /3 be the vector in which the plane of q is intersected by 
 
82. J PEODUCTS AND QUOTIENTS OF VECTORS. 51 
 
 a plane perpendicular to a, we can find other two vectors, 7 and S, 
 one in each of these planes such that 
 
 7 /3 
 
 0,3 
 And, of course, a may be written -~ ; so that 
 
 -ag + g_ag + -.^ 
 
 = aq+ aq. 
 And the conjugate nnay be written 
 
 q' (a' + a') = q'a+q'a (I 55). 
 Hence, generally, 
 
 (a + a)(b + ^) = ab + a^ + ba + al3; 
 or, breaking up a and b each into the sum of two scalars, and a, /3 
 each into the sum of two vectors, 
 
 («. + «,+«. + «,) (K + h+^. + /S,) 
 
 = (a, + a,) {h + h) + («. + «») (/8, + /S,) + iK + &.) («, + «.) 
 
 (by what precedes, all the factors on the right are distributive, so 
 that wo may easily put it in the form) 
 
 = (a. + «,) {K + A) + (a. + «.) (^. + ^.) + (c^8 + «.) (^ + A) 
 
 + K + a2)(6, + /3J. 
 Putting a^ + a^=p, «, + «,, = 9, 6i + /8i = r, 6, + /3, = s, 
 we have {p-\-q) {r-'rs)=pr+ps + qr-\-qs. 
 
 82. Cayley suggests that the laws of quaternion multiplication 
 may be derived more directly from those of vector multiplication, 
 supposed to be already established. Thus, let a be the unit vector 
 perpendicular to the vector parts of q and of q. Then let 
 
 p = q.a, a- = -a.q', 
 as is evidently permissible, and we have 
 
 pa = q.aa = — q; a(T = — aa.q' = q', 
 so that — q . q' = pa . a<T = — p . a. 
 
 The student may easily extend this process. 
 For variety, we shall now for a time forsake the geometrical 
 mode of proof we have hitherto adopted, and deduce some of our 
 
 4—2 
 
52 QUATERNIONS. [83. 
 
 next steps from the analytical expression for a quaternion given in 
 § 80, and the properties of a rectangular system of unit-vectors as 
 in § 71. 
 
 We will commence by proving the result of § 77 anew. 
 
 83. Let a = xi + yj + zk, 
 
 ^ = oc'i + y'j + z'k. ' 
 
 Then, because by § 71 every product or quotient of i,j, k is reducible 
 to one of them or to a number, we are entitled to assume 
 
 B 
 q = -=oi + b + m + ^k, 
 
 where &>, ^, 7^, f are numbers. This is the proposition of § 80. 
 
 [Of course, with this expression for a quaternion, there is no 
 necessity for a formal proof of such equations as 
 
 p + {q+r) = {'p + q) + r, 
 
 where the various sums are to be interpreted as in § 61. 
 
 All such things become obvious in view of the properties of 
 
 i, j, k-] 
 
 84. But it may be interesting to find eo, |, 77, ^ in terms of 
 
 •'■. y, z, «'; y', z'- 
 
 We have /3 = qa., 
 
 or x'i + y'j + z'k = {a> + ^i + r]j + ^k) {cci + yj + zk) 
 
 = — {^«> + Vy + K^) + (cox +riz —^y)i+((oy+ ^—^z)j+(coz+^y—r]x)k, 
 
 as we easily see by the expressions for the powers and products of 
 i,j, k, given in § 71. But the student must pay particular attention 
 to the order of the factors, else he is certain to make mistakes. 
 This (§ 80) resolves itself into the four equations 
 
 = ^x + r)y + ^, 
 
 x' = cox +7)Z — ^y, 
 
 y' = (oy-^z + ^x, 
 
 z' = (cz + ^y — r)x. 
 The three last equations give 
 
 xx + yy' + zz' = (o («" + y' + z''), 
 which determines w. 
 
 Also we have, from the same three, by the help of the first, 
 lx'+riy' + ^z' = 0; 
 
8;.] PRODUCTS AND QUOTIENTS OP VECTOES. 53 
 
 which, combined with the first, gives 
 
 g _ ^ ^ r_ 
 
 yz' — zy zx' — xz' xy' — yx ' 
 
 and the common value of these three fractions is then easily seen 
 to be 
 
 1 
 
 x' + y' + z'- 
 
 It is easy enough to interpret these expressions by means of 
 ordinary coordinate geometry: but a much simpler process will 
 be furnished by quaternions themselves in the next chapter, and, in 
 giving it, we shall refer back to this section. 
 
 85, The associative law of multiplication is now to be proved 
 by means of the distributive (| 81). We leave the proof to the 
 student. He has merely to multiply together the factors 
 
 w + xi + yj + zk, w + x'i + y'j + z'k, and w" + x'i + y"j + z"k, 
 
 as follows : — 
 
 First, multiply the third factor by the second, and then multiply 
 the product by the first ; next, multiply the second factor by the 
 first and employ the product to multiply the third: always re- 
 membering that the multiplier in any product is placed before the 
 multiplicand. He will find the scalar parts and the coefiicients of 
 i,j, k, in these products, respectively equal, each to each. 
 
 86. With the same expressions for a, /3, as in section 83, we 
 have 
 
 a/3 = (xi + yj + zk) (x'i + y'j + z'k) 
 = — (xx'+yy' + zz') + {yz — zy) i + [zx' — xz')j + (xy' — yx') k. 
 But we have also 
 
 /3a = - {xx + yy' + zz) - {yz - zy) i - {zx' - xz')j - {xy - yx) k. 
 The only difference is in the sign of the vector parts. 
 
 Hence 8a/3 = 80a, (1) 
 
 Vafi = -V0a, (2) 
 
 also a/3 + 0oL = 2Sa0, (3) 
 
 a^-/3a = 2Fa/3, (4) 
 
 and, finally, by § 79, a/3 = K.0oi (5). 
 
 87. It a = ^ we have of course (§ 25) 
 
 x = x', y = y', z = ^'' 
 
54 QUATERNIONS. [88. 
 
 and the formulae of last section become 
 
 a/3 = ^a = a^ = -(a;'' + / + ^^); 
 
 which was anticipated in § 73, where we proved the formula 
 
 {ToLr = -a\ 
 and also, to a certain extent, in § 25. 
 
 88. Now let q and r be any quaternions, then 
 
 S.qr = S.{Sq+Vq) {Sr + Vr), 
 
 = S. (Sq 8r + 8r.Vq + 8q.Vr+ VqVr), 
 
 = 8qSr + S. VqVr, 
 
 since the two middle terms are vectors. 
 
 Similarly, 8.rq = Sr8q + 8 . Vr Vq. 
 
 Hence, since by (1) of § 86 we have 
 
 S.VqVr^S.VrVq, 
 
 we see that S.qr = 8.rq, (1) 
 
 a formula of considerable importance. 
 
 It may easily be extended to any number of quaternions, 
 because, r being arbitrary, we may put for it rs. Thus we have 
 
 8 . qrs = 8 . rsq, 
 
 = 8 . sqr 
 
 by a second application of the process. In words, we have the 
 theorem — the scalar of the product of any number of given 
 quaternions depends only upon the cyclical order in which they are 
 arranged. 
 
 89. An important case is that of three factors, each a vector. 
 The formula then becomes 
 
 /S'.a/37 = /Sf./37« = ^.7a/3. 
 But 8.a^y = 8oi (8^y + V^y) 
 
 = 8a V^y, since a8^y is a vector, 
 = -8oLVy^, by (2) of §86, 
 = -8<x{8y^+Vy^) 
 = ~8.ay^. 
 
 Hence the scalar of the product of three vectors changes sign when 
 the cyclical order is altered. 
 
go. J PRODUCTS AND QUOTIENTS OF VECTORS. 55 
 
 By the results of §§ 55, 73, 79 we see that, for any number 
 of vectors, we have 
 
 ^.a^7...(^X=±%0---7^« 
 (the positive sign belonging to the product of an even number of 
 vectors) so that 
 
 /S.a/3...^;^= + >S.;y;(^.../3a. 
 Similarly 
 
 V. a/3... </)%=+ F.%(/)...^a. 
 Thus we may generalize (3) and (4) of § 86 into 
 
 2S. a/3 ... <^X = «/3 ... <^X + %</>••• /3«, 
 2V. a^... (px = «^--- <f>X + Xi> ••• /3«. 
 the upper sign still being used when the number of factors is 
 even. 
 
 Other curious propositions connected with this will be given 
 later (some, indeed, will be found in the Examples appended to 
 this chapter), as we wish to develop the really fundamental 
 formulae in as compact a form as possible. 
 
 90. By (4) of § 86, 
 
 Hence 2F. aF/37= F. a(/S7-7;8) 
 
 (by multiplying both by a, and taking the vector parts of each 
 
 side) 
 
 = F (a/37 + /3a7 — ^'"■J - «7/3) 
 (by introducing the null term ;8a7 — ^ay)- 
 That is 
 2 F. aF/37 = F. (a/3 + ;Sa) 7 - F {^Say + ^Vaj + Say . 13 + Yay . ^) 
 
 = F.(2>Sfa|8)7-2F^&7 
 (if we notice that F (Fa7 . ;8) = - F . ^Va.y, by (2) of § 86). 
 
 Hence V . aV ^y =- ySoi^ - ^Sya (1), 
 
 a formula of constant occurrence. 
 
 Adding aS^y to both sides, we get another most valuable 
 
 formula 
 
 V.a^y=aS^y-^8y(x + ySa^ (2); 
 
 and the form of this shews that we may interchange 7 and a. 
 without altering the right-hand member. This gives 
 
 F.a/87 = F.7/Sa, 
 a formula which may be greatly extended. (See § 89, above.) 
 
5G QUATERNIONS. [9 1. 
 
 Another simple mode of establishing (2) is as follows :— 
 K . a^y = - y/3oi, 
 .-. 2 F. a/97 =a^y-K. a^y (by § 79 (2)) 
 = a/37 + 7^°^ 
 = a {/3y + j/3) - (ay + ja) ^ + y{a^ + ^a) 
 
 = 2aS/3y - 2l38ay + 2y8ix^. 
 
 91. We have also 
 
 VVa^VyS = - VVyBVajS by (2) of § 86 : 
 
 = BSyVa^ - ySB Va/3 = BS . a^y - yS . a/3S 
 
 = - 13 Sol VyB + aS^ VyB = -^S.ayS + a8. ^yB, 
 
 all of these being arrived at by the help of § 90 (1) and of § 89 ; 
 and by treating alternately Fa/3 and F7S as simple vectors. 
 Equating two of these values, we have 
 
 BS.al3y = aS.^yB + ^S.yaB+yS.al3B (3), 
 
 a very useful formula, expressing any vector whatever in terms 
 of three given vectors. [This, of course, presupposes that a, 0, y 
 are not coplanar, § 23. In fact, if they be coplanar, the factor 
 /S. a/37 vanishes, and thus (3) does not give an expression for B. 
 This will be shewn in § 101 below.] 
 
 92. That such an expression as (3) is possible we knew already 
 by § 23. For variety we may seek another expression of a similar 
 character, by a process which differs entirely from that employed 
 in last section. 
 
 a, /3, 7 being any three non-coplanar vectors, we may derive 
 from them three others Fa/3, V^y, Vyix ; and, as these will not be 
 coplanar, any other vector B may be expressed as the sum of the 
 three, each multiplied by some scalar. It is required to find this 
 expression for b. 
 
 Let B = xVa0 + yV^y + zVya. 
 
 Then SyB = x8 . ya^ = xS . 0^87, 
 
 the terms in y and z going out, because 
 
 Sy F/87 = S . 7/S7 = S^y' = y'S0 = 0, 
 for 7^* is (I 73) a number. 
 
 Similarly 80B — z8 . ^ya = z8 . aj3y, 
 and 8aB = y8 . a^y. 
 Thus BS . a/37 = Va/38yB + V^ySaB + VyaS^B (4). 
 
93-] PEODUCTS AND QUOTIENTS OF VECTORS. 57 
 
 93. We conclude the chapter by shewing (as promised in § 64) 
 that the assumption that the product of two parallel vectors is 
 a number, and the product of two perpendicular vectors a third 
 vector perpendicular to both, is not only useful and convenient, 
 but absolutely inevitable, if our system is to deal indifferently with 
 all directions in space. We abridge Hamilton's reasoning. 
 
 Suppose that there is no direction in space pre-eminent, and 
 that the product of two vectors is something which has quantity, 
 so as to vary in amount if the factors are changed, and to have its 
 sign changed if that of one of them is reversed ; if the vectors be 
 parallel, their product cannot be, in whole or in part, a vector 
 inclined to them, for there is nothing to determine the direction in 
 which it must lie. It cannot be a vector parallel to them ; for by 
 changing the signs of both factors the product is unchanged, 
 whereas, as the whole system has been reversed, the product 
 vector ought to have been reversed. Hence it must be a number. 
 Again, the product of two perpendicular vectors cannot be wholly 
 or partly a number, because on inverting one of them the sign of 
 that number ought to change ; but inverting one of them is simply 
 equivalent to a rotation through two right angles about the other, 
 and (from the symmetry of space) ought to leave the number 
 unchanged. Hence the product of two perpendicular vectors must 
 be a vector, and a simple extension of the same reasoning shews 
 that it must be perpendicular to each of the factors. It is easy to 
 carry this farther, but enough has been said to shew the character 
 of the reasoning. 
 
 EXAMPLES TO CHAPTER II. 
 
 ^ 1. It is obvious from the properties of polar triangles that any 
 mode of representing versors by the sides of a spherical triangle 
 must have an equivalent statement in which they are represented 
 by angles in the polar triangle. 
 
 Shew directly that the product of two versors represented 
 by two angles of a spherical triangle is a third versor represented 
 by the supplement of the remaining angle of the triangle ; and 
 determine the rule which connects the directions in which these 
 angles are to be measured. 
 
58 QUATERNIONS. 
 
 2. Hence derive another proof that we have not generally 
 
 pq = qp. 
 
 3. Hence shew that the proof of the associative principle, 
 § 57, may be made to depend upon the fact that if from any point 
 of the sphere tangent arcs be drawn to a spherical conic, and also 
 arcs to the foci, the inclination of either tangent arc to one of the 
 focal arcs is equal to that of the other tangent arc to the other 
 focal arc. 
 
 ^ 4. Prove the formulae 
 
 2,Sf.a;87=a/S7-7^a, 
 2 V . a^y = a^y + y/Ba. 
 
 ^ 5. Shew that, whatever odd number of vectors be represented 
 by a, j8, 7, &c., we have always 
 
 V.a^ySe=V.6Sy^cx, 
 V . a^'ySe^7j= V. rj^ehy^a, &c. 
 ^ 6. Shew that 
 
 S . VoL^ Vpy VyoL = -(S. ot^yf, 
 
 V . Va^ F/37 ry(x= Fa/3 (y'Sa^ 4 8/3ySya) + , ^ | 
 
 and F ( Fa/3 F . F/37 F7a) = i/^Soiy - aSl3y) S . 0,87. 
 
 ^ 7. If a, /S, 7 be any vectors at right angles to each other, shew 
 that 
 
 (a» + 0' + y^)S. a/3y = a* V^y + /3'Vya + y* Va/3. 
 (a'"-' + /3^»-' + 7^"-^ ,Sf . a/37 = «'" Wy + ^^ F7a + 7^" Fa/3. 
 
 8. If a, /8, 7 be non-coplanar vectors, find the relations among 
 the six scalars, x, y, z and ^, 1;, f, which are implied in the 
 equation ccoL + y^ + sy = ^V^y + rj Yyix + f Fa^. 
 
 ^ 9. If a, /3, 7 be any three non-coplanar vectors, express any 
 fourth vector, S, as a linear function of each of the following sets of 
 three derived vectors, 
 
 V.ya^, V.a^y, V . I3y<x, 
 and F. Fa^F/37F7a, F. F;87F7aFa;S, F. F7aFa;SF/37. 
 vj 10. Eliminate p from the equations 
 
 Sap = a, S^p = b, Syp = c, SSp = d, 
 where a, ^, 7, B are vectors, and a, b, c, d scalars. 
 
 ' 11. In any quadrilateral, plane or gauche, the sum of the 
 squares of the diagonals is double the sum of the squares of the 
 lines joining the middle points of opposite sides. 
 
CHAPTER III. 
 
 INTEEPRETATIONS AND TRANSFORMATIONS OP 
 QUATERNION EXPRESSIONS. 
 
 94. Among the most useful characteristics of the Calculus of 
 Quaternions, the ease of interpreting its formulae geometrically, 
 and the extraordinary variety of transformations of which the 
 simplest expressions are susceptible, deserve a prominent place. 
 We devote this Chapter to some of the more simple of these, 
 together with a few of somewhat more complex character but of 
 constant occurrence in geometrical and physical investigations. 
 Others will appear in every succeeding Chapter. It is here, 
 perhaps, that the student is likely to feel most strongly the 
 peculiar difficulties of the new Calculus. But on that very account 
 he should endeavour to master them, for the variety of forms 
 which any one formula may assume, though puzzling to the 
 beginner, is of the utmost advantage to the advanced student, not 
 alone as aiding him in the solution of complex questions, but 
 as affording an invaluable mental discipline. 
 
 95. If we refer again to the figure of § 77 we see that 
 
 00 = OB cos AOB, 
 
 OB = OB sin AOB. 
 
 Hence, if Ol = a, OB = ^, and ^ AOB = 6, we have 
 
 OB=W, OA = Ta, 
 
 00 = W cos e, OB = W sin 0. 
 
 „ ^0 00 W a 
 
 Hence -^a = OZ = f^ ^°'^- 
 
 Similarly ^^?= S = S'^''^- 
 
60 QUATERNIONS. [96. 
 
 Hence, if j; be a unit- vector perpendicular to a and ^, and such 
 that positive rotation about it, through the angle 0, turns a 
 
 towards /3, or 
 
 urn ^,GB „„j8 
 
 »? = 
 
 = U^^ = UV^ 
 
 UOA OA 
 
 6 TB 
 we have V-=^-~ sind .rj. (See, again, § 84.) 
 
 96. In the same way, or by putting 
 = /Sf/3a - V^a, 
 
 8--V 
 a a. 
 
 \ a. a/ 
 
 we may shew that 
 
 
 /Sa/3 = -TolT^ cor 6, 
 
 
 TVal3 = Toi Tj3 sin 6, 
 
 and 
 
 VolB = Ta T/3 BmO.rj 
 
 where 
 
 V = UVa/S =U(- F"/3a) =UV^ 
 
 Thus the scalar of the product of two vectors is the continued 
 product of their tensors and of the cosine of the supplement of the 
 contained angle. 
 
 The tensor of the vector of the product of two vectors is the con- 
 tinued product of their tensors and the sine of the contained angle ; 
 and the versor of the same is a unit-vector perpendicidar to both, 
 and such that the rotation about it from the first vector (i. e. the 
 multiplier) to the second is left-handed or positive. 
 
 Hence also TVa^ is doxible the area of the triangle two of whose 
 sides are a, ^. 
 
 97. (a) In any plane triangle ABG we have 
 , AG=AB + BG. 
 
 \ Hence AC' = S .AG AG = 8 .AG{AB + BG). 
 
 With the usual notation for a plane triangle the interpretation 
 of this formula is 
 
 — 6" = — be cos A — ab cos G, 
 or b= acos (7 + ccos .4. 
 
99- J INTERPRETATIONS AND TRANSFORMATIONS. 61 
 
 (6) Again we have, obviously, 
 
 V.ABAC=V.AB(AB + BG) 
 = r.ABBC, 
 or cb sin A =ca sin B, 
 
 , sin A sin B sin G 
 
 whence = — = — = . 
 
 a c 
 
 These are truths, but not truisms, as we might have been led 
 to fancy from the excessive simplicity of the process employed. 
 
 98. From § 96 it follows that, if a and /3 be both actual (i. e. 
 real and non-evanescent) vectors, the equation 
 
 Saj3 = 
 
 shews that cos ^ = 0, or that a is perpendicular to /3. And, in fact, 
 we know already that the product of two perpendicular vectors is 
 a vector. ap,- Soti-^-VcL^ = V'of^s ./)A(2i^ ir=^/i= a /.t cul.^<nA-,3. 
 
 Again, if Fa^S = 0, 
 
 we must have sin ^ = 0, or a is parallel to ,8. We know already 
 that the product of two parallel vectors is a scalar. a(A- id^-i-Vdfi 
 
 Hence we see that ' , q. / 
 
 SOL^ = ' 
 
 is equivalent to a = F7/8, 
 
 where 7 is an undetermined vector ; and that 
 
 FayS = 
 is equivalent to a = x^, 
 
 where x is an undetermined scalar. 
 
 99. If we write, as in §§ 83, 84, 
 
 a = ix -hjy +kz, 
 /3 = ix' +jy' + Icz, 
 we have, at once, by § 86, 
 
 (Sa/3 = — XX - yy' — zz 
 
 , Ix X y ?/' z z'\ 
 = -rr' -- + ^S^--- 
 \r r r r r r J 
 
 where r = Ja? + f + z\ r = J x" + y" + z'\ 
 
 Also Va^ = rr' j^-^^ i + '^^~'-j + '-^^^ k 
 
 /- rr rr rr 
 
62 QUATERNIONS. [lOO. 
 
 These express in Cartesian coordinates the propositions we have 
 just proved. In commencing the subject it may perhaps assist 
 the student to see these more familiar forms for the quaternion 
 expressions ; and he will doubtless be induced by their appearance 
 to prosecute the subject, since he cannot fail even at this stage to 
 see how much more simple the quaternion expressions are than 
 those to which he has been accustomed. 
 
 100. The expression S . affy 
 may be written SV{oi^) y, 
 
 because the quaternion aj3y may be broken up into 
 
 S{a^)y+r{al3)y 
 of which the first term is a vector. 
 But, by § 96, 
 
 SV (a/3) y==TaW sin 6 Sjjy. 
 Here Tr; = 1, let ^ be the angle between r] and y, then finally 
 8.a^y = -Ta T/S IVsin 0cos^. 
 
 But as 7} is perpendicular to a and /3, Ty cos is the length of the 
 perpendicular from the extremity of 7 upon the plane of a, /3. And 
 as the product of the other three factors is (§ 96) the area of the 
 parallelogram two of whose sides are a, /S, we see that the mag- 
 nitude of S . «/3y, independent of its sign, is the volume of the 
 parallelepiped of which three coordinate edges are a.,^, y: or six 
 times the volume of the pyramid which has a, ^, y for edges. 
 
 101. Hence the equation 
 
 8.a0y = O, 
 
 if we suppose a, /3, 7 to be actual vectors, shews either that 
 
 sin = 0, 
 
 or cos (f) = 0, 
 
 i. e. two of the three vectors are parallel, or all three are parallel to 
 one plane. 
 
 This is consistent with previous results, for if 7 =p^ we have 
 
 S.a^y = pS.aj3'=:0; 
 
 and, if 7 be coplanar with a, y8, we have 7 =pa + q^, and 
 
 S.a^y = S.a/3 (pa + q/3) = 0. 
 
103-] INTEEPRETATIONS AND TRANSFORMATIONS. 63 
 
 102. This property of the expression S . a^y prepares us to 
 find that it is a determinant. And, in fact, if we take a, /3 as in 
 § 83, and in addition 7 = ix" +jy" + hz", 
 we have at once 
 
 s . a/37 ~ ~ "'" (y^ ~ ^y') ~ y" C-^^' ~ ^■^') ~ ^" i^y' ~ y"')- 
 
 X 
 
 y 
 
 z 
 
 x' 
 
 y' 
 
 z' 
 
 n 
 
 ^.'1 
 
 -." 
 
 X 
 
 y 
 
 z 
 
 The determinant changes sign if we make any two rows change 
 places. This is the proposition we met with before (§ 89) in the 
 form (S . ayS7 — — S. ^ay = S . ^yot, &c. 
 
 If we take three new vectors 
 
 a^ = ix +jx' + kx", 
 
 ^i = iy+jy' + h"> 
 
 7, = iz + j/ + kz", 
 we thus see that they are coplanar if a, /S, 7 are so. That is, if 
 
 S. 0^87=0, 
 then (Sf.a,(S,7, = 0. 
 
 103. We -have, by § 52, 
 
 (Tqy = qKq = (Sq + Vq) (Sq - Vq) (§ 79), 
 = {SqY - ( VqY by algebra, 
 = {SqY + (TVqy (173). 
 If 5 = a/3, we have Kq = /8a, and the formula becomes 
 
 a/3 . /3a = a»/3^ = (SaiSy - ( Fa/3)^ 
 In Cartesian coordinates this is 
 (x' + f + z')(x" + y"+z") 
 
 = (xaf + yy' + zzj + {yz' - zyj + {zx' - xzj + {xy' - yxj. 
 More generally we have 
 
 (%r))^ = (rgr(7'rr 
 
 = {B.qry-{V.qT)\ 
 If we write q = w -\-a=w + ix-\-jy-\-kz, 
 
 r = w'-^^ = w' + ix' +jy' + kz'; 
 this becomes 
 
 (w' -^x' + f + ^') («'" + a;'" + y" + ^") 
 
 = {ww' - xx' - yy' - zz'f + {wx' + w'x + yz' - zy'J 
 + {wy + w'y + zx - xz'f + {wz' + w'z + xy' - yx'J, 
 a formula of algebra due to Euler. 
 
64 QUATERNIONS. [1O4. 
 
 104. We have, of course, by multiplication, 
 
 (a + /3y = a^ + a^S + /3a + ^' = a' + 2>Sfa/3 + /3' (§ 86 (3)). 
 
 TrauslatiDg into the usual notation of plane trigonometry, this 
 
 becomes c'^ = a^ — 2ab cos G + ¥, 
 
 the common formula. 
 
 Again, F . (a + y8) (a - /3) = - Fa^S + F/3a = - 2 Fa/3 (§ 86 (2)). 
 Taking tensors of both sides we have the theorem, the paral- 
 lelogram whose sides are parallel and equal to the diagonals of a 
 given parallelogram, has double its area (§ 96). 
 
 Also ^f(a + /3)(a-^) = a''-/3^ 
 
 and vanishes only when a? = /8^, or To. = T/S ; that is, the diagonals 
 of a parallelogram are at right angles to one another, when, and 
 only when, it is a rhombus. 
 
 Later it will be shewn that this contains a proof that the angle 
 in a semicircle is a right angle. 
 
 105. The expression p = a^oT^ 
 
 obviously denotes a vector whose tensor is equal to that of /3. 
 
 But we have S . ^ap = 0, 
 
 so that p is in the plane of a, /3. 
 
 Also we have Sap = (S'a/3, 
 
 so that /3 and p make equal angles with a, evidently on opposite 
 sides of it. Thus if a be the perpendicular to a reflecting surface 
 and yS the path of an incident ray, —p will be the path of the 
 reflected ray. 
 
 Another mode of obtaining these results is to expand the above 
 expression, thus, § 90 (2), 
 
 3 p = 2a-'*Sa/3-/3 
 
 = 2a-'>Sfa/3 - a' {Sol^ + Fa/3) 
 
 = a-(&;8-Fa/3), 
 
 so that iu the figure of § 77 we see that if OA = a, and OB = /3, we 
 have OB = p = ol^oTK 
 
 Or, again, we may get the result at once by transforming the 
 
 equation to - = if («"' p) = K - . 
 
I06.] INTEEPEETATIONS AND TRANSFORMATIONS. 65 
 
 106. For any three coplanar vectors the expression 
 
 is (§ 101) a vector. It is interesting to determine what this vector 
 is. The reader will easily see that if a circle be described about 
 the triangle, two of whose sides are (in order) a and /S, and if from 
 the extremity of )8 a line parallel to 7 be drawn, again cutting the 
 circle, the vector joining the point of intersection with the origin 
 of a is the direction of the vector ci^'y. For we may write it in the 
 form 
 
 p = «/3^^-v = - (r/8)^«^-v = - mr ^ 7. 
 
 which shews that the versor [-p,] which turns /S into a direction 
 
 parallel to a, turns 7 into a direction parallel to p. And this ex- 
 presses the long-known property of opposite angles of a quadri- 
 lateral inscribed in a circle. 
 
 Hence if a, /3, 7 be the sides of a triangle taken in order, the 
 tangents to the circumscribing circle at the angles of the triangle 
 are parallel respectively to 
 
 a/87, /37a, and ja^. 
 Suppose two of these to be parallel, i. e. let 
 8^87 = ir;S7a = a;a7/8 (§ 90), 
 since the expression is a vector. Hence 
 
 ^7 = ocy^, 
 which requires either 
 
 x=l, Vry^ = or 7 II yS, 
 a case not contemplated in the problem ; 
 or a; = -l, S^y=0, 
 
 i. e. the triangle is right-angled. And geometry shews us at once 
 that this is correct. 
 
 Again, if the triangle be isosceles, the tangent at the vertex is 
 parallel to the base. Here we have 
 
 53/3 = a,S7, 
 or a; (a + 7) = a (a -1- 7) 7 ; 
 
 whence x = 'f — a?, or T<y = Ta, as required. 
 
 As an elegant extension of this proposition the reader may 
 T. Q. I. 5 
 
66 QUATERNIONS. [lO?- 
 
 prove that the vector of the continued product a^r^h of the vector- 
 sides of any quadrilateral inscribed in a sphere is parallel to the 
 radius drawn to the corner (a, S). [For, if e be the vector from S, 
 a to y8, 7, a/Se and eyS are (by what precedes) vectors touching the 
 sphere at a, S. And their product (whose vector part must be 
 parallel to the radius at a, S) is 
 
 aySe . €78 = e" . a.^'^h^ 
 
 107. To exemplify the variety of possible transformations 
 even of simple expressions, we will take cases which are of 
 frequent occurrence in applications to geometry. 
 
 Thus T{p+a) = T{p-a), 
 
 [which expresses that if 
 
 OA=a, OA'=-a, and OP = p, 
 we have AP = A'P, 
 
 and thus that P is any point equidistant from two fixed points,] 
 may be written (p + a.y = (p — af, 
 
 or p^ + 2Sap + a? = p^- 2Sap + a^ (§ 104), 
 
 whence Sap = 0. 
 
 This may be changed to 
 
 ap+ pa.= 0, 
 or ap+Kap = 0, 
 
 SU^ = 0, 
 a 
 
 or finally, TW ^ = 1 
 
 a ' 
 
 all of which express properties of a plane. 
 
 Again, Tp = Ta 
 
 may be written T - = 1, 
 
 (p + ay-2Sa(p + a) = 0, 
 p=(p + ay'a(p + a), 
 S{p + a)(p-a)= 0, 
 or finally, T.{p + a) (p-a) = ITVap. 
 
 All of these express properties of a sphere. They will be 
 interpreted when we come to geometrical applications. 
 
I08.] INTEEPRBTATIONS AND TRANSFORMATIONS. 67 
 
 108. To find the space relation among five points. 
 
 A system of five points, so far as its internal relations are 
 concerned, is fully given by the vectors from one to the other four. 
 If three of these be called a, y8, y, the fourth, 8, is necessarily 
 expressible as xa.+y^ + zr^. Hence the relation required must be 
 independent of x, y, z. 
 
 But >SfaS = ««" + yBa^ + zSarf ' 
 
 SyB = xSyoi + ySy^ -H zy' 
 SBS = S^ = xSBa +ySB^ + zSBy ) 
 
 The elimination of x, y, z gives a determinant of the fourth order 
 which may be written 
 
 = 0. 
 
 .(1). 
 
 SoiO. 
 
 8a^ 
 
 Say 
 
 SaB 
 
 S^a 
 
 <S'/3/3 
 
 S^y 
 
 S^B 
 
 Sya 
 
 Sy0 
 
 Syy 
 
 SyB 
 
 SBa 
 
 SB^ 
 
 SBy 
 
 SBB 
 
 Now each term may be put in either of two forms, thus 
 S^y = I {^» + y - (^ - yy} = - T^Ty cos A- 
 
 If the former be taken we have the expression connecting the 
 distances, two and two, of five points in the form given by Muir 
 (Proc. R. S. E. 1889) ; if we use the latter, the tensors divide out 
 (some in rows, some in columns), and we have the relation among 
 the cosines of the sides and diagonals of a spherical quadrilateral. 
 
 We may easily shew (as an exercise in qviaternion manipulation 
 merely) that this is the only condition, by shewing that from it 
 we can get the condition when any other of the points is taken as 
 origin. Thus, let the origin be at a, the vectors are — a, ^ — a, 
 y — a, S— a. But, by changing the signs of the first row, and first 
 column, of the determinant above, and then adding their values 
 term by term to the other rows and columns, it becomes 
 
 ;Sf( -a)(-a) ;Sf( -a)(/3-a) S( - a) (y - a) 
 
 S(^-a)(-a) S(^-a)(^-a) 8(^-a){y-a) 
 
 8(y-ci)(-a) S(y-a)(0-a) 8(j-a)(y-a} 
 
 S(B-a)(-a) S(B-a)(^-a) S(B-a){y-a) 
 
 which, when equated to zero, gives the same relation as before. 
 [See Ex. 10 at the end of this Chapter.] 
 
 5—2 
 
 8( - 
 
 -«) (S- 
 
 -«) 
 
 S(0- 
 
 -a)(B- 
 
 -a) 
 
 S(y- 
 
 -a)(B- 
 
 ■«) 
 
 S{B- 
 
 -a){B- 
 
 ■a) 
 
68 
 
 QUATERNIONS. 
 
 [io8. 
 
 An additional point, with e = as'a + y'/3 + ^'7- gi^^^ ^^^^ additional 
 equations like (1) ; i. e. 
 
 Sae^x'a' +y'SoL^ + z'8ay, 
 S^e = ai'8^a + y'0' +z'8l3y, 
 Sje = a/8ya + y'S'y^ + z'f, 
 8Se = x'8Bci + y'SB^ + z'8By 
 = xS6a +y8e^ +z86<y, 
 e'' = x'8oie +y'8^6+z'Sye, 
 from which corresponding conclusions may be drawn. 
 
 Another mode of solving the problem at the head of this 
 section is to write the identity 
 
 2m (a - ey - l^ma' - 28 . 01ma + e'tm, 
 where the ms are undetermined scalars, and the as are given 
 vectors, while is any vector whatever. 
 
 Now, provided that the number of given vectors exceeds four, we 
 do not completely determine the ms by imposing the conditions 
 
 2m = 0, 2ma = 0. 
 Thus we may write the above identity, for each of five vectors 
 successively, as 
 
 2m (a — aji' = Xina', 
 
 2m (a — a J' = 2ma^ 
 
 2m (a - a^y = 2ma^. 
 Take, with these, 2m = 0, 
 
 and we have six linear equations from which to eliminate the ms. 
 The resulting determinant is 
 
 ■ a„ a, - a; 
 
 ■ a 
 
 a„-a, a„ 
 
 • a' a„ — «„■ 
 
 1 
 
 a.' 1 
 
 a. -a. a. 
 
 «,-«.^ 1 
 
 1 
 
 
 
 2ma'' = 0. 
 
 This is equivalent to the form in which Cayley gave the 
 relation among the mutual distances of five points. {Gamb. Math. 
 Journ. 1841.) 
 
I lO.] INTERPRETATIONS AND TRANSFORMATIONS. 69 
 
 109. We have seen in § 95 that a quaternion may be divided 
 into its scalar and vector parts as follows : — 
 
 ^ = 8^ + V^ = ^(cose + 6sme); 
 a a a Ta.^ ^ ' 
 
 where 6 is the angle between the directions of a and /S, and e = UV- 
 
 a 
 
 is the unit-vector perpendicular to the plane of a. and /3 so situated 
 
 that positive (i. e. left-handed) rotation about it turns a towards /S. 
 
 Similarly we have (§ 96) 
 
 = TaT/S (- cos + 6 sin 0), 
 6 and e having the same signification as before. 
 
 110. Hence, considering the versor parts alone, we have 
 
 U - = cos ^ 4- e sin 6. 
 a 
 
 Similarly ^ 'h~ '^^^ ^ -1- e sin ^ ; 
 
 ^ being the positive angle between the directions of 7 and /S, and e 
 the same vector as before, if a, /8, 7 be coplanar. 
 
 Also we have 
 
 1/ ^ = cos (6* -I- </)) -t- 6 sin {6 + </>). 
 But we have always 
 
 /8"a a' 
 
 and therefore U\.U^ =U'^; 
 
 ^ a a' 
 
 or cos (<^ -I- ^) -1- 6 sin {<j) + 6) = (cos <f>+ esin (p) (cos ^ -F e sin 0) 
 
 = cos <f)cr>s6- sin (j}sin6 + e (sin <pcos0 + cos (/> sin 6), 
 
 from which we have at once the fundamental formulae for the 
 cosine and sine of the sum of two arcs, by equating separately the 
 scalar and vector parts of these quaternions. 
 
 And we see, as an immediate consequence of the expressions 
 above, that 
 
 cos md + e sin mO = (cos ^ -|- e sin 9f\ 
 
 if m be a positive whole number. For the left-hand side is a versor 
 
70 QUATEKNIONS. [ill. 
 
 which turns through the angle md at once, while the right-hand 
 side is a versor which effects the same object by m successive turn- 
 ings each through an angle 6. See §§ 8, 9. 
 
 111. To extend this proposition to fractional indices we have 
 
 a 
 
 only to write - for 6, when we obtain the results as in ordinary 
 trigonometry. 
 
 From De Moivre's Theorem, thus proved, we may of course 
 deduce the rest of Analytical Trigonometry. And as we have 
 already deduced, as interpretations of self-evident quaternion trans- 
 formations (§§ 97, 104), the fundamental formulae for the solution 
 of plane triangles, we will now pass to the consideration of spherical 
 trigonometry, a subject specially adapted for treatment by qua- 
 ternions ; but to which we cannot afford more than a very few 
 sections. (More on this subject will be found in Chap. XI. in con- 
 nexion with the Kinematics of rotation.) The reader is referred to 
 Hamilton's works for the treatment of this subject by quaternion 
 exponentials. 
 
 112. Let a, /3, 7 be unit-vectors drawn from the centre to the 
 corners A, B, G oi sl triangle on the unit-sphere. Then it is evident 
 that, with the usual notation, we have (§ 96), 
 
 Sa^ = — cos c, S/Sy = — cos a, Sya. = — cos b, 
 
 TVoL^= sine, TV^r^= sin a, TV'^ol^ sin 6. 
 
 Also UVa.^, UV^y, UVya are evidently the vectors of the corners 
 of the polar triangle. 
 
 Hence S.UVa^ UV/Sy = cos B, &c., 
 
 TV. Ura^UV/3y = smB, &c. 
 Now (§ 90 (1)) we have 
 
 SVa^V^j = S . aV(^V^y) 
 
 = - Sa^S^y + ^'Say. 
 Remembering that we have 
 
 SVa^Vfiy = TVa^TV/3yS . UVa/3UV^y, 
 we see that the formula just written is equivalent to 
 sin a sin c cos 5 = — cos a cos c + cos b, 
 1 or cos 6 = cos a cos c -I- sin a sin c cos B. 
 
115.] INTEEPRETATIONS AND TRANSFORM ATIONS. 71 
 
 113. Again, V. Val3V^j = - ^Sa^y, 
 which gives 
 
 TV. Fa/3F/37 = TS . oi^j = TS.aV^y = TS. ^Vja = T8 . yVa^, 
 
 or sin a sin c sin 5 = sin a sinp^ = sin b sinp^ = sin c sinp„ ; 
 
 where p^ is the arc drawn from A perpendicular to BC, &c. 
 
 Hence sin p^ = sin c sin jB, 
 
 sin a sin c . „ 
 
 sin »6 = -■ — 7 — sm B, 
 
 ^ sm 
 
 sin p^ = sin a sin B. 
 
 114. Combining the results of the last two sections, we have 
 Va^ . Vfij = sin a sin c cos i? — /3 sin a sin c sin B 
 
 = sin a sin c (cos £ — yS sin B). 
 
 Hence i7 . Fa^ V^y = (cos 5 - y8 sin 5)| 
 
 and U . F7/3 F^a = (cos ^ + /S sin 5) j ' 
 
 These are therefore versors which turn all vectors perpendicular to 
 OB negatively or positively about OB through the angle B. 
 
 [It will be shewn later (§ 119) that, in the combination 
 
 (cos 5 + jS sin 5) ( ) (cos £ - /3 sin B), 
 
 the system operated on is made to rotate, as if rigid, round the 
 vector axis /3 through an angle 25.] 
 
 As another instance, we have 
 
 „ sin B 
 
 tan B = ^ 
 
 cosB 
 
 TV.Va0V^y 
 ~ S.Va/3V^y 
 
 V.Va0V^y 
 ~ ^ S.Va^V^y 
 
 _ S . a-lSy _ „ ,,v 
 
 ~~ 8oLy + S<xl3S^y *" ^ '' 
 
 The interpretation of each of these forms gives a different theorem 
 in spherical trigonometry. 
 
 115. Again, let us square the equal quantities 
 
 V.a^y and aSl3y-^8ay + ySa^, 
 
72 QUATERNIONS. [ll6. 
 
 supposing OL, /3, y to be any unit-vectors whatever. We have 
 
 - (F, a^yf - S'^y + 8'ya + S'a^ + 280y8yaSa^. 
 But the left-hand member may be written as 
 
 T'.al3y-S'.a^y, 
 whence 
 
 1-S\a^y = S'^y -h S'yfx + /S'a/3 + 2S^y8yaSa^, 
 or 1 — cos^a — cos''6 — cos^c + 2 cos a cos 6 cos c 
 
 = sin^'a sin^^„ = &c. 
 
 = sin^'a sin^6 sin^C = &c., 
 all of which are well-known formulae. 
 
 116. Again, for any quaternion, 
 
 q = 8q+Vq, 
 so that, if « be a positive integer, 
 
 3" = (;Sf^)" + n {8qr Vq + ^^^ (^?)"" (F?7 + . . • 
 From this at once 
 
 8.q" = (8qr-'^^{8qrTWq 
 
 + 1.2.3.4 ^^^^ 'T'iVq)- &c.. 
 
 V. q" = Vq ^n (8qr'- "" ' \ l'^ - (-S?)""' T^ Vq + &c. 
 
 If g be a versor we have 
 
 q = cos u+ 6 sin u, 
 so that 
 
 71 '}7 — 1 
 
 8.q" = (cos «)" - • ^ (cos m)"-' (sin it)' -F , 
 
 = cos ?itt ; 
 F.5" = ^sinM 
 
 = sin WM 
 
 11 (cos M) ' ^ n „ (cos U) (siU M/ -|- , 
 
 as we might at once have concluded from § 110. 
 
 Such results may be multiplied indefinitely by any one who has 
 mastered the elements of quaternions. 
 
I 1 7-] INTERPRETATIONS AND TRANSFOBMATIONS. 73 
 
 117. A curious proposition, due to Hamilton, gives us a 
 quaternion expression for the spherical excess in any triangle. 
 The following proof, which is very nearly the same as one of his, 
 though by no means the simplest that can be given, is chosen here 
 because it incidentally gives a good deal of other information. 
 We leave the quaternion proof as an exercise. 
 
 Let the unit-vectors drawn from the centre of the sphere to 
 A, B, G, respectively, be a, y8, 7. It is required to express, as an 
 arc and as an angle on the sphere, the quaternion 
 
 /8«-'7. 
 
 The figure represents an orthographic projection made on a 
 plane perpendicular to 7. Hence G is the centre of the circle DEe. 
 Let the great circle through A, B meet DEe in E, e, and let DE be 
 
 a quadrant. Thus BE represents 7 (§ 72). Also make EF = AB 
 = /3a~'. Then, evidently, 
 
 which gives the arcual representation required. 
 
 Let DF cut Ee in 0. Make Ga = EG, and join D, a, and a, F. 
 Obviously, as D is the pole of Ee, Da is a quadrant ; and since 
 EO = Ca, Ga = EG, a quadrant also. Hence a is the pole of DG, 
 and therefore the quaternion may be represented by the angle 
 DaF. 
 
 Make Gb = Ga, and draw the arcs Pa^, Pba from P, the pole of 
 
74 QUATERNIONS. [ll7- 
 
 AB. Comparing the triangles Eba and ea/S, we see that Ea = e/3. 
 But, since P is the pole of AB, F^a is a right angle: and therefore 
 as Fa is a quadrant, so is F^. Thus AB is the complement of Ea 
 or /3e, and therefore 
 
 a^ = 2AB. 
 
 Join 64 and produce it to c so that Ac = bA; join c, P, cutting 
 AB in 0. Also join c, 5, and B, a. 
 
 Since P is the pole of AB, the angles at o are right angles ; 
 and therefore, by the equal triangles hoA, coA, we have 
 
 aA = Ao. 
 
 But a/8 = 2AB, 
 
 whence oB = B^, 
 
 and therefore the triangles coB and Ba^ are equal, and c, B, a lie 
 on the same great circle. 
 
 Produce cA and cB to meet in H (on the opposite side of the 
 sphere). H and c are diametrically opposite, and therefore cP, 
 produced, passes through H. 
 
 Now Pa = Pb = PH, for they differ from quadrants by the 
 equal arcs a/S, bcc, oc. Hence these arcs divide the triangle Hab 
 into three isosceles triangles. 
 
 But Z PHb + Z PHa = Z aHb = Z bca. 
 
 Also Z Pab = ir- A cab -A PaH, 
 
 Z Pba = z Pa6 = tt - Z c6a - Z PbH. 
 
 Adding, 2 Z Pa6 = 27r — Z ca& — Z c6a — Z 6ca 
 
 = TT — (spherical excess of abc). 
 
 But, as Z Pa/3 and Z jDae are right angles, we have 
 
 angle of /SoT^y = Z FaD = Z jSae = Z Pa6 
 
 = ^ — ^ (spherical excess of abc). 
 
 [Numerous singular geometrical theorems, easily proved ab 
 initio by quaternions, follow from this : e.g. The arc AB, which 
 bisects two sides of a spherical triangle abc, intersects the base at 
 the distance of a quadrant from its middle point. All spherical 
 triangles, with a common side, and having their other sides 
 bisected by the same great circle (i.e. having their vertices in a 
 
119.] INTERPRETATIONS AND TRANSrORMATlONS. 75 
 
 small circle parallel to this great circle) Lave equal areas, &c. 
 
 &C.J 
 
 118. Let Oa = a', 06 = /S', Oc = 7', and we have 
 
 = 6^.BA 
 
 = Ed.FE=FG. 
 
 But FO is the complement of DF. Hence the angle of the 
 quaternion 
 
 (l)'(f)'©* 
 
 is half the spherical excess of the triangle whose angular points are 
 at the extremities of the unit-vectors a', /S', 7'. 
 
 [In seeking a purely quaternion proof of the preceding proposi- 
 tions, the student may commence by shewing that for any three 
 unit-vectors we have 
 
 The angle of the first of these quaternions can be. easily assigned ; 
 and the equation shews how to find that of /Sa"'^. 
 
 Another easy method is to commence afresh by forming from 
 the vectors of the corners of a spherical triangle three new vectors 
 thus : — 
 
 
 &c. 
 
 Then the angle between the planes of a, y3' and 7', a; or of /3, 7' 
 and a, ^ ; or of 7, a' and yS', 7 ; is obviously the spherical excess. 
 
 But a still simpler method of proof is easily derived from the 
 composition of rotations.] 
 
 119. It may be well to introduce here, though it belongs 
 rather to Kinematics than to Geometry, the interpretation of the 
 operator 
 
 9( )?-'• 
 
 By a rotation, about the axis of q, through double the angle of q, 
 the quaternion r becomes the quaternion qrq'\ Its tensor and 
 angle remain unchanged, its plane or axis alone varies. 
 
76 QUATERNIONS. [l20. 
 
 A glance at the figure is sufficient for . „ 
 
 the proof, if we note that of course 
 T . qrq^ = Tr, and therefore that we need 
 consider the versor parts only. Let Q 
 be the pole of q, 
 
 AB = q, AB' = q-\ RG' = r. 
 
 Join C'A, and make AC = C'A. Join 
 CB. 
 
 Then CB is qrq'^, its arc CB is evidently equal in length to that 
 of r, B'C ; and its plane (making the same angle with B'B that 
 that of B'C does) has evidently been made to revolve about Q, the 
 pole of q, through double the angle of q. 
 
 It is obvious, from the nature of the above proof, that this 
 operation is distributive ; i. e. that 
 
 q(r + s) q~^ = qrq~^ + qsq^. 
 
 If r be a vector, = p, then qpq~^ (which is also a vector) is the 
 result of a rotation through double the angle of q about the axis 
 of q. Hence, as Hamilton has expressed it, if B represent a rigid 
 system, or assemblage of vectors, 
 
 qBq-' 
 
 is its new position after rotating through double the angle of q 
 about the axis of q. 
 
 120. To compound such rotations, we have 
 
 r . qBq'' .r'^ = rq.B . (rq)'^. 
 
 To cause rotation through an angle t-iold the double of the angle 
 
 of q we write q'Bq~'. 
 
 To reverse the direction of this rotation write q''Bq'. 
 
 To translate the body B without rotation, each point of it moving 
 through the vector a, we write a + B. 
 
 To produce rotation of the translated body about the same axis, 
 and through the same angle, as before, 
 
 q{cc + B)q-\ 
 
 Had we rotated first, and then translated, we should have had 
 
 a + qBq'^. 
 
COS 2 - X sin ^ j f j ^cos ^ + X sin ^ 
 
 1 2 I.J INTERPBETATIONS AND TRANSFORMATIONS. 11 
 
 From the point of view of those who do not believe in the 
 Moon's rotation, the former of these expressions ought to be 
 
 qdq^ + B 
 instead of 
 
 But to such men quaternions are unintelligible. 
 
 121. The operator above explained finds, of course, some 
 of its most direct applications in the ordinary questions of 
 Astronomy, connected with the apparent diurnal rotation of the 
 stars. If X be a unit-vector parallel to the polar axis, and h the 
 hour angle from the meridian, the operator is 
 
 ^^4)( )' 
 
 or L-^ ( ) L, 
 
 the inverse going first, because the apparent rotation is negative 
 (clockwise). 
 
 If the upward line be i, and the southward j, we have 
 \ — isml— j cos I, 
 
 where I is the latitude of the observer. The meridian equatorial 
 unit vector is 
 
 /u. = i cos I +j sin I ; 
 
 and X, fi, k of course form a rectangular unit system. 
 
 The meridian unit-vector of a heavenly body is 
 
 S = i cos {l — d)+ j sin (I — d), 
 
 = X sin d + fi cos d, 
 
 where d is its declination. 
 
 Hence when its hour-angle is h, its vector is 
 
 S' = L'BL. 
 
 The vertical plane containing it intersects the horizon in 
 
 iViS'=jSjS' + Jc8k8', 
 so that 
 
 tan (azimuth) = -^^ (!)■ 
 
 [This may also be obtained directly from the last formula (1) 
 of § 114.] 
 
78 QUATERNIONS. [l22. 
 
 To find its Amplitude, i.e. its azimuth at rising or setting, 
 the hour-angle must be obtained from the condition 
 
 SiB'==0 (2). 
 
 These relations, with others immediately deducible from them, 
 enable us (at once and for ever) to dispense with the hideous 
 formulae of Spherical Trigonometry. 
 
 122. To shew how readily they can be applied, let us 
 
 translate the expressions above into the ordinary notation. This 
 
 is effected at once by means of the expressions for \, fi, L, and 
 
 8 above, which give by inspection 
 
 S' = X sin 6?+ (/i cos h —ksm h) cos d, 
 
 and we have from (1) and (2) of last section respectively 
 
 , . , . sin ^ cos d ,t , 
 
 tan (azimuth) = ^-^ — 5 ^ — ; ^ ^ (i), 
 
 cos L sin d — sm L cos d cos /i 
 
 cos h + tan I tan d = (2). 
 
 In Capt. Weir's ingenious Azimuth Diagram, these equations 
 
 are represented graphically by the rectangular coordinates of a 
 
 system of confocal conies : — viz. 
 
 X = sin h sec I ) 
 
 .{6) 
 
 y = cos h tan I \ 
 
 The ellipses of this system depend upon I alone, the hyperbolas 
 upon h. Since (1) can, by means of (3), be written as 
 
 tan (azimuth) = , = , 
 
 tan d—y 
 
 we see that the azimuth can be constructed at once by joining 
 with the point 0, — tan d, the intersection of the proper ellipse and 
 hyperbola. 
 
 Equation (2) puts these expressions for the coordinates in the 
 form 
 
 X = sec lj\ — tan" I tan" d 1 
 y = — tan^ I tan d 
 
 The elimination of d gives the ellipse as before, but that of I 
 gives, instead of the hyperbolas, the circles 
 
 a? + y'^ — y (tan d — cot d) = 1. 
 
 The radius is 
 
 I (tan d + coid); 
 and the coordinates of the centre are 
 
 0, ^ (tan d — cot d). 
 
I24.J INTERPRETATIONS AND TRANSFORMATIONS. 79 
 
 123. A scalar equation in p, the vector of an undetermined 
 point, is generally the equation of a surface; since we may use 
 in it the expression p = xa, 
 
 where x is an unknown scalar, and a any assumed unit-vector. 
 The result is an equation to determine x. Thus one or more 
 points are found on the vector xa, whose coordinates satisfy the 
 equation ; and the locus is a surface whose degree is determined 
 by that of the equation which gives the values ofx. 
 
 But a vector equation in p, as we have seen, generally leads to 
 three scalar equations, from which the three rectangular or other 
 components of the sought vector are to be derived. Such a vector 
 equation, then, usually belongs to a definite number of points in 
 space. But in certain cases these may form a li7ie, and even a 
 surface, the vector equation losing as it were one or two of the 
 three scalar equations to which it is usually equivalent. 
 
 Thus while the equation ap = yS 
 gives at once p = a~'/8, 
 
 which is the vector of a definite point (since by making p a vector 
 
 we have evidently assumed 
 
 ,Sa/3 = 0); 
 
 the closely allied equation Vap = 13 
 
 is easily seen to involve Sa^ = 0, 
 
 and to be satisfied by p = oT^^ + xa, 
 
 whatever be x. Hence the vector of any point whatever in the 
 line drawn parallel to a from the extremity of a'^^ satisfies the 
 given equation. [The difference between the results depends 
 upon the fact that Sap is indeterminate in the second form, but 
 definite (= 0) in the first.] 
 
 124. Again, Vap . Vp^ = ( Fa/3)'' 
 
 is equivalent to but two scalar equations. For it shews that Vap 
 and V^p are parallel, i.e. p lies in the same plane as a and ^, and 
 can therefore be written (§ 24) 
 
 p = xa + 2//3, 
 where x and y are scalars as yet undetermined. 
 We have now Vap = yVa^, 
 
 Vpl3 = xVa^, 
 
80 QUATERNIONS. [l^S- 
 
 which, by the given equation, lead to 
 
 xy = l, or 2/ = -, 
 
 or finally p = a;a + -^ ; 
 
 which (§ 40) is the equation of a hyperbola whose asymptotes are 
 in the directions of a and /8. 
 
 125. Again, the equation 
 
 V.Val3Voip = 0, 
 
 though apparently equivalent to three scalar equations, is really 
 
 equivalent to one only. In fact we see by § 91 that it may be 
 
 written 
 
 -aS.al3p = 0, 
 
 whence, if a be not zero, we have 
 
 S.a^p = 0, 
 and thus (§ 101) the only condition is that p is coplanar with a, ^. 
 Hence the equation represents the plane in which a and ^ lie. 
 
 126. Some very curious results are obtained when we extend 
 these processes of interpretation to functions of a quaternion 
 
 q = w + p 
 
 instead of functions of a mere vector p. 
 
 A scalar equation containing such a quaternion, along with 
 quaternion constants, gives, as in last section, the equation of a 
 surface, if we assign a definite value to w. Heace for successive 
 values of w, we have successive surfaces belonging to a system ; 
 and thus when w is indeterminate the equation represents not a 
 surface, as before, but a volume, in the sense that the vector of any 
 point within that volume satisfies the equation. 
 
 Thus the equation {TqY = a^, 
 
 or w" — p" = a", 
 
 or {Tpf = a^-w\ 
 
 represents, for any assigned value of w, not greater than a, a sphere 
 whose radius is J a' - w\ Hence the equation is satisfied by the 
 vector of any point whatever in the volume of a sphere of radius a, 
 whose centre is origin. 
 
128.] INTERPRETATIONS AND TRANSFORMATIONS. 81 
 
 Again, by the same kind of investigation, 
 
 where q = w-\- p, is easily seen to represent the volume of a sphere 
 of radius a described about the extremity of /8 as centre. 
 
 Also 8{<f) = — a^ is the equation of infinite space less the space 
 contained in a sphere of radius a about the origin. 
 
 Similar consequences as to the interpretation of vector equa- 
 tions in quaternions may be readily deduced by the reader. 
 
 127. The following transformation is enuntiated without proof 
 by Hamilton {Lectures, p. 587, and Elements, p. 299). 
 
 r-' (rV)^g-' = U {rq + KrKq). 
 
 To prove it, let r"' (r'Y)^ g"' = t, 
 
 then Tt = 1, 
 
 and therefore Kt = r\ 
 
 But {rY)^ = rtq, 
 
 or r^q' = rtqrtq, 
 
 or rq = tqrt. 
 
 Hence KqKr = f'KrKqt'', 
 
 or KrKq = tKqKrt. 
 
 Thus we have U (rq + KrKq) = tU (qr ± KqKr) t, 
 or, if we put s = U(qr ± KqKr), 
 
 Ks=+ tst. 
 Hence sKs = (Ts)" = 1 = ± stst, 
 
 which, if we take the positive sign, requires 
 
 st=±l, 
 or t= + s''=± UKs, 
 
 which is the required transformation. 
 
 [It is to be noticed that there are other results which might 
 have been arrived at by using the negative sign above; some 
 involving an arbitrary unit-vector, others involving the imaginary 
 of ordinary algebra.] 
 
 128. As a final example, we take a transformation of Hamil- 
 ton's, of great importance in the theory of surfaces of the second 
 order. 
 
 Transform the expression 
 
 (Sapy+is^py+iS'Ypy 
 
 T. Q. I. ^ 
 
8 2 QUATERNIONS. [ 1 2 8, 
 
 in which a, ^, y are any three mutually rectangular vectors, into 
 the form 
 
 \ k'-c' ) ' 
 which involves only two vector-constants, t, k. 
 
 [The student should remark here that (, k, two undetermined 
 vectors, involve six disposable constants : — and that a, ^8, 7, being 
 a rectangular system, involve also only six constants.] 
 
 {T{ip^pK)f={ip^pK){pt + Kp) (§52, 55) 
 
 = {^ + l^)p^ + {ipKp + pKpC) 
 = {l,'+K')p' + 2S.ipKp 
 = (t - Kfp^ + 4>SipSKp. 
 
 Hence (Sapf + (S^pf + (Sypf = |^ p^ + 4 ^^^. 
 
 But a- {Sccpf + ^-' (S/3py + y-' (Sypy = p' (§§ 25, 73). 
 Multiply by /3^ and subtract, we get 
 
 The left side breaks up into two real factors if ^ be intermediate 
 in value to a^ and 7^ : and that the right side may do so the term 
 in p^ must vanish. This condition gives 
 
 ^'' ~ I ' 2_ 2\a j ^"^^ t^6 identity becomes 
 
 «{V(-IVV(f-))^''M-?)-V(f-0}' 
 
 _ . SlpSKp 
 
 Hence we must have 
 
 ^.-^lV(-§)-V(f:- 
 
 where ^ is an undetermined scalar. 
 
 To determine p, substitute in the expression for ^', and we find 
 4 (f - k)' 
 
 p/ ' ' V pJ 
 
 ^^^=(i^=(^-^)V-^^) + (i.+J)V-7=) 
 
 = b'+^)(«^-7=)-2(«=+y) + 4^^ 
 
129-] INTEKPRETATIONS AND TRANSFORMATIONS. 83 
 
 Thus the transformation succeeds if 
 
 which gives p + - = ±2 
 
 Hence (^2 _ ^y = [-. -p') (a' - 7') = ± 4 Ja'y", 
 
 or {k'' - c')-' = ± TocTy. 
 
 , . Ta+Ty 1 Ta-Ty 
 
 Again, p = ■ , - = . , 
 
 and therefore 
 
 Thus we have proved the possibility of the transformation, and 
 determined the transforming vectors t, k. 
 
 129. By differentiating the equation 
 
 (Sapf + (S^pf + (SypT = [^^^) 
 we obtain, as will be seen in Chapter IV, the following, 
 
 SapSap' + S^pS^p' + SypSyp = 8 • i^P + P'c) (.p^ + p U) _ 
 
 [^K — I, ) 
 
 where p' also may be any vector whatever. 
 
 This is another very important formula of transformation ; and 
 it will be a good exercise for the student to prove its truth by 
 processes analogous to those in last section. We may merely 
 observe, what indeed is obvious, that by putting p' = p it becomes 
 the formula of last section. And we see that we may write, with 
 the recent values of i and k in terms of a, y, the identity 
 
 aSap + ^S^p + ySyp = ' J_ .^ — ^ 
 
 _ (t -K^p + 2 (nSVp + k Slp) 
 
 6—2 
 
8 4 QUATERNIONS. [ I 3 O- 
 
 130. In various quaternion investigations, especially in such 
 as involve imaginary intersections of curves and surfaces, the old 
 imaginary of algebra of course appears. But it is to be particularly 
 noticed that this expression is analogous to a scalar and not to a 
 vector, and that like real scalars it is commutative in multipli- 
 cation with all other factors. Thus it appears, by the same proof 
 as in algebra, that any quaternion expression which contains this 
 imaginary can always be broken up into the sum of two parts, one 
 real, the other multiplied by the first power of V — 1. Such an 
 expression, viz. 
 
 q = q' + ^/-lq", 
 
 where q' and q" are real quaternions, is called by Hamilton a 
 BIQTJATEENION. [The student should be warned that the term 
 Biquaternion has since been employed by other writers in the 
 sense sometimes of a "set" of 8 elements, analogous to the 
 Quaternion 4 ; sometimes for an expression g' + dq" where 6 is not 
 the algebraic imaginary. By them Hamilton's Biquaternion is 
 called simply a quaternion with non-real constituents.] Some 
 little care is requisite in the management of these expressions, but 
 there is no new difficulty. The points to be observed are : first, 
 that any biquaternion can be divided into a real and an imaginary 
 part, the latter being the product of '^'^1 by a real quaternion ; 
 second, that this V — 1 is commutative with all other quantities in 
 multiplication ; third, that if two biquaternions be equal, as 
 
 gr' f V^;^ g" = r' + V ~1 r", 
 we have, as in algebra, q' = r\ q" = r"; 
 
 so that an equation between biquaternions involves in general 
 eight equations between scalars. Compare § 80. 
 
 131. We have obviously, since V — 1 is a scalar, 
 
 S (q' + V3i g") = Sq' + V31 Sq", 
 
 F (g' + V ^ g") = Vq' + V^i Vq". 
 Hence (§ 103) 
 
 {T{q' + '^^q")f ^ 
 
 = {Sq' + ^-l Sq"+ Fg' + V - 1 Vq") (Sq +\f'^ Sq" - Vq' 
 
 - V^l Vq") 
 = (Sq' + V^T S^y -(Vq' + V^ VqJ, 
 
 = {Tqj - (Tq'j + 2 V^5f . q'Kq". 
 
1 3 2. J INTERPRETATIONS AND TRANSFORMATIONS. 85 
 
 The only remark which need be made on such formulae is this, that 
 the tensor of a biquaternion may vanish while both of the component 
 quaternions are finite. 
 
 Thus, if Tq = Tq", 
 
 and- 8.qKq" = 0, 
 
 the above formula gives 
 
 T(q+'J^iq") = 0. 
 The condition S . q'Kq" = 
 
 may be written 
 
 Kq" = q'-\ or q" = - aKq'-^ = - j^, , 
 where a is any vector whatever. 
 
 Hence Tq' = Tq" = TKq" = ^, , 
 
 and therefore 
 
 Tq' ( Uq' - \f'^ Ua . Uq) = (1 - V^ Ua) q' 
 is the general form of a biquaternion whose tensor is zero. 
 
 132. More generally we have, q, r, q', r being any four real 
 and non-evanesceut quaternions, 
 
 {q + '^^9'') (r + V - Ir') = qr- q'r + V^H. {qr + q'r). 
 
 That this product may vanish we must have 
 
 qr = q'r', 
 
 and qr' = — q'r. 
 
 Eliminatipg / we have qq'~^qr = — q'r, 
 
 which gives {'I'^lf = ~ !> 
 
 i.e. q = g'a 
 
 where a is some unit-vector. 
 
 And the two equations now agree in giving 
 
 — r = ar, 
 
 so that we have the biquaternion factors in the form 
 
 g' (« + ^^ - 1) ^^^ - (a - V - 1) r' ; 
 and their product is 
 
 -q {0L + V^T) (a - V ^) r, 
 which, of course, vanishes. 
 
86 QUATERNIONS. [l33- 
 
 [A somewhat simpler investigation of the same proposition 
 may be obtained by writing the biquaternions as 
 
 (?' (g'"' q + V^ ) and {rr""^ + V^) r', 
 or q (q" + V^) and {r" + V - 1) r', 
 
 and shewing that 
 
 q" = — r" = a, where Ta = 1.] 
 
 From this it appears that if the product of two bivectors 
 
 p + <T V-1 and p' + 0-' V - i 
 
 is zero, we must have 
 
 a-'p = -p'a'-'=Ua, 
 
 where a may be any vector whatever. But this result is still more 
 easily obtained by means of a direct process. 
 
 133. It may be well to observe here (as we intend to avail our- 
 selves of them in the succeeding Chapters) that certain abbreviated 
 forms of expression may be used when they are not liable to confuse, 
 or lead to error. Thus we may write 
 
 Pq for (Tqf, 
 just as we write cos"^ for (cos 6y, 
 
 although the true meanings of these expressions are 
 
 T(Tq) and cos (cos 61). 
 The former is justifiable, as T{Tq) = Tq, and therefore 'fq is not 
 required to signify the second tensor (or tensor of the tensor) of q. 
 But the trigonometrical usage is defensible only on the score of 
 convenience, and is habitually violated by the employment of 
 cos"' X in its natural and proper sense. 
 Similarly we may write 
 
 S'q for (Sqy, &c., 
 but it may be advisable not to use 
 
 Sq' 
 
 as the equivalent of either of those just written; inasmuch as it 
 might be confounded with the (generally) different quantity 
 
 S.q' or Siq'), 
 although this is rarely written without the point or the brackets. 
 
 The question of the use of points or brackets is one on which 
 no very definite rules can be laid down. A beginner ought to use 
 
134-J INTERPRETATIONS AND TRANSFORMATIONS. 87 
 
 them freely, and he will soon learn by trial which of them are 
 absolutely necessary to prevent ambiguity. 
 
 In the present work this course has been adopted : — the 
 earlier examples in each part of the subject being treated with 
 a free use of points and brackets, while in the later examples 
 superfluous marks of the kind are gradually got rid of. 
 
 It may be well to indicate some general principles which 
 regulate the omission of these marks. Thus in S .a/3 or F. a/3 
 the point is obviously unnecessary : — because Sot = 0, and Va = a, 
 so that the S would annihilate the term if it applied to a alone, 
 while in the same case the V would be superfluous. But in S . gr- 
 and V.qr, the point (or an equivalent) is indispensable, for Sq.r, 
 and Vq . r are usually quite different from the first written 
 quantities. In the case of K, and of d (used for scalar differen- 
 tiation), the omission of the point indicates that the operator acts 
 only on the nearest factor ; — thus 
 
 Kqr = {Kq) r = Kq.r, dqr = (dq) r=dq.r; 
 while, if its action extend farther, we write 
 
 K .qr = K (qr), d.qr=d (qr), &c. 
 
 In more complex cases we must be ruled by the general 
 
 principle of dropping nothing which is essential. Thus, for 
 
 instance 
 
 V{pK{dq)Virq.r)) 
 
 may be written without ambiguity as 
 
 V.pKdqVVqr, 
 
 but nothing more can be dropped without altering its value. 
 
 Another peculiarity of notation, which will occasionally be 
 
 required, shows which portions of a complex product are affected 
 
 by an operator. Thus we write 
 
 if V operates on a and also on t, but 
 
 if it operates on t alone. See, in this connection, the last Example 
 at the end of Chap. IV. below. 
 
 134. The beginner may expect to be at first a little puzzled 
 with this aspect of the notation ; but, as he learns more of the 
 subject, he will soon see clearly the distinction between such an 
 
 expression as 
 
 S.Va/3V/3y, 
 
8 8 QUATERNIONS. [ 1 3 4- 
 
 where we may omit at pleasure either the point or the first V 
 without altering the value, and the very different one 
 
 which admits of no such changes, without alteration of its value. 
 
 All these simplifications of notation are, in fact, merely examples 
 of the transformations of quaternion expressions to which part of 
 this Chapter has been devoted. Thus, to take a very simple ex- 
 ample, we easily see that 
 
 = Sar.( F7/3)/3 = S.a F(7/3)/S = S. V(y^)0a = S V<y^ V^a 
 = ,S . 7/3F/3a = 8.K{0y)V^ci=S.^yKV^a=- 8./3yV0a 
 = S . Vy^V^a, &c., &c. 
 
 The above group does not nearly exhaust the list of even the simpler 
 ways of expressing the given quantity. We recommend it to the 
 careful study of the reader. He will find it advisable, at first, to 
 use stops and brackets pretty freely; but will gradually learn to 
 dispense with those which are not absolutely necessary to prevent 
 ambiguity. 
 
 There is, however, one additional point of notation to which 
 the reader's attention should be most carefully directed. A very 
 simple instance will suffice. Take the expressions 
 
 ^.^ and ^. 
 y a ya. 
 
 The first of these is 
 
 0y-\yor^ = 0ar\ 
 
 and presents no difficulty. But the second, though at first sight 
 it closely resembles the first, is in general totally different in 
 value, being in fact equal to 
 
 /37a"V\ 
 
 For the denominator must be treated as om quaternion. If, 
 then, we write 
 
 we have 
 
 so that, as stated above, 
 
 7a ^ 
 
 /3y = qya, 
 
 q = 0ya-'y-\ 
 
 We see therefore that 
 ^y 
 7 ' a a OLy ' 7a 
 
 '3^-^ = ^^;but„o« = ^. 
 
INTERPEETATIONS AND TEANSI'ORMATIONS. 89 
 
 EXAMPLES TO CHAPTER III. 
 
 1. Investigate, by quaternions, the requisite formulae for 
 changing from any one set of coordinate axes to another ; and 
 derive from your general result, and also from special investiga- 
 tions, the usual expressions for the following cases : — 
 
 (a) Rectangular axes turned about z through any angle. 
 
 (6) Rectangular axes turned into any new position by rota- 
 tion about a line equally inclined to the three. 
 
 (c) Rectangular turned to oblique, one of the new axes 
 lying in each of the former coordinate planes. 
 
 2. Point out the distinction between 
 ='+^Vand^« + ^)^ 
 
 a \a 
 
 a + ^ _ Fa/3 
 a - /3 ~ 1 + >Sa/S ' 
 
 a - /3 Va/3 
 
 a J a. 
 
 and find the value of their difference. 
 
 If r/3/a = l, then [7°''^^-^^^' 
 
 Shew also that 
 and 
 
 a 4- /3 l-Sa/3' 
 
 provided a and /S be unit-vectors. If these conditions are not 
 fulfilled, what are the true values ? 
 
 3. Shew that, whatever quaternion »' may be, the expression 
 
 ar + rfi, 
 in which a and /S are any two unit-vectors, is reducible to the 
 
 form 
 
 l(a + ^)+m{al3- 1), 
 
 where I and m are scalars. 
 
 4. If Tp = Tfx^T^ = 1, and S.a^p = 0, shew by direct trans- 
 formations that 
 
 Interpret this theorem geometrically. 
 
90 QUATERNIONS. 
 
 5. If ,Sf«/3 = 0, ra = y/8=l, shew that 
 
 (1 + a'O/S = 2 cos '^- a^/3 = 2>Sfa « . « 2 ^. 
 
 6. Put in its simplest form the equation 
 
 pS . Fa^ F^7 VyoL = aY. Fva Fa/3 + & F. Fa^S F/37 + c F. F/37 F^a ; 
 and shew that a = S .^yp, &c. 
 
 7. Shew that any quaternion may in general, in one way only, 
 be expressed as a homogeneous linear function of four given 
 quaternions. Point out the nature of the exceptional cases. Also 
 find the simplest form in which any quaternion may generally be 
 expressed in terms of two given quaternions. 
 
 8. Prove the following theorems, and exhibit them as proper- 
 ties of determinants : — 
 
 (a) S.(a+0)(^ + ry)(ry + a) = 2S.a^y, 
 
 (b) S . Va^V^yVya =-(S. a/SyY, 
 
 (c) S.V{a + 0)(/3 + y)V(^ + ry)(y + c)V(y+a)(a + ^) 
 
 (d) S . ViVa^V^y) ViV^r^VriOi) F(F7aFa/3) = -{S. oi^yy, 
 
 (e) S.Be^=-16{S.a0yy, 
 
 where S = F(F(a + yS) (;S + 7) V(/3 + 7) (7 + a)), 
 
 6 = F(F(/3 + 7) (7 + «) F(7 + a) (a + ;8)), 
 ?=F(F(7 + a)(a + ^)F(«+^)(/3 + 7)). 
 
 9. Prove the common formula for the product of two determi- 
 nants of the third order in the form 
 
 ,S.a/37^.«A% = 
 
 Saa^ S^a^ Sya^ 
 &/3, 8^/3^ Sy^^ 
 Say^ S^y, Syy^ 
 
 10. Shew that, whatever be the eight vectors involved, 
 
 Saa^ Sa^^ Say^ SaB, = 8 .a/3yS .0^y^B^Sa^(B - B) = 0. 
 >S/3a, 8^/3^ S/3y^ 8/38, 
 Sya, Sy^, 8yy, 8yB, 
 SBa, 88^, SSy, 8S8, 
 
 If the single term Saa, be changed to Sa,a,, the value of the 
 determinant is 
 
 8.^yBS.^j,8,87^(a,-o^). 
 
INTERPRETATIONS AND TRANSFORMATIONS. 91 
 
 State these as propositions in spherical trigonometry. 
 Form the corresponding null determinant for any two groups 
 of five quaternions : and give its geometrical interpretation. 
 
 11. If, in § 102, a, /S, 7 be three mutually perpendicular 
 vectors, can anything be predicated as to a,, /3,, 7, ? If a, /3, 7 be 
 rectangular mto^ vectors, what of a^, ^^, 7^ ? 
 
 12. If a, ^, 7, a', ^', 7' be two sets of rectangular unit-vectors, 
 shew that 
 
 SaoL' = Sy^'S/3y'-8l3^'Syj', &c., &c. 
 
 13. The lines bisecting pairs of opposite sides of a quadri- 
 lateral (plane or gauche) are perpendicular to each other when the 
 diagonals of the quadrilateral are equal. 
 
 14. Shew that 
 
 {a) S.c/=2S'q-T'q, 
 
 {b) S.q' = S'q~3SqrVq, 
 
 (c) a'^V + /SP.a/37=F^a,87, 
 
 {d) /Sf ( F . a^7 F . /37a F . 7a/3) = 4^S'x^S^y8yaS . a/37, 
 
 (e) V.q'={SS'q-rVq)Vq, 
 if) q UVq'' = -Sq.UVq + TVq; 
 and interpret each as a formula in plane or spherical trigonometry. 
 
 15. If q be an undetermined quaternion, what loci are repre- 
 sented by 
 
 (a) {qory=-a\ 
 
 (b) {qorr = a\ 
 
 (c) 8.iq-ay=a\ 
 
 where a is any given scalar and a any given vector ? 
 
 16. If g" be any quaternion, shew that the equation 
 
 is satisfied, not alone hy Q = ± q, but also by 
 
 Q = ±J-l{Sq.UVq-TVq). 
 
 (Hamilton, Lectures, p. 673.) 
 
 1 7. Wherein consists the difference between the two equations 
 
 P^ = l, andf^V = -l? 
 
92 QUATEBNIONS. 
 
 What is the full interpretation of each, a being a given, and p an 
 undetermined, vector ? 
 
 18. Find the full consequences of each of the following 
 groups of equations, as regards both the unknown vector p and 
 the given vectors a, ^, y : — 
 
 (a) . r^n (^) S.a^p = 0, (c) S.a/3p =0, 
 S.^yp = 0; g^^ ^Q, s.a/3yp = 0. 
 
 19. From §§ 74, 110, shew that, if e be any unit-vector, and m 
 
 , „, WITT . rmr 
 
 any scalar, e = cos -^ + e sin -^ . 
 
 Hence shew that if a, /S, 7 be radii drawn to the corners of a tri- 
 angle on the unit-sphere, whose spherical excess is m right angles, 
 
 g + ZS 7 + « ^ + 7 ^ ^m 
 ;8-|-7"a+/3'7-t-a 
 
 Also that, if ^, i?, be the angles of the triangle, we have 
 
 2cr 25 2^ 
 
 20. Shew that for any three vectors a, /8, 7 we have 
 
 ( Ua/3f + ( f7/S7)' + ( C^a7)' -F ( fT. a/37)'' + 4:Uary. 8Ua/3SU^j = - 2. 
 
 (Hamilton, Elements, p. 388.) 
 
 21. If ttj, a^, a,, « be any four scalars, and p^, p^, p, any three 
 vectors, shew that 
 
 {S . p,p,p,r + (S . a,Vp,p,Y + x' (2 Vp,p,f -x'i%. a, {p, - p,))" 
 
 + 2n {so' + Sp^p^ + a,a^) = 2U {x' + p') + 2na' 
 + ^{{x' + a,' + p/) (( 7p,p3)^ + 2a,a, {x' + Sp,p,) - x" (p, - p/)} ; 
 where Ha^ = a'a,^a^\ 
 
 Verify this formula by a simple process in the particular case 
 
 cii = cfg = a, = « = 0. 
 
 {Ibid.) 
 
 22. Eliminate p from the equations 
 
 F./3pap = 0, ;S7p = 0; 
 and state the problem and its solution in a geometrical form. 
 
INTEKPRETATIONS AND TRANSFORMATIONS. 93 
 
 23. If p, q, r, s be four versors, such that 
 
 qj} = — sr = a, 
 
 rq = -ps = ^, 
 where a and /8 are unit-vectors ; shew that 
 
 S{V. VsrqV.VrVp) = 0. 
 Interpret this as a property of a spherical quadrilateral. 
 
 24. Shew that, ii pq, rs, pr,. and qs be vectors, we have 
 
 SiV.VprsV.VqVr)=0. 
 
 2.5. If a, /3, 7 be unit-vectors, 
 
 r^yS . «^7 = - a (1 - ,Sf=^7) - /3 {SajS^r + Sa^) 
 
 - 7 {Soi^S/3ry + Say). 
 
 26. If i, j, k, i, j', k', be two sets of rectangular unit-vectors, 
 shew that 
 
 S . Vii'Vjj'Vkk' = {SyJ - {Sji'f 
 
 = {Sjk'f - (SkjJ = &c., 
 and find the values of the vector of the same product. 
 
 27. If a, y8, 7 be a rectangular unit-vector system, shew that, 
 whatever be X, fi, v, 
 
 \SHoi + /mS'jj + v8'kl3, 
 
 XS'ky + fiSHj3+vS'ja, 
 and X/Sfy/3 + fiS'kct + vSSj, 
 
 are coplanar vectors. What is the connection between this and 
 the result of the preceding example ? 
 
CHAPTEE IV. 
 
 DIFFERENTIATION OF QUATERNIONS. 
 
 135. In Chapter I. we have already considered as a special 
 case the differentiation of a vector function of a scalar independent 
 variable : and it is easy to see at once that a similar process is 
 applicable to a quaternion function of a scalar independent variable. 
 The differential, or differential coefficient, thus found, is in general 
 another function of the same scalar variable ; and can therefore be 
 differentiated anew by a second, third, &c. application of the same 
 process. And precisely similar remarks apply to partial differentia- 
 tion of a quaternion function of any number of scalar independent 
 variables. In fact, this process is identical with ordinary differ- 
 entiation. 
 
 136. But when we come to differentiate a function of a vector, 
 or of a (juaternion, some caution is requisite ; there is, in general 
 (except, of course, when the independent variable is a mere scalar), 
 nothing which can be called a differential coefficient ; and in fact 
 we require (as already hinted in § 33) to employ a definition of a 
 differential, somewhat different from the ordinary one but, coinciding 
 with it when applied to functions of mere scalar variables. 
 
 137. li r = F (q) be a function of a quaternion q, 
 
 dr = dFq = ^^n{F(q + ^)-F(q)}, 
 
 where n is a scalar which is ultimately to be made infinite, is 
 defined to be the differential of r or Fq. 
 
 Here dq may be any quaternion whatever, and the right-hand 
 member may be written 
 
1 39- J DIFFERENTIATION OF QUATERNIONS. 95 
 
 where / is a new function, depending on the form of F; homo- 
 geneous and of the first degree in dq; but not, in general, capable 
 of being put in the form 
 
 f {q) dq. 
 
 138. To make more clear these last remarks, we may observe 
 that the function 
 
 / i9> dq), 
 thus derived as the differential of F (q), is distnhutive with respect 
 to dq. That is 
 
 /(g, r + s)=/(g,r)+/(g', s), 
 
 r and s being any quaternions. 
 
 For f{q,r + s)=^^n\F{q + ''±?)-F{q)^ 
 
 11. 
 
 <^n\F{. 
 
 =f{<l,r)+f{q,s). 
 And, as a particular case, it is obvious that if x be any scalar, 
 f {q, xr) = xf {q, r). 
 
 139, And if we define in the same way 
 
 dF {q, r, s ) 
 
 as being the value of 
 
 dq dr ds \ -r, , C\ 
 
 where q, r, s, ... dq, dr, ds, are any quaternions whatever ; we 
 
 shall obviously arrive at a result which may be written 
 
 f(q, r, s,...dq, dr, ds, ), 
 
 where / is homogeneous and linear in the system of quaternions 
 
 dq, dr, ds, and distributive with respect to each of them. Thus, 
 
 in differentiating any power, product, &c. of one or more quater- 
 nions, each factor is to be differentiated as if it alone were variable ; 
 and the terms corresponding to these are to be added for the com- 
 plete differential. This differs from the ordinary process of scalar 
 differentiation solely in the fact that, on account of the non-com- 
 mutative property of quaternion multiplication, each factor must in 
 general be differentiated in situ. Thus 
 
 d iqr) = dq.r+ qdr, but not generally = rdq + qdr. 
 
96 QUATERNIONS. [l40- 
 
 140. As Examples we take chiefly those which lead to results 
 which will be of constant use to us in succeeding Chapters. Some 
 of the work will be given at full length as an exercise in quaternion 
 transformations. 
 
 (1) {Tpf = -p\ 
 
 The differential of the left-hand side is simply, since Tp is a 
 scalar, 
 
 2I> dTp. 
 
 That of p" is ^^n\^{p + ^)^-p- 
 
 = <^«g>Spdp + M)(§104) 
 
 = 2Spdp. 
 Hence Tp dTp=^ -Spdp, 
 
 or dTp = -S.Updp=S^, 
 
 Ip p 
 
 (2) Again, p=TpUp 
 
 dp = dTp. Up + TpdUp, 
 
 , dp dTp dUp 
 
 whence -C = ^ + -^ 
 
 p Ip Up 
 
 P Vp ' ^ ' 
 
 Hence dUp^^dp 
 
 Up p 
 
 This may be transformed into F^^ or ^A &c 
 
 p Ip ' 
 
 (3) {TqY = qKq 
 2TqdTq = d{qKq) = ^^n^(q^^)K(q+'kyqKq 
 
 = ^^n{&±ll^Ul.dqKdq), 
 
 = qKdq + dqKq, 
 
 = qKdq + K{qKdq) (§55), 
 
 = 2S. qKdq = 28. dqKq. 
 
140.] DIFFERENTIATION OF QUATERNIONS. 97 
 
 Hence dTq = S. dq UKq = S.dq Uq-' = Tq8^, 
 
 since Tq^TKq, and UKq=Uq-\ 
 
 [If q = p, & vector, Kq = Kp = -p, and the formula becomes 
 dTp = -S. Up dp, asin(l).] 
 
 Again, dq = Tqd Uq + UqdTq, 
 
 which gives dq^dTq dUq_ 
 
 ^ q Tq^ Uq' 
 
 whence, as gdq ^dTq 
 
 q Tq' 
 
 we have ydq^dUq^ 
 
 q Uq 
 
 (4) d{q')=<.n^(q+^J-qj 
 
 — qdq + dq.q 
 
 = 2S . qdq + 2Sq . Vdq + 2Sdq . Vq. 
 If g' be a vector, as p, Sq and Sdq vanish, and we have 
 d(p'') = 2Spdp, as in (1). 
 
 (5) Let q = r^. 
 This gives dr^ = dq. But 
 
 dr = d (q') = qdq +dq.q. 
 This, multiplied by q and into Kq, gives the two equations 
 
 qdr = q^dq + qdq . q, 
 and drKq = dq . Tq' -f qdq . Kq. 
 
 Adding, we have 
 
 qdr + dr.Kq = {q' + Tq' + 2Sq .q)dq = 4<8q . qdq ; 
 
 whence dq, i.e. dr^, is at once found in terms of dr. This process 
 is given by Hamilton, Lectures, p. 628. See also § 193 below, and 
 No. 7 of the Miscellaneous Examples at the end of this work. 
 
 (6) qq-' = l, 
 
 qdq^ + dq.q~^ = Q; 
 . : dq'^ = — q'^dq . q^. 
 If 5 is a vector, = p suppose, 
 
 dq^ = dp~^ = — p'^dp . p"' 
 T. Q. I. 7 
 
98 QUATERNIONS. [14°' 
 
 QUATERNIONS. 
 
 dp 
 
 ~7' 
 
 P P 
 
 -f^p. 
 \p 
 
 pip 
 
 = -K 
 
 fdp\l 
 \p)p' 
 
 (7) q = 8q+Vq, 
 dq = dSq + dVq. 
 
 But dq = 8dq + Vdq. 
 
 Comparing, we have 
 
 dSq = Sdq, dVq=Vdq. 
 Since Kq ==Sq— Vq, we find by a similar process 
 
 dKq = Kdq. 
 
 (8) In the expression qaq~\ where a is any constant quaternion, 
 q may be regarded as a mere versor, so that 
 
 (Tqy=l=qKq = qq-\ 
 Thus S.dqKq^O; 
 
 and hence dqq''^ = — qdq'^, 
 
 as well as 
 
 q'^dq = — dq'^q, 
 
 are vectors. But, if a = a + a, where a is a scalar, qsiq~^ = a, i.e. 
 
 constant, so that we are concerned only with d (qaq'^). 
 Hence d{qaq'^) = dq aq'^ — qaq~^dqq~\ 
 
 = dqq''^ . qaq"^ — qaq^ . dqq'^, 
 
 = 2V. dqq^qoiq^ = — 2V. qdq'^qaq'^. 
 
 (9) With the restriction in (8) above we may write 
 
 q = cos u+ 6 sin u, 
 
 where Te = l; SOdd = 0. 
 
 Hence q''^ = cos m — ^ sin m ; 
 
 — q'^dq = dq'^q = { — (sin u+ 6 cos u) du — dd sin u] (cos m + ^ sin «) 
 
 = — 6 du — dd sin u (cos m + ^ sin m) ; 
 
 — qdq'^ = dq q~^ = 6 du + dd sin u (cos u—d sin tt). 
 
 Both forms are represented as linear functions of the rectangular 
 system of vectors 
 
 e, dd, edo. 
 
I40.] DIFFERENTIATION OF QUATERNIONS, 99 
 
 If the plane of q be fixed, ^ is a constant unit vector, and 
 dq q~^ = - dq-' q = 0du. 
 
 (10) The equation (belonging to a family of spheres) 
 
 p-ct 
 gives Sdp{{p + a)-e'{p-a)] = Q; 
 
 or, by elimination of e, 
 
 Sdp{{p + ar-{p-ay}=0, 
 whose geometrical interpretation gives a well-knovifn theorem. 
 
 If we confine our attention to a plane section through the 
 vector a, viz. 
 
 S.yap = 0, S.'yadp = 0; or S^p = 0, 8^dp = 0, 
 where /Sl|F7a|lFap ; 
 
 we have 
 dp\\ V. ^ {(p + a)-' -(p- «)-'} or V. dp F/3 {(/J + «)-' - (p - a)-} = 0. 
 
 (11) Again, from 
 
 SUP + ^ 
 
 = e 
 
 p-a 
 
 (which is the equation of the family of tores produced by the 
 rotation of a group of circles about their common chord) we have 
 
 SU.(p + a)(p-a) = -e. 
 
 Also this gives FfT" . (p + a) (p - a) = /3 = ^l-e' U . Yap. 
 
 We obtain from the first of these, by differentiation, 
 
 s{yjf-^-U{p + '^)U{p-a)+U{p + a)V^^.U{p-a)yQ, 
 
 or S . ^dp [{p + «)-' - (p - a)-'} = 0. 
 
 If we consider /S to be constant, we limit ourselves to a meridian 
 section of the surface (i.e. a circle) and the form of the equation 
 shews that, as /S is perpendicular to the plane of a, p (and, of 
 course, dp), 
 
 F.c^p{(p + a)--(p-a)-} = 0. 
 
 We leave to the reader the differentiation of the vector form of 
 the equation above. 
 
 These results are useful, not only as elementary proofs of 
 geometrical theorems but, as hints on the integration of various 
 simple forms. 
 
 7—2 
 
Lhi-^I 
 
 100 QUATERNIONS. 
 
 (12) As a final instance, take the equation 
 
 where p' stands for dp/ds, s being the arc of a curve. 
 
 By § 38, a is a unit vector, and the expression shews by its 
 form that it belongs to a plane curve. Let ;8 be a vector in its 
 plane, and perpendicular to a. Operate by >S . /3 and we get 
 
 ^-^Spp'-8l3p'=0, 
 
 whose integral is 
 
 p'-S^p = 0, 
 
 the tensor of /9 being the constant of integration. 
 
 141. Successive differentiation of course presents no new 
 difficulty. 
 
 Thus, we have seen that 
 
 d (q^) = dq.q + qdq. 
 Differentiating again, we have 
 
 d'(q') = d'q.q + 2(dqy + qd'q, 
 and so on for higher orders. 
 
 If 5' be a vector, as p, we have, § 140 (1), 
 d(p') = 28pdp. 
 Hence d' (p") = 2 (dpY + 2Spd'p, and so on. 
 
 Similarly d'Up = -d(^, Vpdp) . 
 
 Tj , ,1 2dTp 2Spdp 
 
 But ^Tp^—t/ = ^' 
 
 and d . Vpdp = V . pd^p. 
 
 Hence d^Up^^^iVpdpf - mA ^^EeIpWr 
 J-P Ip Ip 
 
 = ^, {{Vpdpy + p' Vpd'p - 2 VpdpSpdp]*. 
 
 142. If the first differential of q be considered as a constant 
 quaternion, we have, of course, 
 
 d'q = 0, d'q = 0, &c. 
 and the preceding formulae become considerably simplified. 
 
 * This may be farther simpliaed; but it may be well to caution the student that 
 we cannot, for Buoh a purpose, write the above expression as 
 
 ^^V.p{dpVpdp + d?p. p^-2dpSpdp}. 
 
1 44- J DIFFERENTIATION OF QLTATBBNIONS. 101 
 
 Hamilton has shewn that in this case Taylor's Theorem admits 
 of an easy extension to quaternions. That is, we may write 
 
 f{q + xdq) =f{q) + xdf (q) + ^ dy (3) + 
 
 if <Pq = 0; subject, of course, to particular exceptions and limita- 
 tions as in the ordinary applications to functions of scalar variables. 
 Thus, let 
 
 / (2) — ?'' ^^^ ^® have 
 ^f (?) = Q'^1 + 1^1 -q + dq.q^, 
 d'fiq) = 2dq . qdq + 2q {dqf + 2 (dqfq, 
 dY{q) = 6idqy, 
 and it is easy to verify by multiplication that we have rigorously 
 (q + xdqf = ^ -\-x (q^dq + qdq .q + dq.q') + 
 
 x' (dq .qdq + q {dqf + {dqjq) + eg' {dqf ; 
 which is the value given by the application of the above form of 
 Taylor's Theorem. 
 
 As we shall not have occasion to employ this theorem, and as 
 the demonstrations which have been found are all too laborious for 
 an elementary treatise, we refer the reader to Hamilton's works, 
 where he will find several of them. 
 
 143. To differentiate a function of a function of a quaternion 
 we proceed as with scalar variables, attending to the peculiarities 
 already pointed out. 
 
 144. A case of considerable importance in geometrical and 
 physical applications of quaternions is the differentiation of a scalar 
 function of p, the vector of any point in space. 
 
 Let F(p)=C, 
 
 where ^ is a scalar function and G an arbitrary constant, be the 
 equation of a series of surfaces. Its differential, 
 
 Zip, dp) = 0, 
 is, of course, a scalar function : and, being homogeneous and linear 
 in dp, § 137, may be thus written, 
 
 Svdp = 0, 
 where i' is a vector, in general a function of p. 
 
 This vector, v, is shewn, by the last written equation, to have 
 the direction of the normal to the given surface at the extremity 
 of p. It is, in fact, perpendicular to every tangent line dp; 
 i 36, 98. 
 
102 QUATERNIONS. [H5- 
 
 145. This leads us directly to one of the most remarkable 
 operators peculiar to the Quaternion Calculus ; viz. 
 
 ^-'d-.+^Ty + '^d-, W> 
 
 to whose elementary properties we will devote the remainder of 
 the chapter. The above definition is that originally given by 
 Hamilton, before the calculus had even partially thrown off its 
 early Cartesian trammels. Since i, j, k stand for any system of 
 rectangular unit vectors, while x, y, z are Cartesian co-ordinates 
 referred to these as axes, it is implied in (1) that V is an Invariant. 
 This will presently be justified. Meanwhile it is easy to see that 
 if p be the vector of any point in space, so that 
 
 p = ix + jy + kz, 
 
 we have V/3 = — 3 (2), 
 
 vr,.vy.^T7T7-=^^^'=^^.^. ..,3), 
 
 V {TpY = n {Tpf-^^Tp = n{Tpr-^p (4), 
 
 of which the most important case is 
 
 Tp {Tpf Tp' ^''^■ 
 
 A second application gives 
 
 ^'Tp = -^-^T^-P = ^ C6). 
 
 Again 
 
 Vp = -^ = VTp.Up + Tp.VUp = -\ + Tp.VUp, 
 
 so that VUp = -^ (7). 
 
 By the definition (1) we see that 
 
 -=-{(S"-(|)"H0} («). 
 
 the negative of what has been called Laplace's Operator. Thus 
 (6) is merely a special case of Laplace's equation for the potential 
 in free space. 
 
 Again we see by (2), a being any constant vector, 
 
 SaV .p- S/Sap = r. aVVp = 0, 
 from which 
 
 VVap +VaV.p = (SaV. p - aSVp) + {aSVp - VSap) = 0. 
 
•(3), 
 
 1 46. J DIFFERENTIATION OF QUATERNIONS. 103 
 
 [The student should note here that, in expanding the terms of 
 the vector function on the left by the formula (1) of | 90, the 
 partial terms are written so that V is always to the left of (though 
 not necessarily contiguous to) its subject, p.] 
 
 146. By the help of these elementary results, of which (3) 
 and (7) are specially noteworthy, we easily find the effect of V 
 upon more complex functions. 
 
 For instance, taking different modes of operating, we have with 
 
 a = ia +jb + kc 
 SaV . p = VSap =:-V {ax + by + cz) = -{ia + jb + kc) = - a...(l), 
 or thus VSap = iSai + jSaj + k8a.k = — a ; 
 
 while - FaV.p = VFap = -VFpa = -V(pa-5f/3a) 
 
 = 3a-a = 2a .• (2), 
 
 or V VoLp = i Vai + j Vaj + k Vak = 2 a. 
 
 From the latter of these we have 
 
 Fap _ 2a _ -SpFap 2ap^ + S pVa p _ ap' - SpSap 
 ly~T^' Tp' ~ Tp' ' ~ Tp' 
 
 [where note that the first of these values is obtained thus, 
 „ Fap VVap _ 1 „ 
 
 The order is of vital importance.] 
 This, in its turn, gives 
 
 <? ^^V ^"P - ^'^^P 3>Sfap/gp5p _ . Sap 
 
 S.8pV^--^ 2y -^Tp' ^*^' 
 
 where 8 is a symbol of variation. This is a result of great physical 
 importance, especially in electro-dynamics. We may alter the 
 right-hand member (by § 145, (5)) so as to write the whole in the 
 form 
 
 'S.SpV-^ = S&V.^ = &V.S^ (4'). 
 
 And it is easy to see that 8 may be substituted for V in the left- 
 hand member. [The reason for this may be traced in the result of 
 §145(6).] 
 
 As an addition to these examples, note that (by (2) of § 148, 
 
 below) 
 
 Vap . Fap 
 
104 QUATERNIONS. [H?- 
 
 which may be contrasted with (4) above. The altered position of 
 the point produces a complete change in the meaning of the left- 
 hand member. 
 
 Finally, we see that 
 
 VSa/Sf/3/j = -S/8a (5), 
 
 a result which will be found useful in next Chapter. 
 
 147. Still more important are the results obtained from the 
 operator V when it is applied to 
 
 <T = i^ + jV + k^ (1): 
 
 and functions of this vector. (Here ^, 77, ^ are functions of x, y, z, 
 so that o- is a vector whose value is definite for each point of 
 space.) 
 
 We have at once 
 
 dx dy dzj \dy dz 
 
 -m^iM^-i) «■ 
 
 Those who are acquainted with mathematical physics will 
 recognize at a glance the importance of this expression. For, if a 
 denote the displacement (or the velocity) of a point originally 
 situated at p, it is clear that 
 
 «''=-(l4M) <«)■ 
 
 represents the consequent condensation of the group of points (say 
 particles of a fluid) originally in the neighbourhood of p, while 
 
 represents double the (vector) axis of rotation of the same group. 
 
 Other, and more purely quaternion, methods will be employed 
 later, to deduce these results afresh, and to develop their applica- 
 tions. They are introduced here in their semi-Cartesian form 
 merely to shew the importance of the operator V. 
 
 148. Let us recur to the equation of § 144, viz. 
 
 Fip) = G (1). 
 
 Ordinary complete differentiation gives 
 
 ,„ dF , dF. dF , 
 
I49-] DIFFERENTIATION OF QUATERNIONS. 105 
 
 or, what is obviously the same, 
 
 dF==-SdpVF (2), 
 
 which we may write, if we please, as 
 
 -SdpV .F. 
 Here the point is obviously unnecessary, but we shall soon 
 come to cases in which it, or some equivalent, is indispensable. 
 Thus it appears that the operator 
 
 -SdpV 
 
 is equivalent to total differentiation as involved in the passage 
 from p to p + dp. Hence, of course, as in § 144 
 
 dF ip)=0 = 8vdp = - SdpVF, 
 
 and thus (as dp may have any of an infinite number of values) 
 
 v = -VF (3). 
 
 If we pass from one surface of the series (1) to a consecutive 
 one by the vector Bp, we have 
 
 -8vBp = Ba 
 
 Hence — v'^SC is a value of Sp ; so that the teiisor of v is, at every 
 point, inversely as the normal distance between two consecutive 
 surfaces of the series. 
 
 Thus, if (1) be the equation of a series of equipotential surfaces, 
 V, as given by (3), represents the vector force at the point p ; if 
 (1) be a set of isothermals, v (multiplied by the conductivity, a 
 scalar) is the flux of heat, &c. 
 
 149. We may extend the result (2) of § 148 to vector functions 
 by multiplying both sides into a constant vector, a, and adding 
 three such expressions together. Thus if 
 
 ,7 = aF+^F^ + ryF„ 
 we obtain at once 
 
 da = -8{dpV)a = -SdpV.a- (4). 
 
 But here the brackets, or the point, should (at first, at least) be 
 employed ; otherwise we might confound the expression with 
 
 da- = — 8. dpVa 
 which, as equating a vector to a scalar, is an absurdity (unless both 
 sides vanish). See again, § 148. 
 
 Finally, from (2) and (4), we have for any quaternion 
 
 dq = -SdpV.q (5). 
 
106 QUATERNIONS. [l49- 
 
 The student's attention is particularly called to the simple 
 processes we have adopted in obtaining (4) and (5) from (2) of 
 § 148 ; because, in later chapters, other and more complex results 
 obtained by the same processes will frequently be taken for 
 granted ; especially when other operators than S (dpV) are em- 
 ployed. The precautions necessary in such matters are two-fold, 
 (a) the operator must never be placed anywhere after the operand; 
 (h) its commutative or non-commutative character must be carefully 
 kept in view. 
 
 EXAMPLES TO CHAPTER IV. 
 
 1. Shew that 
 
 (a) d.SUq = S.UqV'^j = -S^^TVUq, 
 
 (6) d.VUq=V.Uq-'V{dq.q-'), 
 
 (0) d.TYUq = 8^^ = 8^^SUq, 
 
 (d) d.a.'' = '^a^*'dx, 
 
 (e) d\Tq = [S' .dqq-'-S. (dqq-J} Tq = -TqV'^. 
 
 2. If Fp = -2<.8apS^p + ^gp' 
 give dFp = Svdp, 
 
 shew that v^tV. ap/3 + {g + tSajS) p. 
 
 3. Find the maximum and minimum values of Tp, when 
 
 (a) (p-af = -a'. 
 
 (b) (p-ay = -a') 
 
 S^p = 0; I 
 
 (c) p' - SocpS/Bp = -a'; 
 
 {d) p' - SapS^p = - a^) 
 Syp = 0. j 
 
 Point out the differences, geometrical and analytical, between 
 (a), (c) on the one hand, and (6), (d) on the other. 
 State each of the problems in words. 
 
DIFFERENTIATION OF QUATERNIONS. 107 
 
 4. With V as in § 145, shew that 
 
 /SV {qoiq-^) = 2S.Vqaq-' = 2S.aq-'Vq. 
 VV (qaq-') = 2qaq-'S.Vq q-' - 2S (qaq-'V) q . q-' 
 where g' is a function of p, and a any constant vector. 
 
 5. Shew that, if a, /3, 7 be a constant rectangular unit- vector 
 system, 
 
 qaq'^d . qaq'^ + q/3q~^d . q^q'^ + qr^q'^d . q^q~^ = idqq'^ 
 
 6. Integrate the differential equations : — 
 
 (a) p + aS^p = 7, 
 
 (6) q + aq = b, 
 
 (c) = Vad. Proc. B. S. E. 1870. 
 
 7. Shew that 
 
 (a) JVadpS^p = ^V.a(p8^p + V. ^JVpdp). 
 
 (b) Jdp Vap = ^{pVap-V. aJVpdp) + 8 . aJVpdp. 
 
 (c) JV . Vadp Vl3p = UaS. ^jVpdp + ^S . aJVpdp - pS . a^p), 
 
 {d) fS . Vadp V^p = J (SupS^p - p'Soi/S - S . a^JVpdp). 
 
 When these integrals are taken round a closed plane curve we 
 have 
 
 JVpdp = 2Ay, 
 
 where A is the area, and 7 a unit vector perpendicular to its plane. 
 In this case 
 
 Jdp Vap = A Vja + 2ASya, 
 
 JV.VadpV^p = A (a8l3y + /3&7), 
 
 JS.VadpV^p = A8.al3y. 
 
 8. State, in words, the geometrical theorems involved in the 
 equations of § 140, (10), (11), (12). 
 
 9. Shew (by means of § 91) that 
 
 V^,S.o-jVo- = ^.V(rVj.t7^, 
 
 where V, V^, operate respectively on a; er^ ; but, after the 
 operations are performed, we put 
 
 V, = V, a^ = <T. 
 
CHAPTEK V. 
 
 THE SOLUTION OF EQUATIONS OP THE FIRST DEGREE. 
 
 150. We have seen that the differentiation of any function 
 whatever of a quaternion, q, leads to an equation of the form 
 
 dr=f(q, dq), 
 where / is linear and homogeneous in dq. To complete the process 
 of differentiation, we must have the means of solving this equation, 
 so as to be able to exhibit directly the value of dq. 
 
 This general equation is not of so much practical importance 
 as the particular case in which dq is a vector ; and, besides, as we 
 proceed to shew^ the solution of the general question may easily be 
 made to depend upon that of the particular case ; so that we shall 
 commence with the latter. 
 
 The most general expression for the function / is easily seen 
 to be 
 
 dr =f(q, dq) = 2 F. adqb + 8 . cdq, 
 
 where a, h, and c may be any quaternion functions of q whatever, 
 including even scalar constants. Every possible term of a linear 
 and homogeneous function is reducible to this form, as the reader 
 may see at once by writing down all the forms he can devise. 
 Taking the scalars of both sides, we have 
 
 Sdr = S. cdq = SdqSc + 8 . VdqVc. 
 But we have also, by taking the vector parts, 
 
 Vdr = tV.adqb = 8dq.tVab + tV.a{Vdq)b. 
 Eliminating Sdq between the equations for Sd7' and Vdr it is 
 obvious that a linear and vector expression in Vdq will remain. 
 Such an expression, so far as it contains Vdq, may always be 
 reduced to the form of a sum of terms of the type aS . /S Vdq, by 
 the help of formulte like those in §§ 90, 91. Solving this, we have 
 Vdq, and Sdq is then found from the preceding equation. 
 
I 5 2. J SOLUTION OF EQUATIONS. 109 
 
 151. The problem may now be stated thus. 
 Find the value of p from the equation 
 
 aSPp + a^S^j} +... = t. aS^p = 7, 
 
 where a, ^8, a^, /3i,...7 are given vectors. 
 
 The most general form of the left-hand member requires but 
 three distinct or independent terms. These, however, in con- 
 sequence of the form of the expression, involve scalar constants 
 only; since the whole can obviously be reduced to terms of the 
 forms AiSip, BiSjp, CjSjp, &c. and there are only nine such forms. 
 In fact we may write the most general form either as 
 
 iSap +jS^p + kSyp, 
 or as a^Sip + ^^Sjp + y^Skp, 
 
 according as we arrange it by the vector, or by the scalar, factors 
 of the several terms. But the form 
 
 aS^p + (x^S^,p + a^Sj^p, 
 is that which, as committing us to no special system of vectors of 
 reference, is most convenient for a discussion of its properties. 
 If we write, with Hamilton, 
 
 cPp = -Z.a8^p (1), 
 
 the given equation may be written 
 
 <jip = j, 
 or p = ^"V, 
 
 and the object of our investigation is to find the value of the 
 inverse function 0~'. 
 
 It is important to remark that the definition (1) shews </> to be 
 distributive, so that 
 
 4> (p + 0-) = 4>P + 4"^- 
 A particular case of this is 
 
 ^ {xp) = cc^p, 
 where a; is a scalar. 
 
 Also, by the statement above, it is clear that ^, in its most 
 general form, essentially involves nine independent scalars. 
 
 152. We have seen that any vector whatever may be expressed 
 linearly in terms of any three non-coplanar vectors. Hence, we 
 should expect d priori that a vector such as <^<^0/3, or ^'p, for 
 instance, should be capable of expression in terms of p, ^p, and 
 
 [This is, of course, on the supposition that p, <pp, and (p^p are 
 
110 QUATERNIONS. [ I 5 2 ■ 
 
 not generally coplanar. But it may easily be seen to extend to 
 that case also. For if these vectors be generally coplanar, so are <f>p, 
 ^'p, and (j)^p, since they may be written o-, 0o-, and (^V. And thus, 
 of course, <f)^p can be expressed as above. If in a particular case, 
 we should have, for some definite vector p, ^p = gp where ^r is a 
 scalar, we shall obviously have ^'p = g^p and 0'/3 = g^p, so that the 
 equation will still subsist. And a similar explanation holds for 
 the particular case when, for some definite value of p, the three 
 vectors p, (j)p, (jy^p are coplanar. For then we have an equation of 
 the form 
 
 ^V = Ap + B4>p, 
 
 which gives <p^p = A<f)p + B<f)^p 
 
 = ABp+(A+B')<jip, 
 
 so that (f>^p is in the same plane.] 
 If, then, we write 
 
 - 4>'p = xp + y^p + z4>^p (1), 
 
 and bear in mind the distributive character of the operator tf), it is 
 evident (if only ex absurdo) that x, y, z are quantities independent 
 of the vector p. 
 
 [The words above, "it is evident," have been objected to by 
 more than one correspondent. But, on full consideration, I not 
 only leave them where they are, but put them in Italics. For 
 they are, of course, addressed to the reader only ; and it is to be 
 presumed that, before he reaches them, he has mastered the 
 contents of at least the more important previous sections which 
 bear on this question, such as §§ 23, 151. If, with these sections 
 in his mind, and a homogeneous linear equation such as (1) before 
 him, he does not see the " evidence," he has begun the study of 
 Quaternions too soon. A formal demonstration, giving the values 
 of X, y, z, will however be found in §| 156 — 9 below.] 
 
 If any three vectors, as i, j, k, be substituted for p, they will in 
 general enable us to assign the values of the three coefEcients on 
 the right side of the equation, and the solution of the problem of 
 §151 is complete. For by putting 0"'p for p and transposing, the 
 equation becomes 
 
 - x^'^p =yp + z^p + (js^p ; 
 
 that is, the unknown inverse function is expressed in terms of 
 direct operations. Should x vanish, while y remains finite, we 
 must substitute ^'^p for p, and have 
 
 - y^"V = 2p + (t>p; 
 
154-] SOLUTION OF EQUATIONS. Ill 
 
 and if x and y both vanish 
 
 - z<^-'p = p. 
 
 [We may remark here that it is in general possible to determine 
 x,y,z by putting one known vector for p in (1). The circumstances 
 in which some particular vector does not stiffice will be clear from 
 the theory to be given below.] 
 
 153. To illustrate this process by a simple example we shall 
 take the very important case in which belongs to a central 
 surface of the second order ; suppose an ellipsoid ; in which case it 
 will be shewn (in Chap. IX.) that we may write 
 
 (/)/3 = - aSSip — b^jSjp — c'JcSkp, 
 where i, j, k are parallel to the principal diameters, and the semi- 
 lengths of these are 1/a, 1/6, 1/c. 
 Here we have 
 
 <f>i = a\ ^S = aH, <})S = a\ 
 
 (j)k = c% (j>'k = 0% (ji'k = c'k. 
 
 Hence, putting separately i, j, k for p in the equation (1) of 
 last section, we have 
 
 — a^= X + ya' + za*, 
 
 -}f = x + yh'+ zb\ 
 
 — c° = x+yc'+ zc'. 
 Hence d\ 6^ c° are the roots of the cubic 
 
 which involves the conditions 
 
 ^^-^a' + b' + c'), 
 
 y = a'b' + bV + c'a', 
 
 x^-a'bV. 
 Thus, with the above value of 0, we have 
 
 (/)> = a'bVp - (aV + bY + cV) </)p + (a' + ¥ + c') <t>y 
 
 154. Putting (fr^a- in place of p (which is any vector what- 
 ever) and changing the order of the terms, we have the desired 
 inversion of the function cj) in the form 
 
 a%V<p-'(r = (aV + b'c' + cV) a-(a' + b' + c') <pa + <^V, 
 where the inverse function is expressed in terms of the direct 
 
112 QUATERNIONS. [ 1 5 5 . 
 
 function. For this particular case the solution we have given 
 is complete, and satisfactory; and it has the advantage of pre- 
 paring the reader to expect a similar form of solution in more 
 complex cases. 
 
 155. It may also be useful as a preparation for what follows, 
 if we put the equation of § 153 in the form 
 
 = $ (p) = (f>'p - (a' + 6= + O ^'p + (aV + b'c' + cV) <f>p - a'bVp 
 = [cp' - (a' + b' + c') 4>' + (a'b' + ¥c' + c'a') 4> - a'bV] p 
 = {(0-ci=)(<^-&^)(<^-c^)}p (2). 
 
 This last transformation is permitted because (§ 151) <f) is com- 
 mutative with scalars like a^, i.e. (p (a'p) = a^<j}p. The explanation 
 of its meaning must, however, be deferred to a later section. 
 
 (§ 177.) 
 
 Here we remark that the equation 
 
 V.p^p = 0, or ^p=gp, 
 
 where <f> is as in § 153, and g is some undetermined scalar, is 
 satisfied, not merely by every vector of null-length, but by the 
 definite system of three rectangular vectors Ai, Bj, Gk whatever 
 be their tensors, the corresponding particular values of g being 
 a\ 6^ c^ 
 
 156. We now give Hamilton's admirable investigation. 
 
 The most general form of a linear and vector function of a 
 vector may of course be written as 
 
 (pp = 'tV .qpr, 
 
 where q and r are any constant quaternions, either or both of which 
 may degrade to a scalar or a vector. 
 
 Hence, operating hj S.a- where o- is any vector whatever, 
 
 S<T^p=-8a^V .qpr = Sp1V.raq = 8p4>'a (1), 
 
 if we agree to write <^'(t = 'tV.ra-q, 
 
 and remember the proposition of § 88. The functions <j) and ^' 
 are thus conjugate to one another, and on this property the whole 
 investigation depends. 
 
 157. Let X, /J, be any two vectors, such that 
 
 (pp = VXfi. 
 Operating hj S .\ and S./jlwg have 
 
 S\<j>p = 0, Sfi<f>p = 0. 
 
I58.J 
 
 SOLUTION OF EQUATIONS. 
 
 113 
 
 But, introducing the conjugate function </>', these become 
 
 8p4)'\ = 0, Sp,p'fi = 0, 
 and give p in the form mp = F^'X^'^, 
 
 where m is a scalar which, as we shall presently see, is independent 
 of X, fi, and p. 
 
 But our original assumption gives 
 
 hence we have m<ji'^V\/i=V(f)'X^'fi (2) 
 
 and the problem of inverting is solved. 
 
 It remains to find the value of the constant m, and to express 
 
 the vector 
 
 as a function of VX/j,. 
 
 V<f>'X(f)'fi 
 
 158. To find the value of m, we may operate on (2) by S. (j)'v, 
 where v is any vector not coplanar with X and fi, and we get 
 mS . ^'z/0-' VX/j, = mS . v(f>(j)-' VXfi (by (1) of § 1 56) 
 = mS . Xfip = 8 . <f)'X^' iji,cj>'p, or 
 
 m = 
 
 8 . <f) X(j)/j,^'v 
 
 8.Xfiv (^^• 
 
 [That this quantity is independent of the particular vectors 
 X, p,, V is evident from the fact that if 
 
 X' = xX + yp, + zv, p,' = x^ + y^n + Zjy, and v =xp\. + yjji, + z^v 
 
 be any other three vectors (which is possible since X, p,, v are not 
 coplanar), we have 
 
 ^'X' = X(j>'X + y(f>'p, + Z(f)'v, &c., &c. ; 
 
 from which we deduce 
 
 8 . <j)'X'^'/i,'(f)'v' = 
 
 and 
 
 .fii 
 
 X 
 
 y 
 
 z 
 
 ^1 
 
 2'i 
 
 ^1 
 
 x^ 
 
 y^ 
 
 ^2 
 
 X 
 
 y 
 
 z 
 
 «=. 
 
 yi 
 
 ^1 
 
 ^^ 
 
 y. 
 
 ^2 
 
 8.<t>'x4>'p,<i>'v, 
 
 8 . Xpv, 
 
 so that the numerator and denominator of the fraction which ex- 
 presses m are altered in the same ratio. Each of these quantities 
 is in fact an Invariant, and the numerical multiplier is the same 
 for both when we pass from any one set of three vectors to another. 
 T. Q. I. 8 
 
114 
 
 QUATERNIONS. [l59- 
 
 A still simpler proof is obtained at once by writing X + «/a for \ 
 in (3), and noticing that neither numerator nor denominator is 
 altered.] 
 
 159. We have next to express 
 
 V(f>'\^'fi 
 as a function of VXfi. For this purpose let us change <f> to ^-g, 
 where g is any scalar. It is evident that ^' becomes ^' -g, and 
 our equation (2) becomes 
 
 'm,{^-grrx,M=V(<l>'-g)\(<l>'-g)iM; 
 
 = (m^"' —gx + /) ^^H'! suppose. 
 In this equation (see (3) above) 
 
 S.(cl,'-g)X{,j,'-g)fi(cl>'-g )v 
 
 = m — m^g + 'm^g^—g^ (4) 
 
 is what m becomes when (f> is changed into (f> — g; m^ and m^ being 
 two new scalar constants whose values are 
 
 S . (X (f>'/j, (])'v + fi ip'i' <j>'\ + V (f)'X 4>'fi) 
 S . Xfiv 
 
 8 . (Xfi (j)'v + fiv (f)'X + vX ^'fi) 
 ""^^ STX/Iv ■ 
 
 If, in these expressions, we put X + x/j, for X, we find that the terms 
 in X vanish identically ; so that they also are invariants. 
 
 Substituting for m^, and equating the coefficients of the 
 various powers of g after operating on both sides by ^ — g, we 
 have two identities and the following two equations, 
 
 ^2 = <}' + X' 
 
 ™i ~ 4'X + in4>~^. 
 [The first determines %, and shews that we were justified in 
 treating V {(f)'X/u, + X^'/n) as a linear and vector function of V . X/j-. 
 The result might have been also obtained thus, 
 
 8 .XxVX/j, = S .X(t)'X ij, = - 8 .Xfi.<})'X = - S .X(f)VXfi, 
 8 . iJi,xVXfjL = S . /iX<j>'/ji = — 8 . fjb^VXfi, 
 8 . VX VXfjL = 8 . (v(j)X fj, + vX4>'fi') 
 = m^8X/ji,v — 8 . XfKJj'v 
 = 8.v {mJ'Xfx-<pVXix); 
 
 m. 
 
l62.j SOLUTION OF EQUATIONS. 115 
 
 and all three (the utmost generality) are satisfied by 
 
 160. Eliminating ;^ from these equations we find 
 
 wij = (m^ — </)) + m4>~\ 
 or m0"* = OTi — TOj (^ + (^^ (5), 
 
 which contains the complete solution of linear and vector equa- 
 tions. 
 
 161. More to satisfy the student as to the validity of the 
 above investigation, about whose logic he may at first feel some 
 difficulties, than to obtain easy solutions, we take a few very 
 simple examples to begin with : we treat them with all desirable 
 prolixity, as useful practice in quaternion analysis ; and we append 
 for comparison easy solutions obtained by methods specially 
 adapted to each case. The advanced student need therefore pay 
 but little attention to the next ten sections. 
 
 162. Example I. 
 Let (f)p=V. ap^ = ry. 
 
 Then <^'/> = V . /3pa = <f)p. 
 
 Hence m = -^ — -Sf ( V. a\^ V. aii/3 V. av^). 
 
 Now X, fi, V are any three non-coplanar vectors ; and we may 
 therefore put for them a, /8, 7, if the latter he non-coplanar. 
 With this proviso 
 
 ^ o . apy 
 
 m, = ^^8(a^V.ay^ + a'^.^.y + cc.a^\y) 
 ^ S.apy 
 
 = -/Sfa/3. 
 
 Hence we have by (5) above 
 
 a.'^^Sa^.<}>-'y = oi'0'Sa^ .p = - o-'^'y +Soi^ V. 07^ + F. a (F. a7/3) /3, 
 
 which is one form of solution. 
 
 8—2 
 
116 QUATERNIONS. [163. 
 
 By expanding the vectors of products we may easily reduce it 
 to the form 
 
 or'S(Xy + l3-'Sl3y-y 
 
 or /3 = 
 
 <Sa/8 
 
 163. To verify this solution, we have 
 
 r. a/D/3 = ^ (^S<xy + aS^y - V. ay^) = y, 
 which is the given equation. 
 
 164. An easier mode of arriving at the same solution, in this 
 simple case, is as follows : — 
 
 Operating by ;S. a and 8. ^ on the given equation 
 F. a/0/8 = 7, 
 we obtain a^S^p = Say, 
 
 ^'8ap = 8^y; 
 and therefore a8/3p = a"' Stxry, 
 
 /38ixp = ^'S/3y. 
 But the given equation may be written 
 
 a8^p-p8a^ + ^8ttp = y. 
 Substituting and transposing we get 
 
 pSoil3 = a-'8ay + ^-'Sl3y - 7, 
 which agrees with the result of § 162. 
 
 [Note that, at first sight, one might think that the value of p 
 should have a term with an arbitrary scalar factor added. But the 
 notation p is limited to a vector. Had the equation been written 
 
 V. aql3 = 7 
 we should have had 
 
 aq^ = x + y, ox q — xcC^^"^ + oT'^y^'^. 
 
 But because q is to be a vector 
 
 8q = 0, or xSa^ + 8.ay^ = 0; 
 and, with this value of x, the expression for q takes the form given 
 above for p.] 
 
 165. If a, ^, 7 be coplanar, the above mode of solution ia 
 applicable, but the result may be deduced much more simply. 
 
 For (§ 101) S . ct/3y = 0, and the equation then gives 8 . a^p = 0, 
 so that p also is coplanar with a, (8, 7. 
 
l66.J SOLUTION OF EQUATIONS. 117 
 
 Hence the equation may be written 
 
 and at once 
 
 p = a-V/3-'; 
 
 and this, being a vector, may be written 
 
 = a-'S^'y + ^-'Sa-'y - ySoT'^-'. 
 This formula is equivalent to that just given, but not equal to 
 it term by term. [The student will find it a good exercise to 
 prove directly that, if a, /3, y are coplanar, we have 
 
 ^ (a-'Say + ^-'S^y -y) = OL-'S^-'y + ^-'So^-'y - ySa-'^~\] 
 
 The conclusion that 
 
 S.OL^p = 0, 
 
 in this case, is not necessarily true if 
 
 8al3 = 0. 
 
 But then the original equation becomes 
 
 aS^p + /8/Sfap = 7, 
 
 which is consistent with 
 
 S.al3y=0. 
 This equation gives 
 
 Say Sa^ 
 
 y(a'/3'-8'a^) = a 
 
 + /3 
 
 ;S'^7 Sa0 
 Say a^ 
 
 S^y 13' 
 
 by comparison of which with the given equation we find 
 
 Sap and S^p. 
 
 The value of p remains therefore with an indeterminate vector 
 part, parallel to ayS ; i.e. it involves one arbitrary scalar. 
 
 166. Example II. 
 
 Let <f>p = V. a^p = y. 
 
 Suppose a, /3, 7 not to be coplanar, and employ them as X, p,, v 
 to calculate the coefficients in the equation for (/>"'. We have 
 
 >Si . a(f)p = S . a-a^p = S.pV. a-a/3 = S . pcp'a: 
 Hence (f)'p = V. pa^ = V. ^ap. 
 
 We have now 
 
 = a'^'Sa^, 
 
118 QUATERNIONS. [167. 
 
 m=^^^-5(a./3a/3.F./8a7 + ;8a^/3.F.|8a7 + ^«'•^«^•7) 
 
 ' S . a/3 J 
 
 ^ . a/37 
 
 Hence by (5) of §160 
 a'fi'Sa^ . 4>-'-/ = oC'lS'Sa^ . p 
 
 = (2 {Sa^y + «^/3') 7 - SSa^r. a/Sry+V. a^ V. al3j, 
 which, by expanding the vectors of products, takes easily the 
 simpler form 
 
 a'^'SoiiS . p = a'^'y - a^'Say + 2^8a^Say - ^a'S^y. 
 
 167. To verify this, operate by V. a/3 on both sides, and we 
 have 
 
 a'^'Sa/3V.a/3p = (x'/SW. a^y-V. a^a/3'Say+2(X^'Sa^Say-aa'/3'S^y 
 = a^/S" (aS^y - ^Say + ySoi/3) - (2a8a0 - ^a'')^'Say 
 
 + 2a^'Sa^Say - aa'^S^y 
 = a'^^8a/3 . 7, 
 
 or V . a/3/3 = 7. 
 
 168. To solve the same equation without employing the 
 ■general method, we may proceed as follows : — 
 
 y=V. a/3p = pSa^ +V.V (a^) p. 
 
 Operating by S. Va^ we have 
 
 S.a^y = S.a^p8a^. 
 
 Divide this by /Sa/3, and add it to the given equation. We thus 
 obtain 
 
 y+^^ = p8a^+V.V{a^)p + 8.V{a^)p, 
 
 = {Sa/3+Va^)p, 
 
 = a/3p- 
 
 a form of solution somewhat simpler than that before obtained. 
 
 To shew that they agree, however, let us multiply by a!'/3^Sa/3, 
 and we get 
 
 a.'^''8a^ . p = ^ay8a/3 + ^aS . a^y. 
 
169.] SOLUTION OF EQUATIONS. 119 
 
 In this form we see at once that the right-hand side is a vector, 
 since its scalar is evidently zero (§ 89). Hence we may write 
 
 a'^'Soc/S .p=V. /3aySa^ - Va^S . a^y. 
 
 But by (3) of § 91, 
 
 -yS.a^Va^ + aS.^{V<x^)ry + l3S.V(a^)(Xy+Va^S.a/3y = 0. 
 Add this to the right-hand side, and we have 
 (x'^'Sa^ . p = 7 ((Sa^f -S.a^Va^)-a {Sa^S^y -S.I3( Va^) y) 
 
 + 13 (Soi/SSay + S.V (ayS) a^). 
 But {Sa^y -S.a^Va^ = (Sa/Sf - ( Va^f = ol'^\ 
 
 S(x^8^y -S.^i Fa/3) 7 = Sa^S^y - S^aS^y + ^'Say = /S'Say, 
 Sa^Say + 8 .V («^) a^ = Sa^S(xy + Sa^Say - a'S^y 
 = 2Sa/3S(xy-oc'S/3y; 
 
 and the substitution of these values renders our equation identical 
 with that of § 166. 
 
 [If a, /3, 7 be coplanar, the simplified forms of the expression 
 for p lead to the equation 
 
 Sa^ . ^'oT'y = 7 - a-' S<xy + 2l3Sa.-'^-\Soiy - ^-'S/3y, 
 
 which, as before, we leave as an exercise to the student.] 
 
 169. Example III. The solution of the equation 
 
 Vep = 7 
 
 leads to the vanishing of some of the quantities m. Before, how- 
 ever, treating it by the general method, we shall deduce its solu- 
 tion from that of 
 
 V. a^p = y 
 
 already given. Our reason for so doing is that we thus have an 
 opportunity of shewing the nature of some of the cases in which 
 one or more of m, m^, m^ vanish ; and also of introducing an 
 example of the use of vanishing fractions in quaternions. Far 
 simpler solutions will be given in the following sections. 
 The solution of the last- written equation is, § 166, 
 
 a'^'Sa^ . p = a'/B'y - a^'Say - /Ba'S/By + 2l3Soi/3Soiy. 
 
 If we now put a^ = e + e 
 
 where e is a scalar, the solution of the first-written equation will 
 evidently be derived from that of the second by making e gradually 
 tend to zero. 
 
120 QUATERNIONS. [l?^- 
 
 We have, for this purpose, the following necessary transforma- 
 tions : — 
 
 a'^2 = a/3 ^. a/3 = (e + e) (e - e) = e= - e', 
 
 a^^8ar^ + /3a''/S/37 = ap . ^Say + l3a . aS^y, 
 
 = (e + e) /SSay + (e - e) aS^y, 
 = e {^Say + aS^y) + eV. 7 Fa/8, 
 = e i^Say + aS^y) + e Vye. 
 Hence the solution becomes 
 
 (e'' _ e') ep = (e' - e") 7 - e {^Soiy + aS/Sy) -eVy6 + 2e/3Say, 
 = {e" -€^)y + eV.y Va^ -eVye, 
 = (e^ — e') y + e Vye + ye' — eSye, 
 = e^y + e Vye — eSye. 
 Dividing by e, and then putting e = 0, we have 
 
 -.>=F7e-e^,(^^). 
 
 Now, by the form of the given equation, we see that 
 
 ^76 = 0. 
 Hence the limit is indeterminate, and we may put for it x, where 
 X is any scalar. Our solution is, therefore, 
 
 p = -V'^^ + xe-'; 
 
 or, as it may be written, since Sye = 0, 
 
 P = e"' (7 + x). 
 The verification is obvious — for we have 
 
 ep = y + X. 
 
 170. This suggests a very simple mode of solution. For we 
 see that the given equation leaves Sep indeterminate. Assume, 
 
 therefore, 
 
 Sep = x 
 
 and add to the given equation. We obtain 
 
 ep = x + y, 
 
 or p = e"' (7 + X), 
 
 if, and only if, p satisfies the equation 
 
 Vep = 7. 
 
 171. To apply the general method, we may take e, 7 and 67 
 (which is a vector) for X, fi, v. 
 
[7 2. J SOLUTION OF EQUATIONS. 121 
 
 We find 
 
 0',o = Vpe. 
 
 Hence 
 
 m = 0, 
 
 
 m, = —-5-5*8.(6.6 
 ' 6 7 
 
 
 m, = 0. 
 
 Hence 
 
 - e'<l> + </,' = 0, 
 
 
 </.-• = J </. + </.-^o. 
 
 it is, 
 
 p = ^ Vej + X€, 
 
 or 
 
 = 6"'ry + a;e, as before. 
 
 Our warrant for putting xe, as the equivalent of <f)'^ is this : — 
 
 The equation ^'o- = 
 
 may be written V. eVea- = = ae' — eSecr. 
 
 Hence, unless o- = 0, we have o- 1| e = xe. 
 
 [Of course it is well to avoid, when possible, the use of expressions 
 such as ^"° &c. but the student must be prepared to meet them ; 
 and it is well that he should gain confidence in using them, by 
 verifying that they lead to correct results in cases where other 
 modes of solution are available.] 
 
 172. Example IV. As a final example let us take the most 
 general form of ^, which, as has been shewn in § 151, may be 
 expressed as follows : — 
 
 <j>p = aS^p + a^S0,p + a^S^^ = y. 
 Here ^'p = ^Sap + l3^Sa^p + 0^Sa^, 
 
 and, consequently, taking a, a^, a^, which are in this case non- 
 coplanar vectors, for \, /jl, v, we have 
 
 
 Solo. SoL^a Sa^a. 
 /Saa, jSa,aj /Sfa^a, 
 Saa^ Sa^a^ Sa^a^ 
 
 (^S<xa^+ ) 
 
 where A = Sa^a^Sa^a^ - Sa^a^Sct^a^ 
 
122 QUATERNIONS. [172. 
 
 J.J = Saa^Sa^a^ — Sa^a^Saa^ 
 = —8. Vaa.^ FajOtg. 
 Hence the value of the determinant is 
 - {SmS . Va^a^Va^a^ + Sa^aS . Va^a Va^a^ + Sa^(xS . Faa, Va^a^) 
 
 ^-8.<x( Va,<x^ . ««.«,) {by § 92 (4)) = -(S. aa^aj\ 
 
 The interpretation of this result in spherical trigonometry is 
 very interesting. (See Ex. (9) p. 90.) 
 By it we see that 
 
 Similarly, 
 
 (/3Saa^ + A^Sfa.a, + /3,*Sfa,aJ + &c.J 
 = o (S . a/3/3, (Saa^Saa - SaaSaa) + ...) 
 
 = ^ (S.a0^^S.aV.a^Va^a^+...} 
 
 S . aajHj 121 
 
 + F/3A'S.FaAF«.«,) 
 + /S . a. ( F/SiS^^S . Faa^ Fa,a + . . . ) 
 + S. a,(F/S/3./S.Faa,Fa«, + ...)] ; 
 or, taking the terms by columns instead of by rows, 
 
 = - ^-^ [S. F/S/3, (a/Sf . Faa, Fa,a, + a,<S. Faa. Fa^a + a^S . Faa^ Faa,) 
 
 + + ....], 
 
 = - S (Faa,F^/3, + Fa.a.F^,^, + Fa.aF^,^). 
 Again, 
 
 or, grouping as before, 
 
 = o ^ [/8 ( Faa,/Sa5(j + Fa^a/Saa^ + Faja^Saa) + . . . J, 
 
174-1 SOLUTION OF EQUATIONS. 
 
 123 
 
 ^ K^, ^ ^^ ("^ • ""^''^^ +•••](§ 92 (4)), 
 
 And the solution is, therefore, 
 
 <t>-'yS . aa,a^ . 0^,0^ = pS . au.a^S . 00^/3, 
 
 = 72/S . Vaa,V/30, + <j)jtSa0 - fy. 
 
 [It will be excellent practice for the student to work out in 
 
 detail the blank portions of the above investigation, and also to 
 
 prove directly that the value of p we have just found satisfies the 
 
 given equation.] 
 
 173. But it is not necessary to go through such a long process 
 to get the solution — though it will be advantageous to the student 
 to read it carefully — for if we operate on the proposed equation by 
 S . a^a^, S . a^a, and S . aa^ we get 
 
 S.a^a^aS0p = S.a^a^y, 
 
 S . ajxa^S^^p = S . a^ay, 
 
 S . aa^a^S/S^ = 8 . aa^y. 
 
 From these, by § 92 (4), we have at once 
 
 pS . a«,a,Sf . 00,&, = F/3/S..S . aa^y + F/3./3,5f . a,a,,7 + V0,08 . <x^a.y. 
 
 The student will find it a useful exercise to prove that this is 
 equivalent to the solution in § 172. 
 
 To verify the present solution we have 
 {oiS^p + a^SlB^p + a^S^^p) 8 . oia^a^8 . 0/3^13^ 
 
 = 0iS. 00^0.^8 . a,a,7 + a^S . 0,0,08 . d^ay + a,8 . 0,00,8 . aa,7 
 =^8.00,0,(y8.CLa,a,),hj^91(S). 
 
 174. It is evident, from these examples, that for special cases 
 we can usually find modes of solution of the linear and vector 
 equation which are simpler in application than the general process 
 of § 160. The real value of that process however consists partly in 
 its enabling us to express inverse functions of (f), such as (</> — g)'^ 
 for instance, in terms of direct operations, a property which will be 
 of great use to us later ; partly in its leading us to the fundamental 
 
 cubic 
 
 <j)^ — m,^ + m,^ — m = 0, 
 
 which is an immediate deduction from the equation of § 160, and 
 whose interpretation is of the utmost importance with reference to 
 the axes of surfaces of the second order, principal axes of inertia. 
 
124 QUATERNIONS. [l75- 
 
 the analysis of strains in a distorted solid, and various similar 
 enquiries. 
 
 We see, of course, that the existence of the cubic renders it a 
 mere question of ordinary algebra to express any rational function 
 whatever of in a rational three-term form such as 
 
 In fact it will be seen to follow, from the results of § 177 below, 
 
 that we have in general three independent scalar equations of the 
 
 form 
 
 F(g) = A+Bg + Cf, 
 
 which determine the values of A, B, C without ambiguity. The 
 result appears in a form closely resembling that known as 
 Lagrange's interpolation formula. 
 
 175. When the function <j) is its own conjugate, that is, when 
 Sp^a = Sa<f)p 
 for all values of p and a; the vectors for which 
 
 i<}>-g)p = 
 form in general a real and definite rectangular system. This, of 
 course, may in particular cases degrade into one definite vector, and 
 any pair of others perpendicular to it ; and cases may occur in 
 which the equation is satisfied for every vector. 
 To prove this, suppose the roots of 
 
 mg = m — m^g + m^g"^ —g^z=Q 
 (§ 159 (4)) to be real and different, then 
 
 where p^, p^, p^ are three definite vectors determined by the 
 constants involved in ^. 
 
 Hence, operating on the first by Sp^^, and on the second by 8p^, 
 we have 
 
 ^■P-2<l>Pi='9iSPiP,' 
 
 S-Pl<PP2=9i^P,Pi- 
 
 The first members of these equations are equal, because ^ is its 
 own conjugate. 
 
 Thus {9.-9,)Sp,p, = 0; 
 
 which, as g^ and g^ are by hypothesis different, requires 
 
 Sp,p, = 0. 
 
177-1 SOLUTION OF EQUATIONS. 125 
 
 Similarly Sp^^ = 0, Sp^p^ = 0. 
 
 If two roots be equal, as g^, g^, we still have, by the above proof, 
 Sp^p^ = and (Sipj/Sg = 0. But there is nothing farther to determine 
 Pj and /3g, which are therefore any vectors perpendicular to p^. 
 
 If all three roots be equal, every real vector satisfies the equation 
 {^-g)p^O. 
 
 176. Next as to the reality of the three directions in this 
 case. 
 
 Suppose g + h'J — 1 to be a root, and let p + cr V — 1 be the 
 corresponding value of p, where g and h are real numbers, p and a 
 real vectors, and V — 1 the old imaginary of algebra. 
 
 Then (p + o- V^) = (^ + /i V^)(p + o- V^), 
 
 and this divides itself, as in algebra, into the two equations 
 
 <f)p = gp- ha, 
 
 <^cr = hp+ ga. 
 
 Operating on these by 8 .a, S.p respectively, and taking the 
 
 difference of the results, remembering our condition as to the 
 
 nature of (/> 
 
 8cr(j>p = Sp(f)cr, 
 
 we have h (o-" + p^) = 0. 
 
 But, as <r and p are both real vectors, the sum of their squares 
 cannot vanish, unless their tensors separately vanish. Hence h 
 vanishes, and with it the impossible part of the root. 
 
 The function need not be self-conjugate, in order that the 
 roots of 
 
 may be all real. For we may take g^, g^, g^ any real scalars, and 
 a, /8, 7 any three real, non-coplanar, vectors. Then if ^ be such that 
 
 8.a^y.^p= g,aS . ^yp + g^^8 . yap + g^yS . a^p, 
 we have obviously 
 
 (<^ - ^.) a = 0, C<^ - 5r,) ^ = 0, (</> - ^r,) 7 = 0. 
 Here cf) is self-conjugate only if a, /3, 7 form a rectangular 
 system. 
 
 177. Thus though we have shewn that the equation 
 
 g' — m^g" + m^g — m = 
 he^s three real roots, in general different from one another, when ^ 
 
126 QITATEENIONS. [178. 
 
 is self-conjugate, the converse is by no means true. This must be 
 most carefully kept in mind. 
 
 In all cases the cubic in <^ may be written 
 
 ('^-5'O(0-^.)(</'-5'3) = O (1). 
 
 and in this form we can easily see its meaning, provided the values 
 of g are real. For there are in every such case three real (and in 
 general non-coplanar) vectors, p^, p^, p^ for which respectively 
 
 (<^-5',)p. = 0, {(t>-9,)p, = 0, (4>-g,)p, = 0. 
 Then, since any vector p may be expressed by the equation 
 pS ■ p,PJ>s = P,S. p,p,p + p^S . p,p,p +p,8. p,p^p (§ 91), 
 we see that when the complex operation, denoted by the left-hand 
 member of the symbolic equation, (1), is performed on p, the 
 first of the three factors makes the term in p^ vanish, the second 
 and third those in p,^ and p^ respectively. In other words, by the 
 successive performance, upon a vector, of the operations ^—g^,^ — g^, 
 (^ —g^, it is deprived successively of its resolved parts in the direc- 
 tions of pj, p^, P3 respectively ; and is thus necessarily reduced to 
 zero, since p^, p^, p^ are (because we have supposed g^, g^, g^ to be 
 distinct) distinct and non-coplanar vectors. 
 
 178. If we take p^, p^, p^ as rectangular wm'^-vectors, we have 
 
 -p= p,8p,p + p.,Sp^ + p,Sp,p, 
 whence </>/> = - g,p,Sp^p - g,p,Sp^ - g,p,Sp,p ; 
 
 or, still more simply, putting i,j, k for p^, p^, p^, we find that any 
 self-conjugate function may be thus expressed 
 
 <^P = - 9iiSip - gJSjp - gJeSkp (2), 
 
 provided, of course, i, j, k be taken as the roots of the equation 
 
 Vp^p = 0. 
 A rectangular unit-vector system requires three scalar quan- 
 tities, only, for its full specification, g^, g^, g^ are other three. 
 Thus any self-conjugate function involves only six independent 
 scalars. 
 
 179. A very important transformation of the self-conjugate 
 linear and vector function is easily derived from this form. 
 
 We have seen that it involves, besides those of the system i,j, k, 
 three scalar constants only, viz. g^, g^, g^ Let us enquire, then, 
 whether it can be reduced to the following form 
 
 ^p=/p + hV.(i + ek) p [i-ek) (3), 
 
1 8 1.] SOLUTION OF EQUATIONS. 127 
 
 which also involves but three scalar constants/, h, e, in addition to 
 those of i, j, k, the roots of 
 
 Vpjtp = 0. 
 
 Substituting for p the equivalent 
 
 p = — iSip - jSjp — kSkp, 
 
 expanding, and equating coefficients of i,j, k in the two expressions 
 (2) and (3) for (})p, we find 
 
 g,=f+h{\-e% 
 
 These give at once 
 
 -(5'i-5'2) = 2A, 
 
 Hence, as we suppose the transformation to be real, and therefore 
 e" to be positive, it is evident that g^ —g^ and g^ — g^ have the same 
 sign ; so that we must choose as auxiliary vectors in the last term 
 of (/)/3 those two of the rectangular directions %j, k for which the 
 coefficients g have respectively the greatest and least values. 
 We have then 
 
 ^ 7 
 
 9 1-9, 
 f^ = -i(9i-9,), 
 and f=^(9t+9,)- 
 
 180. We may, therefore, always determine definitely the 
 vectors X, fi, and the scalar/ in the equation 
 
 4'P =fp + ^- V/* 
 when (ji is self-conjugate, and the corresponding cubic has not equal 
 roots ; subject to the single restriction that 
 
 T.Xfi 
 is known, but not the separate tensors of X and p,. This result is 
 important in the theory of surfaces of the second order, and in that 
 of Fresnel's Wave-Surface, and will be considered in Chapters IX. 
 and XII. 
 
 181. Another important transformation of <j) when self- 
 conjugate is the following, 
 
 <f)p = aaVap + b^S^p, 
 where a and b are scalars, and a and /3 unit-vectors. This, of 
 
128 QUATERNIONS. [182. 
 
 course, involves six scalar constants, and belongs to the most 
 general form 
 
 'f>P = - 9,Pi^PiP - 9^P2^P2P - 9sPs^PsP' 
 where pp /a^, Pa are the rectangular unit-vectors for which p and cj)p 
 are parallel. We merely mention this form in passing, as it 
 belongs to the focal transformation of the equation of surfaces of 
 the second order, which will not be farther alluded to in this work. 
 It will be a good exercise for the student to determine a, ^, a and 
 b, in terms of gr,, g^, g^, and p^, p^, p,. 
 
 182. We cannot afford space for a detailed account of the 
 singular properties of these vector functions, and will therefore 
 content ourselves with the enuntiation and proof of one or two of 
 the more important. 
 
 In the equation m^"'F\ya= F^'\^'/x (§ 157), 
 
 substitute \ for <f>'X and fi, for <j)'fi,, and we have 
 
 mV<f,'-'X(f)'-'iJ,:=<f,VX/^. 
 
 Change (j) to ^—g, and therefore <f)' to 4>' — g, and m to m^, we have 
 
 ^.V{<l>' - gr^ (</>' - ^)"V ={<p-9) VXp, ; 
 a formula which will be found to be of considerable use. 
 
 183. Again, by § 159, 
 
 '^^S.p(<l>-grp=:'^Sprp-Spxp+9P'. 
 
 Similarly ^S.p(<p- Tifp = j Sp<f>-'p - Spxp + hp' . 
 Hence 
 
 'fs.p(^-grp-l^s.pi4>-hrp=(g-h)[p^-'^}. 
 
 That is, the functions 
 
 "^S.pi^,- grp, and ™*/Sf . p (<^ - hyp 
 
 are identical, i.e. when equated to constants represent the same series 
 of surfaces, not merely when 
 
 g = K 
 
 but also, whatever be g and h, if they be scalar functions of p which 
 satisfy the equation 
 
 mS . pjT^p = ghp^. 
 
1 85- J SOLUTION OP EQUATIONS. 
 
 129 
 
 This is a generalization, due to Hamilton, of a singular result ob- 
 tained by the author*. 
 
 184. It is easy to extend these results ; but, for the benefit of 
 beginners, we may somewhat simplify them. Let us confine our 
 attention to cones, with equations such as 
 
 ^•p(</>-5'rv=o,| 
 
 s.p(<j>-hy'p = o,\ (1) 
 
 These are equivalent to mSpcfi'^p - gSpxp + g^p' = 0, 
 
 mSpcj>''p - hSpxp + h'p^ = 0. 
 Hence 
 
 m (1 - x) 8p^-'p -ig- hx) Spxp + (g' - Vx) p' = 0, 
 
 whatever scalar be represented by x. 
 
 That is, the two equations (1) represent the same surface if this 
 
 identity be satisfied. As particular cases let 
 
 (1) x = l, in which case 
 
 -Sp-'xp+g + h = o. 
 
 (2) g — hx=0, in which case 
 
 or mSp~^(l)-^p-gh=0. 
 
 (3) a? = |i, giving 
 
 -„,(l^^^Spcf>-'p + (g-h^^^Spxp = 0, 
 
 or -m(h+g) 8pcf>-^p + ghSpxp = 0. 
 
 185. In various investigations we meet with the quaternion 
 
 q = acfia + I3cf)^ + jifyy (1), 
 
 where a, /3, 7 are three unit-vectors at right angles to each other. 
 It admits of being put in a very simple form, which is occasionally 
 of considerable importance. 
 
 We have, obviously, by the properties of a rectangular unit- 
 system 
 
 q = /370a + ya^lS + a^(j)y. 
 As we have also 
 
 ,S.a^7 = -l (§71 (13)), 
 
 * Note on the Cartesian equation of the Wave-Surface. Quarterly Math. Journal, 
 Oct. 1859. 
 
 T. Q. I. 9 
 
130 QUATERNIONS. [l86. 
 
 a glance at the formulae of § 159 shews that 
 
 Sq = -m^, 
 Sit least if (j) be self-conjugate. Even if it be not, still (as will be 
 shewn in § 186) the term by which it differs from a self-conjugate 
 function is of such a form that it disappears in Sq. 
 Wehavealso;by§90 (2), 
 
 Vq = a {S/3<Pj - 8y<l)l3) + ^ {Sy(l)a - SoL<h) + 7 (/S«0^ - S^cjxx) 
 = aS^ (^ - ^') 7 + ^Sy {<f> -<f>')a + rySa (0 - ^') /3 
 = OiS . ^ey + ^8 . yea + 7-Sf . ae^ (§ 186) 
 = - {aSae + ^S^e + ySye) = e. 
 [We may note in passing that the quaternion (1) admits of 
 being expressed in the remarkable forms 
 
 V4>p, or K.<l>[V)p; 
 
 where(§145) ^ = «| + /3| + 7^, 
 
 and p = ax + ^y + yz. 
 
 We will recur to this towards the end of the work.] 
 
 Many similar singular properties of ^ in connection with a rect- 
 angular system might easily be given ; for instance, 
 
 F(aF(/)/3(^7 -1- ^V<Py<f)OL + yY^ai<i>l3) 
 
 = mV(oL4>'-^a + l3^'-^l3 + 70'"'7) = mV. V<t>'-'p = 4>e ; 
 which the reader may easily verify by a process similar to that just 
 given, or (more directly) by the help of | 157 (2). A few others 
 will be found among the Examples appended to this Chapter. 
 
 186. To conclude, we may remark that, as in many of the 
 immediately preceding investigations we have supposed ^ to be 
 self-conjugate, a very simple step enables us to pass from this to 
 the non-conjugate form. 
 
 For, if (f)' be conjugate to ^, we have 
 Sp<j)'a' = 8<7(f)p, 
 and also Sp^a = 8crcj>'p. 
 
 Adding, we have 
 
 8p{cj> + cj,'}a = 8a{cj> + <j,')p; 
 so that the function (0 -|- ^') is self-conjugate. 
 
 Again, Spifip = Spcfi'p, 
 
 which gives Sp (^ — <j}')p = 0. 
 
 Hence {(f>-<fi')p= Vep, 
 
1 8;.] SOLUTION OF EQUATIONS. 131 
 
 where, if (f> be not self-conjugate, e is some real vector, and 
 therefore 
 
 Thus every no7i-conjugate linear and vector function differs from 
 a conjugate function solely by a term of the form 
 
 Vep. 
 The geometric signification of this will be found in the Chapter on 
 Kinematics. 
 
 The vector e involves, of course, three scalar constants. Hence 
 (§§ 151, 178) the linear and vector function involves, in general, nine. 
 
 187. Before leaving this part of the subject, it may be well 
 to say a word or two as to the conditions for three real vector 
 solutions of the equation 
 
 Vp(f,p = 0. 
 
 This question is very fally treated in Hamilton's Elements, and 
 also by Plarr in the Trans. M. S. E. For variety we adopt a semi- 
 graphic method*, based on the result of last section. By that 
 result we see that the equation to be solved may be written as 
 
 (f)p = 'STp + Vep = xp (1) 
 
 where it is a given self-conjugate function, e a given vector, and x 
 an unknown scalar. 
 
 Let ofj, a^, Hg and g^, g^, g^ (the latter taken in descending order 
 
 of magnitude), be the vector and scalar constants of ot, so that 
 
 (§ 177) 
 
 (^-5'i)«i = 0. ^'c. 
 
 We have obviously, by operating on (1) with S . a^ &c., three 
 equations of the form 
 
 8p{(g,-cc)<x^-re<x^}=0 (2). 
 
 Eliminating p (whose tensor is not involved) we have 
 S {{g, -«;)«,- FeaJ {(g, -x)a,- Yea.} {{g, -x)a^- Vea,] = 0, 
 
 or {«^-9di^-9d ioo-g^)-xe'' + 8e^e = (3). 
 
 From each value of x found from this equation the corre- 
 sponding value of p is given by (2) in the form 
 
 p II r {{g, -x)a^- FeaJ [{g, -x)a^- FeaJ, 
 
 II (S'l - <«) {92 - a^) «3 + ^«« {■^-a:)e- eSa.e. 
 * Proe. R. 8. E., 1879—80. 
 
 9—2 
 
132 QUATERNIONS. [l88. 
 
 The simplest method of dealing with (3) seems to be to find the 
 limiting value of Te, Ue being given, that the roots may be all real. 
 They are obviously real when Te = 0. It is clear from the pro- 
 perties of ■ST that the extreme values of — >S> . UewUe (which will be 
 called I) are g^ and g^. 
 
 Trace the curve 
 
 2/ = («-5'i)(«-5'2)(«'-5'3)> 
 and draw the (unique) tangent to it from the point x=^, y = 0, 
 ^ having any assigned value from ^3 to g^. Let this tangent make 
 an angle — 6 with the axis of x. Suppose a simple shear to be 
 applied to the figure so as to make this tangent turn round the 
 point ^, 0, and become the x axis, while the y axis is unchanged. 
 The value of y will be increased by (x — ^} tan 0. Comparing this 
 with (3) we see that tan 6 is the desired limiting value of {Tef. 
 
 188. We have shewn, at some length, how a linear and vector 
 equation containing an unknown vector is to be solved in the most 
 general case ; and this, by § 150, shews how to find an unknown 
 quaternion from any sufficiently general linear equation containing 
 it. That such an equation may be sufficiently general it must 
 have both scalar and vector parts : the first gives one, and the 
 second three, scalar equations ; and these are required to determine 
 completely the four scalar elements of the unknown quaternion. 
 
 Thus Tq = a 
 
 being but one scalar equation, gives 
 
 q = a Ur, 
 where r is any quaternion whatever. 
 
 Similarly 8q = a 
 
 gives q = a + 6, 
 
 where 6 is any vector whatever. In each of these cases, only one 
 scalar condition being given, the solution contains three scalar in- 
 determinates. A similar remark applies to the following : 
 
 TVq=a 
 gives q = x + ae; 
 
 and 8Uq = COS A, 
 
 gives q = xd^^'", 
 
 in each of which x is any scalar, and 6 any unit vector. 
 
1 92-] SOLUTION OF EQUATIONS. 133 
 
 189. Again, the reader may easily prove that 
 
 V.aVq = ^, 
 where a is a given vector, gives, by putting 8q = x, 
 
 Vaq = ^-\-xa. 
 Hence, assuming Saq = y, 
 
 we have aq = y + xa + ^, 
 
 or q = x + ycL'^ + a'^^S. 
 
 Here, the given equation being equivalent to two scalar con- 
 ditions, the solution contains two scalar indeterminates. 
 
 190. Next take the equation 
 
 Vaq = l3. 
 Operating by 8 . a'', we get 
 
 Sq = Sa-'I3, 
 so that the given equation becomes 
 
 Va(SoL-'^+Vq) = ^, 
 or VaVq = ^- aSoT'^ = a ra-% 
 
 From this, by § 170, we see that 
 
 rq = OL-'(x+aVoL-'^), 
 whence q = Sa.'^^ + a'^ (x + aVa.'^^) 
 
 and, the given equation being equivalent to three scalar conditions, 
 but one iindetermined scalar remains in the value of q. 
 
 This solution might have been obtained at once, since our 
 equation gives merely the vector of the quaternion aq, and leaves 
 its scalar undetermined. 
 
 Hence, taking x for the scalar, we have 
 
 aq = Saq + Vaq 
 = x + /3. 
 
 191. Finally, of course, from 
 
 ag = /3, 
 
 which is equivalent to four scalar equations, we obtain a definite 
 value of the unknown quaternion in the form 
 
 192. Before taking leave of linear equations, we may mention 
 
134 QUATERNIONS. [ 1 9 3 • 
 
 that Hamilton has shewn how to solve any linear equation con- 
 taining an unknown quaternion, by a process analogous to that 
 which he employed to determine an unknown vector from a linear 
 and vector equation ; and to which a large part of this Chapter has 
 been devoted. Besides the increased complexity, the peculiar fea- 
 ture disclosed by this beautiful discovery is that the symbolic 
 equation for a linear quaternion function, corresponding to the cubic 
 in (^ of § 174, is a biquadratic, so that the inverse function is given 
 in terms of the first, second, and third powers of the direct function. 
 In an elementary work like the present the discussion of such a 
 question would be out of place : although it is not very difficult to 
 derive the more general result by an application of processes 
 already explained. But it forms a curious example of the well- 
 known fact that a biquadratic equation depends for its solution 
 upon a cubic. The reader is therefore referred to the Elements of 
 Quaternions, p. 491. 
 
 193. As an example of the solution of the linear equation in 
 quaternions, let us take the problem of finding the differential of 
 the n*'' root of a quaternion. This comes to finding dq in terms of 
 dr when 
 
 g" = r. 
 
 [Here n may obviously be treated as an integer ; for, if it were 
 fractional, both sides could be raised to the power expressed by the 
 denominator of the fraction.] 
 
 This gives 
 
 q^-'dq + q^-'dq .q+...+ dq. q"'' = <j> (dq) = dr (1), 
 
 and from this equation dq is to be found ; <^ being now a linear 
 and quaternion function. 
 
 Multiply by q, and then into q, and subtract. We obtain 
 
 q''dq — dq . q" = qdr — dr.q, 
 
 or 2V .Vq''Vdq=2V .VqVdr (2). 
 
 But, from the equation 
 
 q = Sq+ Vq, 
 we have at once Vr = Vq" = Q^Vq, 
 
 where Q„ = n (Sq)"-' - "'""^ ^'^ ^ (^g)"-' (TVqf + &a 
 
 [The value of Q„ is obvious from § 116, but we keep the 
 present form.] 
 
I94-] SOLUTION OF EQUATIONS. 135 
 
 With this (2) becomes 
 
 Q,V.VqVdq=^V.VqVdr, 
 whence Q^ Vdq =' Vdr + x Vq, 
 
 X being an undetermined scalar. 
 
 Adding another such scalar, so as to introduce Sdq and Sdr, we 
 have 
 
 Qn<iq = iy + ooVq) + dr (3). 
 
 Substitute in (1) and we have 
 
 QJr = nf-^ {y + xVq) + <^ {dr), 
 or, by (3) again, 
 
 QJir = ruf-^ (QJq -dr) + 4> (dr) ; 
 so that, finally, 
 
 QJq = dr+lq'-'^(QJr-cj,{dr}) (4). 
 
 ft 
 
 Thus dq is completely determined. 
 
 It is interesting to form, in this case, an equation for ^. This 
 is easily done by eliminating dr from (4) by the help of (1). We 
 thus obtain 
 
 i<f,-nqn{<l>-Q„) = 0. 
 
 This might have been foreseen from the nature of <f), as defined in 
 (1) ; because it is clear that its effect on a scalar, or on a vector 
 parallel to the axis of q (which is commutative with q), is the same 
 as multiplication by 7iq"'^ ; while for any vector in the plane of q 
 it is equivalent to the scalar factor Q^. 
 
 It is left to the student to solve the equation (1) by putting it 
 in the form 
 
 dr=p + qpq'^ + q'pq"' +...+ q"~'pq~''*^ 
 
 or dr- qdrq~^ =p — 2"P2~"> 
 
 where p = dqq"~^. 
 
 The nature of the operator q { ) g"' was considered in § 119 
 
 above. 
 
 194. The question just treated involves the solution of a 
 particular case only of the following equation : — 
 
 (f) (q) =taqa' = b (1), 
 
 where a, a', &c. are coplanar quaternions. 
 
 Let q = r + p, 
 
136 QUATERNIONS. [l95- 
 
 where r is a quaternion coplanar with the as, and p a vector in 
 their plane. Then, for any a, 
 
 ra = ar, 
 while pa = Ka . p. 
 
 Thus the given equation takes the form 
 
 <l>{q) = t {aa!) . r + 2 {aKa) . p, 
 so that the functional equation becomes in its turn 
 {(^ - 2 (aa')} {<^ - 2 (aKa')} = 0. 
 
 If a be the unit-vector perpendicular to the plane of the as, we 
 have 
 
 - b = aSab + (xVab 
 
 = -(Sb-«Sab) + aVaVb, 
 
 and the required solution is obviously 
 
 q = {t aaV (Sb - aSab) - (t aKa')'' aVa Vb. 
 
 In the case (§ 193) of the differential of the ?i* root of a quaternion, 
 s, we have 
 
 2 (aa') = ns"-', 
 
 2 (aKa') = 2{S. s""' + T'sS. s""' +. . .). 
 
 The last expression (in which, it must be noticed, the last term is 
 not to be doubled when n is odd) is the Q„ of the former solution, 
 though the form in which it is expressed is different. It will 
 be a good exercise for the student to prove directly that they are 
 equal. 
 
 195. The solution of the following frequently-occurring par- 
 ticular form of linear quaternion equation 
 
 aq+qb = c, 
 where a, b, and c are any given quaternions, has been effected by 
 Hamilton by an ingenious process, which was applied in § 140 (5) 
 above to a simple case. 
 
 Multiply the whole by Ka, and (separately) into b, and we 
 have 
 
 T'a.q + Ka.qb = Ka . c, 
 
 and a.qb + qb^=^cb. 
 
 Adding, we have 
 
 q{ra + ¥ + 2Sa.b) = Ka.c + cb, 
 from which q is at once found. 
 
1 96. J SOLUTION OF EQUA.TIONS. 137 
 
 To this form any equation such as 
 
 a'g&' + c'qd' = e' 
 can of course be reduced, by multiplication hy c"^ and into b'~^ 
 
 196. To shew some of the characteristic peculiarities in the 
 solution of quaternion equations even of the first degree when they 
 are not sufficiently general, let us take the very simple one 
 
 aq = qb, 
 and give every step of the solution, as practice in transformations. 
 Apply Hamilton's process (§ 195), and we get 
 ra.q = Ka. qb, 
 qb' = aqb. 
 These give q {fa + b' - 2bSa) = 0, 
 
 so that the equation gives no real finite value for q unless 
 
 ra + ¥-2bSa = 0, 
 or b = 8a + ^Tra, 
 
 where /3 is some unit-vector. This gives Sa = Sb. 
 By a similar process we may evidently shew that 
 a = 8b + aTrb, 
 a being another unit-vector. 
 But, by the given equation, 
 
 Ta = Tb, 
 or S'a + rVa = 8'b + 'PVb; 
 
 from which, and the above values of a and b, we see that we may 
 
 write 
 
 8a 8b 
 
 Thus we may write 
 
 a = a -H a, & = a -f- /3, 
 
 where a and ^ are unit-vectors. 
 
 If, then, we separate q into its scalar and vector parts, thus 
 
 q = u + p, 
 the given equation becomes 
 
 (a-fa)(M-F|o) = ('if + p)(a-h/3) (1). 
 
 Multiplying out we have 
 
 u(a-^)=p^- ap, 
 which gives 8{a-^) p=0, 
 
 and therefore p = F7 (a - /8), 
 
 where 7 is an undetermined vector. 
 
138 QUATERNIONS. [196. 
 
 We have now 
 u (a-^) = p^ - ap 
 
 = Vy{a- 0) . /3-aVry(a- ^) 
 
 = 7 (&/? + !)-(«- ^)Sl3y-y(l + 8a^) - (a-^)Say 
 
 Having thus determined u, we have 
 
 2g=-(a + ^)7-7(a + /3) + 7(a-/3)-(a-/3)7 
 
 = -2ay- 27/3. 
 
 Here, of course, we may change the sign of 7, and write the solu- 
 tion of 
 
 aq = qb 
 
 in the form q = ay + 7/?, 
 
 where 7 is any vector, and 
 
 oL=UVa, l3=UVh. 
 
 To verify this solution, we see by (1) that we require only to 
 shew that 
 
 cnq = q0. 
 
 But their common value is evidently 
 
 - 7 + a7/3. 
 
 An apparent increase of generality of this solution may be 
 obtained by writing 
 
 q = ar + r^ 
 
 where r is any quaternion. But this is easily seen to be equiva- 
 lent to adding to 7 (which is any vector) a term of the form scVa^. 
 It will be excellent practice for the student to represent the 
 terms of this equation by versor-arcs, as in § 54, and to deduce the 
 above solution from the diagram annexed : — 
 
 b 
 
 The vector of the intersection of the plane of q, with that of aq 
 
198-] SOLUTION OF EQUATIONS. 139 
 
 and qh, is evidently symmetrically situated with regard to the 
 great circles of a and h. Hence it is parallel to 
 
 (a+j8) Fa^, i.e. to a-/3. 
 Let 7 be any vector in the plane of a. 
 Then qcc 7 (/S - a), 
 
 X a7+7/3, 
 because Sar/ = 0, and thus — 7a = 07. 
 
 Another simple form of solution consists in writing the equa- 
 tion as 
 
 a = qbq'^, 
 
 and applying the results of §119. 
 
 197. No general quaternion method of solving equations of 
 the second or higher degrees has yet been found ; in fact, as will be 
 shewn immediately, even those of the second degree involve (in 
 their most general form) algebraic equations of the sixteenth degree. 
 Hence, in the few remaining sections of this Chapter we shall con- 
 fine ourselves to one or two of the simpler forms for the treatment 
 of which a definite process has been devised. But first, let us 
 consider how many roots an equation of the second degree in an 
 unknown quaternion must generally have. 
 
 If we substitute for the quaternion the expression 
 
 w^-ix+ jy + kz (§ 80), 
 
 and treat the quaternion constants in the same way, we shall have 
 on development (§ 80) four equations, generally of the second 
 degree, to determine w, x, y, z. The number of roots will therefore 
 be 2^ or 16. And similar reasoning shews us that a quaternion 
 equation of the mth degree has to* roots. It is easy to see, how- 
 ever, from some of the simple examples given above (§§ 188 — 190, 
 &c.) that, unless the given equation is equivalent to four inde- 
 pendent scalar equations, the roots will contain one or more 
 indeterminate quantities. 
 
 198. Hamilton has efifected in a simple way the solution of 
 
 the quadratic 
 
 (f = qa+ h, 
 
 or the following, which is virtually the same (as we see by taking 
 
 the conjugate of each side), 
 
 q^ = axi + b. 
 
140 QUATERNIONS. [ 1 99 • 
 
 He puts q = ^{a + w + p), 
 
 where w is a scalar, and p a vector. 
 
 Substituting this value in the first written form of the equation, 
 we get 
 
 a^ + (w+ pf + 2wa + ap-\- pa = 2{a^ + wa + pa) + 46, 
 
 or (w + pf + ap —pa = a' + 46. 
 
 If we put Va = a, S (a' + 46) = c, Via' + 46) = 27, this becomes 
 
 (w + py+2 Vap = c + 27 ; 
 
 which, by equating separately the scalar and vector parts, may be 
 broken up into the two equations 
 
 w^ + p' = c, 
 
 V(w+ a)|0 =7. 
 
 The latter of these can be solved for p by the process of § 168 ; or 
 more simply by operating at once by S.oc, which gives the value 
 of 8 (w + a) p. If we substitute the resulting value of p in the 
 former we obtain, as the reader may easily prove, the equation 
 
 (w' - a.') K - cw' + 7') - ( Vayf = 0. 
 
 The solution of this scalar cubic gives six values of w, for each of 
 which we find a value of p, and thence a value of q. 
 
 Hamilton shews (Lectures, p. 633) that only two of these values 
 are real quaternions, the remaining four being biquaternions, and 
 the other ten roots of the given equation being infinite. 
 
 Hamilton farther remarks that the above process leads, as the 
 reader may easily see, to the solution of the two simultaneous 
 equations 
 
 q + r = a, 
 
 qr = —b; 
 
 and he connects it also with the evaluation of certain continued 
 fractions with quaternion constituents. (See the Miscellaneous 
 Examples at the end of this volume.) 
 
 199. The equation q' = aq + qb, 
 
 though apparently of the second degree, is easily reduced to the 
 first degree by multiplying by, and into, q^, when it becomes 
 
 1 = q'^a + bq'\ 
 
 and may be treated by the process of § 195. 
 The equation 
 
 V.q{a.+ V^q) = 0, 
 
200.] SOLUTION OF EQUATIONS. 14] 
 
 where a and ^ are given vectors, is easily seen to require for a 
 real (i.e. a non biquaternion) solution that q shall be a vector. 
 Hence we may write it as 
 
 whence, at once, 
 
 a + F/Sp = xp. 
 
 Assume S^p = y, 
 
 and we have 
 
 -{y-o) = {x-l3)p, 
 
 or -{co + ^)(y-a)={x'-^')p. 
 
 The condition that p is a vector gives 
 
 xy - Sa^ = 0, 
 so that the value of p, containing one scalar indeterminate, is 
 
 To determine p completely we require one additional scalar 
 condition. 
 
 If we have, for instance, 
 
 Srjp = e, 
 
 X is given by the cubic equation 
 
 (^^-/3^)e = ^.S«7-^^^->Sf.a/37. 
 
 But if the condition be that p is a vector-radius of the unit 
 sphere (a result which will be required below) we have the 
 biquadratic 
 
 „ „„ o uB „ 
 x' - B' = — f - a^ 
 x" 
 
 This gives two real values of x'', but they have opposite signs ; 
 so that there are always two, and only two, real values of x. 
 
 200. The equation 5" = aqb, 
 
 where a and b are given quaternions, gives 
 
 q(aqb) = (aqb)q; 
 and, by § 54, it is evident that the planes of q and aqb must coin- 
 cide. A little consideration (after the manner of the latter part 
 of § 196) will shew that the solution depends upon drawing two 
 arcs which shall intercept given arcs upon each of two great 
 circles; while one of them bisects the other, and is divided by 
 it in the proportion of m : 1. The equation treated in § 196 is 
 the special case of this when m = 1. 
 
142 QUATERNIONS. 
 
 EXAMPLES TO CHAPTER V. 
 
 1. Solve the following equations : — 
 
 (a) y.apl3=V.a.r^^. 
 
 (6) ap^p = pap^. 
 
 (c) ap + p^ = 'Y. 
 
 (d) S.a/3p + l3Sap - a F/Sp = 7. 
 
 (e) p + ap/3 = a^. 
 if) o.pPp = p^pa. 
 
 Do any of these impose any restriction on the generality of a and /3? 
 
 2. Suppose p = ix +jy + hz, 
 
 and —(f>p = aiSip + bjSjp + cJcSkp ; 
 
 put into Cartesian coordinates the following equations : — 
 
 (a) T<l>p=l. 
 
 (b) 8p<l>'p = -l. 
 
 (c) 8.p(cl>' + pTp = -l- 
 
 (d) Tp = T.<f>Up. 
 
 3. If X, fi, V be any three non-coplanar vectors, and 
 
 q = Vfj,v . (f>X + VvX , (f)/j, + VXfj, . <j)v, 
 
 shew that q is necessarily divisible by 8 . X/mv. 
 Also shew that the quotient is 
 
 m, - 26, 
 
 where Yep is the non-commutative part of (pp. 
 
 Hamilton, Elements, p. 442. 
 
 4. Solve the simultaneous equations : — 
 
 8^p =0; 
 ^"''' 8.apcj}p=0. 
 
 ,,, 8ap =0, 
 ^^ Sp<t>P =0. 
 
 8cip =0; 
 
 8. aipKp = 0. 
 
SOLUTION OF EQUATIONS. 143 
 
 5. If <l,p=-2l38ap+Vrp, 
 where r is a given quaternion, shew that 
 
 m = 2 (fif . a, v,>S . ^3/S,/3 J + %S (r Va^a^ . V^,^,) + SrlS . a/Sr 
 
 - 2 (8arS/3r) + SrTr\ 
 and m^-'a- = t(ra^a^S.^^0^(7) + %V.(xV(V^(r.r)+V(rrSr-VrS(Tr. 
 
 Lectures, p. 561. 
 
 6. If [pq] denote pq — qp, 
 
 ipqr) „ S.plqr], 
 
 [pqr] „ (pqr) + [rq] Sp + [pr] 8q + [qp] 8r, 
 
 and (pqi^s) „ S.plqrs]; 
 
 shew that the following relations exist among any five quaternions 
 
 0=p (qrst) + q (rstp) + r (stpq) + s (tpqr) + 1 (pqrs), 
 and q (prst) = [rst] 8pq — [stp] 8rq + [tpr] 8sq — [prs] 8tq. 
 
 Elements, p. 492. 
 
 7. Shew that if (f), yjr be any linear and vector functions, and 
 a, /S, 7 reetangiilar unit-vectors, the vector 
 
 is an invariant. [This will be immediately seen if we write it in 
 the form 6 = V. ^Vilrp, 
 
 which is independent of the directions of a, ^, y. But it is good 
 practice to dispense with V, when possible.] 
 
 If <^p = %7,8^p, 
 
 and ■^p = %r)^8^^p, 
 
 shew that this invariant may be expressed as 
 -SFj/il^f or "tV^^^^. 
 Shew also that ^^'p ~ '^^'p = — Vdp- 
 
 The scalar of the same quaternion is also an invariant, and may be 
 written as 
 
 -^^,8mM. 
 
 = -\8v,cj>^, 
 
 8. Shew that if ^p = a8ap + 08l3p + ySjp, 
 where a, /S, 7 are any three vectors, then 
 
 (fi-'pS" . aj3y = a,8a^p + ^,8^,p + y,Sy,p, 
 where a^ = V^y, &c. 
 
144 QUATERNIONS. 
 
 9. Shew that any self-conjugate linear and vector function may 
 in general be expressed in terms of two given ones, the expression 
 involving terms of the second order. 
 
 Shew also that we may write 
 
 (p + z = a {'!3 + xY + b (i^ + oo) {w +y) + c (w + yf, 
 where a, b, c, x, y, z are scalars, and ■a and w the two given func- 
 tions. What character of generality is necessary in ■a and « ? How 
 is the solution affected by non-self-conjugation in one or both ? 
 
 10. Solve the equations : — 
 
 {a) <f =?,qi+\Oj. 
 
 (b) q' =2q + i. 
 
 (c) qaq = bq + c. 
 (cf) aq =qr = rb. 
 
 11. Shew that <l)VVcf,p = -mVV^-'p. 
 
 12. If be self-conjugate, and a, ^, y a. rectangular system, 
 
 ,Sf. Va<j)ar^(j)^Vry(l>y= 0. 
 
 13. ^i/r and ■\jr(j) give the same values of the invariants m, 
 
 14. If <f>' be conjugate to (f>, <]}(f}' is self-conjugate. 
 
 15. Shew that ( VaOf + ( V/3ef + ( VyOf = Iff" 
 if a, /8, 7 be rectangular unit-vectors. 
 
 1 6. Prove that V (^ - ^) p = - p V ^ + 2 V^. 
 
 17. Solve the equations : — 
 
 (a) 0' = ^; 
 
 <'> *;^:°; 
 
 where one, or two, unknown linear and vector functions are given 
 in terms of known ones. (Tait, Froc. R. S. E. 1870-71.) 
 
 18. If ^ be a self-conjugate linear and vector function, ^ and rj 
 two vectors, the two following equations are consequences one of 
 the other, viz. : — 
 
 g ^ V. 77<^^7 
 
SOLUTION OF EQUATIONS. 145 
 
 From either of them we obtain the equation 
 
 This, taken along with one of the others, gives a singular theorem 
 when translated into ordinary algebra. What property does it give 
 of the surface 
 
 S.pcPp^'p = l'i [Ibid.] 
 
 19. Solve the equation 
 
 qaq = ^q^. 
 
 Shew that it has a vector solution, involving the trisection of an 
 angle : — and find the condition that it shall admit of a real 
 quaternion solution. 
 
 20. Solve 
 
 bqaq = qbqa, 
 
 and state the corresponding geometrical problem ; shewing that 
 when a and b are equal vectors, q is equal to each. 
 
 21. Given cp, a self-conjugate linear and vector function, and 
 a vector e ; find the cubic in ^jr, where 
 
 ■\lrp = <f>p+ Vep. 
 
 22. Investigate the simplest expressions for any linear and 
 vector function in terms of given ones:— and point out what 
 degree of generality is necessary in the latter. 
 
 Why cannot the conjugate of a linear and vector function be 
 generally expressed in powers of the function itself ? 
 
 T. Q. I. 
 
 10 
 
CHAPTEE VI. 
 
 SKETCH OF THE ANALYTICAL THEORY OF QUATEENIONS. 
 (By Peof. Cayley.) 
 
 (a) Expression, Addition, and Multiplication. 
 
 By what precedes we are led to an analytical theory of the 
 Quaternion q = w+ ix ■]- jy + kz, where the imaginary symbols 
 i, j, k are such that 
 
 i^= — 1, f = — \,¥= — l,jk = — kj = i, ki = - ik =j, ij = -ji = k. 
 The Tensor Tq is = Ji^ + x^ + f + z', and 
 
 the Versor Uq is = , (w + ix + jy + kz), 
 
 ^ VW + X + y + z 
 
 i which, or the quaternion itself when Tq = 1, may be expressed in 
 the form 
 
 cos S + sin 8 {ia +jb + kc) where a^ + 6" + c" = 1 ; 
 
 such a quaternion is a Unit Quaternion. The squared tensor 
 w' + x'-^ y^ + z^ is called the Norm. 
 
 The scalar part S:q is = w, and the vector part Yq, or say a 
 Vector, \& = ix-vjy-^kz. The Length is = ^x^ + 2/' + s', and the 
 
 quotient , a =^ ^"^ + JV + ^^)> o^" say a vector ix + jy + kz 
 
 sj X ~T y -T z 
 
 where x'' + y^-^s?= 1, is a Unit Vector. 
 
 The quaternions w-\-ix+jy +kz and w — ix—jy — kz are said 
 to be Conjugates, each of the other. Conjugate quaternions have 
 the same norm ; and the product of the conjugate quaternions is 
 the norm of either of them. The conjugate of a quaternion is 
 denoted by q, or Kq. 
 
ANALYTICAL THEORY. 147 
 
 Quaternions q = w + ix+jy+ kz, q' = w' + ix' + jy' + kz' are 
 added by the formula 
 
 g- + 2' = w + w' + i (a? + x') + j(y + y') + k(z + z), 
 the operation being commutative and associative. 
 They are multiplied by the formula 
 
 qq = ww' — xx — yy — zz 
 + i (wx' + xfw' + yz' — zy') 
 +j {wy + yw' + zx' - z'x) 
 + k {wz' + zw' + 532/' — aj'i/), 
 where observe that the norm is 
 
 = (w' + x' + y^ + z') (w" + x" + y" + z") 
 the product of the norms of q and q. 
 
 The multiplication is not commutative, q'q 4= qq' ; but it is 
 associative, qq .q" = q. q'q" = qq'q", &c. In combination with 
 addition it is distributive, q {q' + q") =■ qq' + qq", &c. 
 
 (b) Imaginary Quaternions. Nullitats. 
 
 The components w, x, y, z oi a, quaternion are usually real, 
 but they may be imaginary of the form a+bj- 1, where ^-l is 
 the imaginary of ordinary algebra: we cannot (as in ordinary 
 algebra) represent this by the letter i, but when occasion requires 
 another letter, say 6, maybe adopted (the meaning, 6 = J- 1, being 
 explained). An imaginary quaternion is thus a quaternion of the 
 form (w + 6w^ +i{x + dx^ +j(y + Oy^ + k{z + 62^), or, what is the 
 same thing, if q, q^ be the real quaternions 10+ ix+Jy + kz, 
 w^ + ix^+jy^ + kz^, it is a quaternion q + Oq^; this algebraical 
 imaginary = J— 1 is commutative with each of the symbols 
 i, j, k : or, what comes to the same thing, it is not in general 
 necessary to explicitly introduce at all, but we work with the 
 quaternion w + ix +jy + kz, in exactly the same way as if w, x, y, z 
 were real values. A quaternion of the above form, q+dq^, was 
 termed by Hamilton a " biquaternion " but it seems preferable to 
 speak of it simply as a quaternion, using the term biquaternion 
 only for a like expression q + dq^, wherein 9 is not the J— 1 of 
 ordinary algebra. 
 
 It may be noticed that, for an imaginary quaternion, the squared 
 tensor or norm w" + ar' + y^ + z' may be = ; when this is so, the 
 quaternion is said to be a "Nullitat"; the case is one to be 
 separately considered. 
 
 10—2 
 
148 
 
 QUATERNIONS. 
 
 (c) Quaternion as a Matrix. 
 Quaternions have an intimate connection with Matrices. 
 Suppose that 6, = J^^, is the J^^ of ordinary algebra, and in 
 place of i, j, k consider the new imaginaries x, y, z, w which are 
 such that 
 
 x = ^{ 1 — 6i), or conversely 1 = x + w, 
 y = l{ j-Gk), i=.0(x-w), 
 
 z = i(-j-ek), j= (2/-4 
 
 w = i( l+0i), k=e{y + z); 
 
 so that a, h, c, d being scalars, ax+by +cz + dw denotes the 
 imaginary quaternion 
 
 ^ {a + d+ {b -c)j} + i {{- a + d)i+{-b- c)k] 0. 
 We obtain for x, y, z, w the laws of combination 
 X y z w 
 
 X X V Q , that is c^ = x, xy = y, xz=0, xw = 
 
 y i) X y &c., 
 
 z 
 
 w 
 
 and consequently for the product of two linear forms in (x, y, z, w) 
 
 we have 
 
 {ax + by + CZ + dw) (a'x + b'y + c'z + d'w) 
 
 = (aa' + be) X + {ab' + bd') y + (ca! + dc') z + (c6' + dd!) w ; 
 and this is precisely the form for the product of two matrices, viz. 
 we have 
 
 {a, c') (6', d!) 
 
 X 
 
 y 
 
 
 
 
 
 
 
 
 
 X 
 
 y 
 
 z 
 
 w 
 
 
 
 
 
 
 
 
 
 z 
 
 w 
 
 a b a', y = (a, b) „ » "= 1 "'"'' + ^<''' "^' + ^'^' 
 
 c d c', d' (c, d) „ „ I ca + dc', cb' + dd' 
 
 and hence the linear form ax + by + cz + dw, and the matrix 
 
 i ' , may be regarded as equivalent symbols. This identifica- ' 
 
 tion was established by the remark and footnote " Peirce's Linear 
 Associative Algebra," Amer. Math. Jour. t. 4 (1881), p. 132. 
 
 (d) The Quaternion Equation %AqB—C. 
 
 In ordinary algebra, an equation of the first degree, or linear 
 equation with one unknown quantity x, is merely an equation of 
 the form ax = b, and it "ives at once x = a" '6. 
 
ANALYTICAL THEORY. 149 
 
 But the case is very different with quaternions; the general 
 form of a linear equation with one unknown quaternion q is 
 A^qB^ + A^qB^ +• . .= 0, or say tAqB = G, 
 
 where G and the several coefficients A and B are given qua- 
 ternions. 
 
 Considering the expression on the left-hand side, and assuming 
 q = w + ix-\-jy + kz, it is obvious that the expression is in effect of 
 the form 
 
 hw + ax +^ y + yz 
 
 + i (S,w -h OL^x -f ^^y + y^z) 
 
 + j{5^io + a^x + ^^y+y^z) 
 
 + lc(B^w + a^x + ^,y-\-y^z), 
 where the coefficients S, a, /3, 7 &c. are given scalar magnitudes : 
 if then this is equal to a given quaternion G, say this is 
 
 X -I- i\ +jX^ + k\, 
 we have for the determination of w, x, y, z the four equations 
 Z w -\- OL X ■\- ^ y ■\- y z =X, 
 SjW -I- CL^x -H /Sji/ -I- y^z — Xj, 
 \w 4- «,« 4- /3^y + 7./ = \, 
 h^iu+a^x + ^^y + y^z=\, 
 
 and we thence have w, x, y, z, each of them as a fraction with a 
 given numerator, and with the common denominator 
 
 s, 
 
 a, 
 
 /8, 
 
 7 
 
 K 
 
 «i. 
 
 P. 
 
 7, 
 
 K 
 
 %' 
 
 0. 
 
 72 
 
 K 
 
 «,. 
 
 /3.. 
 
 7, 
 
 viz. this is the determinant formed with the coefficients S, a, /8, y, 
 &c. Of course if A = 0, then either the equations are inconsistent, 
 or they reduce themselves to fewer than four independent 
 equations. 
 
 The number of these coefficients is = 16, and it is thus clear 
 that, whatever be the number of the terms A^qB^, A^qB,^, &c. we 
 only in effect introduce into the equation 16 coefficients. A single 
 term such as A^qB^ may be regarded as containing seven coefficients, 
 for we may without loss of generality write it in the form 
 
 ^ (H- ia + jb + kc) q(l+ id +je + kf), 
 and thus we do not obtain the general form of linear equation 
 
150 QUATERNIONS. 
 
 by taking a single term A^qB^ (for this contains seven coefficients 
 only) nor by taking two terms A^qB^, A^qB^ (for these contain 14 
 coefficients only) ; but we do, it would seem, obtain the general 
 form by taking three terms (viz. these contain 21 coefficients, 
 which must in effect reduce themselves to 16): that is, a form 
 '^ilJ^i + A^qB.^ + A^qB^ is, or seems to be, capable of representing 
 the above written quaternion form with any values whatever of the 
 16 coefficients S, a, /3, 7 &c. But the further theory of this reduc- 
 tion to 16 coefficients is not here considered. 
 
 The most simple case of course is that of a single term, say we 
 have AqB = C : here multiplying on the left by A~^ and on the 
 right by £"', we obtain at once q = A~^GB~^. 
 
 (e). The Nivellator, and its Matrix. 
 
 In the general case, a solution, equivalent to the foregoing, but 
 differing from it very much in form may be obtained by means 
 of the following considerations. 
 
 A symbol of the above form XA ( )B, operating upon a 
 quaternion q so as to change it into %A (q) B, is termed by 
 Prof. Sylvester a " Nivellator :" it may be represented by a single 
 letter, say we have = 2^ ( )B; the effect of it, as has just 
 been seen, is to convert the components (w, x, y, z), into four linear 
 functions (w^, x^, y^, z^ which may be expressed by the equation 
 
 (w, X, y, z), 
 
 (w„ «„ 2/„ z^) = 
 
 s, 
 
 a, 
 
 /8, 
 
 7 
 
 
 K, 
 
 «i> 
 
 ^>. 
 
 7i 
 
 
 s.. 
 
 S. 
 
 ^. 
 
 72 
 
 
 Ss> 
 
 «8. 
 
 /^s, 
 
 73 
 
 or say by the multiplication of (w, x, y, z) by a matrix which may 
 be called the matrix of the nivellator; and the theory of the 
 solution of the linear equation in quaternions thus enters into 
 relation with that of the solution of the linear equation in 
 matrices. 
 
 The operation denoted by (f> admits of repetition : we have for 
 instance 
 {A^{ )B, + A,( )B,Y = A,'{ )B,' + A,A,{ )B,B, 
 
 + A,A^{ )B^B, + A:( )5/; 
 and similarly for more than two terms, and for higher powers. 
 
 Considering ^ in connexion with its matrix M, we have 
 ilf {w, X, y, z) for the components of <})'' (q), M^ (w, x, y, z) for 
 
ANALYTICAL THEORY. 
 
 151 
 
 those of (^' {q), and so on. Hence also we have the negative 
 powers <^"', &c. of the operation ^. The mode in which <^"' can 
 be calculated will presently appear: but assuming for the moment 
 that it can be calculated, the given equation is {q) = C, that 
 is we have q = ^~' (C), the solution of the equation. 
 
 A matrix M of any order satisfies identically an equation of 
 the same order : viz. for the foregoing matrix M of the fourth 
 order we have 
 
 
 B-M, a, 
 
 A 7 
 
 
 8, «.-ilf, /3, 7, 
 
 
 K' a^. ^,-M, % 
 
 viz. this is 
 
 M' - eM' +fM-' - gM+ h = 0, 
 
 where h is the before mentioned determinant 
 
 
 S, a, /3, 7 
 
 , say this is,h = L 
 
 
 K «i. /3,. 7i 
 
 
 
 ^2, a^, ^2> 72 
 
 
 
 ^3' «3' /^3. 73 
 
 
 M, in its operation on the components [w, x, y, z) of q, exactly 
 represents ^ in its operation on ^ : we thus have 
 
 0* -e<^^ +/</," -5r0 + /, = o, 
 viz. this means that operating successively with </> on the arbitrary 
 quaternion Q we have identically 
 
 ^' [Q) - e^' (Q) +f4>' {Q)-g<f>{Q)-^hQ = 0; 
 
 where observe that the coefficients e, f, g, h have their foregoing 
 values, calculated by means of the minors of the determinant: but 
 that their values may also be calculated quite independently of 
 this determinant: viz. the equation shews that there is an identical 
 linear relation connecting the values (/>* (Q), cj)' (Q), ^^ (Q), </> (Q) 
 and Q: and from the values (assumed to be known) of these 
 quantities, we can calculate the identical equation which connects 
 them. But in whatever way they are found, the coefficients 
 e> f, 9> h are to be regarded as known scalar functions. 
 Writing in the equation (f>'^ Q in place of Q, we have 
 
 </>' (Q) - ef (Q) +f<l^ (Q) -gQ + hr' (Q) = 0, 
 
 viz. this equation gives <^"' (Q) as a linear function of Q, cj> (Q), <f)^ (Q) 
 and <f)^{Q} : and hence for the arbitrary quaternion Q writing the 
 value C, we have q, = 0"^ (0) given as a linear function of 
 
152 
 
 QUATERNIONS. 
 
 G, (f>((T), (f>'{G) and (f)^(C): we have thus the solution of the 
 given linear equation. 
 
 (/) The Vector Equation lApB = C. 
 
 The theory is similar if, instead of quaternions, we have vectors. 
 As to this observe in the first place that, even if A, q, B are each 
 of them a vector, the product AqB will be in general, not a vector, 
 but a quaternion. Hence in the equation tAqB = C,it C and the 
 several coefficients A and B be all of them vectors, the quantity q 
 as determined by this equation will be in general a quaternion : 
 and even if it should come out to be a vector, still in the process 
 of solution it will be necessary to take account, not only of the 
 vector components, but also of the scalar part; so that there is 
 here no simplification of the foregoing general theory. 
 
 But the several coefficients A, B may be vectors so related to 
 each other that the sum %ApB, where p is an arbitrary vector, 
 is always a vector' ; and in this case, if G be also a vector, the 
 equation 'LApB = G will determine p as a vector : and there is 
 here a material simplification. Writing p = ix +jy + kz, then 
 "StApB is in effect of the form 
 
 i {a^x + ^,y + 7i^) 
 
 viz. we have these three linear functions of {x, y, z) to be equalled 
 to given scalar values \, \, \, and here x, y, z have to be 
 determined by the solution of the three linear equations thus 
 obtained. And for the second form of solution, writing as before 
 j> = %A{ )B, then ^ is connected with the more simple matrix 
 
 if = 
 
 
 7, 
 
 «3. /^S. 73 
 
 and it thus (instead of a biquadratic equation) satisfies the cubic 
 equation 
 
 = 0, 
 
 say 
 
 «3, ^3, 7s -^ 
 
 M'-eM'+fM-g-^O. 
 
 ^ Thus, if A, B are conjugate quaternions, ApB is a vector a : this is in fact 
 the form which presents itself in the theory of rotation. 
 
ANALYTICAL THEORY. 153 
 
 We have therefore for <p the cubic equation 
 
 and thus (^"' (Q) is given as a linear function of Q, (Q), (f)" (Q), or, 
 what is the same thing, 0~'(O) as a linear function of C, ^(C), <f)^{(J) : 
 and (this being so) then for the solution of the given equation 
 ^(p) = G, we have p = <^~'((7), a given linear function of C, <p(G), 
 
 (g) Nullitats. 
 
 Simplifications and specialities present themselves in particular 
 cases, for instance in the cases Aq-^qB = G, and Aq = qB, which 
 are afterwards considered. 
 
 The product of a quaternion into its conjugate is equal to 
 the squared tensor, or norm ; ad = T^ (a); and thus the reciprocal 
 of a quaternion is equal to the conjugate divided by the norm ; 
 hence if the norm be = 0, or say if the quatei-nion be a nullitat, 
 there is no reciprocal. In particular, 0, qua quaternion, is a 
 nullitat. 
 
 The equation aqb = c, where a, b, c are given quaternions, 
 q the quaternion sought for, is at once solvable ; we have 
 q = a~'^cb~^; but the solution fails if a, or b, or each of them, is a 
 nullitat. And when this is so, then whatever be the value of q, 
 we have aqb a nullitat, and thus the equation has no solution 
 unless also c be a nullitat. 
 
 If a and c are nullitats, but b is not a nullitat, then the equa- 
 tion gives aq^cb''^, which is of the form aq = c; and similarly if 
 b and c are nullitats but a is not a nullitat, then the equation 
 gives qb = a'^c, which is of the form qb=c: thus the forms to 
 be considered are aq — c, qb = c, and aqb = c, where in the first 
 equation a and c, in the second equation b and c, and in the 
 third equation a, b, c, are nullitats. 
 
 The equation aq = c, a and c nullitats, does not in general 
 admit of solution, but when it does so, the solution is indeter- 
 minate; viz. if Q be a solution, then Q + dR (where R is an 
 arbitrary quaternion) is also a solution. Similarly for the equation 
 qb=c, if Q be a solution, then Q + Sb{S an arbitrary quaternion) 
 is also a solution : and in like manner for the equation aqb = c, if 
 Q be a solution then also Q + dR + 8b {B, 8 arbitrary quaternions) 
 is a solution. 
 
154 QUATERNIONS. 
 
 (h) Conditions of Consistency, when some Coefficients 
 are Nullitats. 
 
 Consider first the equation aq = c; -fiTciimg a = a^+ia^+ja^+ka^ 
 (a/ + a^'' + <+ < = 0) and c=c,+ic,+jc^^kc^{c,^ + c,^+c^-^c^=0); 
 also q = w +ix +jy + kz, the equation gives 
 
 c, = a,w - a^x - a^y - a^z, 
 
 c, = a^w + a^x — a^y + a^z, 
 
 c^ = a^w + a^x + a^y - a^z, 
 
 C3 = a^w - a^x + a^y + a^z, 
 equations which are only consistent with each other when two 
 of the c's are determinate linear functions of the other two c's ; 
 and when this is so, the equations reduce themselves to two 
 independent equations.' Thus from the first, second and third 
 equations, multiplying by afl^ — a„a^,— afl^ — a^a^, and — a^' — a^', 
 and adding, we obtain 
 
 (a^a, - a.A) c, - (a^a^ + a,a^) c^ - (a/ + a,') c, = ; 
 similarly from the first, second and fourth equations, multiplying 
 by - a^a^ — a^a^, — a^a^ + a^a^, — ra/ — a^, and adding, we have 
 
 - (ai^, + a^a^ c, - (a.a, - a,aj c^ - (a/ + O.c, = 0, 
 and when these two equations are satisfied, the original equations 
 are equivalent to two independent equations ; so that we have for 
 instance a solution Q = w + ix where c^ = a^zy — a^x, c, = a^w + a^x, 
 
 that is w = -^ V , X = — ~ — • and the general solution is 
 
 a^+a^' 0^4 +Cij 
 
 then obtained as above. 
 
 The equations connecting the as and the c's may be presented 
 
 in a variety of different forms, all of them of course equivalent in 
 
 virtue of the relations a/ + a^' + a^ + a^ = 0, c/ + c^^ + c/ + c^ = 0; 
 
 viz. writing 
 
 A^ = a^ + a./ = - a/ - a^ , F^ = a^a^ + afl^, F^ = ajx^ - afl^, 
 A^ = a^ + a^' = -a^-a^, F^ = a,a,+a^a^, F^' = a^a,-a^a^, 
 
 then the relation between any three of the c's may be expressed 
 in three different forms, with coefiicients out of the sets A^, A^, A^; 
 F^, F^, F^; Fl, F^, F^-. Obviously the relation between the c's is 
 satisfied if c = : the equation then is aq = 0, satisfied by g' = aB, 
 R an arbitrary quaternion. 
 
. ANALYTICAL THEORY. 155 
 
 We have a precisely similar theory for the equation qh=c; 
 any two of the c's must be determinate linear functions of the 
 other two of them; and we have then only two independent 
 equations for the determination of the w, x, y, z. 
 
 In the case of the equation ag6 = c (a, h, c all nullitats) the 
 analysis is somewhat more complicated, but the final result is a 
 simple and remarkable one; from the condition that a, b are 
 nullitats, it follows that a 6, aib, ajb, akb are scalar (in general 
 imaginary scalar) multiples of one and the same nullitat, say of 
 ah: the condition to be satisfied by c then is that c shall be a 
 scalar multiple of this same nullitat, say c = \ab; the equation 
 aqb = \ab has then a solution q=\, and the general solution is 
 q = \-]- aR + 8b, where R, S are arbitrary quaternions. 
 
 (i) The Linear Equations, aq — qb = 0, and aq—qb = c. 
 
 The foregoing considerations explain a point which presents 
 itself in regard to the equation aq — qb = 0, (a, i given quaternions, 
 q a quaternion sought for): clearly the equation is not solvable 
 (otherwise than by the value q = 0) unless a condition be satisfied 
 by the given quaternions a, b; but this condition is not (what 
 at first sight it would appear to be) T'a = T'^b. The condition 
 (say £1) may be satisfied although T'a^ T'b; and being satisfied, 
 there exists a determinate quaternion q, which must evidently 
 be a nullitat (for from the given equation aq = qb we have 
 (ra-T'b) T\ = 0, that is T'q = 0). If in addition to the con- 
 dition fl we have also T^a — T'b = 0, then (as will appear) we 
 have an indeterminate solution q, which is not in general a 
 nullitat. 
 
 Take the more general equation aq — qh = c: this may be 
 solved by a process (due to Hamilton) as follows: multiplying 
 on the left hand by a and on the right hand by h, we have 
 aaq — dqb = ac, aqb — qh^ = cb, whence subtracting 
 
 aaq —{a + a)qb + qb^ = dc- cb, 
 
 or since ad,a + a are scalars q{aa — {a + d)b + ¥]=dc—cb: viz. 
 this is an equation of the form qB = G (B, G given quaternions), 
 having a solution q = GB~^. 
 
 Suppose c= 0, then also (7=0; and unless 5 is a nullitat, the 
 equation qB = Q (representing the original equation aq = qh), has 
 only the solution q = 0; vi?. the condition in order that the 
 
156 QUATERNIONS. 
 
 equation aq = qh may have a solution other than g = 0, is jB = nul- 
 litat, that \% ad — {a + a) h + ¥ = nullitat ; viz. we must have 
 
 a^ + a^ + a.^+a^^ 
 -2a,(b, + i\+jb, + kbJ 
 
 + h^ + 26, (j.b, +jb, + kb^) - 6; - &/ - &/ = nullitat, 
 that is 
 
 a/ + a,' + a/ + a/ - 2aJ>, + &/ - 6/ - V - 6/ 
 
 + 2 (6, - aj (i&, + j6, + kb^) = nullitat. 
 
 The condition ft thus is 
 
 (a;+<+a/+<-2aA+&;-V-^'.-^'/)'+4(a-6JXV+&,^+630=0, 
 
 that is 
 
 {{a, - b,y + a/ + a.; + < - b,^-b,'-b^'Y+^a,-bf(b,'+b:+b,')=0, 
 
 or, as this may also be written, 
 
 (a, - b,y + 2 (a, - 6,)^ (a/ + < + a/ + b,' + b^ + 6/) 
 
 Writing herein • 
 
 < + a,' + a,' + < = A\ 6/ + 6,» + 6/ + 6/ = B", 
 
 the condition is 
 
 (a, - b,y + 2 (a,- b,y (A^' + ff- < - 6/) + {A' -B'- < + 6/)' = 0, 
 
 which is easily reduced to 
 
 4 {a, - b,) {a,B' - b,A') + {A" - BJ = 0, 
 
 and, as already noticed, this is different from Pa — T^b = 0, that is 
 A'-B' = 0. 
 
 If the equation A'--B^ = is satisfied, then the condition XI 
 reduces itself to a^—b^ = 0; we then have a = a^ + a, b = a^ + ^, 
 where a, /3 are vectors, and the equation is therefore aq = q^ 
 where (since J ' -i?^ = a^ + a/ + a," - 6^" - 6/ - b^', = 0), the tensors 
 are equal, or we may without loss of generality take a, /3 to be 
 given unit vectors, viz. we have a!' = — '\, /3' = — l : and this being 
 so, we obtain at once the solution q = \{a + ^) + fi(l - a^) (X, (it, 
 arbitrary scalars) : in fact this value gives 
 
 ag = X (- 1 + a/3) + /li (a + ;S) = q^. 
 
 Reverting to the general equation aq — qb = c, the conjugate of 
 ad—{a + d)b + b^ is ad — {a + a)b + 1)\ and we thus obtain the 
 solution 
 
 q {4 {a,- 6J (a,B'- b,A') + {A'-BJ} = (dc - cb) [aa -{a+d)b + b% 
 
ANALYTICAL THEORY. 157 
 
 but this solution fails iiad — {a + a)b + b' is a nuUitat : supposing 
 it to be so, the equation is only solvable when C satisfies the 
 condition which expresses that the equation qB=C is solvable 
 when B, C are nuUitats. 
 
 The equation aq — qb = c, could it is clear be in like manner 
 reduced to the form Aq= C. 
 
 (j) The Quadric Equation q'' — 2aq+b = 0. 
 
 We consider the quadric equation q^ — 2aq + b = 0; a and b 
 given quaternions, q the quaternion sought for. The solution 
 which follows is that given by Prof. Sylvester for a quadric 
 equation in binary matrices. 
 
 In general if q be any quaternion, =w + ix +jy + kz, then 
 {q—wf +x'+ ■y''+z^ = 0, that is q^ -2qw + w'' + x^ + y^ + z' = 0, or say 
 q^ - 2q (seal, q) + norm q = 0: viz. this is an identical relation 
 connecting a quaternion with its scalar and its norm. 
 
 Writing as above q = w-j-ix+ji/ + kz, and t = w' + x^ +y^ + z'' 
 for the norm, we thus have 
 
 q^-2wq + t = 0, 
 and combining this with the given equation 
 
 q^-2aq + b = 0, 
 
 we find 2{a-iu)q-(b-t) = 0, that is 2q = (a- w)"' (6 - 1), 
 
 an expression for q in terms of the scalar and norm w, t, and of the 
 known quaternions a and b. 
 
 2q as thus determined satisfies the identical equation 
 
 {2qf - 2 {2q) seal, {(a - w)'' {b-t)\+ norm {{a - w)"' {b - 1)} = 0, 
 
 and we have 
 
 seal, {(a -^) (^'-*)1=- r^ovmia-w) "' 
 
 norm (6— i) 
 norm {{a - w)' {b-t)}= no7iir(a'-=~w) ' 
 
 (d the conjugate of a). 
 
 The equation thus becomes 
 
 4<q^ norm (a - w) - 4<q {seal, {a -w){b- 1)] + norm (6 - «) = ; 
 
 this must agree with 
 
 o» -2qw + t =0, 
 
158 QUATERNIONS. 
 
 or say the function is = 4\ {(f — 2qw + t); we thus have 
 norm (a — w)= \, 
 seal, (a — w)(b — t) = 2\w, 
 norm {b — t) = 4>\t, 
 three equations for the determination of X, w, t; and then, w, t 
 being determined, the required value of q is 2q={a — w)"' (6 - 1) 
 as above. 
 
 To develope the solution let the values of a, b, c, f, g, h be 
 defined as follows : viz. 
 
 norm (ax + by + z) = (a, b, c, f, g, hja;, y, zf, 
 viz. writing a = a^ + ia^ +ja^ + ka^, 
 
 b = b, + ib^+jb, + kb„ 
 then this equation is 
 
 {a,x + b^ + zf + (a^x + b,yf + {a^x + b^yf + {a,x + b^f 
 
 = (a, b, c, f, g, \i\x, y, zf, 
 that is, a, b, c, f, g, h denote as follows 
 a = a/ + aj' + < + <, f=6,, 
 b = &/ + J,^+V + 6/, g=a^ 
 c = 1, h = ap^ + a,6j + a^^^ + ap^. 
 
 We then have 
 
 norm (a — w) = (a^ — w)^ + a^ + a/ + Wg^ 
 
 seal, (a - w) (6 - i) = (a^ - w) (b^ -t) + afi, + afi^ + ajb^, 
 noTxa {b-t)=(b,-ty + b^'+ b,'+ b,\ 
 
 or expressing these in terms of (a, b, c, f, g, h) the foregoing three 
 equations become 
 
 a — 2gw 4- cw^ = X, 
 
 h— gt— fw + etw = 2X1/;, 
 
 b-2ft +cf =^Xt, 
 
 where c (introduced only for greater symmetry) is = 1. 
 
 Writing moreover A, B, C, F, G, H=he-f, ca-g^ ab-h^ 
 gh — af, hf— bg, fg — ch, and Z" = abc — af^ — bg^ — ch'^ + 2fgh ; also 
 in place of w, t introducing into the equations u =w — g, and 
 v = t—(, the equations become 
 
 u' + B =\, 
 
 uv — H = 2\ (u + g), 
 
 v''+A=4>X{v + f). 
 
ANALYTICAL THEORY. 159 
 
 We deduce u^ = x-B, 
 
 u{v~2X) =H+2Xg, 
 
 (v-2\y = iX' + 't\{-A, 
 
 and we thence obtain, to determine \, the cubic equation 
 
 (X - B) (4\' + 4Xf- A) - (2\g + H)' = 0, 
 viz. this is 
 
 4\'+ 4X' (f- a) + \ {_ be + r + 4 (gh - af)} 
 
 + c (abc - aP - bg' - oh'' + 2fgh) = 0, 
 that is, 4\=' + 4X" (i - a.) +\(- A + iF) + K = 0, 
 
 and, X being determined by this equation, then 
 
 u 
 
 and then w = u + g, t = v + i; consequently 
 
 2q = {a-g-uy'(b-i-v). 
 
 Write for a moment a — g — u = @, then 
 
 ®{@ + 2u) = {a-gy-u' = a'-2ag + a.-B-u\ =-X 
 
 (since a = g + ia^+ja^ + kag, SL = g' + a' + a^' + a^' and thus the 
 identical equation for a is a" — 2ag + a = 0) : that is ©"+ 2m@ +\= 0, 
 or \@~^ = -(@ + 2u) = (a-g + u); that is @~', =(a-g-uy\ 
 
 = - r(a-g+u}; and the value of g' is 2q=— ~ (a- g+u) {b- f-v), 
 K A. 
 
 or say it is 
 
 2q^-l(a-g + u)(b-f-2X-^S), 
 
 where X is determined by the cubic equation, and m is = + JX — B; 
 we have thus six roots of the given quadric equation q^ — 2aq + 6 = 0. 
 
CHAPTEE Vri. 
 
 GEOMETRY OF THE STRAIGHT LINE AND PLANE. 
 
 201. Having, in the preceding Chapters, given a brief ex- 
 position of the theory and properties of quaternions, we intend to 
 devote the rest of the work to examples of their practical appli- 
 cation, commencing, of course, with the simplest curve and surface, 
 the straight line and the plane. In this and the remaining Chapters 
 of the work a few of the earlier examples will be wrought out in 
 their fullest detail, with a reference to the previous part of the 
 book whenever a transformation occurs; but, as each Chapter 
 proceeds, superfluous steps will be gradually omitted, until in 
 the later examples the full value of the quaternion processes is 
 exhibited. 
 
 202. Before proceeding to the proper business of the Chapter 
 we make a digression in order to give a few instances of applica- 
 tions to ordinary plane geometry. These the student may multiply 
 indefinitely with great ease. 
 
 (a) Euclid, I. 5. Let a and yS be the vector sides of an iso- 
 sceles triangle ; /3 — a is the base, and 
 
 ToL = Tl3. 
 The proposition will evidently be proved if we shew that 
 a (a - 13)-' = iT/g (yS - a)"' (§ 52). 
 This gives a (a - l3p = (yS - a)-'l3, 
 
 or (^-a)a = /3(a-;S), 
 
 or -a^ = -^\ 
 
 (b) Euclid, I. 32. Let ABC be the triangle, and let 
 
 AB ' 
 
202.J GEOMETEY OF STRAIGHT LINE AND PLANE. 161 
 
 where 7 is a unit-vector perpendicular to the plane of the triangle. 
 If Z = 1, the angle CAB is a right angle (§ 74). Hence 
 
 A=1'k/2 (§ 74). Let B = m -it 12, G = n it 12. We have 
 
 UAO = iUAB, 
 
 UGB = y"UGA, 
 
 UBA^Y'UM 
 
 Hence UM = 7" . 7" . 7' UAB, 
 
 or _ 1 = ryl+m+n^ 
 
 That is l-'rm + n= 2, 
 
 or A+B+G^rr. 
 
 This iS; properly speaking, Legendre's proof; and might have been 
 given in a far shorter form than that above. In fact we have for 
 any three vectors whatever, 
 
 which contains Euclid's proposition as a mere particular case. 
 
 (c) Euclid, 1. 35. Let /3 be the common vector-base of the 
 parallelograms, a the conterminous vector-side of any one of them. 
 For any other the vector-side is a + x^ (§ 28), and the proposition 
 appears as 
 
 TF/3 (a + x/S) = TF/Sa (§§ 96, 98), 
 
 which is obviously true. 
 
 (d) In the base of a triangle find the point from which lines, 
 drawn parallel to the sides and limited by them, are equal. 
 
 If a, /8 be the sides, any point in the base has the vector 
 
 p = (l — w)a + 05^. 
 For the required point 
 
 which determines x. 
 
 Hence the point lies on the line 
 
 which bisects the vertical ang;le of the triangle. 
 
 This is not the only solution, for we should have written 
 
 T{l-x)Ta = Txm 
 
 instead of the less general form above which tacitly assumes that 
 l-x and X have the same sign. We leave this to the student. 
 T. Q. I. 11 
 
162 QUATERNIONS. L^OS- 
 
 (e) If perpendiculars be erected outwards at the middle 
 points of the sides of a triangle, each being proportional to the 
 corresponding side, the mean point of the triangle formed by their 
 extremities coincides with that of the original triangle. Find the 
 ratio of each perpendicular to half the corresponding side of the 
 old triangle that the new triangle may be equilateral. 
 
 Let 2a, 2^, and 2 (a + /8) be the vector-sides of the triangle, 
 i a unit-vector perpendicular to its plane, e the ratio in question. 
 The vectors of the corners of the new triangle are (taking the 
 corner opposite to 2^ as origin) 
 
 Pj = a + eia, 
 
 p^ = 2a + ^ + ei^, 
 
 /33 = a-fyS-ei(a + /S)- 
 From these 
 
 4 (Px + P. + P,) = i (4« + 2/3) = i {2a + 2 (a + ^)}, 
 
 which proves the first part of the proposition. 
 For the second part, we must have 
 
 T(p,-p,) = Tip,-p,) = T(p,-p,). 
 
 Substituting, expanding, and erasing terms common to all, the 
 student will easily find 
 
 3e' = 1. 
 
 Hence, if equilateral triangles be described on the sides of any 
 triangle, their mean points form an equilateral triangle. 
 
 203. Such applications of quaternions as those just made are 
 of course legitimate, but they are not always profitable. In fact, 
 when applied to plane problems, quaternions often degenerate into 
 mere scalars, and become (| 33) Cartesian coordinates of some 
 kind, so that nothing is gained (though nothing is lost) by their 
 use. Before leaving this class of questions we take, as an 
 additional example, the investigation of some properties of the 
 ellipse. 
 
 204. We have already seen (§ 31 [k)) that the equation 
 
 p = a cos + yS sin ^ 
 
 represents an ellipse, being a scalar which may have any value. 
 Hence, for the vector-tangent at the extremity of p we have 
 
 OT = -^ = — a sin ^ -i- /3 cos ^, 
 
207-] GEOMETRY OP STBAIGHT LINE AND PLANE. 163 
 
 which is easily seen to be the value of p when 6 is increased by 
 7r/2. Thus it appears that any two values of p, for which 6 
 differs by ■7r/2, are conjugate diameters. The area of the 
 parallelogram circumscribed to the ellipse and touching it at 
 the extremities of these diameters is, therefore, by § 96, 
 
 4>TVp ^=4,TV{a cos 61 + /S sin 6) (- a sin 6l + ;S cos d) 
 
 = 4>TVal3, 
 a constant, as is well known. 
 
 205. For equal conjugate diameters we must have 
 
 T(a cos 6* + /8 sin 6) = T(-asm0 + ^ cos 6), 
 or (a= - j8') (cos' d - sin' 61) + 4>8ix0 cos 6* sin 6" = 0, 
 
 tan2^ = -^. 
 
 The square of the common length of these diameters is of course 
 
 2 ' 
 
 because we see at once from § 204 that the sum of the squares of 
 conjugate diameters is constant. 
 
 206. The maximum or minimum of p is thus found ; 
 
 dTp _ 1 ^ dp 
 'W~~Tp Pd0' 
 
 = _ ^ {_ („^ _ ^^) cos 6* sin 61 + Sa^ (cos" 6 - sin' 0)]. 
 For a maximum or minimum this must vanish*, hence 
 tan 20 = -i — Tsi > 
 
 and therefore the longest and shortest diameters are equally 
 inclined to each of the equal conjugate diameters (§ 205). Hence, 
 also, they are at right angles to each other. 
 
 207. Suppose for a moment a and ^ to be the greatest and 
 least semidiameters, so that 
 
 ;Sfa,S = 0. 
 
 * The student must carefully notice that here we put -^-0, and not ^=0. 
 A little reflection will shew him that the latter equation involves an absurdity. 
 
 11—2 
 
164 QUATERNIONS. [208. 
 
 Then the equations of any two tangent-lines are 
 
 p = a cos ^ + /3 sin ^ + oc {-asmO + /8 cos 6), 
 p = acosd^ + ^ sin 6', + x^(-a sin 6, + ^ cos 6^). 
 
 If these tangent-lines be at right angles to each other 
 
 8 {-a sine + 13 cos 6) (- a sin 0^ + ^ cos ^j) = 
 
 or a^ sin ^ sin 9^ + /S^ cos 6 cos ^j = 0. 
 
 Also, for their point of intersection we have, by comparing 
 
 coefficients of a, /3 in the above values of p, 
 
 cos6 — X sin 6 = cos 6^ — *•, sin ^j, 
 sin ^ + a; cos d = sin 6^ + x^ cos 6^. 
 
 Determining x^ from these equations, we easily find 
 
 the equation of a circle ; if we take account of the above relation 
 between 6 and 6^ 
 
 Also, as the equations above give x= — x^, the tangents are 
 equal multiples of the diameters parallel to them ; so that the line 
 joining the points of contact is parallel to that joining the 
 extremities of these diameters. 
 
 208. Finally, w^hen the tangents 
 
 p = a cos + ^ sin ^ +x (—a sin 6 + ^ cos 0), 
 
 p=a cos ^j + /3 sin 0^ + x^(—a. sin ^,-1-/8 cos 0^, 
 
 meet in a given point 
 
 p = aa + 6jS, 
 
 we have a = cos — a; sin ^ = cos 0^ — x^ sin 0^, 
 
 b = sm0 + xcos6 = sin 0^ + x^ cos 0^ 
 
 Hence ar" = a^ + 6" - 1 = aSj^ 
 
 and a cos ^ + 6 sin = 1 = a cos ^j -I- 6 sin 0, 
 
 determine the values of and x for the directions and lengths of 
 the two tangents. The equation of the chord of contact is 
 
 p = y(cLcos0 + ^ sin ^) + (1 - y) (a cos 0^ + ^ sin 0,). 
 
 If this pass through the point 
 
 p=pct + q^, 
 
 we have p = y cos + {I - y) cos 0^, 
 
 q = y sin 9 + (l — y) sin 0^, 
 
211.] GEOMETRY OF STRAIGHT LINE AND PLANE. 165 
 
 from which, by the equations which determine 6 and d^, we get 
 
 ap + bq = y + 1 — y = 1. 
 
 Thus if either a and b, or p and q, be given, a linear relation 
 connects the others. This, by 1 80, gives all the ordinary properties 
 of poles and polars. 
 
 209. Although, in || 28 — 30, we have already given some of 
 the equations of the line and plane, these were adduced merely for 
 their applications to anharmonic coordinates and transversals ; 
 and not for investigations of a higher order. Now that we are 
 prepared to determine the lengths and inclinations of lines we 
 may investigate these and other similar forms anew. 
 
 210. The equation of the indefinite line drawn through the 
 origin 0, of which the vector OA, = a, forms a part, is evidently 
 
 p = xa., 
 or p II a, 
 
 or Vap = 0, 
 
 or Up=Ua; 
 
 the essential characteristic of these equations being that they are 
 linear, and involve one indeterminate scalar in the value of p. 
 
 We may put this perhaps more clearly if we take any two 
 vectors, /8, 7, which, along with a, form a non-coplanar system. 
 Operating with ;S . Fa/3 and S . Vay upon any of the preceding 
 equations (except the third, and on it by S. /S and ^S. 7) we get 
 
 8.a^p = 0) 
 
 and S.ayp = 0} ^ ^' 
 
 Separately, these are the equations of the planes containing a, /3, 
 and a, 7 ; together, of course, they denote the line of intersection. 
 
 211. Conversely, to solve equations (1), or to find p in terms 
 of known quantities, we see that they may be written 
 
 S.pVa^^O] 
 S.pVay^O: 
 
 so that p is perpendicular to Fa/3 and Fa7, and is therefore 
 parallel to the vector of their product. That is, 
 
 pl|F.Fa/3Fa7, 
 W-aS.a/Sy, 
 or P = <^<^- 
 
166 QUATERNIONS. [212. 
 
 212. By putting p — 13 for /j we change the origin to a point 
 
 B where OB = — ^, or BO = /3 ; so that the equation of a line 
 
 parallel to a, and passing through the extremity of a vector yS 
 
 drawn from the origin, is 
 
 p — ^ = xa, 
 
 or p = yS + ««. 
 
 Of course any two parallel lines may be represented as 
 
 p = ^ + a;a, 
 
 or VoL(p-^) = 0, 
 
 Fa{p-/3J=0. 
 
 213. The equation of a line, drawn through the extremity of ^, 
 and meeting a perpendicularly, is thus found. Suppose it to be 
 parallel to 7, its equation is 
 
 p = ^ +Xrf. 
 
 . To determine 7 we know, first, that it is perpendicular to a, 
 which gives 
 
 Sari = 0. 
 
 Secondly, a, /3, and 7 are in one plane, which gives 
 
 S.a0y = O. 
 These two equations give 
 
 7l|7.aFa^, 
 whence we have p = ^ + a;aFa/3. 
 
 This might have been obtained in many other ways; for 
 instance, we see at once that 
 
 /3 = «-' a^ = a' Sa^ + a' Va/3. 
 This shews that a"' Fa/3 (which is evidently perpendicular to a) is 
 coplanar with a and /8, and is therefore the direction of the 
 required line ; so that its equation is 
 
 p = ^ + 2/a-'Fa/3, 
 the same as before if we put — ~-^ for a;. 
 
 214. By means of the last investigation we see that 
 
 -a-'Fa/3 
 
 is the vector perpendicular drawn from the extremity of /3 to the 
 line 
 
 p = XOL, 
 
2 1 6.] GEOMETRY OP STRAIGHT LINE AND PLANE. 167 
 Changing the origin, we see that 
 
 is the vector perpendicular from the extremity of /S upon the line 
 
 p = 7 + ««. 
 
 215. The vector joining B (where OB = /3) with any point in 
 
 p = 7 + «« 
 
 is ry + xa — j3. 
 
 Its length is least when 
 
 dT{j + xa-l3)=^0, 
 or Sa (7 + «a - /3) = 0, 
 
 i.e. when it is perpendicular to a. 
 The last equation gives 
 
 or xa = — cT^ 8a (7 - /8). 
 
 Hence the vector perpendicular is 
 
 or «-' 7a (7 - ;8) = - a' Va (/3 - 7), 
 
 which agrees with the result of last section. 
 
 216. To find the shortest vector distance between two lines in 
 space 
 
 p = ^ + xa, 
 
 ,nd Pi = ^i+ «!«! ; 
 
 we must put dT (p — p^ = 0, 
 
 or S(p-p,)(dp-dp,) = 0, 
 
 or ■ 8(p — p^ {adx — a^dx^ = 0. 
 
 Since x and x^ are independent, this breaks up into the two 
 
 conditions 
 
 8a(p-p,) = 0, 
 
 proving the well-known truth that the required line is perpendicular 
 to each of the given lines. 
 
 Hence it is parallel to Vaa^, and therefore we have 
 
 p - p^ = ^ + xa- ^,- x^a, = yVaa, (1). 
 
 Operate by 8 . aaj and we get 
 
 8.aa,(fi-^J = y(Vaa,y. 
 
 ai 
 
168 QUATERNIONS . [217- 
 
 This determines y, and the shortest distance required is 
 
 \_Note. In the two last expressions T before 8 is inserted simply 
 to ensure that the length be taken positively. If 
 
 S. HMj (/8 — /SJ be negative, 
 
 then (§ 89) S . a^a (/3 - /SJ is positive. 
 
 If we omit the T, we must use in the text that one of these two 
 expressions which is positive.] 
 
 To find the extremities of this shortest distance, we must operate 
 on (1) with S .a and S .a^. We thus obtain two equations, which 
 determine x and a;,, as y is already known. 
 
 A somewhat different mode of treating this problem will be 
 discussed presently. 
 
 217. In a given tetrahedron to find a set of rectangular coordi- 
 nate axes, such that each axis shall pass through a pair of opposite 
 edges. 
 
 Let a, ^, <y be three (vector) edges of the tetrahedron, one 
 corner being the origin. Let p be the vector of the origin of the 
 sought rectangular system, which may be called i, j, k (unknown 
 vectors). The condition that i, drawn from p, intersects a is 
 
 8.iap = (1). 
 
 That it intersects the opposite edge, whose equation is 
 
 ■ay = ^ + 00 (^ - y), 
 the condition is 
 
 S. i (^ - ry) (p - ^) = 0, or Si {{^ -y)p-^j} = ...(2). 
 
 There are two other equations like (1), and two like (2), which can 
 be at once written down. 
 
 Put /3-7 = aj, 7-a = /3„ a-/3 = 7„ 
 
 F/37 = a„ Vya = 0„ Fa/3 = 7,, 
 
 Vcc,a = a„ F/3,^ = ^3, F7,7 = 73; 
 
 and the six become 
 
 S . iap = 0, S . ia^p — Sia^ = 0, 
 
 S.j^p = 0, S.j^,p-Sj^, = 0, 
 
 S.kyp = 0, 8.ky^p-Sk% = 0. 
 
 The two in i give i \\ ocSa^ - p {Saa^ + 8a^p). 
 
219-] GEOMETRY OF STRAIGHT LINE AND PLANE. 169 
 
 Similarly, 
 
 j II ^S^jy - p (S/3^, + S^^p), and k \\ ySy^p - p (S77, + Syj,). 
 
 The conditions of rectangularity, viz., 
 
 Sij = 0, Sjk = 0, Ski = 0, 
 
 at once give three equations of the fourth order, the first of which 
 is 
 
 = Sa^8a,j}Sl3j> - SapSot^p (S/3/8, + S^^) - 8^pS^,p {Sua, + 8cc,p) 
 
 + p' (Saa, + Sci^p) (S/3^^ + S^^p). 
 
 The required origin of the rectangular system is thus given as 
 the intersection of three surfaces of the fourth order. 
 
 218. The equation Sap = 
 
 imposes on p the sole condition of being perpendicular to a ; and 
 therefore, being satisfied by the vector drawn from the origin to 
 any point in a plane through the origin and perpendicular to a, is 
 the equation of that plane. 
 
 To find this equation by a direct process similar to that usually 
 employed in coordinate geometry, we may remark that, by § 29, 
 we may write 
 
 p = x^ + yy, 
 
 where /3 and 7 are any two vectors perpendicular to a. In this 
 form the equation contains two indeterminates, and is often useful; 
 but it is more usual to eliminate them, which may be done at 
 once by operating by S .a, when we obtain the equation first 
 written. 
 
 It may also be written, by eliminating one of the indeter- 
 minates only, as 
 
 F/3p = yV^y = za, 
 
 where the form of the equation shews that Sa^ = 0. 
 Similarly we see that 
 
 Sa{p-^) = 
 
 represents a plane drawn through the extremity of /3 and perpen- 
 dicular to a. This, of course, may, like the last, be put into various 
 equivalent forms. 
 
 219. The line of intersection of the two planes 
 
 S.a{p-^)=0\ (1) 
 
 and ;S.a.(p-/3,) = ^ 
 
170 QUATERNIONS. [22O. 
 
 contains all points whose value of p satisfies both conditions. But 
 we may write (§ 92), since a, a^, and Volo.^ are not coplanar, 
 
 p8 . aa^VoiOL^ = Vaa^S.aa^p + V. OL^VaoL^Sap + V. V{a.a.^ aSa^p, 
 
 or, by the given equations, 
 
 -pT'Vaa^ = V. a^Vaa.^Saj3 + V. V{aa,) aSa,^^ + a;Faa,...(2), 
 
 where x, a scalar indeterminate, is put for 8 . aa,p which may have 
 any value. In practice, however, the two definite given scalar 
 equations are generally more useful than the partially indeter- 
 minate vector-form which we have derived from them. 
 
 When both planes pass through the origin we have ^ = y8j = 0, 
 and obtain at once 
 
 p=xVaa^ 
 
 as the equation of the line of intersection. 
 
 220. The plane passing through the origin, and through the 
 line of intersection of the two planes (1), is easily seen to have the 
 equation 
 
 8a^/3^Sap - 8a^8aj} = 0, 
 
 or 8 (a8u,^^ - a,8a^) p = 0. 
 
 For this is evidently the equation of a plane passing through the 
 origin. And, if p be such that 
 
 8oLp = 8oL^, 
 we also have 8(x^p = Sa^^^, 
 
 which are equations (1). 
 
 Hence we see that the vector 
 
 a8a^l3^ - a^Sa^ 
 
 is perpendicular to the vector-line of intersection (2) of the two 
 planes (1), and to every vector joining the origin with a point in 
 that line. 
 
 The student may verify these statements as an exercise. 
 
 221. To find the vector-perpendicular from the extremity of /3 
 on the plane 
 
 Sap = 0, 
 
 we must note that it is necessarily parallel to a, and hence that the 
 value of p for its foot is 
 
 /3 = /S -f- a;a, 
 where xa. is the vector-perpendicular in question. 
 Hence 8fx (/8 -t- xa.) = 0, 
 
223. J GEOMETRY OF STRAIGHT LINE AND PLANE. 171 
 
 which gives x^ = - Sa^, 
 
 or ocu =-a-'Sa0. 
 
 Similarly the vector-perpendicular from the extremity of /3 on the 
 plane 
 
 Sa(p-y) = 
 may easily be shewn to be 
 
 222. The equation of the plane which passes through the ex- 
 tremities of a, yS, 7 may be thus found. If p be the vector of any 
 point in it, p - a, a - ^, and /3-y lie in the plane, and therefore 
 (§ 101) 
 
 8.(p-ci)(a-^)(^-y) = 0, 
 
 or Sp{Voi.^+V^'y+Vya)-S.a^y = 0. 
 
 Hence, if S = « ( Va^ + V/3y + Vya) 
 
 be the vector-perpendicular from the origin on the plane containing 
 the extremities of a, /3, y, we have 
 
 S = ( Fa/3 + F/37 + Fyk)"^ S . ajSy. 
 From this formula, whose interpretation is easy, many curious pro- 
 perties of a tetrahedron may be deduced by the reader. Thus, for 
 instance, if we take the tensor of each side, and remember the 
 result of § 100, we see that 
 
 T{Va^+V^y+Vya.) 
 
 is twice the area of the base of the tetrahedron. This may be 
 more simply proved thus. The vector area of the base is 
 
 1 F(a - ^) (7 - /3) = - f ( Fa^ + F/37 + F^a). 
 
 Hence the sum of the vector areas of the faces of a tetrahedron, 
 and therefore of any solid whatever, is zero. This is the hydrostatic 
 proposition for translational equilibrium of solids immersed in a 
 fluid subject to no external forces. 
 
 223. Taking any two lines whose equations are 
 p = ^ + xa., 
 
 we see that 8.aa^{p — h) = 
 
 is the equation of a plane parallel to both. Which plane, of course, 
 depends on the value of S. 
 
172 QUATERNIONS. [224. 
 
 Now if S = /8, the plane contains the first line ; if 8 = /3j, the 
 second. 
 
 Hence, if yFaa, be the shortest vector distance between the 
 
 lines, we have 
 
 S.aa,(/3-/3.-2/FaaJ = 0, 
 
 or T (y VaaJ = TS.(^- ^,) UVaa^, 
 
 the result of § 216. 
 
 224. Find the equation of the 'plane, passing through the origin, 
 
 which makes equal angles with three given lines. Also find the angles 
 
 in question. 
 
 Let a, /3, 7 be unit-vectors in the directions of the lines, and let 
 
 the equation of the plane be 
 
 Shp = 0. 
 
 Then we have evidently 
 
 (SaS = /S/8S = /S178 = X, suppose, 
 
 cc 
 where — 7p^ 
 
 is the sine of each of the required angles. 
 But (§ 92) we have 
 
 SS.al3y = x{ FayS + V^y + Vya). 
 Hence S.p{Val3+ V/3y + F7a) = 
 is the required equation ; and the required sine is 
 S.alSy 
 
 225. Find the locus of the middle points of a series of straight 
 lines, each parallel to a given plane and having its extremities in two 
 iixed straight lines. 
 
 Let 8yp = 
 
 be the plane, and 
 
 p = j3 + xa, /3 = ^, + ajjMj, 
 
 the fixed lines. Also let x and x^ correspond to the extremities of 
 one of the variable lines, ot being the vector of its middle point. 
 Then, obviously, 2'S7 = /3 + a:a + /3, + a;,a,. 
 Also Sy (/8 - /3, + «« - x^a^) = 0. 
 
 This gives a linear relation between x and a;,, so that, if we sub- 
 stitute for ajj in the preceding equation, we obtain a result of the 
 form 
 
 ■ST = B ■+ xe, 
 
227.J GEOMETRY OP STRAIGHT LINE AND PLANE. 173 
 
 where 8 and e are known vectors. The required locus is, therefore, 
 a straight line. 
 
 226. Three planes meet in a point, and through the line of 
 intersection of each pair a plane is drawn perpendicular to the 
 third ; prove that these planes pass through the same line. 
 
 Let the point be taken as origin, and let the equations of the 
 planes be 
 
 SoLp = 0, S^p = 0, Syp = 0. 
 
 The line of intersection of the first two is || Va^, and therefore the 
 normal to the first of the new planes is 
 
 F. 7Fa^. 
 
 Hence the equation of this plane is 
 
 8.pr.ryVa^ = 0, 
 
 or S^pSay-SapSl3y = 0, 
 
 and those of the other two planes may be easily formed from this 
 by cyclical permutation of a, /S, 7. 
 
 We see at once that any two of these equations give the third 
 by addition or subtraction, which is the proof of the theorem. 
 
 227. Given any number of points A, B, G, Jhc, whose vectors 
 {from the origin) are a,, a^, a.^, &c., find the plane through the origin 
 for which the sum of the squares of the perpendiculars let fall upon 
 it from these points is a maximum or minimum. 
 
 Let (Sfnr/3 = 
 
 be the required equation, with the condition (evidently allowable) 
 
 yi!7 = L 
 
 The perpendiculars are (§ 221) — ■ux'^S-sra^, &c. 
 
 Hence 2/SVa 
 
 is a maximum. This gives 
 
 % . S's^aSad'ST = ; 
 and the condition that ot is a unit-vector gives 
 
 8-md'ST = 0. 
 
 Hence, as din may have any of an infinite number of values, 
 these equations cannot be consistent unless 
 
 2 . aScLTs = flJ^, 
 where a; is a scalar. 
 
174 QUATERNIONS. [228 
 
 The values of a. are known, so that if we put 
 2 . ajSittCT = (^OT, 
 (/) is a given self-conjugate linear and vector function, and therefore 
 X has three values {g^, g^, g^, § 175) which correspond to three 
 mutually perpendicular values of ■sr. For one of these there is a 
 maximum, for another a minimum, for the third a maximum- 
 minimum, in the most general case when g^, g^, g, are all different. 
 
 228. The following beautiful problem is due to Maccullagh. 
 Of a system of three rectangular vectors, passing through the origin, 
 two lie on given planes, find the locus of the third. 
 
 Let the rectangular vectors be ot, p, a. Then by the conditions 
 of the problem 
 
 S^p = Spa = Saw = 0, 
 
 and Sa.w = 0, Sl3p = 0. 
 
 The solution depends on the elimination of p and ot among these 
 five equations. [This would, in general, be impossible, as p and or 
 between them involve six unknown scalars ; but, as the tensors are 
 (by the very form of the equations) not involved, the five given 
 equations are necessary and sufficient to eliminate the four unknown 
 scalars which are really involved. Formally to complete the requisite 
 number of equations we might write 
 
 Tw==a, Tp = b, 
 but a and b may have any values whatever.] 
 From Saw = 0, Saw = 0, 
 
 we have w = a; Vaa. 
 
 Similarly, from S^p = 0, Sap = 0, 
 
 we have p=y F/Scr. 
 
 Substitute in the remaining equation 
 
 Swp = 0, 
 and we have S . VaaV^a = 0, 
 
 or SaaS/3a-a'Sot^ = 0, 
 
 the required equation. As will be seen in next Chapter, this is a 
 cone of the second degree whose circular sections are perpendicular 
 to a and /3. [The disappearance of x and y in the elimination 
 instructively illustrates the note above.] 
 
GEOMETRY OF STRAIOHT LINE AND PLANE. 175 
 
 EXAMPLES TO CHAPTER VII. 
 
 1. What propositions of Euclid are proved by the mere /orm 
 of the equation 
 
 p = {l—x) a + x^, 
 
 which denotes the line joining any two points in space ? 
 
 2. Shew that the chord of contact, of tangents to a parabola 
 which meet at right angles, passes through a fixed point. 
 
 3. Prove the chief properties of the circle (as in Euclid, III.) 
 from the equation 
 
 p = a cos ^ + /3 sin ^ ; 
 
 where Ta = 7/3, and <Sfa/3 = 0. 
 
 4. What locus is represented by the equation 
 
 S'ap + p' = 0,\ 7^ '^ 
 where ra = l? '■^^ Tf^^-6~S?ff--o V -^ 
 
 5. What is the condition that the lines ^ '^ 
 
 intersect ? If this is not satisfied, what is the shortest distance 
 between them ? 
 
 6 Find the equation of the plane which contains the two 
 parallel lines r/-Vo,^-.'Xf-^D - <; 
 
 7. Find the equation of the plane which contains ; 
 
 F«(p-/3) = o,o)r;^('^;-fi^ 
 
 and is perpendicular to >Sfw = 0. i'> f/^r{f-J)^'o^^ 
 
 '^ 8. Find the equation of a straight line passing through a given 
 
 point, and making a given angle with a given plane. 
 
 X Hence form the general equation of a right cone. 
 
 ^^' 9. What conditions must be satisfied with regard to a number 
 of given lines in space that it may be possible to draw through 
 each of them a plane in such a way that these planes may intersect 
 in a common line ? 
 
 -^ 10. Find the equation of the locus of a point the sum of the 
 squares of whose distances from a number of given planes is 
 constant. 
 
176 QUATERNIONS. 
 
 11. Substitute " lines " for " planes " in (10). 
 
 .^it^ 12. Find the equation of the plane which bisects, at right 
 angles, the shortest distance between two given lines. 
 
 Find the locus of a point in this plane which is equidistant from 
 the given lines. 
 
 i 13. Find the conditions that the simultaneous equations 
 Sap = a, S^p = b, Syp = c, 
 may represent a line, and not a point. 
 
 •^ 14. What is represented by the equations 
 {8apf = (S^pY = (Sjpy, 
 where a, /S, y are any three vectors ? 
 
 15. Find the equation of the plane which passes through two 
 given points and makes a given angle with a given plane. 
 
 ^ 16. Find the area of the triangle whose corners have the 
 vectors a, /3, 7. 
 
 Hence form the equation of a circular cylinder whose axis and 
 radius are given. 
 
 -I 17. (Hamilton, Bishop Law's Premium Ex., 1858.) 
 
 (a) Assign some of the transformations of the expression 
 
 Fa/3 
 /8-a" 
 
 where a and /3 are the vectors of two given points A and B. 
 
 (b) The expression represents the vector 7, or 00, of a point 
 in the straight line AB. 
 
 (c) Assign the position of this point 0. 
 ^ 18. (Ibid.) 
 
 (a) If a, ^, 7, 8 be the vectors of four points, A,B,0, D, what 
 is the condition for those points being in one plane ? 
 
 (6) When these four vectors from one origin do not thus 
 terminate upon one plane, what is the expression for the volume 
 of the pyramid, of which the four points are the corners ? 
 
 (c) Express the perpendicular S let fall from the origin on 
 the plane ABO, in terms of a, /3, 7. 
 
GEOMETRY OF STRAIGHT LINE AND PLANE. 177 
 
 19. Find the locus of a point equidistant from the three 
 planes ' ^'-^U-'^H-- 
 
 8oip = 0, S^p = 0, Syp = 0. /_iu -^0^%-: f^ I 
 
 20. If three mutually perpendicular vectors be drawn from a 
 point to a plane, the sum of the reciprocals of the squares of their 
 lengths is independent of their directions. 
 
 21. Find the general form of the equation of a plane from the 
 condition (which is to be assumed as a definition) that any two 
 planes intersect in a single straight line. 
 
 22. Prove that the sum of the vector areas of the faces of any 
 polyhedron is zero. ~__^ir^ § ^ ^ -^ 
 
 T. Q. I. 12 
 
CHAPTER VIII. 
 
 THE SPHEEE AND CYCLIC CONE. 
 
 229. After that of the plane the equations next in order of 
 simplicity are those of the sphere, and of the cone of the second 
 order. To these we devote a short Chapter as a valuable prepara- 
 tion for the study of surfaces of the second order in general. 
 
 230. The equation 
 
 or p" = a^ 
 
 denotes that the length of p is the same as that of a given vector a, 
 and therefore belongs to a sphere of radius Ta whose centre is the 
 origin. In § 107 several transformations of this equation were ob- 
 tained, some of which we will repeat here with their interpretations. 
 
 Thus fif(p + a)(p-a) = 
 
 shews that the chords drawn from any point on the sphere to the 
 extremities of a diameter (whose vectors are a and — a) are at right 
 angles to each other. J Wf-/cyCf-c(j = CF-i-=<Kf-<K) [.'tU i- 
 
 T{p + a) (p-a) = 2TVap^ ^= l/^AMV.,.f»>. 
 
 shews that the rectangle under these chords is four times the area 
 of the triangle two of whose sides are a and p. 
 
 p = {p + a)"^ a(p + a) (see § 105) -ft^ 
 
 shews that the angle at the centre in any circle is double that at 
 the circumference standing on the same arc. All these are easy 
 consequences of the processes already explained for the interpreta- 
 tion of quaternion expressions, 
 
232.] THE SPHERE AND CYCLIC CONE. 179 
 
 231. If the centre of a sphere be at the extremity of a, the 
 equation may be written 
 
 T{p-a) = W, 
 which is the most general form. 
 
 If ra=r/3, 
 
 or 0L^ = ^\ 
 
 in which case the origin is a point on the surface of the sphere, this 
 becomes 
 
 p' - 28ap = 0. 
 From this, in the form 
 
 Sp (p - 2a) = 
 
 another proof that the angle in a semicircle is a right angle is 
 derived at once. 
 
 232. The converse problem is — Find the locus of the feet of 
 perpendiculars let fall from a given point, p=^, on planes passing 
 through the origin. 
 
 Let Sap = 
 
 be one of the planes, then (§221) the vector-perpendicular is 
 
 - a'ScLl3, 
 and, for the locus of its foot, 
 
 p = yS - a-'>Sfa/3, t.c£^. -^ - / <*' i) ^ ^~' I 
 = o-'Fa,8. 
 
 [This is an example of a peculiar form in which quaternions some- 
 times give us the equation of a surface. The equation is a vector 
 one, or equivalent to three scalar equations; but it involves the 
 undetermined vector a in such a way as to be equivalent to only 
 two indeterminates (as the tensor of a is evidently not involved). 
 To put the equation in a more immediately interpretable form, a 
 must be eliminated, and the remarks just made shew this to be 
 possible.] 
 
 Now (p-^f = a'S'al3, 
 
 and (operating hy 8.^ on the value of p above) ( f- 3 - - c/ t> '^/^ 
 
 S^p-^^ = -a-^8\^. ^S^if-^^^^u's^ 
 
 Adding these equations, we get 
 
 p'-80p = O, 
 
 12—2 
 
180 QUATERNIONS. [233- 
 
 SO that, as is evident, the locus is the sphere of which ^ is a 
 diameter. 
 
 233. To find the intersection of the two spheres 
 
 T(p-ci) = T/3, 
 and T{p-ci,) = W„ 
 
 square the equations, and subtract, and we have 
 
 28{a-a,)p = ce-oi,'-{0^-^,'), 
 which is the equation of a plane, perpendicular to a — a,, the vector 
 joining the centres of the spheres. This is always a real plane 
 whether the spheres intersect or not. It is, in fact, what is called 
 their Radical Plane. 
 
 234. Find the locus of a point the ratio of whose distances from 
 two given points is constant. 
 
 Let the given points be and A, the extremities of the vector a. 
 Also let P be the required point in any of its positions, and OP = p. 
 
 Then, at once, if n be the ratio of the lengths of the two lines, 
 Tip-a) = nTp. 
 This gives p" — 2Sap + a' = n^p^, 
 
 or, by an easy transformation, 
 
 Thus the locus is a sphere whose radius is T (= ^j , and whose 
 
 centre is at B, where OB = " ,, , a definite point in the line OA. 
 
 1—n '^ 
 
 235. If in any line, OP, drawn from the origin to a given plane, 
 OQ be taken such that OQ . OP is constant, find the locus of Q. 
 
 Let Sap = a^ 
 
 be the equation of the plane, vr a vector of the required surface. 
 Then, by the conditions, 
 
 Ttj^Tp = constant = V (suppose), 
 
 and U'ST = Up. 
 
 From these p = „'^ = ^ . 
 
 Substituting in the equation of the plane, we have 
 
238. J THE SPHERE AND CYCLIC CONE. 181 
 
 which shews that the locus is a sphere, the origin being situated 
 on it at the point farthest from the given plane. 
 
 236. Find the locus of points the sum of the squares of whose 
 distances from a set of given points is a constant quantity. Find 
 also the least value of this constant, and the corresponding locus. 
 
 Let the vectors from the origin to the given points be a^, a^, 
 
 o„, and to the sought point p, then 
 
 -c' = {p-a,r + {p-a:f + +{p-aj, 
 
 = V-2/Sp2a + 2(a^). 
 
 Otherwise (p -l^") = -l^lM ^<M , 
 
 \ n J n n 
 
 the equation of a sphere the vector of whose centre is — , i.e. 
 
 n 
 
 whose centre is the mean of the system of given points. 
 
 Suppose the origin to be placed at the mean point, the equation 
 becomes 
 
 p2 = _ <^' + '^(.^') (because Sa = 0, § 31 (e)). 
 
 The right-hand side is negative, and therefore the equation denotes 
 a real surface, if 
 
 as might have been expected. When these quantities are equal, 
 the locus becomes a point, viz. the new origin, or the mean point 
 of the system. 
 
 237. If we differentiate the equation 
 
 Tp = Ta 
 we get Spdp = 0. 
 
 Hence (§ 144), p is normal to the surface at its extremity, a well- 
 known property of the sphere. 
 
 If CT be any point in the plane which touches the sphere at the 
 extremity oi p, ■sr — p is a. line in the tangent plane, and therefore 
 perpendicular to p. So that 
 
 8p(l!T-p) = 0, 
 
 or St!yp = -Tp' = a' 
 
 is the equation of the tangent plane. 
 
 238. If this plane pass through a given point B, whose vector 
 
 is 8, we have 
 
 S^p = a\ 
 
182 QUATERNIONS. [239- 
 
 This is the equation of a plane, perpendicular to ^, and cutting 
 from it a portion whose length is 
 
 If this plane pass through a fixed point whose vector is 7 we must 
 have 
 
 so that the locus of /8 is a plane. These results contain all the 
 ordinary properties of poles and polars with regard to a sphere. 
 
 239. A line drawn parallel to 7, from the extremity of y8, has 
 
 the equation 
 
 p = l3 + xy. 
 This meets the sphere 
 
 p' = a' 
 
 in points for which x has the values given by the equation 
 
 l3' + 2xS^y + a,y = ci\ 
 
 The values of x are imaginary, that is, there is no intersection, if 
 
 ay+F^^7<0. 
 The values are equal, or the line touches the sphere, if 
 
 ay+F''/37=0, 
 
 or S'^j = y\^'-OL'). 
 
 This is the equation of a cone similar and similarly situated to the 
 cone of tangent-lines drawn to the sphere, but its vertex is at the 
 centre. That the equation represents a cone is obvious from the 
 fact that it is homogeiieous in Ty, i.e. that it is independent of the 
 length of the vector 7. 
 
 [It may be remarked that from the form of the above equation 
 we see that, if x and x' be its roots, we have 
 
 (xTry){x'Tj) = a'-l3\ 
 
 which is Euclid, III. 35, 36, extended to a sphere.] 
 
 240. Find the locus oftliefoot of the perpendicular let fall from 
 a given point of a sphere on any tangent-plane. 
 
 Taking the centre as origin, the equation of any tangent-plane 
 
 may be written 
 
 Sisyp — 0?. 
 
 The perpendicular must be parallel to p, so that, if we suppose it 
 
241.] THE SPHERE AND CYCLIC CONE. 183 
 
 drawn from the extremity of a (which is a point on the sphere) we 
 have as one value of v: 
 
 ■57 = a + a;p. 
 
 From these equations, with the help of that of the sphere 
 
 we must eliminate p and x. 
 
 We have by operating on the vector equation by (S . or 
 
 ■57^ = (Shot + xS'B^p 
 
 = 8as! + xo?. 
 
 „ CT — a mVot — a) 
 Hence p = = —^ — -. — ^ . 
 
 X OT — OttOT 
 
 Taking the tensors, we have 
 
 the required equation. It may be put in the form 
 
 and the interpretation of this (viz. that the projection of ts on 
 {■m — a) is of constant length) gives at once a characteristic 
 property of the surface formed by the rotation of the Gardioid 
 about its axis of sj'mmetry. 
 
 If the perpendiculars be let fall from a point, /3, not on the 
 sphere, it is easy to see that the equation of the locus is 
 
 ,SVf/■(1^-/3) = -a^ 
 whose interpretation is equally easy. 
 
 241. We have seen that a sphere, referred to any point what- 
 ever as origin, has the equation 
 
 T{p-a) = W. 
 Hence, to find the rectangle under the segments of a chord drawn 
 through any point, we may put 
 
 p = xy; 
 where 7 is any unit-vector whatever. This gives 
 
 xY - 2xSary + 0" = /S^ 
 and the product of the two values of x is 
 
 r 
 
 This is positive, or the vector-chords are drawn in the same direc- 
 tion, if 
 
 T^ < Ta, 
 
 i.e. if the origin is outside the sphere. 
 
184 QUATERNIONS. [242. 
 
 242. A, B are fixed points ; and, being the origin and P a 
 
 point in space, 
 
 AP' + Br^Or; 
 
 find the locus of P, and explain the result when Z A OB is a right, or 
 
 an obtuse, angle. 
 
 Let Ol = a, 05 = A OP = p, then 
 
 or p'-28(a + ^)p = -{a' + ^'), 
 
 or T{p-{oi + ^)] = ^/{-28oL^). 
 
 While 8a^ is negative, that is, while zAOB is acute, the locus 
 
 is a sphere whose centre has the vector a + /3. If Sa.^ = 0, or 
 
 Z.AOB = '7rj2, the locus is reduced to the point 
 
 p = a + /3. 
 If Z AOB > 7r/2 there is no point which satisfies the conditions. 
 
 243. Describe a sphere, with its centre in a given line, so as to 
 pass through a given point and touch a given plane. 
 
 Let xoL, where x is an undetermined scalar, be the vector of 
 
 the centre, r the radius of the sphere, /3 the vector of the given 
 
 point, and 
 
 8<yp — a 
 
 the equation of the given plane. 
 
 The vector-perpendicular from the point xa on the given plane 
 is (§ 221) 
 
 (a - xS'^oL) 7 \ 
 
 Hence, to determine x and r we have the equations 
 
 T.{a- xSya) 7"' = T(xa-^)= r, 
 so that there are, in general, two solutions. It will be a good 
 exercise for the student to find from these equations the condition 
 that there may be no solution, or two coincident ones. 
 
 244. Describe a sphere whose centre is in a given line, and 
 which passes through two given points. 
 
 Let the vector of the centre be xa, as in last section, and let 
 the vectors of the points be /3 and 7. Then, at once, 
 
 T{rf-xoL) = T{^-xa.)=^r. 
 Here there is but one sphere, except in the particular case when 
 we have 
 
 Try = T^, and Say = 8a^, 
 in which case there is an infinite number. 
 
245-] THE SPHERE AND CYCLIC CONE. 185 
 
 The student should carefully compare the results of this 
 section with those of the last, so as to discover why in general two 
 solutions are indicated as possible in the one problem, and only 
 one in the other. 
 
 245. A sphere touches each of two straight lines, which do not 
 meet : find the locus of its centre. 
 
 We may take the origin at the middle point of the shortest 
 distance (§ 216) between the given lines, and their equations will 
 then be 
 
 p = a + a!/3, 
 
 where we have, of course. 
 
 Let o- be the vector of the centre, p that of any point, of one 
 of the spheres, and r its radius ; its equation is 
 
 r (p — cr) = r. 
 
 Since the two given lines are tangents, the following equations in 
 X and x^ must have pairs of equal roots, 
 
 T(a + a;/3-o-) = r, 
 
 y(-a + a;^/3,-o-) = r. 
 
 The equality of the roots in each gives us the conditions 
 
 Eliminating r we obtain 
 
 ^'S-'^a - I3;'S%<T = (a - <ry - (a + cr)^ = - 4/Sa<7, 
 which is the equation of the required locus. 
 
 [As we have not, so far, entered on the consideration of the 
 quaternion form of the equations of the various surfaces of the 
 second order, we may translate this into Cartesian coordinates to 
 find its meaning. If we take coordinate axes of x, y, z respectively 
 parallel to )8, ySj, a, it becomes at once 
 
 {x + myj — (2/ + rnxf = pz, 
 where m and p are constants; and shews that the locus is a 
 hyperbolic paraboloid. Such transformations, which are exceed- 
 ingly simple in all cases, will be of frequent use to the student 
 who is proficient in Cartesian geometry, in the early stages of his 
 study of quaternions. As he acquires a practical knowledge of 
 
186 QUATERNIONS. [246. 
 
 the new calculus, the need of such assistance will gradually cease 
 to be felt] 
 
 Simple as the above solution is, quaternions enable us to give one 
 vastly simpler. For the problem may be thus stated — Find the 
 locus of the point whose distances from two given lines are equal. 
 And, with the above notatioi^ the equality of the perpendiculars is 
 expressed (§ 214) by 
 
 TV. {a- a) U^ = TV. (a + a) U^^, 
 which is easily seen to be equivalent to the equation obtained above. 
 
 246. Two spheres being given, shew that spheres which cut them 
 at given angles cut at right angles another fixed sphere. 
 
 If c be the distance between the centres of two spheres whose 
 radii are a and b, the cosine of the angle of intersection is evidently 
 
 g^ + y - c^ 
 2ab 
 
 Hence, if a, a,, and p be the vectors of the centres, and a, a^, r the 
 radii, of the two fixed, and of one of the variable, spheres ; A and 
 J. J the angles of intersection, we have 
 
 {p - (xf + a^ + r^ = 2ar cos A, 
 
 (p - a^y + a^ + r-^ = la^ cos A^. 
 
 Eliminating the first power of r, we evidently must obtain a result 
 such as 
 
 (|O-e)He^ + r-'' = 0, 
 
 where (by what precedes) e is the vector of the centre, and e the 
 radius, of a fixed sphere 
 
 (p - if + e^ = 0, 
 
 which is cut at right angles by all the varying spheres. By effect- 
 ing the elimination exactly we easily find e and e in terms of 
 given quantities. 
 
 247. If two vectors divide one another into parts a and — ea, 
 p and -p, respectively, the lines joining the free ends meet in 
 
 (1 + e) p - lea. 
 
 CT= . 
 
 1 — e 
 
 We may write this as 
 
 e,p = CT + aj, 
 
 ■f 1 + e 2ea 
 
 " e, = , a = : 
 
 1-e' ^ 1-, 
 
247-] THE SPHERE AND CYCLIC CONE. 187 
 
 For another such pair of vectors, passing through the same 
 point, we have, say, 
 
 /^or = ^ + /3,. 
 
 Thus the plane quadrilateral, whose diagonals are made up 
 respectively of the complete a and ^ vectors, will be projected (by 
 lines from the point ■bt) as a square on the plane of p, a; provided 
 Tp=T<r, and Spa = 0. 
 
 That is, provided 
 
 /,M^ + a,r = e/(^ + ^/ (1), 
 
 >S.(^ + aJ(tT + ^J=0 (2). 
 
 These equations obviously belong to spheres which intersect 
 one another at right angles. For the centres are at 
 
 -J^^^fX-^^.) and -i(a. + ;8,). 
 Thus the distance between the centres is 
 
 and this is obviously less than the sum, and greater than the 
 difference, of the radii 
 
 T.^4^M-^.)> and r.i(a,-/3J. 
 
 And because its square is equal to the sum of their squares, the 
 spheres intersect at right angles. Hence 
 
 Tiie locus of the points, from which a plane {uncrossed) quadri- 
 lateral can he projected as a square, is a circle whose centre is in 
 the plane of the quadrilateral, and whose plane is perpendicular to 
 that plane. 
 
 To find the points (if any) of this circle from which the quadri- 
 lateral is seen as a square, we must introduce the additional 
 conditions 
 
 S'mp = 0, S'STO- = 0, 
 
 or ^ti7 (ot + a,) = 0, »SV(t!7-l-/3J=0 (3). 
 
 Hence the points lie on each of two spheres which pass through 
 the origin : — i.e. the intersection of the diagonals. 
 
 [If we eliminate ■bj among the four equations (1), (2), (3), we 
 find the condition 
 
 This we leave to the student.] 
 
188 QUATERNIONS. [248. 
 
 Another mode of solving the last problem, viz. to find the points 
 from which a given plane quadrilateral is seen as a square, consists 
 in expressing that the four portions of the diagonals subtend equal 
 angles, and that the planes containing them are at right angles to 
 one another. 
 
 The first condition gives, with the notation of the beginning of 
 this section, 
 
 S . uU {-ST - a) = 8 . 'stU {i!7 + ea) 
 
 = S . t!7U {zr - ^) = S .nrU {■ST +fl3) (4). 
 
 The second condition is 
 
 y8.FcraFCT/3 = 0, 
 
 or ^'Sa^ - Sa^S^'ST = (5), 
 
 the cone whose cyclic normals are a, /3. 
 
 [It will be excellent practice for the student to shew that (4) 
 and (5) are equivalent to (1), (2), (3). Thus, in particular, the 
 first equality in (4) 
 
 /S.crf7'(cr-a)=/Si.o7C/'(t3- + ea), 
 is equivalent to the first of (3), viz. 
 
 /S . OT (ot + aj = 0.] 
 It is obvious that the solution of the first problem in this 
 section gives at once the means of solving the problem oi projecting 
 an ellipse into a circle, so that any given (internal) point may be 
 projected as the centre of the circle. And numerous other con- 
 sequences follow, which may be left to the reader. 
 
 248. To inscribe in a given sphere a closed polygon, plane or 
 gauche, whose sides shall be parallel respectively to each of a series 
 of given vectors. 
 
 Let 2> = 1 
 
 be the sphere, a, /3, 7, ,7], 6 the vectors, n in number, and let 
 
 Pi, P3, p„, be the vector-radii drawn to the angles of the polygon. 
 
 Then p^ — p^ = x^a, &c., &c. 
 
 From this, by operating by 8.{p^ + pX we get 
 P2 - Pi' = = Sap^ + Sap,. 
 Also = Vap^ - Vap,. 
 
 Adding, we get = ap^ + Kap, = a/a, + p,a. 
 
 Hence Ps = - «">,«, 
 
 or, if we please, Pa = - aPiM"'. 
 
249-] THE SPHERE AND CYCLIC CONE. 189 
 
 [This might have been writteu down at once from the result of 
 § 105.] 
 
 Similarly p, = - ^''p^ = ^-'aT'p^a^, &c. 
 
 Thus, finally, since the polygon is closed, 
 
 P«« = /'. = (-)" ^" V /8-«-V.ai3 vO- 
 
 We may suppose the tensors of a, /S rj, 6 to be each unity. 
 
 Hence, if 
 
 a = a/3 7)6, 
 
 we have a~* = 6'^i]"^ ^'^cC'\ 
 
 which is a known quaternion ; and thus our condition becomes 
 
 Pi = (-T «">.«• 
 This divides itself into two cases, according as n is an even or an 
 odd number. 
 
 If n be even, we have 
 
 ap, = p,a. 
 
 Removing the common part p^Sa, we have 
 
 Vp,ra = 0. 
 This gives one determinate direction, + Va, for p, ; and shews that 
 there are two, and only two, solutions. 
 If n be odd, we have 
 
 «Pi = - Pi"'' 
 which (operating, for instance, by S.p^) requires that we have 
 
 8a = 0, 
 i.e. that there may be a solution, a must be a vector. 
 Hence Sap^ = 0, 
 
 and therefore pj may be drawn to any point in the great circle of 
 the unit-sphere whose pole is on the vector a. 
 
 249. To illustrate these results, let us take first the case of 
 
 n = 3. We must have 
 
 -S.a/37 = 0, 
 
 or the three given vectors must (as is obvious on other grounds) be 
 parallel to one plane. Here a^y, which lies in this plane, is (§ 106) 
 the vector-tangent at the first corner of each of the inscribed tri- 
 angles ; and is obviously perpendicular to the vector drawn from 
 the centre to that corner. 
 
 If n = 4, we have 
 
 pJV.ci^yS, 
 
 as might have been at once seen from § 106. 
 
190 QUATERNIONS. [25O. 
 
 250. Hamilton has given (Lectures, p. 674 and Appendix C), 
 an ingenious process by which the above investigation is rendered 
 applicable to the more difficult problem in which each side of the 
 inscribed polygon is to pass through a given point instead of being 
 parallel to a given line. His process, which (see his Life, Vol. III., 
 pp. 88, 426) he evidently considered as a specially tough piece of 
 analysis, depends upon the integration of a linear equation in 
 finite differences. 
 
 The gist of Hamilton's method is (briefly) as follows (Lectures, 
 § 676) : 
 
 Let the (unit) vectors to the corners of the polygon be, as 
 
 above, p„ p^, p„. Also let a,, a^, a„ be the points through 
 
 which the successive sides are to pass. The sides are respectively 
 parallel to the vectors * 
 
 which correspond to a, /3, ^, of § 248. Hence, if we write 
 
 ?. = «i-Pi> 
 
 ?3 = («3-/'3)9'2. &C. 
 
 we have (as in that section), since the expressions are independent 
 of the tensors of the qs, 
 
 Ps = + 92P,92~^ 
 
 Pi = -1sPA~'>^(^- 
 
 These give, generally, (with the condition p^^ =■. — !) 
 
 9m = r., + (-irs^P, (1), 
 
 where r — a r -\- s 
 
 m m-1 m m-l' 
 
 [We may easily eliminate s, by the use of the separable symbol D 
 or 1 + A, but this leads to a troublesome species of equation of 
 second differences. Hamilton ingeniously avoids this by the use 
 of biquaternions.J 
 
 Putting I for the algebraic V — 1, we have 
 
 fm + is^ = («„. + 1) (r^_^ - ts^_,), 
 
 (where, as usual, we have a second equation by changing through- 
 out the sign of i). 
 
 The complete solution of this equation is, of course, obtained 
 
250.] THE SPHERE AND CYCLIC CONE. 191 
 
 at once in the form of a finite product. But it is sufficient to 
 know some of its characteristic properties only. 
 The squared tensor is 
 
 so that, by equating real and imaginary parts, we have 
 rr^ - T^s^ = (T^«„ - 1) {T\^_, - T\_,), 
 
 But, by the value of q^ above, we have r^ = a^, s, = 1, so that 
 
 rr„.-r\=(TV-l)(ra„.,-l) (T\-l), 
 
 S.r^Ks^ = 0. 
 
 Thus it appears that we may write 
 
 q. = h + ^ + {-lT{c+y)p, (2), 
 
 with the condition 
 
 hc = Sl3y (3). 
 
 But, if we write, putting i instead of p^ in q„, 
 
 Q = 6 + ^+(-l)»(c + 7)i, 
 
 V 
 we have -^Q = \ + fn, suppose, 
 
 where X and /x are real vectors whose values can be calculated 
 from the data. And we now have 
 
 P»« = Pa = (- 1)" (1 + ^ + Wi) Px (1 + ^ + f^P^y' 
 
 ^ ^ l + X + fip, 
 When n is odd, this gives at once 
 
 {l+8/jip,)p, + S\p, = 0; 
 which, since p^ does not vanish, leads to the two equations 
 
 S\p, = 0, 
 
 These planes intersect in a line, whose intersections with the unit- 
 sphere give the possible extremities of the required first radius. 
 When n is even, we have 
 
 F\p, = fi+ p,S/J-p, = V. p, F/ipj, 
 or V.p,{X + Vf,p,) = 0. (§199.) 
 
 With the notation of (2) the condition (3) becomes 
 
192 QUATERNIONS. [250* 
 
 For further details, see especially Appendices B and C to the 
 Lectures. 
 
 By an immediate application of the linear and vector function 
 of Chapter V., the above solutions may be at once extended to any 
 central surface of the second order. 
 
 250*- The quaternions which Hamilton employed (as above) 
 were such as change the radius to one corner of the polygon into 
 that to the next by a conical rotation. It may be interesting and 
 useful to the student to compare with Hamilton's solution the 
 following, which employs the quaternions which directly turn one 
 side of the polygon to lie along the next. The successive sides are 
 expressed as ratios of one of these quaternions to the next. 
 
 Let Pj, p^, &c., p^ be (unit) vectors drawn from the centre of 
 the sphere to the corners of the polygon; a^, a^,...a„, the points 
 through which the successive sides are to pass. Then (by Euclid) 
 we have 
 
 (P, - «i) (Pi -«.) = !+ «,' = A' suppose. 
 (Ps - "2) (P2 - a^) = 1 + a/ = A^, 
 &c. = &c. 
 
 iP»» - aj (Pn-Cin) = 1 + «„'= A„. 
 
 These equations ensure that if the tensor of any one of the ps be 
 unit, those of all the others shall also be units. Thus we have 
 merely to eliminate p^, ■■■, p„\ and then remark that (for the 
 closure of the polygon) we must have 
 
 Pn^■l = Pi- 
 
 That this elimination is possible we see from the fact already 
 mentioned, which shews that the unknowns are virtually mere 
 unit-vectors ; while each separate equation contains coplanar 
 vectors only. In other words, when p^ and a„ are given, p„,^, is 
 determinate without ambiguity. 
 
 We may now write the first of the equations thus : — 
 
 (P^ - «=) (Pi - «i) = ^, + (a, - aJ (/3, - a,) = q^, suppose. 
 Thus the angle of q^ is the angle of the polygon itself, and in the 
 same plane. By the help of the second of the above equations 
 this becomes 
 
 A^(p,-a^) = (p^-ajq^; 
 whence 
 
 32 = A (ft - «i) + («2 - "a) ?i = (Pa - «a) 9,- 
 
2 50*.j THE SPHERE AND CYCLIC CONE. 193 
 
 By the third, this becomes 
 
 whence 
 
 (p4 - "4) ?2 = ^a^-i + ("s - «4) ?2 = 1r 
 The law of formation is now obvious ; and, if we write 
 
 1o = Pi- «i. /3i = «! - a^. ^2 = a^ - «3' ^c., 
 we have 
 
 <l, = ^A + ^29i - (!)• 
 
 &c., 
 We have also, generally, 
 
 p —„ — %n=l 
 fm "m ) 
 
 im-2 
 
 or 
 
 O — "m-\ ' '^mlm-i _. ""-m-l^m-S "*" '^m-llm-i _ -Pin-g SUBTDOSe f2) 
 "m-2 2m-2 lni-2 
 
 From (1), and the value of q^, we see that all the values of q 
 are linear functions of p, of the form 
 
 ?,™ = 'm+«»P. (3). 
 
 By (2) p^_, = ^„^,„_, + «„?„_, 
 
 = (1 + ol) g„_, + a„ {^,„_,g„_3 + (a„._j - «„) g,„_J 
 
 = ?,„-2 + «» (^„.-,5',„-3 + «..-l?m-2) 
 
 Similarly q^_^ = _p„_, - a„ g-^., ] 
 
 But the first equations in (1) give at once 
 
 ' P'= ^ +«'/'• I whence ^o = +PoP.: 
 g„ = -a, + pj Po = -9oPi. 
 
 ajjd ^^1= a2-«.+ (l+«2«i)/'.l or ^' P'P' 
 
 g, = 1 + a^a, - (a - a,) pj p^ = + q,p, I 
 
 This suggests that 
 
 i«„=(-r^.pj ^'^- 
 
 By (4) we have 
 
 Pm-l ~ 5'm-2 "'' '^mPm-i' 
 ?m-l /'to-2 "■m 2m-2" 
 
 Let m be odd, then we should have by (5) 
 
 Pn.-. = ^ + ^P.' 
 
 T.Q.I. 13 
 
194 QUATERNIONS. [2 5°*- 
 
 whence p„,_j = B- Ap^ + a„(A + Bp^), 
 
 q^^, = A+Bp^-a„(B-Apy, 
 
 or 
 
 P„,-^=S+a„^A-(A-a^B)p„ 
 
 q^_,^A-a„,B+(B+a,„A)p,. 
 These agree with (5), because m - 1 is even. And similarly we 
 may prove the proposition when m is even. 
 If now, in (2), we put n + 1 for m, we have 
 
 G + Dp^., , 
 
 = ^:r — 7^' if n be odd, 
 D + Cp, 
 
 G and D being quaternions to be calculated (as above) from the 
 
 data. The two cases require to be developed separately. 
 
 Take first the odd polygon : — 
 
 then pJ) + pfip^ = G-Dp^, 
 
 or /3i(c^ + 8) + /3,(c + 7)/'i = c + 7-('^+S)pi. 
 
 if we exhibit the scalar and vector parts of the quaternions G and 
 D. Cutting out the parts which cancel one another, and dividing 
 by 2, this becomes 
 
 dp^ + 8hp^ + p^Syp^ - c = 0, 
 which, as p is finite, divides itself at once into the two equations 
 
 Sjp^ + cZ = 0, 
 
 SSp, - c = 0. 
 
 These planes intersect in a line which, by its intersections (if real) 
 
 with the sphere, gives two possible positions of the first corner of 
 
 the polygon. 
 
 For the even polygon we have 
 
 p,D-p,Gp, = G + np^, 
 or Fp,S - 7 - p^Syp^ = ; 
 
 which may be written 
 
 V.p,iB-Vryp,) = 0. 
 This equation gives, as in § 199 above, 
 
 , ,^ = (, + ^)-^(^^g), 
 
 where x is to be found from 
 
2 5 3- J THE SPHERE AND CYCLIC CONE. 195 
 
 The two values of a;' have opposite signs. Hence there are two 
 real values of x, equal and with opposite signs, giving two real 
 points on the sphere. Thus this case of the problem is always 
 possible. 
 
 251. To find the equation of a cone of revolution, whose vertex 
 is the origin. 
 
 Suppose a, where Ta= 1, to be its axis, and e the cosine of-its 
 semi-vertical angle ; then, if p be the vector of any point in the 
 cone, 
 
 SccUp = — e, 
 
 or S^ap = - ey. 
 
 2.52. Change the origin to the point in the axis whose vector 
 is xa, and the equation becomes 
 
 (-X + Sa-sry = - e" (xa + zrf. 
 
 Let the radius of the section of the cone made by 
 
 Sa.'UT = 
 
 retain a constant value h, while x changes ; this necessitates 
 
 X 
 
 = e, 
 J^ + x' 
 
 so that when x is infinite, e is unity. In this case the equation 
 becomes 
 
 S'azT+^'' + ¥ = 0, 
 
 which must therefore be the equation of a circular cylinder of 
 radius 6, whose axis is the vector a. To verify this we have only 
 to notice that if ct be the vector of a point of such a cylinder we 
 must (§ 214) have 
 
 TVa-sT = b, 
 
 which is the same equation as that above. 
 
 253. To find, generally, the equation of a cone which has a 
 circular section : — 
 
 Take the origin as vertex, and let the circular section be the 
 intersection of the plane 
 
 Sap = l 0) 
 
 with the sphere (passing through the origin) 
 
 p' = S^p.i2) \ A ^^Lc ■ 
 
 13—2 
 
196 QUATEENIONS. [2 54- 
 
 These equations may be written thus, 
 8a.Up= Yp' 
 
 S^Up = -Tp. 
 Hence, eliminating Tp by multiplying the right, and left, members 
 together, we find the following equation which Up must satisfy — 
 
 SaUpS^Up = -\, 
 or p" - SapSl3p = 0, [^<^ t- i^^^^^^^ ©--^ 
 
 which is therefore the required equation of the cone. ^ 
 
 As a and /3 are similarly involved, the mere fcn^m of this 
 equation proves the existence of the subcontrary section dis- 
 covered by Apollonius. 
 
 254. The equation just obtained may be written 
 
 8.UaUpS.U^Up = -y^, 
 
 or, since a and /8 are perpendicular to the cyclic planes (§ 59*), 
 
 sin p svnp = constant, 
 
 where p and p are arcs drawn from any point of a spherical conic 
 perpendicular to the cyclic arcs. This is a well-known property of 
 such curves. 
 
 255. If we cut the cyclic cone by any plane passing through 
 the origin, as 
 
 Srip = 0, 
 
 then Fa7 and V^y are the traces on the cyclic planes, so that 
 
 p = x UVay + y UV^y (§ 24). 
 
 Substitute in the equation of the cone, and we get 
 
 -x'-y^ + Pccy^O, 
 
 where P is a known scalar. Hence the values of x and y are the 
 same pair of numbers. This is a very elementary proof of the 
 
 proposition in § 59*, that PL = MQ (in the last figure of that 
 section). 
 
 256. When x and y are eqiml, the transversal arc becomes a 
 tangent to the spherical conic, and is evidently bisected at the 
 point of contact. Here we have 
 
 (S.a^yf 
 
 P = 2 = 2S.UVciyUV^y + 
 
 T.VayV^y- 
 
26o.] THE SPHERE AND CYCLIC CONE. 197 
 
 This is the equation of the cone whose sides are perpendiculars 
 (through the origin) to the planes which touch the cyclic cone, 
 and from this property the same equation may readily he deduced. 
 
 2.57. It may be well to observe that the property of the 
 Stereographic projection of the sphere, viz. that the projection of 
 a circle is a circle, is an immediate consequence of the above form 
 of the equation of a cyclic cone. 
 
 258. That § 253 gives the most general form of the equation 
 of a cone of the second degree, when the vertex is taken as origin, 
 follows from the early results of next Chapter. For it is shewn 
 in § 263 that the equation of a cone of the second degree can 
 always be put in the form 
 
 2l,.SapS0p+Ap'=O. 
 This may be written ^P'pp = 0, 
 
 where ^ is the self-conjugate linear and vector function 
 
 (f>p='ZV. ap/3 +{A+ 2/Sfa;8) p. 
 By § 180 this may be transformed to 
 
 4>p=pp+ V.Xp/J,,:r^^^XS^f>-tSY'^^^' 
 
 and the general equation of the cone becomes AVLi^~Wj.e-^ vLu ^ f ^ 
 
 ( p - sxp) p' + 28xp sp,p = o,^5^:tr^iiiir:r^ 
 
 which is the form obtained in § 253. 
 
 259. Taking the form 
 
 Sp4>p = 
 as the simplest, we find by differentiation 
 
 Sdp^p + Spdipp = 0, 
 or 28dp^p = 0. 
 
 Hence 0/o is perpendicular to the tangent-plane at the extremity 
 of p. The equation of this plane is therefore {m being the vector 
 
 of any point in it) 
 
 8cl>p (■a7-p) = 0, 
 
 or, by the equation of the cone, 
 
 Sucf)p = 0. 
 
 260. The equation of the cone of normals to the tangent-planes 
 of a given cone can he easily formed from that of the cone itself. 
 For we may write the equation of the cone in the form 
 
198 QUATERNIONS. [26 1. 
 
 and if we put (fsp — a, a vector of the new cone, the equation 
 
 becomes 
 
 Sa-<j)''a- = 0. 
 
 Numerous curious properties of these connected cones, and of the 
 corresponding spherical conies, follow at once from these equations. 
 But we must leave them to the reader. 
 
 261. As a final example, let us find the equation of a cyclic 
 cone when five of its vector-sides are given — i.e. find the cone of the 
 second degree whose vertex is the origin, and on whose surface lie 
 the vectors a, /8, 7, S, e. 
 
 If we write, after Hamilton, 
 
 () = S.V{VaPVBe)V{V^r^Vep)V(y'yWpa) (1) 
 
 we have the equation of a cone whose vertex is the origin — for the 
 equation is not altered by putting xp for p. Also it is the equation 
 of a cone of the second degree, since p occurs only twice. Moreover 
 the vectors a, /S, 7, S, e are sides of the cone, because if any one 
 of them be put for p the equation is satisfied. Thus if we put 
 yS for p the equation becomes 
 
 Q = S.V{Va^Vhe)V{ F/87 Fe/3) F { F78 V^a) 
 = S . F ( Fa/3 FSe) { F^Sa/Sf . F7S F^7 Fe/3- V^h8 . V/3a V^y Fe^SK 
 The first term vanishes because ['-' ^ci\/^Y = ~/iSru + ri '^/i'J^ 
 
 /Sf.F(Fa;8FSe)Fy8a = 0, 
 and the second because 
 
 ,S.F^aF;87Fe;8 = 0, 
 
 since the three vectors F/3a, F/87, Fe/?, being each at right angles 
 to /S, must be in one plane. 
 
 As is remarked by Hamilton, this is a very simple proof of 
 Pascal's Theorem — for (1) is the condition that the intersections of 
 the planes of a, /3 and S, e ; /3, 7 and €, p; 7, S and p, a ; shall lie 
 in one plane ; or, making the statement for any plane section of 
 the cone : — /« order that the points of intersection of the three pairs 
 of opposite sides, of a hexagon inscribed in a curve, may always lie 
 in one straight line, the curve must he a conic section. 
 
THE SPHERE AND CYCLIC COxiB. 199 
 
 EXAMPLES TO CHAPTER VIII. 
 1. On the vector of a point P in the plane 
 
 a point Q is taken, such that QO . OP is constant ; find the equation 
 of the locus of Q. 
 
 '' 2. What spheres cut the loci of P and Q in (1) so that both 
 circles of intersection lie on a cone whose vertex is ? 
 
 I 3. A sphere touches a fixed plane, and cuts a fixed sphere. 
 If the point of contact with the plane be given, the plane of the 
 intersection of the spheres contains a fixed line. 
 
 Find the locus of the centre of the variable sphere, if the plane 
 of its intersection with the fixed sphere passes through a given 
 point. 
 
 4. Find the radii of the spheres which touch, simultaneously, 
 the four given planes 
 
 Sap = 0, S^p = 0, 8yp = 0, SBp = l. 
 
 "' [What is the volume of the tetrahedron enclosed by these planes ?] 
 
 ^ 5. If a moveable line, passing through the origin, make with 
 any number of fixed lines angles d, 0^, 6^, &c., such that 
 
 a cos + a^ cos 6^ + = constant, 
 
 where a, a^, are constant scalars, the line describes a right cone. 
 
 6. Determine the conditions that 
 
 Spcjip = 
 may represent a right cone, ^ being as in § 258. 
 
 7. What property of a cone (or of a spherical conic) is given 
 directly by the following particular form of its equation, 
 
 S . ipKp = 01 
 
 8. What are the conditions that the surfaces represented by 
 
 8p<f>p = 0, and S . ipKp = 0, 
 may degenerate into pairs of planes ? 
 
200 QUATERNIONS. 
 
 9. If arcs of great circles, drawn from any given point of a 
 sphere to a fixed great circle, be bisected, find the locus of these 
 'middle points ; and shew that the arcs drawn from the pole of this 
 fixed great circle, to that of which the given point is pole, are also 
 bisected by the same locus. 
 
 10. Find the locus of the vertices of all right cones which 
 have a common ellipse as base. 
 
 11. Two right circular cones have their axes parallel. Find 
 the orthogonal projection of their curve of intersection on the 
 plane containing their axe.s. 
 
 12. Two spheres being given in magnitude and position, every 
 sphere which intersects them in given angles will touch two other 
 fixed spheres and cut a third at right angles. 
 
 13. If a sphere be placed on a table, the breadth of the 
 elliptic shadow formed by rays diverging from a fixed point is 
 independent of the position of the sphere. 
 
 14. Form the equation of the cylinder which has a given 
 circular section, and a given axis. Find the direction of the 
 normal to the subcontrary section. 
 
 15. Given the base of a spherical triangle, and the product of 
 th6 cosines of the sides, the locus of the vertex is a spherical conic, 
 the poles of whose cyclic arcs are the extremities of the given 
 base. 
 
 16. (Hamilton, Bishop Law's Premium Ex., 1858.) 
 
 {a) What property of a sphero-conic is most immediately 
 indicated by the equation 
 
 a. p 
 
 (b) The equation 
 
 {V\pf + {Sfipy = 
 
 also represents a cone of the second order ; \ is a focal line, and 
 fi is perpendicular to the director-plane corresponding. 
 
 (c) What property of a sphero-conic does the equation most 
 immediately indicate ? 
 
 17. Shew that the areas of all triangles, bounded by a tangent 
 to a spherica,l conic and by the cyclic arcs, are equal. 
 
THE SPHERE AND CYCLIC CONE. 201 
 
 18. Shew that the locus of a point, the sum of whose arcual 
 distances from two given points on a sphere is constant, is a 
 spherical conic. 
 
 19. If two tangent planes be drawn to a cyclic cone, the four 
 lines in which they intersect the cyclic planes are sides of a right 
 cone. 
 
 20. Find the equation of the cone whose sides are the 
 intersections of pairs of mutually perpendicular tangent planes 
 to a given cyclic cone. 
 
 21. Find the condition that five given points may lie on a 
 sphere. 
 
 22. What is the surface denoted by the equation 
 
 p° = xo^ + y/S" + Z'f, 
 where p = xa + y^ + ^7, 
 
 a, /3, 7 being given vectors, and x, y, z variable scalars ? 
 
 Express the equation of the surface in terms of p, a, /3, 7 alone. 
 
 23. Find the equation of the cone whose sides bisect the 
 angles between a fixed line, and any line in a given plane, which 
 meets the fixed line. 
 
 What property of a spherical conic is most directly given 
 by this result ? 
 
CHAPTER IX. 
 
 SURFACES or THE SECOND DEGREE. 
 
 262. The general scalar equation of the second degree in a 
 vector p must evidently contain a term independent of p, terms of 
 the form S.apb involving p to the first degree, and others of the 
 form S.apbpc involving p to the second degree, a, b, c, &c. being 
 constant quaternions. Nov? the term S.apb may be v?ritten as 
 
 SpV{ba), 
 or as 
 
 S.(Sa+Va)p(Sb+Vb) = SaSpVb + 8bSpVa + S.pVbVa, 
 
 each of which may evidently be put in the form Syp, where 7 is 
 a known vector. 
 
 Similarly* the term S . apbpc may be reduced to a set of terms, 
 each of which has one of the forms 
 
 Ap\ {SapY, SapS/Sp, 
 
 the second being merely a particular case of the third. Thus (the 
 numerical factors 2 being introduced for convenience) we may 
 write the general scalar equation of the second degree as follows: — 
 
 21..Sa.pSl3p + Ap' + 2Syp = G (1). 
 
 263. Change the origin to I) where OD = 8, then p becomes 
 p + B, and the equation takes the form 
 
 2% . SoipSl3p + Ap' + 2S {8oipSl3B + S^pSaB) + 2ASBp + 2Sjp 
 
 + 2% . SaSS^B + AB' + 28yB -0 = 0; 
 
 * For S . apbpc = S . capbp = S . a!pb£^= {2Sa'Sb - Sa'b) p^ + 2Sa'pSbp ; and in parti- ■ 
 cular cases we may have Va' = VbX^ t^^^^O^Xodou s .3,' + (/^' m^ O.' /^ 
 
264.] SURFACES OP THE SECOND DEGREE. 203 
 
 from which the first power of p disappears, that is the surface is 
 re/erred to its centre, if 
 
 S(a(Sf/3S + /3/Sfa8) + 4S + 7 = (2), 
 
 a vector equation of the first degree, which in general gives 
 a single definite value for B, by the processes of Chapter V. [It 
 would lead us beyond the limits of an elementary treatise to 
 consider the special cases in which (2) represents a line, or a plane, 
 any point of which is a centre of the surface. The processes to be 
 employed in such special cases have been amply illustrated in the 
 Chapter referred to.] 
 
 With this value of S, and putting 
 
 n = C- 2SjS - AB^ - 2t . SclBS^B, 
 the equation becomes 
 
 22 . SapS^p + Ap' = D. 
 
 li D = 0, the surface is conical (a case treated in last Chapter) ; 
 if not, it is an ellipsoid or hyperboloid. Unless expressly stated 
 not to be, the surface will, when D is not zero, be considered an 
 ellipsoid. By this we avoid for the time some rather delicate 
 considerations. 
 
 By dividing by — B, and thus -altering only the tensors of the 
 constants, we see that the equation of central surfaces of the 
 second degree, referred to the centre, is (excluding cones) 
 
 2ti8ap8l3p)+gp' = -l (3). 
 
 [It is convenient to use the negative sign in the right-hand 
 member, as this ensures that the important vector ^p (which we 
 must soon introduce) shall make an acute angle with p ; i.e. be 
 drawn, on the whole, towards the same parts.] 
 
 264. Differentiating, we obtain 
 
 2t {SoLdpS/3p + SapS^dp} + 2gSpdp = 0, 
 or S.dp {2 {aS/3p + ^Sap) + gp] = 0, 
 
 and therefore, by § 144, the tangent plane is 
 
 >Sf (t;7 - /,) (2 {aSPp + /S&p) + gp] = 0, 
 i.e. 8.^ {2 (a8l3p + ^Sap) +gp} = - 1, by (3). 
 
 Hence, if v = 'Z(cl8^p + ^8dcp) + gp (4), 
 
 the tangent plane is 8v'!^ = — 1, 
 
 aad the surface itself is 8vp = — 1. 
 
204 QTJATEENI0N8. [265. 
 
 And, as —v~^ (being perpendicular to the tangent plane, and 
 satisfying its equation) is evidently the vector-perpendicular from 
 the origin on the tangent plane, v is called the vector of proximity. 
 
 265. Hamilton uses for v, which is obviously a linear and 
 vector function of p, the notation (f>p, expressing a functional 
 operator, as in Chapter V. But, for the sake of clearness, we will 
 go over part of the ground again, especially in the interests of 
 students who have mastered only the more elementary parts of 
 that Chapter. 
 
 We have, then, ^p = 1, (aS^p + ^Sap) + gp. 
 
 With this definition of ^, it is easy to see that 
 
 (a) <f) (p + (t) = (j>p + (j)(T, &c., for any two or more vectors. 
 
 {b) (j) (xp) = Xifip, a particular case of (a), x being a scalar. 
 
 (c) d^p = 4, {dp). 
 
 {d) S(T(f>p = S (SacrS^p + S^crSap) + gSpa = Sp(t><T, 
 
 or (f) is, in this case, self-conjugate. 
 
 This last property is of great importance in what follows. 
 
 266. Thus the general equation of central surfaces of the 
 second degree (excluding cones) may now be written 
 
 8p<lyp = -l (1). 
 
 ' Differentiating, Sdpijjp + Spd^p = 0, 
 
 which, by applying (c) and then (d) to the last term on the left, 
 gives 
 
 284pdp = 0, 
 
 and therefore, as in § 264, though now much more simply, the 
 tangent plane at the extremity of p is 
 
 S (t!t - p) (fjp = 0, 
 
 or 8-!^(j>p = Sp(f>p = - 1. 
 
 If this pass through A (OA = a), we have 
 
 Sa<pp = — 1, 
 
 or, by (d), 8p(j>a = — 1, 
 
 for all possible points of contact. 
 
 This is therefore the equation of the plane of contact of tangent 
 planes drawn from A. 
 
268.] SURFACES OF THE SECOND DEGREE. 205 
 
 267. To find the enveloping cone whose vertex is A, notice that 
 
 (Spcf>p + l)+p{8p<pa + iy = 0, 
 where p is any scalar, is the equation of a surface of the second 
 degree touching the ellipsoid along its intersection with the plane. 
 If this pass through A we have 
 
 {8a(t>a + 1) + ^ (8a<j)0L + 1)' = 0, 
 
 and p is found. Then our equation becomes 
 
 {Spcf>p + 1) (&'«</,« + 1) - {Sp(f,a + 1)' = (1), 
 
 which is the cone required. To assure ourselves of this, transfer 
 the origin to A, by putting p + a for p. The result is, using (a) 
 and (d), 
 
 (Sp<l}p + 2>Sfp(^a + Soi^a + 1) {Sa(}>a + 1) - {Sp(f>oi + Satf^ct + 1)' = 0, 
 
 or Sp^yp iScc(t>a + 1) - {Sp^ccY = 0, 
 
 which is homogeneous in Tp, and is therefore the equation of a 
 cone. 
 
 [In the special case when A lies on the surface, we have 
 Sa(j)a +1 = 0, 
 
 and the value of p is infinite. But this is not a case of failure, for 
 the enveloping cone degenerates into the tangent plane 
 
 /S/3(/)a+l = 0.] 
 
 Suppose A infinitely distant, then we may put in (1) xa for a, 
 where x is infinitely great, and, omitting all but the higher terms, 
 the equatioii of the cylinder formed by tangent lines parallel to a is 
 
 (Sp(f)p + 1) Scufya - {Sp(j>OLy = 0. 
 
 See, on this matter, Ex. 21 at end of Chapter. 
 
 268. To study the nature of the surface more closely, let us , 
 find the locus of the middle points of a system of parallel chords. 
 
 Let them be parallel to a, then, if ct be the vector of the middle 
 point of one of them, ■^ + xa and ct — «a are values of p which 
 ought simultaneously to satisfy (1) of § 266. 
 
 That is S .(■sy ± xa) (f){'07 ± xa.) = — 1. 
 
 Hence, by (a) and (d), as before, 
 
 Ssy^tsr + x^Sa(f)a = — 1, 
 
 S^<l>a = Q ...(1). 
 
206 QUATERNIONS. [269. 
 
 The latter equation shews that the locus of the extremity of tsr, 
 the middle point of a chord parallel to a, is a plane through the 
 centre, whose normal is ^a ; that is, a plane parallel to the tangent 
 plane at the point where OA cuts the surface. And (d) shews that 
 this relation is reciprocal^ — so that if j8 be any value of ot, i.e. be 
 any vector in the plane (1), a will be a vector in a diametral plane 
 which bisects all chords parallel to /3. The equations of these 
 
 planes are 
 
 Sis^a = 0, 
 
 so that if Y. ^a<f)^ = 7 (suppose) is their line of intersection, we have 
 
 Sy<f)a = = Sa^y ) 
 
 Sy(j}^ = = S/3<l>y\ (2). 
 
 and (1) gives >S/?(^a = = Sacf)^} 
 
 Hence there is an infinite number of sets of three vectors a, /3, 7, 
 such that all chords parallel to any one are bisected by the diametral 
 plane containing the other two. 
 
 269. It is evident from § 23 that any vector may be expressed 
 as a linear function of any three others not in the same plane ; let 
 then 
 
 p = xa-\-y^ + zy, 
 where, by last section, 
 
 Sa<^^ = S^^a = 0, 
 
 /Sa^7 = Sy<})a = 0, 
 
 6'/307 = Sy(l>/3 = 0. 
 
 And let Satfia = - 1 
 
 S04>I3 = - 1 
 Sy^y = — 1 
 
 so that a, A and 7 are vector conjugate semi-diameters of the 
 surface we are engaged on. 
 
 Substituting the above value of p in the equation of the surface, 
 and attending to the equations in a, /3, 7 and to (a), (6), and (d), 
 we have 
 
 8p4>p = S{xa + 2//8 + zy) (j) {xoi + y^ + zy), 
 
 = -(x' + f + z') = -l. 
 To transform this equation to Cartesian coordinates, we notice that 
 X is the ratio which the projection of p on a bears to a itself, &c. 
 
2 7 1. J SURFACES OF THE SECOND DEaEEE. 207 
 
 If therefore we take the conjugate diameters as axes of ^, tj, ^, and 
 their lengths as a, b, c, the above equation becomes at once 
 
 the ordinary equation of the ellipsoid referred to conjugate 
 diameters. 
 
 270. If we write yjr^ instead of <j), these equations assume an 
 interesting form. We take for granted, what we shall afterwards 
 prove, that this extraction of the square root of the vector function 
 is lawful, and that the new linear and vector function has the same 
 properties (a), (h), (c), (d) (§ 265) as the old. The equation of the 
 surface now becomes 
 
 Spyjr'p = - 1, l^mO= ii^f^f^ ^ft 
 
 or S'yjrpyirp = — 1, 
 
 or, finally, Tyjrp = 1. 
 
 If we compare this with the equation of the unit-sphere 
 
 Tp = l, 
 we see at once the analogy between the two surfaces. The sphere 
 can be changed into the ellipsoid, or vice versd, by a linear deforma- 
 tion of each vector, the operator being the function i^r or its inverse. 
 See the Chapter on Kinematics. 
 
 271. Equations (2) § 268, by § 270 become 
 
 /Sfat''/3 = = /Sfi|ra^/ry8, &c (1), 
 
 so that i/ra, i/f/S, i^, the vectors of the unit-sphere which correspond 
 to semi-conjugate diameters of the ellipsoid, form a rectangular 
 system. 
 
 We may remark here, that, as the equation of the ellipsoid 
 referred to its principal axes is a case of § 269, we may now suppose 
 i, j, and k to have these directions, and the equation is 
 
 -4-^ +- = 1 
 
 which, in quaternions, is 
 
 (Sipy {SjpY (8kpr__, 
 Sp<l>P = — -. p ^- A- 
 
 We here tacitly assume the existence of such axes, but in all cases, 
 by the help of Hamilton's method, developed in Chapter V., we at 
 once arrive at the cubic equation which gives them. 
 
208 QUATERNIONS. [272. 
 
 It is evident from the last-written equation that 
 
 which latter may be easily proved by shewing that 
 
 f'p = ^P- 
 And this expression enables us to verify the assertion of last section 
 about the properties of ■y^. 
 
 As 8ip = — x, &c., X, y, z being the Cartesian coordinates 
 referred to the principal axes, we have now the means of at once 
 transforming any quaternion result connected with the ellipsoid 
 into the ordinary one. 
 
 272. Before proceeding to other forms of the equation of the 
 ellipsoid, we may use those already given in solving a few problems. 
 
 Find ike locus of a point when the perpendicular froTn the centre 
 on its polar plane is of constant length. 
 
 If ■07 be the vector of the point, the polar plane is 
 8p^'ui = — 1, 
 
 and the length of the perpendicular from is ^g— (§ 264). 
 
 Hence the required locus is 
 
 r^oj = c 
 
 or S^^V = - C, 
 
 a concentric ellipsoid, with its axes in the same directions as those 
 of the first. By § 271 its Cartesian equation is 
 
 a c 
 
 273. Find the locus of a point whose distance from a given point 
 is always in a given ratio to its distance from a given line 
 
 Let p=x^he the given line, and A (OA = a.) the given point, 
 
 and choose the origin so that Sal3 = 0. Then for any one of the 
 
 required points 
 
 T(p-a) = eTV^p. 
 
 This is the equation of a surface of the second degree, which may 
 
 be written 
 
 p' - 2Sap + a' = e' (S'^p - ^Y). 
 
2 74- J SDRFACES OF THE SECOND DEGREE. 209 
 
 Let the centre be at S, and make it the origin, then 
 
 and, that the first power of p may disappear, 
 
 (S-a) = e=(/S/Sf;SS-i8^8), 
 
 a linear equation for B. To solve it, note that Sa^ = 0; operate by 
 8 . /3, and we get 
 
 (1 - e'^' + e'^') S/3S = S^S = 0. 
 
 Hence S - a = - e'^% 
 
 Referred to this point as origin the equation becomes 
 
 (l + e^^^yp^-e'S^^p + ^^^^O, 
 
 which shews that it belongs to a surface of revolution (of the 
 second degree) whose axis is parallel to y8, since its intersection 
 with a plane S^p = a, perpendicular to that axis, lies also on the 
 sphere 
 
 <" 1 + e'^' {l + e'^y 
 
 In fact, if the point be the focus of any meridian section of an 
 oblate spheroid, the line is the directrix of the same. 
 
 274. A sphere, passing through the centre of an ellipsoid, is cut 
 by a series of spheres whose centres are on the ellipsoid and which 
 pass through the centre thereof; find the envelop of the planes of 
 intersection. 
 
 Let (p — of = 0^ be the first sphere, i.e. 
 p^-2Sap = Q. 
 One of the others is p^ — ^S-i^p = 0, 
 
 where ^Sfiir^tn- = — 1. 
 
 The plane of intersection is 
 
 8{'UT-a)p = 0. ® 
 Hence, for the envelop (see next Chapter), _ , • 
 
 8^'^^ = 0| _ ^^^^^ ^' = d^X^^^f ■■• f"^ = > 
 
 013- p = 0) 
 or ( /; <^in = xp, {Vx = 0], 
 
 i.e. C2) OT=«^~'/3. 
 
 T. Q. I. 
 
210 QUATERNIONS. [275- 
 
 ■Hence ^s) ai'8p(j)'^p = -l \ 
 
 and i^Lf.) osSpcjT^p = Sap j ' 
 
 and, eliminating x, 
 
 Sp4>-'p = -(SapJ, 
 
 a cone of the second degree. 
 
 275. From a point in the outer of two concentric ellipsoids a 
 tangent cone is drawn to the inner, find the envelop of the plane of 
 contact. 
 
 If S-BT^T^ = — 1 be the outer, and Sp^^p = — 1 be the inner, ^ 
 and i|f being any two self-conjugate linear and vector functions, 
 the plane of contact is 
 
 8'!^'^p = — 1. 
 
 Hence, for the envelop, S'sy'^xlrp = 01 l!^i 
 
 (StB-^tn- = Oj [-^ 
 
 therefore ^ot = wylrp, 
 
 or tn- = xcpT^yjrp. 
 
 This gives xS . ■^p^'^-^p = — 1) 
 
 and x^S . ■^p^'^-^p = — 1 J ' 
 
 and therefore, eliminating x, 
 
 S . flrp<f)''\jrp = — 1, 
 or S . pyjr(f)~^-\{rp = — 1 , 
 
 another concentric ellipsoid, as yjrcjf^yjr is a linear and vector function 
 = 'x^ suppose ; so that the equation may be written 
 
 276. Find the locus of intersection of tangent planes at the 
 extremities of conjugate diameters. 
 
 If a, /S, 7 be the vector semi-diameters, the planes are 
 
 
 S^yfr'a.^-1] 
 
 STST^Jr^y = — 1 1 
 with the conditions § 271. ^M$f-<- f^+ P ^i'ff'^-^ fifr^-i 
 
 Hence — -^tuSi . yjrayjr^yjry = yjr^ = -i/ra + -\|r/S -|- 1^7, by § 92, /'.' S-i''^ Y' 
 therefore T-\^'!s = V3, 
 
 since -^a, t/t^S, 1^7 form a rectangular system of unit-vectors. 
 
 This may also evidently be written 
 
 Sm-yJr^'ST = — 3, 
 
2 78. J SURFACES OF THE SECOND DEGREE. 211 
 
 shewing that the locus is similar and similarly situated to the given 
 ellipsoid, but larger in the ratio V^ '• !• 
 
 277. Find the locus of intersection of three tangent planes 
 mutually at right angles. 
 
 If p be the point of contact, 
 
 SzTyfr'p = - 1 
 
 is the equation of the tangent plane. 
 
 The vector perpendicular from the origin is (§ 264) 
 
 1 
 
 — yj- = xa, suppose, 
 
 where a is a unit-vector. This gives 
 whence, by the equation of the ellipsoid, 
 
 Thus the perpendicular is 
 
 The sum of the squares of these, corresponding to a rectangular 
 unit-system, is 
 
 — 2 Sa^jr^a = m^, 
 
 by § 185. See also § 279. 
 
 278. Find the locus of the interseetion of three spheres whose 
 diameters are semi-conjugate diameters of an ellipsoid. 
 If a be one of the semi-conjugate diameters 
 Saf'a = - 1. 
 And the corresponding sphere is 
 
 p" - Sap = 0, 
 or p" — S-fp-ayjr'^p = 0, 
 
 with similar equations in /8 and 7. Hence, by § 92, 
 
 ylr'^pS . yjrayjr^ylry = — yjr'^p = p'{->^a + i/r/S -|- i/r-y), 
 and, taking tensors, T'^^'^p = >i/STp\ 
 
 or ri/r-y=V3, 
 
 or, finally, 8pyjr'''^p = — 3/3*. 
 
 This is Fresnel's Surface of Elasticity, Chap. XII. 
 
 14—2 
 
212 
 
 QUATERNIONS. 
 
 [279. 
 
 279. Before going farther we may prove some useful properties 
 of the function (f) in the form we are at present using — viz. 
 
 ,'iSip jSjp k8kp\ 
 
 We have P~- ^^'^P —J^JP ~ kSlcp, 
 
 and it is evident that 
 
 ,- * J- i J 7 ^ 
 
 Hence <^V = " f^ +-^ + 'f ^) ■ 
 
 Also (j}-'p = - (aSSip + b'jSjp + c'kSkp), 
 
 and so on. 
 
 Again, if a, 0, 7 be any rectangular unit-vectors 
 
 Sa.<j)a = — ( 
 
 &c. = &c, 
 
 (Siaf (Sjoif jSkay 
 
 {Sipy-^(Sjpy + (Skpy=-p\ 
 
 Sa4>a. + S0j>0 + ,S7</)7 = - {^2 + p + I, 
 
 But as 
 
 we have 
 Similarly 
 
 Sa<i>\ + 804,^0 + S70V = (<^«)' + (<^^)^ + {Hf = - (^ + i + 35) ■ 
 
 Again, 
 
 (iSia \ /iSi0 \ fiSiy 
 
 S . <l>a(f>/3(f>y = - S 
 
 Sia 
 
 8i0 
 
 Sja 
 
 Siy Sjy 
 
 a 
 
 Ska 
 Sk^ 
 Sky 
 
 \ a'' 
 
 1 
 
 d'bY 
 
 Sia, 
 Si/3, 
 
 Sja, Ska 
 Sj0, Sk/3 
 
 1 
 
 
 Siy, 
 
 Sjy, Sky 
 
 
 
 
 
 = Soi ir^- 
 
 l^in 
 
 J. 1 
 
 tnrl 
 
 And so on. These elementary investigations are given here foi 
 the benefit of those who have not read Chapter V. The student 
 may easily obtain all such results in a far more simple manner by 
 means of the formulae of that Chapter. 
 
 280. Find the locus of intersection of a rectangular system of 
 three tangents to an ellipsoid. 
 
 If TO- be the vector of the point of intersection, a, /3, 7 the 
 
282.] SUEPACES OF THE SECOND DEGEEE. 213 
 
 tangents, then, since ot + xa must give equal values of x when 
 substituted in the equation of the surface, so that 
 
 S(-sT + xa) <f>('UT + xa) = — 1, 
 
 or x^Sa^ia + 2xS-!s<^a + ((Stn-e^ra- + 1) = 0, 
 
 we have (S-ujcfiay = Sa(f)a {Si:!(^ + 1). 
 
 Adding this to the two similar equations in /S and 7, we have 
 
 (8a4)^y+ (S0<jit^y+ (Sy<f>^y= (Sacj,a + S^(f)^ + Sj(f,j) (SvT^zj + 1), 
 
 or (<fmy = (-^ + ^^ + -,) (5f^</,^ + 1), 
 
 or ^-{(^ + ^. + ^.)</'-«^j- = -(^.+F + ^.). 
 
 an ellipsoid concentric with the first. 
 
 281. If a rectangular system of chords be drawn through any 
 point within an ellipsoid, the sum of the reciprocals of the rectangles 
 under the segments into which they are divided is constant. 
 
 With the notation of the solution of the preceding problem, to- 
 giving the intersection of the vectors, it is evident that the 
 product of the values of x is one of the rectangles in question 
 taken negatively. 
 
 Hence the required sum is 
 
 1 i 1 
 
 Szs'ip'ST + 1 (StB-^sr + 1 " 
 
 This evidently depends on Sist^'m only and not on the particular 
 directions of a, /3, 7 : and is therefore unaltered if ts be the vector 
 of any point of an ellipsoid similar, and similarly situated, to the 
 given one. [The expression is interpretable even if the point be 
 exterior to the ellipsoid.] 
 
 282. Shew that if any rectangular system of three vectors he 
 drawn from a point of an ellipsoid, the plane containing their other 
 extremities passes through a fixed point. Find the locus of the 
 latter point as the former varies. 
 
 With the same notation as before, we have 
 
 /Sra'^'sr = — 1, 
 
 and S {■m + xa) <f) (-ar + xa) = - 1 ; 
 
 therefore oc = — ^r^ — ■ 
 
 ISa<pa 
 
214 QUATERNIONS. [283. 
 
 Hence the required plane passes through the extremity of 
 
 Sacjia 
 and those of two other vectors similarly determined. It therefore 
 (see § 30) passes through the point whose vector is 
 
 ^ = '^-'^ Sact>a^8^<f}^ + Sy<py ' 
 or = ^_?^ (§185). 
 
 Thus the first part of the proposition is proved. 
 But we have also ot = — -^-M — -^j 0, 
 whence by the equation of the ellipsoid we obtain 
 
 the equation of a concentric ellipsoid. 
 
 283. Find the directions of the three vectors which are parallel 
 to a set of conjugate diameters in each of two central surfaces of the 
 second degree. 
 
 Transferring the centres of both to the origin, let their equations 
 be 
 
 Sp<f)p =-1 or 0] 
 
 and Spylrp = — 1 or [ 
 
 If a, /3, 7 be vectors in the required directions, we must have (§ 268) 
 
 Sa^/3=0, 8af/3=0-\ 
 
 /Sf^^7=0, <S;St7=o[ (2)- 
 
 Sr/cfxx. = 0, SyfoL = ] 
 
 From these equations ifxx || V^y || yjra, &c. 
 
 Hence the three required directions are the roots of 
 
 V.<l,pylrp = (3). 
 
 This is evident on other grounds, for it means that if one of the 
 surfaces expand or contract uniformly till it meets the other, it will 
 touch it successively at points on the three sought vectors. 
 We may put (3) in either of the following forms — 
 
 V. ptff'fp = ] 
 
 or r.p^lr''(},p = 0^ '^*^' 
 
284.] SURFACES OF THE SECOND DEGEEE. 215 
 
 and, as <^ and ■^ are given functions, we find the solutions (when 
 they are all real, so that the problem is possible) by the processes 
 of Chapter V. 
 
 [Note. As <f>~^y}r and yfr'^cj) are not, in general, self-conjugate 
 functions, equations (4) do not signify that a, /8, 7 are vectors 
 parallel to the principal axes of the surfaces 
 
 In i^ese equations it does not matter whether ^'^yjr is self-conjugate 
 or not ; but it does most particularly matter when, as in (4), they 
 are involved in such a manner that their non-conjugate parts do 
 not vanish.] 
 
 Given two surfaces of the second degree, which have parallel 
 conjugate diameters, every surface of the second degree passing 
 through their intersection has conjugate diameters parallel to these. 
 
 For any surface of the second degree through the intersection 
 of 
 
 Sp(pp = — 1 and 8 {p — a) y^ {p — a) = — e, 
 
 is fSp<f>p -S(p-a)yjf(p-a) = e-f, 
 
 where e and / are scalars, of which / is variable. 
 The axes of this depend only on the term 
 
 Sp(fcf>-f)p. 
 
 Hence the set of conjugate diameters which are the same in all 
 are parallel to the roots of 
 
 V{fcl>-^lr)p{f<l>-y!r)p = 0, or Vcf>p^p-p = 0, 
 
 as we might have seen without analysis. 
 
 The locus of the centres is given by the equation 
 
 where /is a scalar variable. 
 
 284. Find the equation of the ellipsoid of which three conjugate 
 semi-diameters are given. 
 
 Let the vector semi-diameters be a, ^, 7, and let 
 
 Sp(f>p = - 1 
 be the equation of the ellipsoid. Then (§ 269) we have 
 ^a(^a=-l, 8a<l>l3 = 0, 
 
 8y^y = - 1, 8y(j)a = ; 
 
216 QUATERNIONS. [285. 
 
 the six scalar conditions requisite (|178) for the determination of 
 the self-conjugate linear and vector function (j>. 
 
 They give a || V(f)^(j)y, 
 
 or xa = ^~W^y. 
 
 Hence —x = xScupa = /S . a^y, 
 
 and similarly for the other combinations. Thus, as we have 
 
 pS. a/37 = "'S'- i^yp +^S . yap + yS .a^p, 
 we find at once 
 
 - 4)pS'' . al3y = V^yS . ^yp + VyaS . yap + Va^S . a^p ; 
 and the required equation may be put in the form 
 S' . a./3y = S'.a^p + S' . /3yp + S\ yap. 
 
 The immediate interpretation is that if four tetrahedra be formed 
 by grouping, three and three, a set of semi-conjugate vector axes of 
 an ellipsoid and any fourth vector of the surface, the sum of the 
 squares of the volumes of three of these tetrahedra is equal to the 
 square of the volume of the fourth. 
 
 285. A line moves with three of its points in given planes, find 
 the locus of any fourth point. 
 
 Let a, b, c be the distances of the three points from the fourth, 
 a, p, y unit-vectors perpendicular to the planes respectively. If p 
 be the vector of the fourth point, referred to the point of inter- 
 section of the planes, and a a unit-vector parallel to the line, we 
 have at once 
 
 S .a (p — aa) = 0, 
 
 S.^{p~ba) = Q, 
 
 S.y{p-ca) = 0. 
 Thus 
 
 aS. a^y=y^ Sap + ^^S^p+^^Syp. 
 
 The condition Ta=^l gives the equation of an ellipsoid referred to 
 its centre. 
 
 We may write the equation in the form 
 
 pSa^y = aV^ySaa + bVyaS/Sa + cVa/SSya 
 = (f)cr . Sa^y, suppose, 
 and from this we find at once for the volume of the ellipsoid 
 
 S.^'aS'^A'y 
 
 ^ -r -r^y ' =T.abc] 
 
 b . apy 
 
287.] SURFACES OF THE SECOND DEGREE. 217 
 
 altogether independent of the relative inclinations of the three 
 planes. This curious extension of a theorem of Monge is due to 
 Booth. 
 
 286. We see from § 270 that (as in § 31 (m)) we can write the 
 equation of an ellipsoid in the elegant form 
 
 where ^ is a self-conjugate linear and vector function, and we 
 impose the condition 
 
 Te = l. 
 
 Hence, when the same ellipsoid is displaced by translation and 
 rotation, by § 119 we may write its equation as 
 
 p = S + q<l>eq'\ 
 with the condition that e is still a unit-vector. 
 
 Where it touches a plane perpendicular to i, we must have, 
 simultaneously, 
 
 = S.iq<i>e'q-' = S.q-Hq<}y6' 
 and = See. 
 
 Hence e = — U<f) (q'Hq) 
 
 at the point of contact ; and, if the plane touched be that of ^'^, 
 
 = SiS-8. q'Hq^ U(f) (q-Hq), 
 or = SiB+T^(q-Sq). 
 
 Thus, if we write 
 
 q'Hq = a, q''jq = ^, q'^kq = 7, 
 we have 
 
 S = -I- iT(i>a +jT^^ + kT<})j, 
 
 which gives the possible positions of the centre of a given ellipsoid 
 when it is made to touch the fixed coordinate planes. 
 
 We see by § 279 that TS is constant. And it forms an 
 interesting, though very simple, problem, to find the region of 
 this spherical surface to which the position of the centre of the 
 ellipsoid is confined. This, of course, involves giving to a, /3, 7 
 all possible values as a rectangular unit system. 
 
 287. For an investigation of the regions, on each of the 
 coordinate planes, within which the point of contact is confined, 
 see a quaternion paper by Plarr (Trans. R. S. E. 1887). The 
 difiiculty of this question lies almost entirely in the eliminations, 
 which are of a very formidable character. Subjoined is a mere 
 sketch of one mode of solution, based on the preceding section. 
 
218 QUATERNIONS. [287. 
 
 The value of p, at the point of coatact with 
 
 Sip = 0, 
 is S-q(j)(U(f)a)q~'; 
 
 or + tiT^a + XiS . q'Sq^ ( U^ol), 
 
 or, finally, 
 
 Hence, in ordinary polar coordinates, the point of contact with 
 the plane oijk is 
 
 a rp.Q , S.4)a.(l)/3' 
 rcosd = T<f>p + — 
 
 .(1). 
 
 To find the boundary of the region within which the point of 
 contact must lie, we must make r a maximum or minimum, 6 
 being constant, and a, /3, 7 being connected by the relations 
 
 r« =T/3 =77=1) .,. 
 
 Sa/3 = 8^j = Sya=^0\ ''^''• 
 
 Differentiating (1) and (2), with the conditions 
 
 d0 = O, 
 (because is taken as constant) and 
 
 dr = 
 
 as the criterion of the maximum, we have eight equations which 
 are linear and homogeneous in da, djS, dy. Eliminating the two 
 latter among the seven equations which contain them, we have 
 
 = 8da [(^Sa/3, + ^,) S^y, + (ySay^ + 7,) ^7^^ (3) 
 
 where /81 = «^ ( D""^" - U(j>l3), 
 
 ^. = - 
 
 j^^F.^a^-'7, 
 
 m 
 
 But we have also, as yet unemployed, 
 
 Sada = (4). 
 
 Conditions (3) and (4) give the two scalar equations 
 <= (fif^,/3 - 8/3,<t) Sy,^ + Sy^^S^^y ) 
 O^S^,ySy^^ + (Sy.^y~Sy^oc)S^^y J W' 
 
288.] SURFACES OF THE SECOND DEGREE. 219 
 
 Theoretically, the ten equations (1), (2), and (5), suffice to 
 eliminate the nine scalars involved in a, /3, 7 : — and thus to leave 
 a single equation in r, 0, which is that of the boundary of the 
 region in question. 
 
 The student will find it a useful exercise to work out fully the 
 steps required for the deduction of (5). 
 
 288. When the equation of a surface of the second order can 
 be put in the form 
 
 8pcl>''p = -l (1), 
 
 where (,^ _ ^r) (,^ _ ^r J (,^ _ ^r J = 0, 
 
 we know that g, g^, g^ are the squares of the principal semi- 
 diameters. Hence, if we put (/> + ^ for ^ we have a second surface, 
 the differences of the squares of whose principal semi- axes are the 
 same as for the first. That is, 
 
 ^p(0 + /O-> = -l (2), 
 
 is a surface confocal with (1). From this simple modification of 
 the equation all the properties of a series of confocal surfaces may 
 easily be deduced. We give a couple of examples. 
 
 Any two confocal surfaces of the second degree, which meet, 
 intersect at right angles. 
 
 For the normal to (2) is evidently parallel to 
 
 and that to another of the series, if it passes through the common 
 point whose vector is p, is parallel to 
 
 But S.{<l> + hrp{c}> + hrP = S.p ^^^^^^^^^^y 
 
 and this evidently vanishes if h and A, are different, as they must 
 be unless the surfaces are identical. 
 
 To find the locus of the points of contact of a series of confocal 
 surfaces with a series of parallel planes. 
 
 Here the direction of the normal at the point, p, of contact is 
 
 and is parallel to the common normal of the planes, say a. 
 
 Thus 
 
 {cl> + h)-'p\\a. 
 
220 QUATERNIONS. [289. 
 
 Thus the locus has the equation of a plane curve 
 
 p = xa + ycjja, 
 
 and the relation between x and y is, by the general equation of 
 
 the confocals, 
 
 1 + y^Sa^a + xya^ = 0. 
 
 Hence the locus is a hyperbola. 
 
 289. To find the conditions of similarity of two central surfaces 
 of the second degree. 
 
 Referring them to their centres, let their equations be 
 
 8p^p-=-\\ ^j^ 
 
 Spyjrp = — 1 J 
 
 Now the obvious conditions are that the axes of the one are 
 proportional to those of the other. Hence, if 
 
 g^ -m^ +m,g -m=0] .„ 
 
 g''-m'J' + m\g'-m=0] ^ '' 
 
 be the equations for determining the squares of the reciprocals of 
 the semi-axes, we must have 
 
 —^ = IJ', — * = /*, — = /^ (^). 
 
 where fi is an undetermined scalar. Thus it appears that there 
 are but two scalar conditions necessary. Eliminating p, we have 
 
 mV^m/j mW, ^ m7 
 
 m^ mj ' mm^ m^ 
 
 which are equivalent to the ordinary conditions. 
 
 290. Find the greatest and least semi-diameters of a central 
 plane section of an ellipsoid. 
 
 Here Spcf)p = — 1 
 
 8oLp= 0| ^^^ 
 
 together represent the elliptic section; and our additional condition 
 is that Tp is a maxim\xm or minimum. 
 
 Dififerentiating the equations of the ellipse, we have 
 8<j>pdp = 0, 
 Sadp = 0, 
 and the maximum condition gives 
 
 dTp = 0, 
 or Spdp = 0. 
 
291.] SURFACES OF THE SECOND DEGREE. 221 
 
 Eliminating the indeterminate vector dp we have 
 
 S.ap(t)p = (2). 
 
 This shews that the maodmum or minimum vector, the normal at 
 its eodremity, and the perpendicular to the plane of section, lie in 
 one plane. It also shews that there are but two vector-directions 
 which satisfy the conditions, and that they are perpendicular to 
 each other, for (2) is unchanged if ap be substituted for p. 
 
 We have now to solve the three equations (1) and (2), to find 
 the vectors of the two (four) points in which the ellipse (1) inter- 
 sects the cone (2). We obtain at once 
 
 ^p = xV.{^-'oL)VcLp. 
 Operating by /S . p we have 
 
 - 1 = xp^8a(f)~^ a. 
 
 Hence _p^^p = p_«M^ 
 
 ^=l^:(i+^^'^)"^« (^); 
 
 from which S . a (1 + p'<f>)-' ct = (4); 
 
 a quadratic equation in p^, from which the lengths of the maximum 
 and minimum vectors are to be determined. By § 184 it may be 
 written 
 
 mp*8a^-'a + p'8.a(m^-(f))a + a' = (5). 
 
 [If we had operated by S . ^""' a or by S . ^~' p, instead of by 
 S.p, we should have obtained an equation apparently different 
 from this, but easily reducible to it. To prove their identity is a 
 good exercise for the student.] 
 
 Substituting the values of p' given by (5) in (3) we obtain the 
 vectors of the required diameters. [The student may easily prove 
 directly that 
 
 (H-p.=</.ra and (l+/),Vr« 
 
 are necessarily perpendicular to each other, if both be perpen- 
 dicular to a, and if />/ and p/ be different. See § 288.] 
 
 291. By (5) of last section we see that 
 
 P'P'' ~ mSa<^-'a 
 Hence the area of the ellipse (1) is 
 
 irTa 
 J— miSa^'^ a 
 
222 QUATERNIONS. [292. 
 
 Also the locus of central normals to all diametral sections of an 
 ellipsoid, whose areas are equal, is the cone 
 
 When the roots of (5) are equal, i.e. when 
 
 im^a' - 8a<l>af = 4>ma'8a(f>-' a (6), 
 
 the section is a circle. It is not difficult to prove that this 
 equation is satisfied by only two values of Ua, but another 
 quaternion form of the equation gives the solution of this and 
 similar problems by inspection. (See § 292 below.) 
 
 292. By § 180 we may write the equation 
 
 Spcfip = - 1 
 in the new form S . Xpfip + pp^ = — 1, 
 
 where p is a known scalar, and X and fi are definitely known (with 
 the exception of their tensors, whose product alone is given) in 
 terms of the constants involved in (f). [The reader is referred 
 again also to §§ 128, 129.] This may be written 
 
 2SXpS/ip + {p-S\/j,)p' = -l (1). 
 
 From this form it is obvious that the surface is cut by any plane 
 perpendicular to X or yu, in a circle. For, if we put 
 
 SXp = a, 
 we have 2aSfip + {p — SXfi) p^ = — 1, 
 
 the equation of a sphere which passes through the plane curve of 
 intersection. 
 
 Hence \ and ytt of § 180 are the values of a in equation (6) of 
 the preceding section. 
 
 293. Any two drcidar sections of a central surface of the 
 second degree, whose planes are not parallel, lie on a sphere. 
 
 For the equation 
 
 (SXp-a)(8fip-b) = 0, 
 where a and b are any scalar constants whatever, is that of a 
 system of two non-parallel planes, cutting the surface in circles. 
 Eliminating the product SXpSfip between this and equation (1) of 
 last section, there remains the equation of a sphere. 
 
 294. To find the generating lines of a central surface of the 
 second degree. 
 
 Let the equation be 
 
 Spcjip = - 1 ; 
 
295-] SURFACES OF THE SECOND DEGREE. 223 
 
 then, if a be the vector of any point on the surface, and ot a vector 
 parallel to a generating line, we must have 
 
 p = a+ X'^s 
 for all values of the scalar x. 
 
 Hence S (a + X's:') ^ (a + aw) = — 1 
 
 gives the two equations 
 
 Stm^ym = J 
 
 The first is the equation of a plane through the origin parallel 
 to the tangent plane at the extremity of a, the second is the 
 equation of the asymptotic cone. The generating lines are there- 
 fore parallel to the intersections of these two surfaces, as is well 
 known. 
 
 From these equations we have 
 
 y(jyaT = VwoT 
 where y is a scalar to be determined. Operating on this hy S.^ 
 and S.y, where /3 and 7 are any two vectors not coplanar with a, 
 we have 
 
 S-ar (y(j>^ +Va^) = 0, S'^ (y<py -Vja) = (1). 
 
 Hence S .^a{y(f>^ + Va/3) (y(f)y-Vya) = 0, 
 
 or my^S . a^j — Sacf>aS . a^y = 0. 
 
 Thus we have the two values 
 
 
 belonging to the two generating lines. That they may be real 
 it is clear that m must be negative : — i.e. the surface must be the 
 one-sheeted hyperboloid. 
 
 295. But by equations (1) we have 
 
 Z'ST = V. (2/^j8 -I- Fa/8) (1/(^7 - Vya) 
 
 = mf<f)-W^y + yV. (j>aV^y - aS . aV^y ; 
 which, according to the sign of y, gives one or other generating 
 line. 
 
 Here F/87 may be any vector whatever, provided it is not 
 perpendicular to a (a condition assumed in last section), and we 
 may write for it 0. 
 
 Substituting the value of y before found, we have 
 
 zzT = - (j,-' - a8a0 ± a/ - - • ^^'^«' 
 
224 QUATERNIONS. [295- 
 
 or, as we may evidently write it, 
 
 = </,-'(F.aF</,a^)±y^-~.F^<^« (2). 
 
 Put r=V6(j)a, 
 
 and we have ^ot = — dT^Var ± a / . t, 
 
 with the condition ST(j>a = 0. 
 
 [Any one of these sets of values forms the complete solution of the 
 problem ; but more than one have been given, on account of their 
 singular nature and the many properties of surfaces of the second 
 degree which immediately follow from them. It will be excellent 
 practice for the student to shew that 
 
 fe = u(v. ^aVd^-'e ± J - ^.F6'^a) 
 
 is an invariant. This may most easily be done by proving that 
 
 F. -^e^Q^ = identically.] 
 
 Perhaps, however, it is simpler to write a for V^j, and we thus 
 obtain 
 
 Z'S! = — <^"' F. o Va(f)a + * / . Fa^a. 
 
 [The reader need hardly be reminded that we are dealing with the 
 general equation of the central surfaces of the second degree — the 
 centre being origin.] 
 
 EXAMPLES TO CHAPTEE IX. 
 
 1. Find the locus of points on the surface 
 
 where the generating Knes are at right angles to one another. 
 
 2. Find the equation of the surface described by a straight 
 line which revolves about an axis, which it does not meet, but 
 with which it is rigidly connected. 
 
SURFACES OF THE SECOND DEGREE. 225 
 
 3. Find the conditions that 
 
 Sp<j)p = -1 
 may be a surface of revolution, with axis parallel to a given vector. 
 
 4. Find the equations of the right cylinders which circum- 
 scribe a given ellipsoid. 
 
 5. Find the equation of the locus of the extremities of per- 
 pendiculars to central plane sections of an ellipsoid, erected at the 
 centre, their lengths being the principal semi-axes of the sections. 
 [Fresnel's Wave-Surface. See Chap. Xll.] 
 
 6. The cone touching central plane sections of an ellipsoid, 
 which are of equal area, is asymptotic to a confocal hyperboloid. 
 
 7. Find the envelop of all non-central plane sections of an 
 ellipsoid when their area is constant. 
 
 8. Find the locus of the intersection of three planes, perpen- 
 dicular to each other, and touching, respectively, each of three 
 confocal surfaces of the second degree. 
 
 9. Find the locus of the foot of the perpendicular from the 
 centre of an ellipsoid upon the plane passing through the extremi- 
 ties of a set of conjugate diameters. 
 
 10. Find the points in an ellipsoid where the inclination 
 of the normal to the radius-vector is greatest. 
 
 11. If four similar and similarly situated surfaces of the 
 second degree intersect, the planes of intersection of each pair pass 
 through a common point. 
 
 12. If a parallelepiped be inscribed in a central surface of the 
 second degree its edges are parallel to a system of conjugate 
 diameters. 
 
 13. Shew that there is an infinite number of sets of axes for 
 which the Cartesian equation of an ellipsoid becomes 
 
 a;'- + y' + 0' = e\ 
 
 14. Find the equation of the surface of the second degree 
 which circumscribes a given tetrahedron so that the tangent plane 
 at each angular point is parallel to the opposite face ; and shew 
 that its centre is the mean point of the tetrahedron. 
 
 T. Q, I, 15 
 
226 QUATERNIONS. 
 
 15. Two similar and similarly situated surfaces of the second 
 degree intersect in a plane curve, whose plane is conjugate to the 
 vector joining their centres. 
 
 16. Find the locus of all points on 
 
 Spcf>p = - 1, 
 where the normals meet the normal at a given point. 
 
 Also the locus of points on the surface, the normals at which 
 meet a given line in space. 
 
 17. Normals drawn at points situated on a generating line 
 are parallel to a fixed plane. 
 
 18. Find the envelop of the planes of contact of tangent 
 planes drawn to an ellipsoid from points of a concentric sphere. 
 Find the locus of the point from which the tangent planes are 
 drawn if the envelop of the planes of contact is a sphere. 
 
 19. The sum of the reciprocals of the squares of the perpen- 
 diculars from the centre upon three conjugate tangent planes is 
 constant. 
 
 20. Cones are drawn, touching an ellipsoid, from any two 
 points of a similar, similarly situated, and concentric ellipsoid. 
 Shew that they intersect in two plane curves. 
 
 Find the locus of the vertices of the cones when these plane 
 sections are at right angles to one another. 
 
 21. Any two tangent cylinders to an ellipsoid intersect in two 
 plane ellipses, and no other tangent cylinder can be drawn through 
 either of these. 
 
 Find the locus of these ellipses : — 
 
 (a) When the axes of the two cylinders are conjugate to each 
 other, and to a given diameter. 
 
 (b) When they are conjugate to each other, and to diameters 
 lying in one plane. 
 
 (c) When they are conjugate to each other, and to any 
 diameter whatever. 
 
 22. If a, /3, 7 be unit vectors parallel to conjugate semi- 
 diameters of an ellipsoid, what is the vector 
 
 J - Sa4>a J - S/3'cf)^ J - Sy<}>ry ' 
 and what the locus of its extremity ? 
 
SURFACES OF THE SECOND DEGREE. 227 
 
 23. Find the locus of the points of contact of tangent planes 
 which are equidistant from the centre of a surface of the second 
 degree. 
 
 24. From a fixed point A, on the surface of a given sphere, 
 draw any chord AD; let D' be the second point of intersection of 
 the sphere with the secant BD drawn from any point B ; and 
 take a radius vector AE, equal in length to BD', and in direction 
 either coincident with, or opposite to, the chord AD : the locus 
 of E is an ellipsoid, whose centre is A, and which passes through 
 B. (Hamilton, Elements, p. 227.) 
 
 25. Shew that the equation 
 
 P (e' - 1) (e + Saa') = {Sapf - 2e8ap8oLp + {SoLpf + (1 - e') p\ 
 where e is a variable (scalar) parameter, and a, a' unit-vectors, 
 represents a system of confocal surfaces. {Ihid. p. 644.) 
 
 26. Shew that the locus of the diameters of 
 
 8p<l>p = - 1, 
 
 which are parallel to the chords bisected by the tangent planes to 
 
 the cone 
 
 Spyjrp = 0, 
 
 is the cone /S.p0i|r~'^/3 = 0. 
 
 27. Find the equation of a cone, whose vertex is one summit 
 of a given tetrahedron, and which passes through the circle 
 circumscribing the opposite side. 
 
 28. Shew that the locus of points on the surface 
 
 Sp4>p = - 1, 
 
 the normals at which meet that drawn at the point /s = ■nr, is on 
 
 the cone 
 
 S. (p — tn-) <f>i!7(j}p = 0. 
 
 29. Find the equation of the locus of a point the square 
 of whose distance from a given line is proportional to its distance 
 from a given plane. 
 
 30. Shew that the locus of the pole of the plane 
 
 Sap = l, 
 with respect to the surface 
 
 Sp<f)p = — 1, 
 is a sphere, if a be subject to the condition 
 
 Sa(l>-'a = a 
 
 15—2 
 
228 QUATERNIONS. 
 
 31. Shew that the equation of the surface generated by lines 
 drawn through the origin parallel to the normals to 
 
 8p4>-'p = - 1 
 along its lines of intersection with 
 
 Spi<t> + h)-'p = -l, 
 is OT^-A/Sfro(<^ + A)-V = 0. 
 
 32. Common tangent planes are drawn to 
 
 2S\pSfip + (p-SXfi)p' = -l, and Tp = h, 
 find the value of h that the lines of contact with the former surface 
 may be plane curves. What are they, in this case, on the sphere ? 
 Discuss the case of 
 
 / - ^ V = 0. 
 
 33. If tangent cones be drawn to 
 
 8p(f>'p = -l. 
 from every point of 
 
 Spcfip = - 1, 
 
 the envelop of their planes of contact is 
 
 Sp4>'p = - 1. 
 
 34. Tangent cones are drawn from every point of 
 
 S{p-a)4)(p-a) = - n\ 
 to the similar and similarly situated surface 
 
 Sp<f)p = - 1, 
 shew that their planes of contact envelop the surface 
 
 (8a<pp + iy = -ri'8p<pp. 
 
 35. Find the envelop of planes which touch the parabolas 
 
 p = af + ^t, p = au' + 7M, 
 where a, /8, 7 form a rectangular system, and t and u are scalars. 
 
 36. Find the equation of the surface on which lie the lines of 
 contact of tangent cones drawn from a fixed point to a series of 
 similar, similarly situated, and concentric ellipsoids. 
 
 37. Discuss the surfaces whose equations are 
 
 8ap8^p = Syp, 
 and S'ap + 8.al3p = -l. 
 
 38. Shew that the locus of the vertices of the right cones 
 which touch an ellipsoid is a hyperbola. 
 
SUEFACES OP THE SECOND DEGREE. 229 
 
 39. If a,, Kj, ttg be vector coBJugate diameters of 
 
 8p4>p = - 1, 
 where </>' — m.jfi' + m^<f) — m = 0, 
 
 shew that 
 
 \ / ^ \12/ ^ 128^. 
 
 and 2 (^a)" = — m^. 
 
 40. Find the locus of the lines of contact of tangent planes 
 from a given point to a series of spheres, whose centres are in one 
 line and which pass through a given point in that line. 
 
 41. Find the locus of a circle of variable radius, whose plane 
 is always parallel to a given plane, and which passes through each 
 of three given lines in space. 
 
CHAPTER X. 
 
 GEOMETRY OF CURVES AND SURFACES. 
 
 296. We have already seen (§ 31 (I)) that the equations 
 
 and p = (j){t, u) = 'Z . af(t, u), 
 
 where a represents one of a set of given vectors, and / a scalar 
 function of scalars t and u, represent respectively a curve and a 
 surface. We commence the present too brief Chapter with a few 
 of the immediate deductions from these forms of expression. We 
 shall then give a number of examples, with little attempt at 
 systematic development or even arrangement. 
 
 297. What may be denoted by t and u in these equations is, 
 of course, quite immaterial : but in the case of curves, considered 
 geometrically, t is most conveniently taken as the length, s, of the 
 curve, measured from some fixed point. In the Kinematical 
 investigations of the next Chapter t may, with great convenience, 
 be employed to denote time. 
 
 298. Thus we may write the equation of any curve in 
 space as 
 
 p = 4>s, 
 
 where ^ is a vector function of the length, s, of the curve. Of 
 course it is a linear function when, and only when, the equation 
 (as in § 31 [1)) represents a straight line. 
 
 If be a periodic function, such that 
 
 <^{l + s) = 4>{s), 
 
 the curve is a reentrant one, generally a Knot in space. 
 
30I.] GEOMETRY OF CURVES AND SURFACES. 231 
 
 [In § 306 it is shewn that when s is the arc certain forms of (^ 
 are not admissible.] 
 
 299. We have also seen (§§ 38, 39) that 
 
 is a vector of unit length in the direction of the tangent at the 
 extremity of p. 
 
 At the proximate point, denoted by s + Ss, this unit tangent 
 vector becomes 
 
 0's + ^"sSs + &c. 
 
 But, because y^'s = 1, 
 
 we have S . cl>'scj)"s = 0. 
 
 Hence (f)"s, which is a vector in the osculating plane of the curve, 
 is also perpendicular to the tangent. 
 
 Also, if 89 be the angle between the successive tangents (j)'s 
 and (f>'s + ^"sSs + , we have 
 
 SO that the tensor of ^"s is the reciprocal of the radius of absolute 
 curvature at the point s. 
 
 300. Thus, if OP = ^s be the vector of any point P of the 
 curve, and if G be the centre of curvature at P, we have 
 
 and thus 00 = (j>s — -rn- 
 
 q) s 
 
 is the equation of the locus of the centre of curvature. 
 
 Hence also V. (j}'s<f>"s or ^'s^"s 
 
 is a vector perpendicular to the osculating plane ; and therefore 
 
 T^^(<l>'sU<l^"s) 
 
 is the tortuosity of the given curve, or the rate of rotation of 
 its osculating plane per unit of length. 
 
 301. As an example of the use of these expressions let us find 
 the curve whose curvature and tortuosity are both constant. 
 
 We have curvature = T(\)"s = Tp" = c. 
 
232 QUATERNIONS. [^OI. 
 
 Hence <j)'s^"s = p'p" = ca, 
 
 where a is a unit vector perpendicular to the osculating plane. 
 
 This gives 
 
 p'p"' + p"' = c^^£ = cc,Up" = c,p", 
 
 if Cj represent the tortuosity. 
 Integrating we get 
 
 pp" = c,p' + ^ (1)' 
 
 where /3 is a constant vector. Squaring both sides of this equation, 
 
 we get 
 
 c' = c/ - /3' - 2c,8^p' 
 
 (for by operating with S.p' upon (1) we get \-c^ = S^p'), 
 or T^ = J^+c,\ 
 
 Multiply (1) by p', remembering that 
 
 To =1, 
 and we obtain — p" = — c^ + p'^, 
 
 or, by integration, p' = c^s — p^ + a (2), 
 
 where a is a constant quaternion. Eliminating p', we have 
 
 -p" = -c, + c,sl3-p^' + al3, 
 of which the vector part is 
 
 p" -p^'=- c,s/3 - Va^. 
 The complete integral of this equation is evidently 
 
 p = lcos.sW + r,sm. sT^ - ~i (c,s^ + Fa/3) ....... (3), 
 
 f and 7) being any two constant vectors. "We have also by (2), 
 
 80 p = CjS + Sa, 
 which requires that 80^ = 0, S^rj = 0. 
 The farther test, that Tp' = 1, gives us 
 -l = Tl3\^sm\sT/3+r)'cos\sTl3-28^V^in.sT0cos.sW)-^^,. 
 
 C + Cj 
 
 This requires, of course, 
 
 8^V = 0, T^=Tn = ^„ 
 
 so that (3) becomes the general equation of a helix traced on a 
 right cylinder. (Compare § 31 (?«,).) 
 
303.J GEOMETRY OF CURVES AND SURFACES. 233 
 
 302. The vector perpendicular from the origin on the tangent 
 to the curve 
 
 is, of course, -Vp'p, or p'Vpp 
 
 (since p' is a unit vector). 
 
 To find a common property of curves whose tangents are all 
 equidistant from the origin. 
 
 Here 
 
 TVpp' = c, 
 
 which may be written — p" — S^pp' = (? (1). 
 
 This equation shews that, as is otherwise evident, every curve 
 on a sphere whose centre is the origin satisfies the condition. For 
 obviously 
 
 — jo" = c" gives Spp' = 0, 
 
 and these satisfy (1). 
 
 li Spp' does not vanish, the integral of (1) is 
 
 jT^c' = s (2), 
 
 an arbitrary constant not being necessary, as we may measure s 
 from any point of the curve. The equation of an involute which 
 commences at this assumed point is 
 
 -ST = p — sp'. 
 
 This gives 2V'' = Tp" + s' + 2s8pp' 
 
 = Tp' + s'-2s jTp' - c^ by (1), 
 
 = c^ by (2). 
 
 This includes all curves whose involutes lie on a sphere about the 
 origin. 
 
 303. Find the locus of the foot of the perpendicular drawn to a 
 tangent to a right helix from a point in the axis. 
 
 The equation of the helix is 
 
 p = a cos - + /S sin - + 7s, 
 
 '^ a a ' 
 
 where the vectors a, /3, 7 are at right angles to each other, and 
 
 Ta = T^ = b, while aTj = J a' - b\ 
 (The latter condition is from Tp = 1.) 
 
234 QUATERNIONS. [304- 
 
 The equation of the required locus is, by last section, 
 
 •S7 = p' Vpp' 
 
 This curve lies on the hyperboloid whose equation is 
 
 as the reader may easily prove for himself. 
 
 304. To find the least distance between consecutive tangents to a 
 tortuous curve. 
 
 Let one tangent be tij = p + xp', 
 then a consecutive one, at a distance Ss along the curve, is 
 
 ^ = p + p'Ss + p"^^ + &c. + y{p' + p"Bs + p"'^+...). 
 
 The magnitude of the least distance between these lines is, by 
 §§ 216, 223, 
 
 S.{p'Bs + p"^^^ + p"'^+ ...)uV.p'{p' + p"Ss + p"'^l + ...) 
 
 TVp'p"Ss ' 
 
 if we neglect terms of higher orders. 
 
 It may be written, since p'p" is a vector, and Tp' = 1, 
 
 ^^^S.Up"Vpp"'. 
 But (§ 140 (2)) ^^^' =V^Ss=^,p'S. p'p"p"'. 
 
 Hence pr,8 .Up"Vp'p"' 
 
 is the small angle, S^, between the two successive positions of the 
 osculating plane. [See also § 300.] 
 
 Thus the shortest distance between two consecutive tangents is 
 expressed by the formula 
 
 12r ' 
 
 where r, = y^j-,, , is the radius of absolute curvature of the tortuous 
 curve. 
 
306.] GEOMETRY OP CURVES AND SURFACES. 235 
 
 305. Let us recur for a moment to the equation of the parabola 
 (§31(/)) 
 
 Here p' = („ + ^()^^^ 
 
 whence, if we assume Sa^ = 0, 
 
 from which the length of the arc of the curve can be derived in 
 terms of t by integration. 
 
 Again, p" = (« + ^i)g+/3f|y. 
 
 ds^ \ds, 
 
 R t <il^d^ 1 _ di S .0ia + ^t) 
 
 " ds' ds'T(a + 0t) '^ ds T{a + l3ty ' 
 
 Hence „__{a±mVcc^ 
 
 and therefore, for the vector of the centre of curvature we have 
 (§ 300), 
 
 w = a« + ^ - (a^ + ISHJ (- 13a!' + a^H)-\ 
 
 
 which is the quaternion equation of the evolute. 
 
 306. One of the simplest forms of the equation of a tortuous 
 curve is 
 
 where a, /3, 7 are any three non-coplanar vectors, and the numerical 
 factors are introduced for convenience. This curve lies on a para- 
 bolic cylinder whose generating lines are parallel to j; and also on 
 cylinders whose bases are a cubical and a semi-cubical parabola, 
 their generating lines being parallel to /3 and a respectively. We 
 have by the equation of the curve 
 
 from which, by Tp = 1, the length of the curve can be found in 
 terms of t ; and 
 
236 QUATERNIONS. [S^?- 
 
 from which p" can be expressed in terms of s. The investigation 
 of various properties of this curve is very easy, and will be of great 
 use to the student. 
 
 [N'ote. — It is to be observed that in this equation t cannot stand 
 for s, the length of the curve. It is a good exercise for the student 
 to shew that such an equation as 
 
 or even the simpler form 
 
 p = as + I3s\ 
 involves an absurdity.] 
 
 307. The equation p = <j)*e, 
 
 where ^ is a given self-conjugate linear and vector function, t a 
 scalar variable, and e an arbitrary vector constant, belongs to a 
 curious class of curves. 
 
 We have at once j7 — 4>* los "^ ^' 
 
 where log is another self-conjugate linear and vector function, 
 which we may denote by j^. These functions are obviously com- 
 mutative, as they have the same principal set of rectangular vectors, 
 hence we may write 
 
 dp 
 
 which of course gives ~ = 'x^p, &c., 
 
 since x does not involve t. 
 
 As a verification, we should have 
 
 = (l + % + j^X^ + )p 
 
 = ^'^p, 
 
 where e is the base of Napier's Logarithms. 
 
 This is obviously true if ^^* — e*^, 
 or (^ = ex^ 
 
 or log^ = 5(;, 
 
 which is our assumption. See § 337, below. 
 
 [The above process is, at first sight, rather startling, but the 
 
3 1 O.J GEOMETRY OE CURVES AND SURFACES. 237 
 
 student may easily verify it by writing, in accordance with the 
 results of Chapter V, 
 
 <^e = - g.aSae - g^S^e - g^/Sye, 
 
 whence ^'e = - g\ aSae — g\^8^e — g\jSye. 
 
 He will find at once 
 
 Xe = - log g^ aSae - log g^S^e - log g^ySye, 
 
 and the results just given follow immediately.] 
 
 308. That the equation 
 
 p = (j) {t, u) = li . of (t, u) 
 
 represents a surface is obvious from the fact that it becomes the 
 equation of a definite curve whenever either t or u has a particular 
 value assigned to it. Hence the equation at once furnishes us with 
 two systems of curves, lying wholly on the surface, and such that 
 one of each system can, in general, be drawn through any assigned 
 point on the surface. Tangents drawn to these curves a,t their 
 point of intersection must, of course, lie in the tangent plane, whose 
 equation we have thus the means of forming. [Of course, there 
 may occasionally be cases of indeterminatenesa, as when the curves 
 happen to touch one another. But the general consideration of 
 singular points on surfaces is beyond the scope of this work.] 
 
 309. By the equation we have 
 
 where the brackets are inserted to indicate partial differential 
 coefficients. If we write this as 
 
 dp = (j)\dt+ip'„du, 
 the normal to the tangent plane is evidently 
 
 and the equation of that plane 
 
 810. Thus, as a simple example, let 
 
 p = ta + u^ + tuy. 
 
 This surface is evidently to be constructed by drawing through each 
 point ta, of the line a, a line parallel to ^ + ty; or through u^, a 
 line parallel to a + wy. 
 
238 QUATERNIONS. [S^^- 
 
 We may easily eliminate t and u, and obtain 
 >Sf . P'^pS . yap = S . a^jS . a^p ; 
 and the methods of last chapter enable us to recognise a hyperbolic 
 paraboloid. 
 
 Again, suppose a straight line to move along a fixed straight 
 line, remaining always perpendicular to it, while rotating about it 
 through an angle proportional to the space it has advanced ; the 
 equation of the ruled surface described will evidently be 
 
 p = at + u(l3cofit + ysint) (1 ) , 
 
 where a, /8, 7 are rectangular vectors, and 
 
 W = Ty. 
 This surface evidently intersects the right cylinder 
 
 p = a{l3 cost + y sin t) + va, 
 in a helix (§§ 31 (m), 301) whose equation is 
 p = at + a{j3cost + y sin t). 
 
 These equations illustrate very well the remarks made in §§ 31 (l), 
 308, as to the curves or surfaces represented by a vector equation 
 according as it contains one or two scalar variables. 
 From (1) we have 
 
 dp = [a — u(^ sin t — y cos t)] dt + (/8 cos ^ + 7 sin t) du, 
 
 so that the normal at the extremity of p is 
 
 Ta (7 cost-^ sin t)-u W Uol. 
 
 Hence, as we proceed along a generating line of the surface, for 
 which t is constant, we see that the direction of the normal changes. 
 This, of course, proves that the surface is not developable. 
 
 « 
 
 311. Hence the criterion for a developable surface is that if it 
 be expressed by an equation of the form 
 
 p = <f)t + u-<Jrt, 
 
 where (f)t and yjrt are vector functions, we must have the direction 
 of the normal 
 
 V{<f)'t + uylr't}yjrt 
 independent of u. 
 
 This requires either FA|rfi|r'i = 0, 
 
 which would reduce the surface to a cylinder, all the generating 
 lines being parallel to each other ; or 
 
312.] GEOMETRY OF CURVES AND SORE ACES. 239 
 
 This is the criterion we seek, and it shews that we may write, for a 
 developable surface in general, the equation 
 
 p = 4)t+u<j)'t (1). 
 
 Evidently p = ^t 
 
 is a curve (generally tortuous) and 4>'t is a tangent vector. Hence 
 a developable surface is the locus of all tangent lines to a tortuous 
 curve. 
 
 Of course the tangent plane to the surface is the osculating 
 plane at the corresponding point of the curve; and this is indicated 
 by the fact that the normal to (1) is parallel to 
 
 V4>'t<^"t. (See § 300.) 
 
 To find the form of the section of the surface made by a normal 
 plane through a point in the curve. 
 
 The equation of the surface in the neighbourhood of the 
 extremity of p is approximately 
 
 -S7 = p + s/)' + -^ p" + &c. + a; {p' + sp" + &c.). 
 
 The part of in- — p which is parallel to p is 
 
 -p'S(^-p)p' = -p'|-(s + *)-p-(^ + f) + ...}; 
 
 therefore -ur - p = Ap' ^ U^ xs \ p" - {-^ + -^\ p'Vp'p'" + ... . 
 
 And, when A=(i, i.e. in the normal section, we have approximately 
 
 x = — s, 
 
 so that '!^ — p=—^p +^pvpp . 
 
 Hence the curve has an equation of the form 
 
 a semicubical parabola. 
 
 312. A Geodetic line is a curve drawn on a surface so that its 
 osculating plane at any point contains the normal to the surface. 
 Hence, if v be the normal at the extremity of p, p and p" the first 
 and second differentials of the vector of the geodetic, 
 
 8 . vp'p" = 0, 
 
 which may be easily transformed into 
 
 V.vdUp' = 0. 
 
240 QUATERNIONS. [S^S- 
 
 313. In the sphere Tp = a-vfe have 
 
 v\\p, 
 hence S . pp'p" = 0, 
 
 which shews of course that p is confined to a plane passing through 
 the origin, the centre of the sphere. 
 
 For a formal proof, we may proceed as follows — 
 The above equation is equivalent to the three 
 
 8dp = 0, Sep' = 0, 8dp" = 0, 
 from which we see at once that ^ is a constant vector, and 
 therefore the first expression, which includes the others, is the 
 complete integral. 
 
 Or we may proceed thus — 
 
 = -pS. pp'p" + p"S . pY = V. Vpp' Vpp" = V. Vpp'd Vpp, 
 whence by § 140 (2) we have at once 
 
 TJVpp = const. = 9 suppose, 
 which gives the same results as before. 
 
 314. In any cone, when the vertex is taken as origin, we 
 have, of course, 
 
 8vp = 0, 
 
 since p lies iu the tangent plane. But we have also 
 
 8vp' = 0. 
 Hence, by the general equation of § 312, eliminating v we get 
 = 8. pp'Vp'p" = 8pdUp' by § 140 (2). 
 
 Integrating G = 8pUp' -1 8dp Up' = 8p Up' + j Tdp. 
 
 The interpretation of this is, that the length of any arc of the 
 geodetic is equal to the projection of the side of the cone (drawn 
 to its extremity) upon the tangent to the geodetic. In other 
 words, when the cone is developed on a plane the geodetic becomes a 
 straight line. A similar result may easily be obtained for the 
 geodetic lines on any developable surface whatever. 
 
 31.5. To find the shortest line connecting two points on a given 
 surface. 
 
 Here I Tdp is to be a minimum, subject to the condition that 
 dp lies in the given surface. [We employ 8, though (in the 
 
3 1 7-] GEOMETRY OF CURVES ANB SURFACES. 241 
 
 notation we employ) it would naturally denote a vector, as the 
 symbol of variation.] 
 
 Now SJTdp=jSTdp = -j^^^ = -js.Udpd8p 
 
 = -[S.UdpBp] + J8. Spd Udp, 
 
 where the term in brackets vanishes at the limits, as the extreme 
 points are fixed, and therefore 8/3 = at each. 
 Hence our only conditions are 
 
 I S . Spdlldp = 0, and SvSp = 0, giving 
 
 V.vdUdp = 0, asin§ 312. 
 
 If the extremities of the curve are not given, but are to lie on 
 given curves, we must refer to the integrated portion of the 
 expression for the variation of the length of the arc. And its 
 form 
 
 S.UdpBp 
 
 shews that the shortest line cuts each of the given curves at right 
 angles. 
 
 316. The osculating plane of the curve 
 
 p=(fit 
 
 is 8.<^'tj>"t{tz-p) = (1), 
 
 and is, of course, the tangent plane to the surface 
 
 p=(f>t + ucj>'t (2). 
 
 Let us attempt the converse of the process we have, so far, 
 pursued, and endeavour to find (2) as the envelop of the variable 
 plane (1). ^ 
 
 Differentiating (1) with respect to t only, we have 
 
 By this equation, combined with (1), we have 
 
 ^-pllF.Fff F<^y"||f, 
 or tn- = p + ucj)' = + u(f)', 
 
 which is equation (2). 
 
 317. This leads us to the consideration of envelops generally, 
 and the process just employed may easily be extended to the 
 problem of finding the envelop of a series of surfaces whose 
 equation contains one scalar parameter. 
 
 T. Q. I. 16 
 
242 QUATERNIONS. [SiS. 
 
 When the given equation is a scalar one, the process of finding 
 the envelop is precisely the same as that employed in ordinary 
 Cartesian geometry, though the vs^ork is often shorter and simpler. 
 
 If the equation be given in the form 
 
 p = ylr(t, u, v), 
 
 where i/r is a vector function, t and u the scalar variables for any 
 one surface, v the scalar parameter, we have for a proximate surface 
 
 Hence at all points on the intersection of two successive surfaces 
 of the series we have 
 
 fJt+yjr'^Bu + '^'Jv-=0, 
 
 which is equivalent to the following scalar equation connecting the 
 quantities t, u and v ; 
 
 S . yfr'^ -»|r ^ ->|r'„ = 0. 
 
 This equation, along with 
 
 p=-^{t, u, v), 
 
 enables us to eliminate t, u, v, and the resulting scalar equation is 
 that of the required envelop. 
 
 318. As an example, let us find the envelop of the osculating 
 plane of a tortuous curve. Here the equation of the plane is 
 (§ 316) 
 
 S.{zT-p)(f>'t(f>"t=0, 
 
 or CT = <^i + x^'t + y<ji"t = yjr {x-, y, t), 
 
 if p = 4)t 
 
 be the equation of the curve. 
 
 Our condition is, by last section, 
 
 or S . (fi't 4>"t [4>'t + x<f>"t + y4/"t] = 0, 
 
 or yS.^'t4)"t^"'t = 0. 
 
 Now the second factor cannot vanish, unless the given curve 
 be plane, so that we must have 
 
 y = o, 
 
 and the envelop is ■st= <f)t + x^'t 
 
 the developable surface, of which the given curve is the edge of 
 regression, as in § 31(j. 
 
320.J GEOMETRY OE CURVES AND SURFACES. 243 
 
 319. When the equation contains two scalar parameters, its 
 differential coefficients with respect to them must vanish, and we 
 have thus three equations from which to eliminate two numerical 
 quantities. 
 
 A very common form in which these two scalar parameters 
 appear in quaternions is that of an arbitrary unit-vector. In this 
 case the problem may be thus stated ; — 
 
 Find the envelop of the surface whose scalar equation is 
 Fip,oL) = 0, 
 where a. is subject to the one condition 
 
 Ta=l. 
 Differentiating with respect to a alone, we have 
 Svda = 0, Sada = 0, 
 
 where i" is a known vector function of p and a. Since da. may have 
 any of an infinite number of values, these equations shew that 
 
 Vav = 0. 
 
 This is equivalent to two scalar conditions only, and these, in 
 addition to the two given scalar equations, enable us to eliminate a. 
 With the brief explanation we have given, and the examples 
 which follow, the student will easily see how to treat any other set 
 of data he may meet with in a question of envelops. 
 
 320. Fitid the envelop of a plane whose distance from the 
 origin is constant. 
 
 Here Sap = — c, 
 
 with the condition Ta = 1. 
 
 Hence, by last section, Vpa = 0, 
 and therefore p = ca, 
 
 or Tp = c, 
 
 the sphere of radius c, as was to be expected. 
 
 If we seek the envelop of those only of the planes which are 
 parallel to a given vector ^, we have the additional relation 
 
 8a^ = 0. 
 
 In this case the three differentiated equations are 
 
 8pda=0, Sa.da = 0, S^da = 0, 
 
 and they give S . a/3p = 0. 
 
 16—2 
 
244 QUATERNIONS. [32 1- 
 
 Hence a=U.0V^p, 
 
 and the envelop is TV^p = cT^. 
 
 the circular cylinder of radius c and axis coinciding with /3. 
 
 By putting Sa.^ = e, where e is a constant different from zero, 
 we pick out all the planes of the series which have a definite 
 inclination to /S, and of course get as their envelop a right cone. 
 
 321. The equation *Sf'ajO + 28.a^p = b 
 represents a parabolic cylinder, whose generating lines are parallel 
 to the vector aVal3. For the equation is of the second degree, and 
 is not altered by increasing p by the vector a;aFa/3; also the 
 surface cuts planes perpendicular to a in one line, and planes 
 perpendicular to Val3 in two parallel lines. Its form and position 
 of course depend upon the values of a, 13, and b. It is required to 
 find its envelop if ^ and b be constant, and a be subject to the one 
 
 scalar condition 
 
 Ta=l. 
 
 The process of § 319 gives, by inspection, 
 
 pSap + V^p = xa. 
 
 Operating hy S .a, we get 
 
 S^ap + 8 . a/3/3 = -x, 
 
 which gives S . a/3p = x + b. 
 
 But, by operating successively by 8 .V^p and by 8. p, we have 
 
 (r0py=co8.a0p, 
 
 and (p' — x) 8ap = 0. 
 
 Omitting, for the present, the factor 8ap, these three equations 
 
 give, by elimination of x and a, 
 
 (VM = p'(p' + b), 
 
 which is the equation of the envelop required. 
 
 This is evidently a surface of revolution of the fourth degree 
 whose axis is /S ; but, to get a clearer idea of its nature, put 
 
 and the equation becomes 
 
 (V^^y = c' + b^\ 
 
 which is obviously a surface of revolution of the second degree, 
 referred to its centre. Hence the required envelop is the reciprocal 
 of such a surface, in the sense that the rectangle under the lengths 
 
323.] GEOMETRY OF CURVES AND SURFACES. 245 
 
 of condirectional radii of the two is constant : i.e. it is the Electric 
 Image. 
 
 We have a curious particular case if the constants are so 
 related that 
 
 6 + ;S^ = 0, 
 
 for then the envelop breaks up into the two equal spheres, 
 touching each other at the origin, 
 
 while the corresponding surface of the second order becomes the 
 two parallel planes 
 
 S^^ = ± c\ 
 
 322. The particular solution above met with, viz. 
 
 Sap = 0, 
 
 limits the original problem, which now becomes one of finding the 
 envelop of a line instead of a surface. In fact this equation, taken 
 in conjunction with that of the parabolic cylinder, belongs to that 
 generating line of the cylinder which is the locus of the vertices 
 of the principal parabolic sections. 
 
 Our equations become 
 
 2S.aL^p = h, 
 Sap = 0, 
 Ta = \; 
 whence F/3/3 = xa ; 
 
 giving x = -S.aPp = - ^, 
 
 and thence TV^p = ^ ; 
 
 so that the envelop is a circular cylinder whose axis is /S. [It is 
 to be remarked that the equations above require that 
 
 Sa0 = 0, 
 so that the problem now solved is merely that of the envelop of a 
 parabolic cylinder which rotates about its focal line. This discussion 
 has been entered into merely for the sake of explaining a peculiarity 
 in a former result, because of course the present results can be 
 obtained immediately by an exceedingly simple process.] 
 
 323. The equation 8apS.a0p = a^, 
 with the condition Ta = 1, 
 
246 QUATERNIONS. 
 
 [324- 
 
 represents a series of hyperbolic cylinders. It is required to find 
 their envelop. 
 
 As before, we have 
 
 p8 . a^p + V/3pSap = xa, 
 which by operating hj S.a, S.p, and S.V^p, gives 
 
 2a' = -a;, 
 p''S . a/Sp = xSap, 
 {V^pfSap = x8.a^p. 
 Eliminating a and x we have, as the equation of the envelop, 
 
 p'iV^pY^^a'. 
 Comparing this with the equations 
 
 p' = - 2a^ 
 and {V^pY = -2a\ 
 
 which represent a sphere and one of its circumscribing cylinders, 
 we see that, if condirectional radii of the three surfaces be drawn 
 from the origin, that of the new surface is a geometric mean 
 between those of the two others. 
 
 324. Find the envelop of all spheres which touch one given line 
 and have their centres in another. 
 
 Let p = ^ + yy 
 
 be the line touched by all the spheres, and let xa be the vector of 
 the centre of any one of them, the equation is (by § 213, or § 214) 
 
 ry\p-xar = -{V.j{0-xoi)}\ 
 or, putting for simplicity, but without loss of generality, 
 
 Ty=l, Saj3 = 0, Sl3y=0, 
 so that /3 is the least vector distance between the given lines, 
 
 {p - xaf = {13- xaf + x'S'ay, 
 and, finally, p' — ^"- 2xSap = x^S'ay. 
 
 Hence, by § 317, - 2Sap = 2x8'ay. 
 
 [This gives no definite envelop, except the point p = ^, if 
 
 *Sa7 = 0, 
 i.e. if the line of centres is perpendicular to the line touched by 
 all the spheres.] 
 
 Eliminating x, we have for the equation of the envelop 
 S'ap + 8'ay{p''^^') = 0, 
 
326.J GEOMETRY OF CURVES AND SURFACES. 247 
 
 which denotes a surface of revolution of the second degree, whose 
 axis is a. 
 
 Since, from the form of the equation, Tp may have any 
 magnitude not less than T0, and since the section by the plane 
 
 Sap = 
 is a real circle, on the sphere 
 
 the surface is a hyperboloid of one sheet. 
 
 [It will be instructive to the student to find the signs of the 
 values of g^, g^, g^ as in § 177, and thence to prove the above 
 conclusion.] 
 
 325. As a final example of this kind let us find the envelop 
 of the hyperbolic cylinder 
 
 8apSl3p - c = 0, 
 where the vectors a and /3 are subject to the conditions 
 
 Ta = Tl3 = l, 
 Say = 0, S/3S = 0, 
 7 and B being given vectors. 
 
 [It will be easily seen that two of the six scalars involved 
 in a, still remain as variable parameters.] 
 
 We have Sada = 0, ;Sf7C^a = 0, 
 
 so that da = ooVay. 
 
 Similarly d/S = yV/3B. 
 
 But, by the equation of the cylinders, 
 
 SapSpdl3 + SpdaS^p = 0, 
 or ySapS . 0Sp + xS . aypS^p = 0. 
 
 Now by the nature of the given equation, neither Sap nor S^p can 
 vanish, so that the independence of da and d^ requires 
 
 S.ayp^O, S.l3Bp = 0. 
 Hence a= U .j Vyp, = U.S VSp, 
 
 and the envelop is T . Vyp VBp — cTyh = 0, 
 
 a surface of the fourth degree, which may be constructed by laying 
 off mean proportionals between the lengths of condirectional radii 
 of two equal right cylinders whose axes meet in the origin. 
 
 326. We may now easily see the truth of the following 
 general statement. 
 
248 QUATERNIONS. [326. 
 
 Suppose the given equation of the series of surfaces, whose 
 envelop is required, to contain m vector, and n scalar, parameters ; 
 and that these are subject to p vector, and q scalar, conditions. 
 
 In all there are Sm + n scalar parameters, subject to Sp + q 
 scalar conditions. 
 
 That there may be an envelop we must therefore in general 
 have 
 
 (3m + n)- (Sp + q) = l, or = 2. 
 
 In the former case the enveloping surface is given as the locus of 
 a series of curves, in the latter of a series of points. 
 
 Differentiation of the equations gives us 3p + q + 1 equations, 
 linear and homogeneous in the 3m + n differentials of the scalar 
 parameters, so that by the elimination of these we have one final 
 scalar equation in the first case, two in the second ; and thus in 
 each case we have just equations enough to eliminate all the 
 arbitrary parameters. 
 
 Sometimes a very simple consideration renders laborious cal- 
 culation unnecessary. Thus a rectangular system turns about the 
 centre of an ellipsoid. Find the envelop of the plane which passes 
 through the three points of intersection. 
 
 If a, /8, 7 be the rectangular unit-system, the points of 
 intersection with 
 
 Sp<f>p = — 1 
 
 are at the extremities of 
 
 - " ^ 7 
 
 And if these lie in the plane 
 
 Sep = ^ (1), 
 
 we must have e = — 1,aj —Sa.<l)a (2). 
 
 It would be troublesome to work out the envelop of (1), with 
 (2) and the conditions of a rectangular unit-system as the data, 
 but we may proceed as follows. 
 
 The length of the perpendicular from the centre on the 
 plane (1) is 
 
 J - (/S'a0a + 8^(j)^ + Sy4,j) ' 
 
 a constant, by § 185. Hence the envelop is a sphere of which this 
 is the radius ; and it has the same property for all the ellipsoids 
 
328.] GEOMETRY OP CURVES AND SaRPACES. 249 
 
 which, having their axes in the same lines as the first, intersect it 
 at the point on 
 
 i+j + k 
 
 We may obtain the result in another way. By § 281 the sum 
 of the reciprocals of the squares of three rectangular central vectors 
 of an ellipsoid is constant ; while it is easily shewn (see Ex. 20 to 
 Chap. VII.) that the same sum with regard to a plane is the 
 reciprocal of the square of its distance from the origin. 
 
 327. To find the locus of the foot of the perpendicular drawn 
 from the origin to a tangent plane to any surface. 
 
 If Svdp = 
 
 be the differentiated equation of the surface, the equation of the 
 
 tangent plane is 
 
 8{^-p)v = 0. 
 
 We may introduce the condition 
 
 Svp = — 1, 
 which in general alters the tensor of v, so that — v~^ becomes the 
 required vector perpendicular, as it satisfies the equation 
 
 S-STV = — 1. 
 It remains that we eliminate p between the equation of the 
 given surface, and the vector equation 
 
 -ST = — v~^. 
 The result is the scalar equation (in ■ct) required. 
 For example, if the given surface be the ellipsoid 
 8p(l>p = - 1, 
 we have ot"* = — v = — (j)p, 
 
 so that the required equation is 
 
 S'uy (f) 'ST = — 1, 
 or >S-sj-(^~'i!r = — OT , 
 
 which is Fresnel's Surface of Elasticity. (§ 278.) 
 
 It is well to remark that this equation is derived from that of 
 
 the reciprocal ellipsoid 
 
 Spcf>-'p = - 1 
 
 by putting ot"' for p. 
 
 328. To find the reciprocal of a given surface with respect to 
 the unit sphere whose centre is the origin. 
 
 With the condition Spv = — 1, 
 
250 QUATERNIONS. [S^Q- 
 
 of last section, we see that v is the vector of the pole of the 
 tangent plane 
 
 Hence we must put zj = v, 
 
 and eliminate p by the help of the equation of the given surface. 
 Take the ellipsoid of last section, and we have 
 
 ■m = <pp, 
 so that the reciprocal surface is represented by 
 
 ,Sfl!7(/)-V=-l. 
 
 It is obvious that the former ellipsoid can be produced from 
 this by a second application of the process. 
 
 And the property is general, for 
 Spv = -l 
 gives, by differentiation, and attention to the condition 
 
 Svdp-^0 
 the new relation Spdv = 0, 
 
 so that p and v are corresponding vectors of the two surfaces : 
 either being that of the pole of a tangent plane drawn at the 
 extremity of the other. 
 
 329. If the given surface be a cone with its vertex at the 
 origin, we have a peculiar case. For here every tangent plane 
 passes through the origin, and therefore the required locus is 
 wholly at an infinite distance. The difficulty consists in Spv 
 becoming in this case a numerical multiple of the quantity which 
 is equated to zero in the equation of the cone, so that of course 
 we cannot put as above 
 
 8pv = -l. 
 
 330. The properties of the normal vector v enable us to write 
 the partial differential equations of families of surfaces in a very 
 simple form. 
 
 Thus the distinguishing property of Cylinders is that all their 
 generating lines are parallel. Hence all positions of v must be 
 parallel to a given plane — or 
 
 Sav = 0, 
 
 which is the quaternion form of the well-known equation 
 
 , dF dF dF , 
 
 i T — h m -, - -I- n ,-=(). 
 dx dy dz 
 
33 2. J GEOMETRY OF CURVES AND SURFACES. 251 
 
 To integrate it, remember that we have always 
 
 Svdp = 0, 
 
 and that as v is perpendicular to a it may be expressed in terms 
 of any two vectors, /3 and 7, each perpendicular to a. 
 
 Hence v = xj3 + yy, 
 
 and osS^dp + ySydp = 0. 
 
 This shews that S0p and Syp are together constant or together 
 variable, so that 
 
 S0p=f(Syp), 
 
 where /is any scalar function whatever. 
 
 331. In Surfaces of RevUhition the normal intersects the axis. 
 Hence, taking the origin in the axis a, we have 
 
 S.apv = 0, 
 
 or v = !jca + yp. 
 
 Hence xSadp + ySpdp = 0, 
 
 whence the integral Tp =f(Sap). 
 
 The more common form, which is easily derived from that just 
 written, is 
 
 TVap = F{Sap). 
 
 In Cones we have Svp = 0, 
 
 and therefore 
 
 Svdp = S.v {Tpd Up + UpdTp) = TpSvd Up. 
 Hence Svd Up = 0, 
 
 so that V must be a function of Up, and therefore the integral is 
 
 f{Up) = 0, 
 
 which simply expi-esses the fact that the equation does not involve 
 the tensor of p, i.e. that in Cartesian coordinates it is homogeneous. 
 
 332. If equal lengths he laid off on the normals drawn to any 
 surface, the new surface formed by their extremities is normal to the 
 same lines. 
 
 For we have zj = p + aUv, 
 
 and Svd'ST = Svdp + aSvd Uv = 0, 
 
 which proves the proposition. 
 Take, for example, the surface 
 
 Spj>p = - 1 ; 
 
252 QUATERNIONS. [333- 
 
 the above equation becomes 
 
 ad>p 
 
 so that p = (7^ + 1 j OT, 
 
 a,nd the equation of the new surface is to be found by eliminating 
 
 fjf-r- (written x) between the equations 
 1 (pp .p' . 
 
 - 1 = ,Sf . (a;^ + 1)-^^ (a;(f> + 1)"^, 
 
 and -^, = 8.<p{x<p + l)-'zT(l>{x4> + iy''S7. 
 
 333. It appears from last sStsWon that if one orthogonal 
 surface can be drawn cutting a given system of straight lines, an 
 indefinitely great number may be drawn : and that the portions of 
 these lines intercepted between any two selected surfaces of the 
 series are all equal. 
 
 Let p = a- +XT, 
 
 where a and t are vector functions of p, and x is any scalar, be 
 the general equation of a system of lines : we have 
 STdp = = S{p-cr)dp 
 
 as the differentiated equation of the series of orthogonal surfaces, 
 if it exist. Hence the following problem. _^ 
 
 334. It is required to find the criterion of integrahility of the 
 equation 
 
 Svdp = (1) 
 
 as the complete differential of the equation of a series of surfaces. 
 
 Hamilton has given {Elements, p. 702) an extremely elegant 
 solution of this problem, by means of the properties of linear and 
 vector functions. We adopt a different and somewhat less rapid 
 process, on account of some results it offers which will be useful to 
 us later ; and also because it will shew the student the connection 
 of our methods with those of ordinary differential equations. 
 
 If we assume Fp = G 
 
 to be the integral, we have by § 144, 
 
 SdpVF=0. 
 
 Comparing with the given equation, (1), we see that the latter 
 represents a series of surfaces if v, or a scalar multiple of it, can be 
 expressed as VF. 
 
335-] GEOMETRY OF CURVES AND SURFACES. 253 
 
 If v = VF, 
 
 and the last-written quantities are necessarily scalars, so that the 
 only requisite condition of the integrability of (1) is 
 
 VVu = (2). 
 
 If V do not satisfy this criterion, it may when multiplied by a 
 scalar. Hence the farther condition 
 
 7V (luv) = 0, 
 
 which may be written 
 
 VvVw-wWv^O (3). 
 
 This requires that 
 
 SvVv = (4). 
 
 If then (2) be not satisfied, we must try (4).^ If (4) be satisfied w 
 will be found from (3) ; and in either case (1) is at once integrable. 
 
 [If we put dv = c})dp, 
 
 where ^ is a linear and vector function, not necessarily self- 
 conjugate, we have 
 
 by § 185. Thus, if (f> be self-conjugate, 6 = 0, and the criterion (2) 
 is satisfied. If <j) be not self-conjugate we have by (4) for the 
 criterion 
 
 >Sfei' = 0. 
 
 These results accord with Hamilton's, lately referred to, but the 
 mode of obtaining them is quite different from his.] 
 
 335. As a simple example let us first take lines diverging 
 from a point. Here v \\ p, and we see that ii v = p 
 
 Vv = -3, 
 so that (2) is satisfied. And the equation is 
 
 Spdp = 0, 
 whose integral Tp'' = C 
 
 gives a series of concentric spheres. 
 
 Lines perpendicular to, and intersecting, a fixed line. 
 If a be the fixed line, /3 any of the others, we have 
 S.a^p = 0, Sa^ = 0, 8^dp=-0. 
 
254 QUATERNIONS. [335- 
 
 Here v\\a.Vap, 
 
 and therefore equal to it, because (2) is satisfied. 
 
 Hence S .dpa Vap = 0, 
 
 or S.VapVadp = 0, 
 
 whose integi-al is the equation of a series of right cylinders 
 
 r'VoLp = c. 
 
 To find the orthogonal trajectories of a series of circles whose 
 centres are in, and their planes perpendicular to, a given line. 
 
 Let a be a unit-vector in the direction of the line, then one of 
 the circles has the equations 
 
 Tp = C] 
 Sap = G'\' 
 
 where C and C" are any constant scalars whatever. 
 Hence, for the required surfaces 
 
 v\\d,p\\Vap, 
 where d^p is an element of one of the circles, v the normal to the 
 orthogonal surface. Now let dp be an element of a tangent to 
 the orthogonal surface, and we have 
 
 Svdp = 8 . apdp = 0. 
 
 This shews that dp is in the same plane as a and p, i.e. that the 
 orthogonal surfaces are planes passing through the common ax 
 [To integrate the equation 
 
 S . apdp = 
 evidently requires, by § 334, the introduction of a factor. Foi 
 V.VVap = -aSVp + SaV.p (§90,(1)) 
 = 2a, 
 so that the first criterion is not satisfied. But 
 
 8. VapV.Vrap=2S.aVap=^0, 
 so that the second criterion holds. It gives, by (3) of § 334, 
 
 - V. VapVw + 2wa = 0, 
 or pSaVw — aSpVw + Iwa — 0. 
 
 m, ^ ■ 8aVw = 
 
 That is cr t7 o 
 
 These equations are satisfied by 
 
 1 
 
 Vap 
 
336.] GEOMETRY OP CURVES AND SURFACES. 255 
 
 But a simpler mode of integration is easily seen. Our equation 
 may be written 
 
 P Up ^13' 
 
 which is immediately integrable, l3 being an arbitrary but constant 
 vector. 
 
 As we have not introduced into this work the logarithms of 
 versors, nor the corresponding angles of quaternions, we must refer 
 to Hamilton's Elements for a further development of this point.] 
 
 336. As another example, let us find series of surfaces which, 
 together, divide space into cubes. 
 
 If p be the vector of one series which has the required property, 
 (T that of a second, it is clear that (ii being a scalar) 
 
 da = uq'^dpq (1), 
 
 where u and q are functions of p. For, to values of dp belonging 
 to edges of one cube correspond values of da belonging to edges of 
 another. Operate by »Sf . a, where a is any constant vector, then 
 
 Sada = uS . qaq'^dp. 
 
 As the left-hand member is a complete differential, we have by 
 §334 
 
 FV {uqaq-') = 0. 
 
 This is easily put in the form {Chap. IV., Ex. (4), and § 140 (8)} 
 
 V. — qaq-' = - 2qaq-' S . Vqq'' + 2S (qaq-'V) q.q'\. .(2). 
 
 Multiply by qa.q'^, and add together three equations of the resulting 
 form, in which the values of a form a rectangular unit system. 
 Then 
 
 + 2 — = + 6S .Vfl g-' - 2Va q'\ 
 u 
 
 This shews that 
 
 S.Vqq-' = 0. 
 
 Take account of this result in (2), and put dp for qaq~\ which 
 may be any vector. Thus 
 
 V.dp^ = 2dq.q-^ = 2V^ (3). 
 
 From this we see at once that 
 
 q = a Up, 
 
256 QUATERNIONS. [33^- 
 
 where a is any constant veisor. Then (3) gives 
 
 
 
 
 u 
 
 2 Up 
 - Tp 
 
 du 
 , or — = 
 
 u 
 
 28pdp 
 Tp' 
 
 so 
 
 that 
 Thus, 
 
 from 
 
 (1), we 
 
 u = 
 have 
 
 C 
 ' Tp'- 
 
 P 
 
 
 This gives the Electric Image transformation, with any subsequent 
 rotation, followed (or, as is easily seen, preceded) by a translation. 
 Hence the only series of surfaces which satisfy the question, are 
 mutually perpendicular planes ; and their images, which are series 
 of spheres, passing through a common point and having their 
 centres on three rectangular lines passing through that point. 
 [For another mode of solution see Proc. R. S. E., Dec. 1877.] 
 
 In some respects analogous to this is the celebrated physical 
 problem of finding series of Orthogonal Isothermal Surfaces. We 
 give a slight sketch of it here. 
 
 If three such series of surfaces be denoted by their temperatures 
 
 thus : — 
 
 F =T F = T F =T 
 
 1 1' 2 2' S -^3' 
 
 the conditions of orthogonality are fully expressed by putting for 
 the respective values of the flux of heat in each series 
 
 VF, = u^qocq-\ V . F, = u.^q^q-\ VF^ = u^qyq-\ 
 
 where a, ^, y form a constant rectangular unit vector system. 
 But the isothermal conditions are simply 
 
 V'F^ = VF, = V'F^ = 0. 
 
 Hence we have three simultaneous equations, of which the 
 first is 
 
 V . u^qaq^ = {a). 
 
 [In the previous problem a might be any vector whatever, the 
 values of u were equal, and the vector part only of the left-hand 
 member was equated to zero. These conditions led to an unique 
 form of solution. Nothing of the kind is to be expected here.] 
 
 From equation (a) we have at once 
 
 S.oLVv^ + 28.0L'^qq-'=-0\ 
 
 V.aVv^-2oi'Syqq-' + 2S{a.'V)qq-' = 0^ ^^^ 
 
 where v^ has been put for logu^, and a' for qaq'^. 
 
337-] GEOMETRY OF CURVES AND SURFACES. 257 
 
 From the group of six equations, of which (6) gives two, we 
 have 
 
 with three of the type 
 
 8.a:8{aV)qq-' = 0. 
 We also obtain without difficulty 
 
 tVv + 4V^g"' = 0, 
 which may be put in the form 
 
 But when we attempt to find the value of dq we are led to 
 expressions such as 
 
 27. a';Sf (a'dp) Vu, + 2dqq-' = 0, 
 which, in consequence of the three different values of v, are 
 comparatively unmanageable. (See Ex. 24 at end of Chapter.) 
 They become, however, comparatively simple when one of the 
 three families is assumed. The student will find it useful to work 
 out the problem when F^ represents a series of parallel planes, so 
 that the others are cylinders ; or when F^ represents planes passing 
 through a line, the others being surfaces of revolution ; &c. 
 
 337. To find the orthogonal trajectories of a given series of 
 surfaces. 
 
 If the equation Fp = G 
 
 give 8vdp'= 0, 
 
 the equation of the orthogonal curves is 
 
 Vvdp =0. 
 
 This is equivalent to two scalar differential equations (§ 210), 
 which, when the problem is possible, belong to surfaces on each of 
 which the required lines lie. The finding of the requisite criterion 
 we leave to the student. [He has only to operate on the last- 
 written equation by 8. a, where a is any constant vector ; and, 
 bearing this italicized word in mind, proceed as in § 334.] 
 
 Let the surfaces be concentric spheres. 
 
 Here p' = G, 
 
 and therefore Vpdp = 0. 
 
 Hence Tp'd Up = -Up Vpdp = 0, 
 
 and the integral is Up = constant, 
 
 denoting straight lines through the origin. 
 
 T. Q. I. 17 
 
258 QUATERNIONS. [SS^- 
 
 Let the surfaces be spheres touching each other at a common 
 point. The equation is (§ 235) 
 
 Sap-' = C, 
 whence V. papdp = 0. 
 
 The integrals may be written 
 
 &.oi^p = 0, p' + hTVap^O, 
 the first (^ being any vector) is a plane through the common 
 diameter ; the second represents a series of rings or tores (§ 340) 
 formed by the revolution, about a, of circles touching that line 
 at the point common to the spheres. 
 
 Let the surfaces be similar, similarly situated, and concentric, 
 surfaces of the second degree. 
 
 Here Spxp = G, 
 
 therefore ^XP^P = ^■ 
 
 But, by § 307, the integral of this equation is 
 
 p = e*^e 
 
 where <j> and % are related to each other, as in § 307; and e is any 
 constant vector. 
 
 338. To integrate the linear partial differential equation of a 
 family of surfaces. 
 
 The equation (see § 330) 
 
 ax dy dz 
 may be put in the very simple form 
 
 <S(aV)M = (1), 
 
 if we write (T = iP +jQ + kB, 
 
 — . d . d , d 
 
 and ^ ='''j- +J J- +'<' J- ■ 
 
 doc •' dy dz 
 
 [From this we see that the meaning of the differential 
 equation is that at every point of the surface 
 
 u = const. 
 
 the corresponding vector, a, is a tangent line. Thus we have a 
 suggestion of the ordinary method of solving such equations.] 
 
 (1) gives, at once, • Vu = mVdar, 
 
338.] GEOMETRY OF CURVES AND SURFACES. 259 
 
 where m is a scalar and a vector (in whose tensor m might have 
 been included, but is kept separate for a special purpose). Hence 
 
 du = — 8(dpV)u 
 
 = — mS . dadp 
 
 = -8edT, 
 
 if we put dT = m Vcrdp 
 
 so that m is an integrating factor of V . adp. If a value of m can 
 be found, it is obvious, from the form of the above equation, that 
 6 must be a function of t alone ; and the integral is therefore 
 
 u = F(t) = const. 
 
 where F is an arbitrary scalar function. 
 
 Thus the differential equation of Cylinders is 
 
 S(aV)u = 0, 
 
 where a is a constant vector. Here m = 1, and 
 
 u = F{ Vap) = const. 
 
 That of Cones referred to the vertex is 
 
 S{pS7)u = 0. 
 
 Here the expression to be made integrable is 
 
 Vpdp. 
 
 But Hamilton long ago shewed that (§ 140 (2)) 
 
 dUp _ y-dp _ Vpdp 
 
 -u^- j-jTpr' 
 
 which indicates the value of m, and gives 
 
 u = F (Up) = const. 
 
 It is obvious that the above is only one of a great number of 
 different processes which may be applied to integrate the differen- 
 tial equation. It is quite easy, for instance, to pass from it to the 
 assumption of a vector integrating factor instead of the scalar m, 
 and to derive the usual criterion of integrability. There is no 
 difficulty in modifying the process to suit the case when the right- 
 hand member is a multiple of u. In fact it seems to throw a very 
 clear light upon the whole subject of the integration of partial 
 differential equations. If, instead of 8 (o-V), we employ other 
 operators as 8 (o-V) 8 (tV), 8 . aVrV, &c. (where V may or may not 
 operate on w alone), we can pass to linear partial differential 
 
 17—2 
 
260 QUATERNIONS. [339- 
 
 equations of the second and higher orders. Similar theorems ca.n 
 be obtained from vector operators, as F(crV)*- 
 
 339. Find the general equation of surfaces described by a line 
 which always meets, at right angles, a fixed line. 
 
 If a be the fixed line, yS and 7 forming with it a rectangular 
 unit system, then 
 
 p — xa + y{^ + zy), 
 
 where y may have all values, but x and z are mutually dependent, 
 is one form of the equation. 
 
 Another, expressing the arbitrary relation between x and z, is 
 
 But we may also write 
 
 p = ciF{x) + ya'^, 
 
 as it obviously expresses the same conditions. 
 
 The simplest case is when F{x) = hx. The surface is one which 
 cuts, in a right helix, every cylinder which has a. for its axis. 
 
 340. The centre of a sphere moves in a given circle, find the 
 equation of the ring described. 
 
 Let a be the unit-vector axis of the circle, its centre the origin, 
 r its radius, a that of the sphere. 
 
 Then (^p-^y^^a' 
 
 is the equation of the sphere in any position, where 
 
 -Say8 = 0, T^ = r. 
 
 These give (§ 326) S . a^p = 0, and /8 must now be eliminated. 
 The immediate result is that 
 
 l3 = raUVap, 
 giving ip' -r' + a''Y = 4<r'T' Yap, 
 
 = ir'{-p'-8'ap), 
 which is the required equation. It may easily be changed to 
 
 {p'-a' + ry = -4>ay-4>r'8'ap (1), 
 
 and in this form it enables us to give a very simple proof of the 
 singular property of the ring (or tore) discovered by Villarceau. 
 
 For the planes Sp (a ± ^^^-^j = 0, 
 
 * Tait, Froc. B. S. E., 1869-70. 
 
34I-J GEOMETRY OP CURVES AND SURFACES. 261 
 
 which together are represented by 
 
 r' (r' - a") S'ap - a'S'^p = 0, 
 
 evidently pass through the origin and touch (and cut) the ring. 
 
 The latter equation may be written 
 
 r'8'ap - a" {S'ap + S'p U^) = 0, 
 
 or r'8'ap + a'(p' + 8\oipU^) = (2). 
 
 The plane intersections of (1) and (2) lie obviously on the new 
 
 surface 
 
 (p'-a' + ry = 4^a'8\apU^, 
 
 which consists of two spheres of radius r, as we see by writing its 
 separate factors in the form 
 
 {p±aoLUfiy + r'' = 0. 
 
 341. It may be instructive to work out this problem from a 
 different point of view, especially as it affords excellent practice in 
 transformations. 
 
 A circle revolves about an axis passing within it, the perpen- 
 dicular from the centre on the axis lying in the plane of the circle: 
 shew that, for a certain position of the axis, the same solid may he 
 traced out by a circle revolving about an external axis in its own 
 plane. 
 
 Let a = Jb'' + c' be the radius of the circle, i the vector axis of 
 rotation, — ca (where To. = 1) the vector perpendicular from the 
 centre on the axis i, and let the vector 
 
 bi + cia 
 
 be perpendicular to the plane of the circle. 
 The equations of the circle are 
 
 (p-cay + b' + c' = 
 
 s(i + ^ia.jp = 0'' 
 Also - p" = 8Sp + 8'ap-\-^. iap, 
 
 = S'ip + 8'ap + Ks'ip 
 
 G 
 
 by the second of the equations of the circle. But, by the first, 
 
 (p' + by = 4,c'S'ap = - 4 {cy + a'SHp), 
 which is easily transformed into 
 
 {p'-by = -ia'{p' + 8Sp), 
 or p^-W^-'iaTVip. 
 
262 QUATERNIONS. [342- 
 
 If we put this in the forms 
 
 and {p-a^y + c' = 0, 
 
 where /3 is a unit-vector perpendicular to i and in the plane of i 
 and p, we see at once that the surface will be traced out by, a 
 circle of radius c, revolving about i, an axis in its own plane, 
 distant a from its centre. 
 
 [This problem is not well adapted to shew the gain in brevity 
 and distinctness which generally attends the use of quaternions ; 
 as, from its very nature, it hints at the adoption of rectangular 
 axes and scalar equations for its treatment, so that the solution 
 we have given is but little different from an ordinary Cartesian 
 one.J 
 
 342. A surface is generated by a straight line which intersects 
 two fixed straight lines : find the general equation. 
 
 If the given lines intersect, there is no surface but the plane 
 containing them. 
 
 Let then their equations be, 
 
 p = a + x^, p = ttj + ajj/Sj. 
 Hence every point of the surface satisfies the condition, § 30, 
 
 p = y(a + x^) + (l-y)(a, + x^^;) (1). 
 
 Obviously y may have any value whatever : so that to specify a 
 particular surface we must have a relation between x and aj^. By 
 the help of this, x^ may be eliminated from (1), which then takes 
 the usual form of the equation of a surface 
 
 p = (f>{x, y). 
 Or we may operate on (1) by V. (a+x^ — a^ — x^^^, so that we 
 get a vector equation equivalent to two scalar equations (§§ 98, 123), 
 and not containing y. From this x and a;, may easily be foiind in 
 terms of p, and the general equation of the possible surfaces may 
 be written 
 
 /{«=, 00,) = 0, 
 where / is an arbitrary scalar function, and the values of x and a;, 
 are expressed in terms of p. 
 
 This process is obviously applicable if we have, instead of two 
 straight lines, any two given curves through which the line must 
 pass; and even when the tracing line is itself a given curve, 
 situated in a given manner. But an example or two will make the 
 whole process clear. 
 
344-J GEOMETRY OF CURVES AND SURFACES. 263 
 
 343. Suppose the moveable line to be restricted by the condition 
 that it is always parallel to a fixed plane. 
 
 Then, in addition to (l),-we have the condition 
 Sj («! + «j/Qj — a — x^) = 0, 
 
 y being a vector perpendicular to the fixed plane. 
 
 We lose no generality by assuming a and a^, which are any 
 vectors drawn from the origin to the fixed lines, to be each per- 
 pendicular to 7 ; for, if for instance we could not assume Sya = 0, 
 it would follow that S'y^=0,sind the required surface would either 
 be impossible, or would be a plane, cases which we need not con- 
 sider. Hence 
 
 Eliminating w^, by the help of this equation, from (1) of last section, 
 we have 
 
 p=y(a + x^) + (l-y)(a, + x^^^y 
 
 Operating by any three non-coplanar vectors and with the charac- 
 teristic 8, we obtain three equations from which to eliminate x and 
 y. Operating by /S . 7 we find 
 
 8yp = xS^y. 
 
 Eliminating x by means of this, we have finally 
 
 which appears to be of the third degree. It is really, however, only 
 of the second degree : since, in consequence of our assumptions, 
 we have 
 
 FaMj II 7, 
 
 and therefore Syp is a spurious factor of the left-hand side. 
 
 344. Let the fixed lines be perpendicular to each other, and let 
 the moveable line pass through the circumference of a circle, whose 
 centre is in the common perpendicular, and whose plane bisects that 
 line at right angles. 
 
 Here the equations of the fixed lines may be written 
 p = a + x^, p = -a + x^y, 
 where a, ^, 7 form a rectangular system, and we may assume the 
 two latter to be unit-vectors. 
 
 The circle has the equations 
 
 p'=--a\ Scip = 0. 
 
264 QUATERNIONS. [345- 
 
 Equation (1) of § 342 becomes 
 
 p = y (a + «/3) + (1 - 2/) (- « + a;.7)- 
 Hence Sa'^ p = y-0--y) = 0, or y = ^. 
 
 Also p' = -a' = {2y - Vf a" - x'f - x^ (1 - y)\ 
 
 ■or 4a'' = (a;'' + 0, 
 
 so that if we now suppose the tensors of ^ and 7 to be each la, we 
 may put x = cos Q, x^ = sin B, from which 
 
 p = (2y-l)a + 2//3cos^+(l-y)7sin^; 
 
 and finally (iT^r^. + ^^-Z^ pf = ^'^^^ 
 
 For this specially simple case the solution is not better than the 
 ordinary Cartesian one ; but the student will easily see that we 
 may by very slight changes adapt the above to data far less sym- 
 metrical than those from which we started. Suppose, for instance, 
 /S and 7 not to be at right angles to one another ; and suppose the 
 plane of the circle not to be parallel to their plane, &c., &c. But 
 farther, operate on every line in space by the linear and vector 
 function 0, and we distort the circle into an ellipse, the straight 
 lines remaining straight. If we choose a form of whose principal 
 axes are parallel to a, /8, 7, the data will remain symmetrical, but 
 not unless. This subject will be considered again in the next 
 Chapter. 
 
 345. To find the curvature of a normal section of a central 
 surface of the second degree. 
 
 In this, and the few similar investigations which follow, it will 
 be simpler to employ infinitesimals than differentials ; though for 
 a thorough treatment of the subject the latter method, as it may 
 be seen in Hamilton's Elements, is preferable. 
 
 We have, of course, ^P'l'P = ~ !> 
 
 and, if jO + 8/3 be also a vector of the surface, we have rigorously, 
 whatever he the tensor of Sp, 
 
 S(p + Sp)cj>(p + Sp) = -L 
 
 Hence 2SBp^p + 8Bp<})8p = (1). 
 
 Now (f)p is normal to the tangent plane at the extremity of p, 
 so that if t denote the distance of the point p + Bp from that plane 
 
 t = -8BpU<l>p, 
 
346.] GEOMETRY OE CURVES AND SURFACES. 265 
 
 and (1) may therefore be written 
 
 2tT<])p - rSpS . USp4>UBp = 0. 
 But the curvature of the section is evidently 
 
 ^T'Bp- 
 or, by the last equation, 
 
 ~^8.uspci>mp. 
 
 In the limit, Bp is a vector in the tangent plane ; let ot be the 
 vector semidiameter of the surface which is parallel to it, and the 
 equation of the surface gives 
 
 so that the curvature of the normal section, at the point p, in the 
 direction of ■ar, is 
 
 1 
 
 T<ppr^' 
 
 directly as the perpendicular from the centre on the tangent plane, 
 and inversely as the square of the semidiameter parallel to the 
 tangent line, a well-known theorem. 
 
 346. By the help of the known properties of the central 
 section parallel to the tangent plane, this theorem gives us all 
 the ordinary properties of the directions of maximum and mini- 
 mum curvature, their being at right angles to each other, the 
 curvature in any normal section in terms of the chief curvatures 
 and the inclination to their planes, &c., &c., without farther 
 analysis. And when, in a future section, we shew how to find 
 an osculating surface of the second degree at any point of a given 
 surface, the same properties will be at once established for surfaces 
 in general. Meanwhile we may prove another curious property 
 of the surfaces of the second degree, which similar reasoning 
 extends to all surfaces. 
 
 The equation of the normal at the point p + Bp on the surface 
 treated in last section is 
 
 ■sr = p + Bp + xcp (p + Bp) (1). 
 
 This intersects the normal at p if (§§ 216, 223) 
 
 S . Bp(f)p(pBp = 0, 
 
 that is, by the result of § 290, if Bp be parallel to the maximum or 
 
266 QUATERNIONS. [34^- 
 
 minimum diameter of the central section parallel to the tangent 
 plane. 
 
 Let o-j and cr^ be those diameters, then we may write in general 
 
 where p and q are scalars, infinitely small. 
 
 If we draw through a point P in the normal at p a line parallel 
 to o-j, we may write its equation 
 
 sj = p + a(j)p + ycr^. 
 
 The proximate normal (1) passes this line at a distance (see § 216) 
 
 S.{a<l>p-^p)UYa,4>{p + hp), 
 
 or, neglecting terms of the second order, 
 
 The first term in the bracket vanishes because o-j is a principal 
 vector of the section parallel to the tangent plane, and thus the 
 expression becomes 
 
 Hence, if we take a = Ta^, the distance of the normal from the 
 new line is of the second order only. This makes the distance of 
 P from the point of contact T(f)pT(r^, i.e. the principal radius of 
 curvature along the tangent line parallel to a^. That is, the group 
 of normals drawn near a point of a central surface of the second 
 degree pass ultimately through two lines each parallel to the tangent 
 to one principal section, and drawn through the centre of curvature 
 of the other. The student may form a notion of the nature of this 
 proposition by considering a small square plate, with normals 
 drawn at every point, to be slightly bent, but by different amounts, 
 in planes perpendicular to its edges. The first bending will make 
 all the normals pass through the axis of the cylinder of which the 
 plate now forms part; the second bending will not sensibly disturb 
 this arrangement, except by lengthening or shortening the line in 
 which the normals meet, but it will make them meet also in the 
 axis of the new cylinder, at right angles to the first. A small 
 pencil of light, with its focal lines, presents this appearance, due 
 to the fact that a series of rays originally normal to a surface 
 remain normals to a surface after any number of reflections and 
 (ordinary) refractions. (See § 332.) 
 
348.] GEOMETRY OP CURVES AND SURFACES. 267 
 
 347. To extend these theorems to surfaces in general, it is 
 only necessary, as Hamilton has shewn, to prove that if we write 
 
 dv = cfidp, 
 
 ^ is a self-conjugate function ; and then the properties of <p, as ex- 
 plained in preceding Chapters, are applicable to the question. 
 
 As the reader will easily see, this is merely another form of the 
 investigation contained in § 334. But it is again cited here to 
 shew what a number of very simple demonstrations may be given 
 of almost all quaternion theorems. 
 
 The vector v is defined by an equation of the form 
 
 dfp = Svdp, 
 
 where / is a scalar function. Operating on this by another inde- 
 pendent symbol of differentiation, S, we have 
 
 Mfp = SSvdp + SvMp. 
 In the same way we have 
 
 dBfp = SdvBp + SvdBp. 
 
 But, as d and S are independent, the left-hand members of these 
 equations, as well as the second terms on the right (if these exist 
 at all), are equal, so that we have 
 
 SdvSp = SSvdp. 
 This becomes, putting dv = ^dp, 
 
 and therefore hv = <^hp, 
 
 SSptpdp = 8dp<f>Bp, 
 which proves the proposition. 
 
 348. If we write the differential of the equation of a surface 
 in the form 
 
 dfp = 2Svdp, 
 
 then it is easy to see that 
 
 f(p + dp) =fp + 28vdp + Sdvdp + &c., 
 the remaining terms containing as factors the third and higher 
 powers of Tdp. To the second order, then, we may write, except 
 for certain singular points, 
 
 = 28vdp + Sdvdp, 
 and, as before, (§ 345), the curvature of the normal section whose 
 
 tangent line is dp is 
 
 l_ qdv 
 Tv Tp' 
 
268 QUATERNIONS. 
 
 [349- 
 
 349. The step taken in last section, although a very simple 
 one, virtually implies that the first three terms of the expansion of 
 f{p + dp) are to be formed in accordance with Taylor's Theorem, 
 whose applicability to the expansion of scalar functions of quater- 
 nions has not been proved in this work (see § 142) ; we therefore 
 give another investigation of the curvature of a normal section, 
 employing for that purpose the formulae of § 299. 
 
 We have, treating dp as an element of a curve, 
 
 Svdp = 0, 
 or, making s the independent variable, 
 
 Svp' = 0. 
 From this, by a second differentiation. 
 
 The curvature is, therefore, since v \\ p" and Tp' = 1, 
 
 350. Since we have shewn that 
 
 dv = (f>dp 
 where (^ is a self-conjugate linear and vector function, whose con- 
 stants depend only upon the nature of the surface, and the position 
 of the point of contact of the tangent plane ; so long as we do not 
 alter these we must consider (f) as possessing the properties explained 
 in Chapter V. 
 
 Hence, as the expression for Tp" does not involve the tensor of 
 dp, we may put for dp any unit-vector t, subject of course to the 
 condition 
 
 &T = (1). 
 
 And the curvature of the normal section whose tangent is t is 
 
 If we consider the central section of the surface of the second degree 
 
 made by the plane Svsr = 0, 
 
 we see at once that the curvature of the given surface along the 
 normal section touched by r is inversely as the square of the 
 parallel radius in the auxiliary surface. This, of course, includes 
 Euler's and other well-known Theorems. 
 
352.] GEOMETRY OF CUBVBS AND SURFACES. 269 
 
 351. To find the directions of maximum and minimum 
 curvature, we have 
 
 8T(f}T = max. or min. 
 
 with the conditions, Svt = 0, 
 
 Tt= 1. 
 
 By differentiation, as in § 290, we obtain the farther equation 
 
 S.vrcf>T = (1). 
 
 If T be one of the two required directions, t' = tUv is the other, 
 for the last-written equation may be put in the form 
 
 S.TUvci>(vTUv)=0, 
 
 i.e. S.r'<l>(vT') = 0, 
 
 or S.vt'^t' = 0. 
 
 Hence the sections of greatest and least curvature are perpendicular 
 to one another. 
 
 We easily obtain, as in § 290, the following equation 
 
 which gives two values of 8t<^t, and these divided by — Tv are 
 the required curvatures. 
 
 352. Before leaving this very brief introduction to a subject, 
 an exhaustive treatment of which will be found in Hamilton's 
 Elements, we may make a remark on equation (1) of last section 
 
 8 . VT(pT = 0, 
 
 or, as it may be written, by returning to the notation of § 350, 
 
 8 . vdpdv = 0. 
 
 This is the general equation of lines of curvature. For, if we 
 define a line of curvature on any surface as a line such that 
 normals drawn at contiguous points in it intersect, then, hp being 
 an element of such a line, the normals 
 
 ■u!- = p + wv and ■!!T = p + Sp + y(y + Bv) 
 
 must intersect. This gives, by § 216, the condition 
 
 8 . BpvSv = 0, 
 as above. 
 
270 QUATERNIONS. 
 
 EXAMPLES TO CHAPTER X. 
 
 1. Find the length of any arc of a curve drawn on a sphere 
 so as to make a constant angle with a fixed diameter. 
 
 2. Shew that, if the normal plane of a curve always contains 
 a fixed line, the curve is a circle. 
 
 3. Find the radius of spherical curvature of the curve 
 
 Also find the equation of the locus of the centre of spherical 
 curvature. 
 
 4. (Hamilton, Bishop Law's Premium Examination, 1854.) 
 
 (a) If p be the variable vector of a curve in space, and if the 
 differential dK be treated as = 0, then the equation 
 
 dT(p-K) = 
 
 obliges K to be the vector of some point in the normal plane to 
 the curve. 
 
 (6) In like manner the system of two equations, where dx 
 and d^K are each =0, 
 
 dT(p-K) = 0, d'T(p-K) = 0, 
 represents the axis of the element, or the right line drawn through 
 the centre of the osculating circle, perpendicular to the osculating 
 plane. 
 
 (c) The system of the three equations, in which k is treated 
 as constant, 
 
 dT{p-K) = 0, d^T{p-x) = 0, d'T(p-K) = 0, 
 
 determines the vector k of the centre of the osculating sphere. 
 
 (d) For the three last equations we may substitute the 
 following : 
 
 S.(p-K)dp==0, 
 
 S.(p-K)d'p + dp''==0, 
 8.(p-ic)d'p + SS.dpd'p = 0. 
 
 (e) Hence, generally, whatever the independent and scalar 
 variable may be, on which the variable vector p of the curve 
 
GEOMETRY OE CURVES AND SURFACES. 271 
 
 depends, the vector k of the centre of the osculating sphere admits 
 of being thus expressed : 
 
 3 V. dpd'pS . dpd'p - dpW. dpd'p 
 " '' S.dpd'pd'p 
 
 (/) In general 
 
 d ip-' V.dpUp) = d (Tp-'V. pdp) 
 
 = Tp-' (3F. pdpS.pdp-pW.pd'p) ; 
 whence, 
 
 sr.pdpS.pdp-p'V.pd^p = p'Tpd(p-'V.dpUp); 
 
 and, therefore, the recent expression for « admits of being thus 
 transformed, 
 
 dp*d(dp-^V.d'pUdp) 
 "-P^ S.d'pd'pUdp ■ 
 
 (g) If the length of the element of the curve be constant, 
 dTdp = 0, this last expression for the vector of the centre of the 
 osculating sphere to a curve of double curvature becomes, more 
 
 d . d^pdp^ _ 
 
 K = p + 
 or K = p + 
 
 8.dpd:'pd'p' 
 V. d^pdp' 
 
 S.dpd'pd'p- 
 
 {h) Verify that this expression gives k = 0, for a curve 
 described on a sphere which has its centre at the origin of vectors ; 
 or shew that whenever dTp = 0, d^Tp = 0, d^Tp = 0, as well as 
 dTdp = 0, then 
 
 p8.dp-'d'pd'p=V.dpd^p. 
 
 5. Find the curve from every point of which three given 
 spheres appear of equal magnitude. 
 
 6. Shew that the locus of a point, the difference of whose 
 distances from each two of three given points is constant, is a 
 plane curve. 
 
 7. Find the equation of the curve which cuts at a given angle 
 all the sides of a cone of the second degree. 
 
 Find the length of any arc of this curve in terms of the 
 distances of its extremities from the vertex. 
 
 8. Why is the centre of spherical curvature, of a curve 
 described on a sphere, not necessarily the centre of the sphere ? 
 
272 QUATERNIONS. 
 
 9. Find the equation of the developable surface whose gene- 
 rating lines are the intersections of successive normal planes to a 
 given tortuous curve. 
 
 10. Find the length of an arc of a tortuous curve whose 
 normal planes are equidistant from the origin. 
 
 11. The reciprocals of the perpendiculars from the origin on 
 the tangent planes to a developable surface are vectors of a 
 tortuous curve ; from whose osculating planes the cusp-edge of 
 the original surface may be reproduced by the same process. 
 
 12. The equation p=Vcf^, 
 
 where a is a unit-vector not perpendicular to /3, represents an 
 ellipse. If we put 7 = Fa/S, shew that the equations of the locus 
 of the centre of curvature are 
 
 S^l3p + ^'yp = (^SUal3)*. 
 
 13. Find the radius of absolute curvature of a spherical 
 conic. 
 
 14. If a cone be cut in a circle by a plane perpendicular to a 
 side, the axis of the right cone which osculates it, along that side, 
 passes through the centre of the section. 
 
 15. Shew how to find the vector of an umbilicus. Apply 
 your method to the surfaces whose equations are 
 
 Sp<l>p = -1, 
 and SapS^pSyp = - 1. 
 
 16. Find the locus of the umbilici of the surfaces represented 
 by the equation 
 
 sp{cj>+hrp=^-i, 
 
 where h is an arbitrary parameter. 
 
 IT. Shew how to find the equation of a tangent plane which 
 touches a surface along a line, straight or curved. Find such 
 planes for the following surfaces 
 
 8p(l)p = - 1, 
 
 Sp{cI>-pYp = -1, 
 
 and (p^-a%hy + 4>(ay + b'S'ap) = 0. 
 
GEOMETRY OF CURVES AND SURFACES. 273 
 
 18. Find the condition that the equation 
 
 where <^ is a self-conjugate linear and vector function, may 
 represent a cone. 
 
 19. Shew from the general equation that cones and cylinders 
 are the only developable surfaces of the second degree. 
 
 20. Find the equation of the envelop of planes drawn at each 
 point of an ellipsoid perpendicular to the radius vector from the 
 centre. 
 
 21. Find the equation of the envelop of spheres whose centres 
 lie on a given sphere, and which pass through a given point. 
 
 22. Find the locus of the, foot of the perpendicular from the 
 centre to the tangent plane of a hyperboloid of one, or of two, 
 sheets. 
 
 23. Hamilton, Bishop Law's Premium Examination, 1852. 
 
 (a) If p be the vector of a curve in space, the length of the 
 element of that curve is Tdp ; and the variation of the length of a 
 finite arc of the curve is 
 
 BJTdp = -JSUdpSdp = - ASUdpSp + JSdUdpBp. 
 
 (b) Hence, if the curve be a shortest line on a given surface, 
 for which the normal vector is v, so that SvBp = 0, this shortest or 
 geodetic curve must satisfy the differential equation, 
 
 VvdUdp = 0. 
 
 Also, for the extremities of the arc, we have the limiting 
 
 equations, 
 
 SUdp„Bp, = ; SUdpM = 0. 
 
 Interpret these results. 
 
 (c) For a spheric surface, Vvp = 0, V.pdlTdp^O; the inte- 
 grated equation of the geodetics is VpUdp = 'SJ, giving S-sTp = 
 (great circle). 
 
 For an arbitrary cylindric surface, 
 
 Sa.v = 0, 8.adUdp = 0; 
 the integral shews that the geodetic is generally a helix, making 
 a constant angle with the generating lines of the cylinder. 
 
 (d) For an arbitrary conic surface, 
 
 Svp = 0, SpdUdp = 0; 
 T, Q. I. 18 
 
274 QUATERNIONS. 
 
 integrate this differential equation, so as to deduce from it, 
 TVpUdp = const. 
 
 Interpret this result; shew that the perpendicular from the 
 vertex of the cone on the tangent to a given, geodetic line is 
 constant ; this gives the rectilinear development. 
 
 When the cone is of the second degree, the same property is a 
 particular case of a theorem respecting confocal surfaces. 
 
 (e) For a surface of revolution, 
 
 S . apv = 0, 8 . apdJJdp = ; 
 integration gives, 
 
 const. = 8.apUdp=TVapSU(Vap.dpy, 
 the perpendicular distance of a point on a geodetic line from the 
 axis of revolution varies inversely as the cosine of the angle under 
 which the geodetic crosses a parallel (or circle) on the surface. 
 
 (/) The differential equntion, S .apdUdp = 0, is satisfied not 
 only by the geodetics, but also by the circles, on a surface of 
 revolution; give the explanation of this fact of calculation, and 
 shew that it arises from the coincidence between the normal plane 
 to the circle and the plane of the meridian of the surface. 
 
 (g) For any arbitrary surface, th'e equation of the geodetic 
 may be thus transformed, S . vdpd^p = ; deduce this form, and 
 shew that it expresses the normal property of the osculating plane. 
 
 (h) If the element of the geodetic be constant, dTdp = 0, 
 then the general equation formerly assigned may be reduced to 
 V. vd'p = 0. 
 
 Under the same condition, d^p = — v'^Sdvdp. 
 
 (i) If the equation of a central surface of the second order be 
 put under the form fp = 1, where the function / is scalar, and 
 homogeneous of the second dimension, then the differential of that 
 function is of the form dfp = 28 . vdp, where the normal vector, 
 V = ^p, is a distributive function of p (homogeneous of the first 
 dimension), dv = d^p = ^dp. 
 
 This normal vector v may be called the vector of proximity 
 (namely, of the element of the surface to the centre) ; because its 
 reciprocal, v~\ represents in length and in direction the perpen- 
 dicular let fall from the centre on the tangent plane to the surface. 
 
 (Jc) If we make 8(T^p=f(a, p) this function / is commu- 
 tative with respect to the two vectors on which it depends, 
 
GEOMETRY OF CURVES AND SURFACES. 275 
 
 f{p, o) =/(o", p) ; it is also connected with the former function /, 
 of a single vector p, by the relation, /(p, p) =fp : so that 
 
 fp = 8p<^p. 
 
 fdp = Sdpdv ; dfdp = 28 . dvd^p ; for a geodetic, with constant 
 element, 
 
 this equation is immediately integrable, and gives const. 
 = Tv'J (fUdp) = reciprocal of Joachimstal's product, PD. 
 
 (I) If we give the name of " Didonia" to the curve (discussed 
 by Delaunay) which, on a given surface and with a given perimeter, 
 contains the greatest area, then for such a Didonian curve we have 
 by quaternions the formula, 
 
 fS.UvdpSp + cSJTdp = 0, 
 
 where c is an arbitrary constant. 
 
 Derive hence the differential equation of the second order, 
 equivalent (through the constant c) to one of the third order, 
 
 c-'dp=r.UvdUdp. 
 
 Geodetics are, therefore, 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 general form, is 
 
 p = OT + c JJvdp, 
 
 ■ST being vector of centre, and c being radius of circle. 
 
 (m) Operating by S. Udp, the general differential equation of 
 the Didonia takes easily the following forms : 
 
 c-'Tdp =8{Uvdp.dUdp); 
 
 c-'Tdp' = 8{Uvdp.d'p); 
 
 c-'Tdp^ = 8. Uvdpd^p; 
 
 ^-x^g^pdp^^ 
 Uvdp 
 
 [n) The vector a, of the centre of the osculating circle to a 
 curve in space, of which the element Tdp is constant, has for 
 expression, 
 
 a, = p + ^. 
 
 d^ 
 
 'P 
 
 18—2 
 
276 QUATERNIONS. 
 
 Hence for the general Didonia, 
 
 .-1 _ a (<" - pr . 
 
 " ~'^ Uvdp ' 
 
 "^ vap 
 
 (o) Hence, the radius of curvature of any one Didonia varies, 
 in general, proportionally to the cosine of the inclination of the 
 osculating 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. 
 
 (p) 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 
 
 -. _ vdp' 
 
 ^~''^ ti.vdpd'p- 
 
 Hence, for the general Didonia, with the same signification of the 
 symbols, 
 
 ^ = p — c Uvdp ; 
 
 and the constant c expresses the length of the interval p — ^, 
 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, the radius of this sphere shall always be equal to c. 
 
 (r) The recent expression for |, combined with the first form 
 of the general differential equation of the Didonia, gives 
 
 d^ = -cVdUvUdp; Vvd^ = (i. 
 
 (s) Hence, or from the geometrical signification of the con- 
 stant c, the known property may be proved, that if a developable 
 surface 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 be flattened (generally) 
 into a circular arc, with radius = c. 
 
GEOMETEY OP CURVES AND SURE ACES. 277 
 
 24. Find the condition that the equation 
 
 may give three real values of/ for any given value of p. If /be a 
 function of a scalar parameter f, shew how to find the form of this 
 function in order that we may have 
 
 ^ daf^ df ^ dz' "• 
 Prove that the following is the relation between / and ^, 
 
 ^J{9.+f)i.9.+f)(.9,+f) -'Vm/ 
 in the notation of § 159. (Tait, Trans. R. S. E. 1873.) 
 
 25. Shew, after Hamilton, that the proof of Dupin's theorem, 
 that "each member of one of three series of orthogonal surfaces 
 cuts each member of each of the other series along its lines of 
 curvature," may be expressed in quaternion notation as follows : 
 
 If Svdp = 0, Sv'dp = 0, S.vv'dp = 
 
 be integrable, and if 
 
 8vv' = 0, then Vv'dp = 0, makes 8 . vv'dv = 0. 
 
 Or, as follows, 
 
 If SvVv = 0, 8v'Vv' = 0, 8v"Vv" = 0, and V.vvv" = 0, 
 
 then 8.v"(SvV)v = 0, 
 
 , -, . d . d J d 
 
 where V=t-=- + i ^f- + k -^. 
 
 da; •' dy dz 
 
 26. Shew that the equation 
 
 Vap=pV^p 
 represents the line of intersection of a cylinder and cone, of the 
 second order, which have /3 as a common generating line. 
 
 27. Two spheres are described, with centres at A, B, where 
 OA = a, 0B = /3, and radii a, b. Any line, OPQ, drawn from the 
 origin, cuts them in P, Q respectively. Shew that the equation 
 of the locus of intersection of AP, BQ has the form 
 
 V{a + aU(p-a)]{^ + bU{p-^)} = 0. 
 
 Shew that this involves 8 . a^p = 0, 
 
 and therefore that the left side is a scalar multiple of V .a.^, so 
 that the locus is a plane curve. 
 
278 QUATERNIONS. 
 
 Also shew that in the particular case 
 
 Fa/3 = 0, 
 the locus is the surface formed by the revolution of a Cartesian 
 oval about its axis. 
 
 28. Integrate the equations 
 
 vdp{(p-oir-(p+ar}=o, 
 
 V.dpV^{{p-ay-(p + ar} = 0. 
 
 Shew that each represents a series of circles in space. What 
 is the common property of the circles of each series ? [See § 140, 
 (10), (11).] 
 
 29. Express the general equation of a knot of any kind, on 
 an endless cord, in the form 
 
 p = <l> (s), 
 
 pointing out precisely the nature of the function <^. 
 
 What are the conditions to which <f) must be subject, when 
 possible distortions of the knot are to be represented ? What are 
 the conditions that the string may be capable of being brought 
 into a mere ring form ? 
 
 30. Find the envelop of the planes of equilateral triangles 
 whose vertices are situated in three given lines in space. What 
 does it become when two, or all three, of these lines intersect ? 
 
 31. Form the equation of the surface described by a circle, 
 when two given points in its axis are constrained to move on given 
 straight lines. Also when the constraining lines are two concentric 
 circles in one plane. 
 
CHAPTER XI. 
 
 KINEMATICS. 
 
 353. In the present Chapter it is not proposed to give a 
 connected account of even the elements of so extensive a subject 
 as that indicated by the Title. All that is contemplated is to 
 treat a few branches of the subject in such a way as to shew the 
 student how to apply the processes of Quaternions. 
 
 And, with a view to the next Chapter, the portions selected 
 for treatment will be those of most direct interest in their physical 
 applications. 
 
 A. Kinematics of a Point. 
 
 354. When a point's vector, p, is a function of the time t, we 
 have seen (§ 36) that its vector-velocity is expressed by -w- or, in 
 Newton's notation, by p. 
 
 That is, if p = ^t 
 
 be the equation of an orbit, containing (as the reader may see) not 
 
 merely the form of the orbit, hut the law of its description also, 
 
 then 
 
 p = 4>'t 
 
 gives at once the form of the Hodograph and the law of its 
 
 description. 
 
 This shews immediately that the vector-acceleration of a point's 
 
 motion, 
 
 d'p 
 
 i '' P' 
 
280 QUATEENIONS. [355- 
 
 is the vector-velocity in the hodograph. Thus the fundamental pro- 
 perties of the hodograph are proved almost intuitively. 
 
 355. Changing the independent variable, we have 
 
 dp ds , 
 P=dsdt^'P' 
 
 if we employ the dash, as before, to denote -^ . 
 
 This merely shews, in another form, that p expresses the 
 velocity in magnitude and direction. But a second differentiation 
 gives 
 
 p = vp' + i?p". 
 
 This shews that the vector-acceleration can be resolved into two 
 components, the first, vp , being in the direction of motion and 
 
 equal in magnitude to the acceleration of the speed, v or -j; 
 
 the second, '^p" , being in the direction of the radius of absolute 
 curvature, and having for its amount the square of the speed 
 multiplied by the curvature. 
 
 [It is scarcely conceivable that this important fundamental 
 proposition can be proved more elegantly than by the process 
 just given.] 
 
 356. If the motion be in a plane curve, we may write the 
 equation as follows, so as to introduce the usual polar coordinates, 
 r and Q, 
 
 where a is a unit-vector perpendicular to, /3 a unit-vector in, the 
 plane of the curve. 
 
 Here, of course, r and 6 may be considered as connected by one 
 scalar equation ; or better, each may be looked on as a function of 
 t. By differentiation we get 
 
 p = ra'"'' ^ + react''" ^, 
 which shews at once that r is the velocity along, r0 that perpen- 
 pendicular to, the radius vector. Again, 
 
 p = (r - r6') 0^'" ^ + (2re + rO) ao?"' yS, 
 which gives, by inspection, the components of acceleration along, 
 and perpendicular to, the radius vector. 
 
 357. For uniform acceleration in a constant direction, we have 
 at once 
 
358- J KINEMATICS. 281 
 
 Whence p = a< + /3, 
 
 where /S is the vector-velocity at epoch. This shews that the 
 hodograph is a straight line described uniformly. 
 
 Also p = -Y + ^t, 
 
 no constant being added if the origin be assumed to be the position 
 of the moving point at epoch. 
 
 Since the resolved parts of p, parallel to /3 and a, vary respect- 
 ively as the first and second powers of t, the curve is evidently a 
 parabola (§ 31 (/)). 
 
 But we may easily deduce from the equation the following 
 result, 
 
 Tip + ^/3a-'^) = - SUoi (p + I' a-) , 
 
 the equation of a paraboloid of revolution, whose axis is a. Also 
 
 S . a/3p = 0, 
 and therefore the distance of any point in the path from the 
 point — ^jSa~'jS is equal to its distance from the line whose 
 equation is 
 
 p = -^ a' + xaVajS. 
 
 Thus we recognise the focus and directrix property. [The 
 student should remark here how the distances of the point of 
 projection (which may, of course, be any point of the path) from 
 the focus and from the directrix are represented in Magnitude and 
 direction by the two similar but different expressions 
 
 -\^a'^ and -i/3V^ 
 or -i(/3a-0/S and -|/3(/3a-). 
 
 This is an excellent example of the non-commutative character of 
 quaternion multiplication.] 
 
 358. That the moving point may reach a point 7 (where 7 is, 
 of course, coplanar with a. and ;S) we must have, for some real 
 value of t, 
 
 Now suppose TyS, the speed of projection, to be given = v, and, 
 for shortness, write -as for f//3. 
 
 Then y = p' + vt^ (a). 
 
282 QUATERNIONS. [359- 
 
 Since rtB- = l, ' 
 
 we have ±^ - (v' - Say) f + Trf = 0. 
 
 The values of f are real if 
 
 iv' - Sayf - Ta'Tr/ 
 is positive. Now, as TaTy is never less than Say, this condition 
 evidently requires that v" — Say also shall be positive. Hence, 
 when they are real, both values of f are positive. Thus we have 
 four values of t which satisfy the conditions, and it is easy to see 
 that since, disregarding the signs, they are equal two and two, 
 each pair refer to the same path, but described in opposite direc- 
 tions between the origin and the extremity of 7. There are 
 therefore, if any, in general two parabolas which satisfy the 
 conditions. The directions of projection are (of course) given by 
 the corresponding values of ot. These, in turn, are obtained at 
 once from (a) in the form 
 
 1 t 
 
 where t has one or other of the values previously found. 
 
 359. The envelop of all the trajectories possible, with a given 
 speed, evidently corresponds to 
 
 (v'-8ayy-Ta'Ty' = 0, 
 
 for then 7 is the vector of intersection of two indefinitely close 
 paths in the same vertical plane. 
 
 Now v" - Say = TaTy 
 
 is evidently the equation of a paraboloid of revolution of which 
 the origin is the focus, the axis parallel to a, and the directrix 
 
 plane at a distance y^r- . 
 la 
 
 All the ordinary problems connected with parabolic motion are 
 
 easily solved by means of the above formulse. Some, however, are 
 
 even more easily treated by assuming a horizontal unit-vector in 
 
 the plane of motion, and expressing /3 in terms of it and of a. 
 
 But this must be left to the student. 
 
 360. For acceleration directed to or from a fixed point, we 
 have, taking that point as origin, and putting P for the magnitude 
 of the central acceleration, 
 
 ■p = PUp. 
 
361.] KINEMATICS. 283 
 
 From this, at once, 
 
 Vpp = 0. 
 Integrating Vpp = -y = a constant vector. 
 
 The interpretation of this simple formula is— first, p and p are 
 in a plane perpendicular to 7, hence the path is in a plane (of 
 course passing through the origin) ; second, the doubled area of 
 the triangle, two of whose sides are p and p (that is, the moment 
 of the velocity) is constant. 
 
 [It is scarcely possible to imagine that a more simple proof 
 than this can be given of the fundamental facts, that a central 
 orbit is a plane curve, and that equal areas are described by the 
 radius vector in equal times.] 
 
 3G1. When the law of acceleration to or from the origin is 
 that of the inverse square of the distance, we have 
 
 Tp" 
 
 where m is negative if the acceleration be directed to the origin. 
 
 ■■ inUp 
 Hence p = -^rt ■ 
 
 The following beautiful method of integration is due to Hamil- 
 ton. (See § 140, (2).) 
 
 Generallv ^ = -^-IpI - -^IR^ 
 
 generally, ^^ ^y - y^, , 
 
 thereiore pry = — m —^ , 
 
 and py = e — m Up, 
 
 where e is a constant vector, perpendicular to 7, because 
 
 Syp = 0. 
 Hence, in this case, we have for the hodograph, 
 p = 67"^ — mUp. 7"^ 
 
 Of the two parts of this expression, which are both vectors, the 
 first is constant, and the second is constant in length. Hence the 
 locus of the extremity of p is a circle in a plane perpendicular to 7 
 (i.e. parallel to the plane of the orbit), whose radius is T.my~\ 
 and whose centre is at the extremity of the vector 67"'. 
 
 [This equation contains the whole theory of the Circular 
 Hodograph. Its consequences are developed at length in Hamil- 
 ton's Elements.] 
 
284 QUATERNIONS. [3^2. 
 
 362. We may write the equations of this circle in the form 
 2'(p-67-') = y.wi7"', 
 (a sphere), and Syp = 
 
 (a plane through the origin, and through the centre of the sphere). 
 
 The equation of the orbit is found by operating by F.p upon 
 that of the hodograph. We thus obtain 
 
 y=V. pey''- + mTpy'^, 
 or rf = 8ep + mTp, 
 
 or mTp = Se (r/e'' - p) ; 
 
 in which last form we at once recognise the focus and directrix 
 property. This is in fact the equation of a conicoid of revolution 
 about its principal axis (e), and the origin is one of the foci. The 
 orbit is found by combining it with the equation of its plane, 
 
 8yp = 0. 
 We see at once that 7'''6~' is the vector distance of the directrix 
 from the focus ; and similarly that the excentricity is T. em''-, and 
 
 the major axis 21 
 
 vf + 6' 
 
 363. To take a simpler case : let the acceleration vary as the 
 distance from the origin. 
 
 Then p=± m^p, 
 
 the upper or lower sign being used according as the acceleration is 
 from or to the centre. 
 
 This is (^, + m") p = 0. 
 
 Hence p = ae"'* + )8e-'»*; 
 
 or p = acosmt + ^ sin mt, 
 
 where a and /3 are arbitrary, but constant, vectors ; and e is the 
 
 base of Napier's logarithms. 
 
 The first is the equation of a hyperbola (§ 31, A;) of which a 
 and /3 are the directions of the asymptotes ; the second, that of an 
 ellipse of which a and j8 are semi-conjugate diameters. 
 
 Since p = m {ae™* - /Se^*"*}, 
 
 or =m{— asinmi + /Scosmi}, 
 
 the hodograph is again a hyperbola or ellipse. But in the first 
 case it is, if we neglect the change of dimensions indicated by the 
 
364.] KINEMATICS. 285 
 
 scalar factor m, conjugate to the orbit ; in the case of the ellipse 
 it is similar and similarly situated. 
 
 364. Again, let the acceleration be as the inverse third power of 
 the distance, we have 
 
 .._mUp 
 
 Of course, we have, as usual, 
 
 Vpp = 7. 
 
 Also, operating hy S. p, 
 
 „... mSpp 
 
 >^PP = -TiKf > 
 r 
 of which the integral is 
 
 P -^-^' 
 the equation of energy. 
 
 Again, 8pp = -^. 
 
 Hence Spp + p^ = C, 
 
 or Spp = Gt, 
 
 no constant being added if we reckon the time from the passage 
 through the apse, where Spp = 0. 
 
 We have, therefore, by a second integration, 
 
 p' = Cf + G' (1). 
 
 [To determine C, remark that 
 
 pp = Ct + 7, 
 
 or py=cr-y\ 
 
 But p^/j" = Gp' — m (by the equation of energy), 
 
 = C'f+GG-m,hY(l). 
 Hence CG'^m-j^] 
 
 To complete the solution, we have, by § 140 (2), 
 
 where /3 is a unit-vector in the plane of the orbit. 
 
 But V^ = 
 
 p _ 7 
 
 ,2' 
 
 P P 
 
 dt 
 
 Hence ^'^^ ~^ ^ ~ ^ j i 
 
 /3 " 'JCf + G 
 
286 QUATERNIONS. \_3^5- 
 
 The elimination of t between this equation and (1) gives Tp in 
 terms of Up, or the required equation of the path. 
 
 We may remark that if 6 be the ordinary polar angle in the 
 orbit, 
 
 Hence we have 6 — — Ty I „ ^ — ^, I 
 
 and r' = -{Cf + C') J 
 
 from which the ordinary equations of Cotes' spirals can be at once 
 
 found. [See Tait and Steele's Dynamics of a Particle, Appendix 
 
 (A).] 
 
 365. To find the conditions that a given curve may he the hodo- 
 graph corresponding to a central orbit. 
 
 If CT be its vector, given as a function of the time, Ji^dt is that 
 of the orbit ; hence the requisite conditions are given by 
 
 F'37/OTdi! = 7 (1), 
 
 where 7 is a constant vector. 
 
 We may transform this into other shapes more resembling the 
 Cartesian ones. 
 
 Thus r^f'^dt^O (2), 
 
 and V^JTsdt + V-ut'ot = 0. 
 
 From (2) /ardi = «*, 
 
 and therefore by (1) a; Fot-ot = y, 
 
 or the curve is plane. And 
 
 ajFra-sj- + Fotot = ; 
 
 or eliminating x, 7 Fctto- = — ( F•^JOT)^ 
 
 Now if v' be the velocity in the hodograph, B' its radius of curva- 
 ture, p' the perpendicular on the tangent ; this equation gives at 
 once 
 
 hv' = R'p'\ 
 which agrees with known results. 
 
 366. The equation of an epitrochoid or hypotrochoid, referred 
 to the centre of the fixed circle, is evidently 
 
 p = ai^^'''a + bi'"^"''a, 
 where a is a unit vector in the plane of the curve and i another 
 
367.] KINEMATICS. 287 
 
 perpendicular to it. Here co and «j are the angular velocities in 
 the two circles, and t is the time elapsed since the tracing point 
 and the centres of the two circles were in one straight line. 
 Hence, for the length of an arc of such a curve, 
 
 s = JTpdt = Jdt sj{(i!'a? + 2a)&),a?» cos (w - wj i + (o^V], 
 
 = jdt i\/\{(oa + (cfiy ± 4(»(B,a6 
 
 cos 
 sin^ 
 
 «-«.^ 
 
 which is, of course, an elliptic function. 
 
 But when the curve becomes an epicycloid or a hypocyeloid, 
 <oa + (Op = 0, and 
 
 s = 2^i±..fll)\dt^^--^t, 
 
 which can be expressed in finite terms, as was first shewn by 
 Newton. 
 
 The hodograph is another curve of the same class, whose equa- 
 tion is 
 
 latlir 1 7,,, -SiOii/ir 
 
 'p = i{am^"''a + h(oJ 
 
 a 
 
 and the acceleration is denoted in magnitude and direction by the 
 vector 
 
 p = — ami a — oio^i a. 
 
 Of course the equations of the common Cycloid and Trochoid 
 may be easily deduced from these forms by making a indefinitely 
 great and w indefinitely small, but the product aco finite ; and 
 transferring the origin to the point 
 
 p = aa. 
 
 B. Kinematics of a Rigid System. 
 
 367. Let i be the normal-vector to any plane. 
 Let ID- and p be the vectors of any two points in a rigid plate 
 in contact with the plane. 
 
 After any small displacement of the rigid plate in its plane, let 
 dra- and dp be the increments of ot and p. 
 
 Then Sid'^s = 0, Sidp = ; and, since T (zr-p) is constant, 
 
 8(^-p){d^-dp) = 0. 
 
 And we may evidently assume, consistently with these equations, 
 
 dp = o}i{p — t), 
 cZot = (oi (ot — t) ; 
 
288 QUATERNIONS. [S^S. 
 
 where of course t is the vector of some point in the plane, to a 
 rotation m about which the displacement is therefore equivalent. 
 
 Eliminating t, we have 
 
 . dl-Ts —p) 
 
 cci = , 
 
 ZT — p 
 
 which gives a, and thence r is at once found. 
 For any other point a in the plane figure 
 
 Sida = 0, 
 S(p-a) (dp - da) = 0. Hence dp-d(T = w^ i {p-<r). 
 S(a — 'ax) (d-u7 — da-) = 0. Hence da — d'!!T = co^ i (o- — ot). 
 From which, at once, «, = w^ = &>, and 
 
 d<r = toi {a — t), 
 or this point also is displaced by a rotation « about an axis through 
 the extremity of t and parallel to i. 
 
 368. In the case of a rigid body moving about a fixed point 
 let tn-, p, a denote the vectors of any three points of the body ; the 
 fixed point being origin. 
 
 Then ot'', p"^, cr" are constant, and so are S-a^p, Spa, and Sam. 
 
 After any small displacement we have, for m and p, 
 
 Spdp = oi (1). 
 
 S'zirdp + SpdtiT = oj 
 Now these three equations are satisfied by 
 d-!^ = FttOT, dp = Vap, 
 where a is any vector whatever. But if cZot and dp are given, then 
 
 VdT^dp = V. VuT^Vap = aS . ap-cr. 
 Operate by S .Vzrp, and remember (1), 
 
 S'-i^dp = S^pdvT = S^ . oLp-sT. 
 
 TT VdTjrdp VdpdvT 
 
 ^^°^^ «=;W^= %fo (2>- 
 
 Now consider a, Sada- = 
 
 Spda = — Sadp 
 
 ;Si'CTdcr,= — Sad-urj 
 
 da = Vaa satisfies them all, by (2), and we have thus the proposi- 
 tion that any small displacement of a rigid body about a fixed 
 point is equivalent to a rotation. 
 
370-J KTISrEMATlCS. 289 
 
 369. To represent the rotation of a rigid body about a given 
 axis, through a given finite angle. [This is a work of supererogation, 
 if we consider the results of § 119. But it may be interesting to 
 obtain these results in another manner.] 
 
 Let a be a unit-vector in the direction of the axis, p the vector 
 of any point in the body with reference to a fixed point in the axis, 
 and 6 the angle of rotation. 
 
 Then p = «"• Sap + a"' Vap, 
 
 = — aSap — aVap. 
 
 The rotation leaves, of course, the first part unaffected, but the 
 second evidently becomes 
 
 -a'^'aVoLp, 
 
 or - a Vap cos + Vap sin 6. 
 
 Hence p becomes 
 
 Pi = — aSap — aVap cos + Vap sin 0, 
 
 = (cos 0/2 + a sin 61/2) p (cos (9/2 - a sin 0/2), 
 
 = a pa 
 
 370. Hence to compound two rotations about axes which meet, 
 we may evidently write, as the effect of an additional rotation <p 
 about the unit vector /8, 
 
 p, = ^^''p,/3-'^l\ 
 
 Hence p, = ^*^'a*^>a"'^'/3"*^ 
 
 If the /S-rotation had been first, and then the o-rotation, we should 
 have had 
 
 p^ = a' ^'" p^ '^' a ' , 
 and the non-commutative property of quaternion multiplication 
 shews that we have not, in general, 
 
 P'-> = P',- 
 If a, /S, 7 be radii of the unit sphere to the corners of a spherical 
 triangle whose angles are 0/2, <]>/2, ■\Jr/2, we know that 
 
 J*'" 0"'" a*'" = - 1. (Hamilton, Lectures, p. 267.) 
 
 Hence ^l" a"" ^-y'*'", 
 
 and we may write p^ = -y"''''" py''''", 
 
 or, successive rotations about radii to two corners of a spherical 
 
 triangle, and through angles double of those of the tnangle, are 
 
 T. Q. I, 19 
 
290 QUATERNIONS. {.37^- 
 
 equivalent to a single rotation about the radius to the third corner, 
 and through an angle double of the exterior angle of the triangle. 
 
 Thus any number of successive finite rotations of a system, of 
 which one point is fixed, may be compounded into a single rotation 
 about a definite axis. 
 
 371. When the rotations are indefinitely small, the effect of 
 one is, by § 369, 
 
 /3, = jo + aFa/3, 
 
 and for the two, neglecting products of small quantities, 
 
 /j, = p+aFa/j + 6F/3/9, 
 
 a and 6 representing the angles of rotation about the unit-vectors 
 a and /3 respectively. 
 
 But this is equivalent to 
 
 P, = p + T{aa + m VU{m + b/3) p, 
 representing a rotation through an angle ^T (aa + 6/8), about the 
 unit-vector U (aa -I- b/3). Now the latter is the direction, and the 
 former the length, of the diagonal of the parallelogram whose sides 
 are aa and b/3. 
 
 We may write these results more simply, by putting a for aa, 
 )8 for b/S, where a and ^ are now no longer unit-vectors, but repre- 
 sent by their versors the axes, and by their tensors the angles 
 (small), of rotation. 
 
 Thus p^ = p+VaLp, 
 
 p, = p+Vap+V^p, 
 = p + V{a + ^)p. 
 
 372. Given the instantansous axis in terms of the time, it is 
 required to find the single rotation which will bring the body from 
 any initial position to its position at a given time. 
 
 If a be the initial vector of any point of the body, tn- the value 
 of the same at time t, and q the required quaternion, we have 
 by § 119 
 
 •^ = 9«?~' (!)• 
 
 Differentiating with respect to t, this gives 
 
 •57 = qaq'^ — qaq'^qq'^, 
 = qq-\qaq-'~qaq-\qq-\ 
 = 2V.(Vqq-\qaq-'). 
 But •ET = Vev!- = V. eqaq~^. 
 
373-J 
 
 KINEMATICS. 
 
 291 
 
 Hence, as qaq ' may be any vector whatever in the displaced 
 body, we must have 
 
 e = 2Vqq'^ (2). 
 
 This result may be stated in even a simpler form than (2), for 
 we have always, whatever quaternion q may be, 
 
 8qq-^^^{Tqr, 
 
 and, therefore, if we suppose the tensor of q, which, as it is not 
 involved in g'( )q'^, may have any value whatever, to be a 
 constant (unity, for instance), we may write (2) in the form 
 
 n = M (3). 
 
 An immediate consequence, which will be of use to us later, is 
 q.q-'€q=2q (4). 
 
 373. To express q in terms of the usual angles yfr, 6, (j). 
 
 Here the vectors i, j, k in the original position of the body 
 correspond to OA, OB, 00, re- 
 spectively, at time t. The trans- 
 position is defined to be effected 
 by — first, a rotation y^ about k ; 
 second, a rotation 6 about the new 
 position of the line originally 
 coinciding with j ; third, a rotation 
 j> about the final position of the 
 line at first coinciding with k. 
 
 This selection of angles, in 
 terms of which the quaternion is 
 to be expressed, is essentially 
 
 unsymmetrical, and therefore the results cannot be expected to be 
 simple. 
 
 The rotation i|r about k has the operator 
 
 k^l^'i )k-'^'\ 
 
 This converts J into 17, where 
 
 -q = i*'" jk~*''' =j cos ■>Jr — i sin i|r. 
 
 The body next rotates about iy through an angle d. This has 
 the operator 
 
 19—2 
 
292 QUATERNIONS. [374- 
 
 It converts k into 
 0G = ^= rj^'kr)-^'" = (cos 6/2 + r, sin 6/2) k (cos 0/2 - v sin 0/2) 
 — k cos ^ + sin 6 (i cos ■^+j sin t^). 
 
 The body now turns through the angle ^ about f, the operator 
 being 
 
 Hence, omitting a few reductions, which we leave as excellent 
 practice for the reader, we find 
 
 q = ^l^rj^l-k*''' 
 
 = (cos ^/2 + ^sin 4>/2) (cos 0/2 + j? sin 0/2) (cos i|r/2 + k sin f/2) " 
 
 = cos (<^ + yjr)/2 . cos ^/2 + i sin (f - -«|r)/2 . sin 0/2 + 
 
 ' ' j cos (<^ -,f)/2 . sin 61/2 + k sin (0 + •t)/2 . cos 0/2, 
 
 which is, of course, essentially unsymmetrical. 
 
 374. To find the usual equations connecting yjr, 0, (p with the 
 angular velocities about three rectangular axes fixed in the body. 
 
 Having the value of q in last section in terms of the three 
 angles, it may be useful to employ it, in conjunction with equation 
 (3) of § 372, partly as a verification of that equation. Of course, 
 this is an exceedingly roundabout process, and does not in the 
 least resemble the simple one which is immediately suggested by 
 quaternions. 
 
 We have 2q = eq={eo^0A + co^aB + (o^OG}q, 
 
 whence 2q'^q = q'^ {w^ OA + a^ OB + w^ OG] q, 
 
 or 2^ = g (ift), +j(o^ + k(o^). 
 
 This breaks up into the four (equivalent to three independent) 
 equations 
 
 2 ^ [cos (^ + ^|r)/2 . cos 0/2] 
 
 = - Wj sin {(f) - i/r)/2 . sin 0/2 - tu, cos {(f> - -f)/2 . sin 6/2 
 
 - ft)g sin (</) + ■>|r)/2 . cos 0/2, 
 2 ^ [sin (^ - f)/2 . sin 0/2] 
 
 = &), cos (j) + >^)/2 . cos 0/2 - m.^ sin (^ + •x|r)/2 . cos 0/2 
 
 + Wg cos (^ - •\|f)/2 . sin 0/2, 
 
3 75- J KINEMATICS. 293 
 
 2j[cos(<f>-f)/2.smd/2] 
 
 = «j sin {<j) + ■f)/2 . cos 6/2 + w, cos {<j) + ■^)I2 . cos dj-l 
 
 - o), siu (<^ - ■«/r)/2 . sin ^/2, 
 2|[sin((^ + V^)/2.cos6'/2J 
 
 = - ft)j cos (^ - i|r)/2 . sin ^/2 + m^ sin (^ - T/r)/2 . sin ^/2 
 
 + cBj cos (^ + •v|r)/2 . cos 612. 
 
 From the second and third, eliminate ^ — t^, and we get by 
 inspection 
 
 cos 6j2 . 6 = {co^ sin ^ + a>^ cos ^) cos 6/2, 
 
 or ^ = WjSin ^+a)jC0S(^ (1). 
 
 Similarly, by eliminating 6 between the same two equations, 
 
 sin 6/2 . (^ — i/f) = 0), sin 6/2 + co^ cos <j) cos 6/2 — co^ sin (}> cos ^/2. 
 
 And from the first and last of the group of four 
 
 cos 6/2 . (0 + 1^) = Wg cos 6/2 — co^ cos ^ sin 6/2 + a^ sin ^ sin 6/2. 
 
 These last two equations give 
 
 (j) + yjr cos 6 = CO ^ (2). 
 
 ^ cos 6 + yjr = { — a)i cos 4> + (o^ sin ^) sin ^ + Wg cos ^. 
 
 From the last two we have 
 
 ■yjr sin 6 = — 0)^ cos </) + «2 sin (^ (3). 
 
 (1), (2), (3) are the forms in which the equations are usually given. 
 
 375. To deduce expressions for the direction-cosines of a set of 
 rectangular axes, in any position, in terms of rational functions of 
 three quantities only. 
 
 Let a, /3, 7 be unit-vectors in the directions of these axes. Let q 
 be, as in § 372, the requisite quaternion operator for turning the 
 coordinate axes into the position of this rectangular system. Then 
 
 q = w-'rxi + yj + zk, 
 
 where, as in § 372, we may write 
 
 \=w' + as' + f + z\ 
 
 Then we have q'^ = w — xi — yj — zk, 
 
 and therefore 
 
 a = qiq~^ = (wi ^x — yk + zj) {w — xi — yj — zk) 
 
 = {w" + a;^ - y' ~ 2^)i+2 (wz + xy)j 4 2{xz- wy) k, 
 
294 QUATERNIONS. [37^- 
 
 where the coefficients of i, j, k are the direction-cosines of a as 
 required. A similar process gives by inspection those of ^ and 7. 
 
 As given by Cayley*, after Rodrigues, they have a slightly 
 different and somewhat less simple form — to which, however, they 
 are easily reduced by putting 
 
 w = cc/X = ylfi = zjv = l/«i 
 The geometrical interpretation of either set is obvious from the 
 nature of quaternions. For (taking Cayley's notation) if be the 
 angle of rotation : cos/, cos g, cos h, the direction-cosines of the 
 axis, we have 
 
 q = w + xi + yj + zk = cos 6/2 + sin 6/2 . (i cosf+j cosg + k cos A), 
 so that • w = cos 6/2, 
 
 a; = sin 6/2.cosf, 
 y = sin 6/2 . cos g, 
 z = sin 6/2 . cos h. 
 From these we pass at once to Rodrigues' subsidiary formulae, 
 K = l/vf = sec'6/2, 
 X = x/w = tan 6/2 . cos f, 
 &c. = &c. 
 
 G. Kinematics of a Deformable System. 
 
 376. By the definition of Homogeneous Strain, it is evident 
 that if we take any three (non-co planar) unit-vectors a, /3, 7 in 
 an unstrained mass, they become after the strain other vectors, 
 not necessarily unit-vectors, a^, /8j, 7,. 
 
 Hence any other given vector, which of course may be thus 
 
 expressed, p = xa + y^ + zy, 
 
 becomes p^ = xa^ + y^^ + zy^, 
 
 and is therefore known if a^, /8j, 7, be given. 
 
 More precisely 
 
 pS . a^y = aS . /37/3 + ^S . yap 4 yS . a/3jo 
 becomes 
 
 p^S . a/87 = ^pS • a/37 = a^S . ^yp -(- /S,S . yap -1- 7^^ . a^p. 
 Thus the properties of ^, as in Chapter V., enable us to study 
 with great simplicity homogeneous strains in a solid or liquid. 
 
 * Camb. and Dub. Math. Journal. Vol. i. (1846). 
 
37^-] KINEMATICS. 295 
 
 For instance, to find a vector whose direction is unchanged by 
 the strain, is to solve the equation 
 
 Vp<l>p = 0, or (j)p=gp (1), 
 
 where ^r is a scalar unknown. 
 
 [This vector equation is equivalent to thr'ee scalar equations, and 
 contains only three unknown quantities ; viz. two for the direction 
 of p (the tensor does not enter, or, rather, is a factor of each side), 
 and the scalar g.] 
 
 We have seen that every such equation leads to a cubic in g 
 which may be written 
 
 where m^, m^, m are scalars depending in a known manner on the 
 constant vectors involved in <p. This must have one real root, and 
 may have three. 
 
 377. For simplicity let us assume that a, jS, 7 form a rectangular 
 system, then we may operate on (1) hy S.a, S.^, and S.j; and 
 thus at once obtain the equation for g, in the form 
 
 Saa^+g, Sa^^, Saj^ =0. 
 
 S^a„ S^^,+g, 8^ J, 
 
 8ya^, 8y^^, S^yy^ + g 
 
 To reduce this we have, for the term independent of g, 
 
 Saa^, Sa^„ Say, =-8. a^y8 . a,;S,7. (Ex. 9, Chap. III.), 
 8^a„ S^^„ S^y, 
 Sya,, Sy^,, 8yy, 
 which, if the mass be rigid, becomes — 1. 
 The coefficient of the first power of g is, 
 
 % (8^Myyi - S^y^Sy^^) = - 2^. ^^7^/3.7, = - 2^f . a/3.7,. 
 
 Thus we can at once form the equation; which becomes, 
 for the special case of a rigid system, 
 
 - 1 - ^ i8aa, + 8^^, + 8yy,) + g' (Sm, + 8^^, + 8yy,) + g' = 0, 
 or (g - 1) {g' + g{l + 8aa, + 8/3^, + 8yy,) + 1} = 0. 
 
 378. If we take Tp = G we consider a portion of the mass 
 initially spherical. This becomes of course 
 
 T^p.^C, 
 an ellipsoid, in the strained state of the body. 
 
296 QUATERNIONS. [379- 
 
 Or if we consider a portion which is spherical after the strain, i.e. 
 
 its initial form was T<^p — G, 
 
 another ellipsoid. The relation between these ellipsoids is obvious 
 
 from their equations. (See § 327.) 
 
 In either case the axes of the ellipsoid correspond to a rect- 
 angular set of three diameters of the sphere (§ 271). But we must 
 carefully separate the cases in which these corresponding lines in 
 the two surfaces are, and are not, coincident. For, in the former 
 case there is pibre strain, in the latter the strain is accompanied 
 by rotation. Here we have at once the distinction pointed out by 
 Stokes* and Helmholtz-f- between the cases of fluid motion in 
 which there is, or is not, a velocity-potential. In ordinary fluid 
 motion the distortion is of the nature of a pure strain, i.e. is differ- 
 entially non-rotational ; while in vortex motion it is essentially 
 accompanied by rotation. But the resultant of two pure strains is 
 generally a strain accompanied by rotation. The question before 
 us beautifully illustrates the properties of the linear and vector 
 function. 
 
 379. To find the criterion of a pure strain. Take a, /3, 7 now 
 as unit-vectors parallel to the axes of the strain-ellipsoid, they 
 become after the strain, aa, h^, cy, provided the strain be pure. 
 
 Hence p^ = (f>p = — aaSap — b^S^p — cySyp. 
 
 And we have, for the criterion of a pure strain, the property of 
 the function <p, that it is self-conjugate, i.e. 
 
 8p<f}(r = 8a'<pp. 
 
 380. Two pure strains, in succession, generally give a strain 
 accompanied by rotation. For if (f), i/r represent the strains, since 
 they are pure we have 
 
 Sp<j)cr = 8a(j)p] 
 
 Spyira- = So-ijrp) ^ '■ 
 
 But for the compound strain we have 
 
 and we have not generally 
 
 Spxa- = Sa-xp. 
 
 * Cambridge Phil. Trans. 1845. 
 
 t Crelle, vol. Iv. 1857. See also Phil. Mag. (Supplement) June 1867. 
 
38 I.J KINEMATICS. 297 
 
 For Sp-^^a = Sa-^-\lrp, 
 
 by (1), and i|r^ is not generally the same as (j>-f. (See Ex. 7 to 
 Chapter V.) 
 
 To find the lines which are most altered in length by the strain. 
 
 Here Tcjyp is a maximum or minimum, while Tp is constant ; 
 so that 
 
 Sdp(ji'(j)p = 0, Spdp = 0. 
 
 Hence ^'^/o = ^i", 
 
 and the required lines are the principal vectors of ^'(/>, which 
 
 (§ 381) is obviously self-conjugate ; i.e. denotes a pure strain. 
 
 381. The simplicity of this view of the question leads us to 
 suppose that we may easily separate the pure strain from the 
 rotation in any case, and exhibit the corresponding functions. 
 
 When the linear and vector function expressing a strain is 
 self-conjugate the strain is pure. When not self-conjugate, it may 
 be broken up into pure and rotational parts in various ways (ana- 
 logous to the separation of a quaternion into the sum of a scalar 
 and a vector part, or into the product of a tensor and a versor 
 part), of which two are particularly noticeable. Denoting by a 
 bar a self-conjugate function, we have thus either 
 
 = t+^-e( ), 
 
 ^ = q^{ )q-\ or (f>=^.q( )q-\ 
 
 where e is a vector, and q a quaternion (which may obviously be 
 regarded as a mere versor). [The student must remark that, 
 although the same letters have been employed (from habit) in 
 writing the two last formulae, one is not a transformation of the 
 other. In the first a pure strain is succeeded by a rotation, in the 
 second the rotation is followed by the pure strain.] 
 
 That this is possible is seen from the fact that <^ involves nine 
 independent constants, while yjr and ot each involve six, and e and q 
 each three. If (/>' be the function conjugate to cp, we have 
 
 4>' = f-V.e( ), 
 so that 2ylr = <j) + <j)', 
 
 and 2V.e{ ) = ^-^', 
 
 which completely determine the first decomposition. This is, of 
 course, perfectly well known in quaternions, but it does not seem 
 to have been noticed as a theorem in the kinematics of strains that 
 
298 QUATERNIONS. [S^^- 
 
 there is always one, and but one, mode of resolving a strain into 
 the geometrical composition of the separate effects of (1) a pure 
 strain, and (2) a rotation accompanied by uniform dilatation 
 perpendicular to its axis, the dilatation being measured by 
 (sec. ^ - 1) where 9 is the angle of rotation. 
 
 In the second form (whose solution does not appear to have 
 been attempted), we have 
 
 (j} = qW( ) q-\ 
 where the pure strain precedes the rotation, and from this 
 <f>'=^.q-^( )q, 
 
 or in the conjugate strain the rotation (reversed) is followed by the 
 pure strain. From these 
 
 4,'<^ = ^.q-'{q^{ )q~']q 
 
 — :^ 
 
 — ^ , 
 
 and OT is to be found by the solution of a biquadratic equation*. 
 It is evident, indeed, from the identical equation 
 
 S . <7^'<pp = S . p(j)'^cr 
 that the operator ^'(j) is self-conjugate. 
 In the same way 
 
 <^f( ) = q^'{q-H )q]q\ 
 or q-' {^(f>'p) q = ^ (q~'pq) = f (q~'pq). 
 
 * Suppose the cubic in ra^ to be 
 
 Now W^ is equal to 0'0, a known function, which we may call oi. Thus 
 
 S^ = (<), 
 and therefore ct and u are commutative in multiplication. 
 Eliminating ^ between these equations we have, first, 
 
 (w-g2)a + gi^-g = = w(oi + g^)-g^to-g, 
 andfinaUy '•'^ + (^gi- g2'')<^^ + {gi''-2gg^) <^-g''=0. 
 
 This must agree with the (known) cubic in a, 
 
 u' - mju^ + miw - m = 0, suppose ; 
 so that, by comparison of coefficients we have 
 
 2ffi-</2^=-nt2. gi'-^ggi=mi, g^=m; 
 and thus g is known, and ffi -'"h 
 
 where 2p,-(^^=_,^. 
 
 The values of the quantities g being found, ct is given in terms of u by the equation 
 
383-] KINEMATICS. 299 
 
 which shew the relations between ^(/)' , <^'0, ard q. 
 
 To determine q we have 
 
 (l>p.q=q^p 
 whatever be p, so that 
 
 S. Vq{<j>-^)p = 0, 
 or S.p {<!>'- S) Vq = 0, 
 
 which gives (<^' — ^)Vq = 0. 
 
 The former equation gives evidently 
 
 Vq\\ r.{<f)-^)a{<j)-^)l3 
 whatever be a and $ ; and the rest of the solution follows at once. 
 A similar process gives us the solution when the rotation precedes 
 the pure strain. [Proc. R. 8. E. 1870— l.J 
 
 382. In general, if 
 
 Pi = 4^9 = - "^iSt^P - /3.»S/3/) - ry.Syp, 
 the angle between any two lines, say p and a, becomes in the 
 altered state of the body 
 
 cos-^-^f. U^pU<i><7). 
 The plane S^p = becomes (with the notation of § 157) 
 8cj)'-'^p = S^cji-'p = 0. 
 [For if \, /J, be any two vectors in it, 
 
 ? II T^V- 
 But they become ^\ (j)fi, and the line perpendicular to both is 
 
 Hence the angle between the planes S^p = 0, and Srjp = 0, which 
 is cos"^(— (S. U^Ut]), becomes 
 
 cos-'{-S.U<l>"'^U<i>'-'7i). 
 
 The locus of lines equally elongated is, of course, 
 
 T4,Up = e, 
 
 or T<j>p = eTp, 
 
 a cone of the second degree. 
 
 383. In the case of a Simple Shear, we have, obviously, 
 
 p, = ^/3 = /> + ^8ap, 
 where a is a unit vector, and 
 
300 QUATERNIONS. [384- 
 
 The vectors which are unaltered in length are given by 
 
 or 28l3p8oip + ^'S'ap = 0, 
 
 which breaks up into S .ap=0, 
 
 and 8p (2^ + ^'a) = 0. 
 
 The intersection of this plane with the plane of a, ^ is perpen- 
 dicular to 2y8 + /S'a. Let it be a + «/8, then 
 
 >Sf.(2/3 + ;S^a)(a + a;/3) = 0, 
 i.e. 2a;- 1=0. 
 
 Hence the intersection required is 
 
 a+2- 
 
 For the axes of the strain, one is of course a/3, and the others 
 are found by making T(f>V'p a maximum and minimum. 
 
 Let p = a + x^, 
 
 then p^=j>p = a + xp-^, 
 
 Tp, 
 and tf = '^^^- °^ "i^"^-' 
 
 J-P 
 
 gives x' -x + -^ = 0, 
 
 from which the values of x (say x-^ and x^ are found. 
 
 Also, as a verification, we must show that the lines of the 
 body which become most altered in length are perpendicular to 
 one another both before and after the shear. 
 
 Thus /S . (a + a;,/3) (a + a;,/3) = - 1 + ^\x^, 
 
 should be = 0. It is so, since, by the equation in x, 
 
 _ 1 
 
 * A — Qi • 
 
 Again 
 
 8[0L + {x,-l)^]\a^{x.^-\)^}=-l+^'[x,X,-{x^+X,) + l], 
 
 ought also to be zero. And, in fact, 
 
 «i + a?2 = 1 
 by the equation for x ; so that this also is verified. 
 
 384. We regret that our limits do not allow us to enter farther 
 upon this very beautiful application. [The reader is referred to 
 
385.] KINEMATICS. 301 
 
 Chapter X. of Kelland and Tait's Introduction to Quaternions ; in 
 which the treatment of linear and vector equations is based upon 
 the theory of homogeneous strain ; which, in its turn, is much more 
 fully developed than in the present work.] 
 
 But it may be interesting to consider briefly the effects of any 
 continuous displacements (of the particles of a body) by the help of 
 the operator V. 
 
 We have seen (§ 148) that the effect of the operator - Sdp^, 
 upon any scalar function of the vector of a point, is to produce 
 total differentiation due to the passage from p to p + dp. 
 
 Hence if cr be the displacement of p, that oi p + Bp is 
 ,7-S(SpV)a. 
 
 Thus the strain of the group of particles near p is such that 
 
 <})T = T-8 (tV) 0-. 
 
 [Here we virtually assume that cr is a continuous function of p.] 
 But if this correspond to a linear dilatation e, combined with 
 a rotation whose vector-axis is e, both being infinitesimal, 
 
 ^T = T (1 + e) + Fer. 
 
 Thus, for all values of r, each with its proper e, 
 
 V{e + e)T = -S{rV)a. 
 
 This gives at once (for instance by putting in succession for t 
 any three rectangular unit vectors) 
 
 26 - Se = Vo-, 
 
 from which we conclude as follows : — 
 
 If o- (a continuous function of p) represent the vector displace- 
 ment of a point situated at the extremity of the vector p (drawn 
 from the origin) 
 
 jSVo- represents the consequent cubical compression of the 
 group of points in the vicinity of that considered, and 
 
 FV(7 represents twice the vector axis of rotation of the same 
 group of points. 
 
 385. As an illustration, suppose we fix our attention upon a 
 group of points which originally filled a small sphere about the 
 extremity of p as centre, whose equation referred to that point is 
 
 Ta> = c (1). 
 
 After displacement p becomes p + cr, and, by last section, p + w 
 becomes p + co^- a — (SoN) cr. Hence the vector of the new surface 
 
302 QUATERNIONS. [3^6. 
 
 which encloses the group of points (drawn from the extremity of 
 
 P + (t) is 
 
 (o, = co-{Sa.V)a (2). 
 
 Hence w is a homogeneous linear and vector function of «, ; or 
 
 and therefore, by (1), ^"^^i = c> 
 
 the equation of the new surface, which is evidently a central surface 
 
 of the second degree, and therefore, of course, an ellipsoid. 
 
 We may solve (2) with great ease by approximation, if we as- 
 sume that TV a- is very small, and therefore that in the small term 
 we may put «, for co ; i.e. omit squares of small quantities ; thus 
 ft) = a)j + (/Sci),V) a. 
 
 386. If the vector displacement of each point of a medium is in 
 the direction of, and proportional to, the attraction exerted at that 
 point by any system of material masses, the displacement is effected 
 without rotation. 
 
 For ii Fp = G be the potential surface, we have Sadp a complete 
 
 differential ; and, by § 334, 
 
 FVo- = 0. 
 
 Conversely, if there he no rotation, the displacements are in the 
 direction of and proportional to, the normal vectors to a series of 
 surfaces. 
 
 For = -V.dpVVa = -(SdpV)<7 + VS,Tdp, 
 
 where, in the last term, V acts on a alone. 
 
 Now, of the two terms on the right, the first is (§ 149, (4)) 
 the complete differential da, and therefore the remaining term 
 must be a complete differential. This, of course, means that 
 
 Sa-dp 
 is a complete differential. 
 
 Thus, in a distorted system, there is no compression if 
 /SVo- = 0, 
 and no rotation if Wa = ; 
 
 and evidently merely transference if a = a= a constant vector, 
 which is one case of 
 
 Va- = 0. 
 
 In the important case of o- = VFp 
 there is (as proved already) no rotation, since 
 
 Vo- = V'Fp 
 
387.] KINEMATICS. 303 
 
 is evidently a scalar. In this case, then, there are only translation 
 and compression, and the latter is at each point proportional to the 
 density of a distribution of matter, which would give the potential 
 Fp. For if r be such density, we have at once 
 
 V^Fp = 47rr* 
 
 D. Axes of Inertia. 
 
 387. The Moment of Inertia of a body about a unit vector a 
 
 as axis is evidently 
 
 Mh' = -tm(rapy, 
 
 where p is the vector of the element m of the mass, and the origin 
 of p is in the axis. [The letter h has, for an obvious reason, been 
 put here in place of the k which is usually employed for the radius 
 of gyration.] 
 
 Hence if we put ^ = e^a/h, where e is constant, we have, as 
 locus of the extremity of /3, 
 
 Me' = -tm {V^pf = - MS^(j>^ (suppose), 
 the well-known ellipsoid. The linear and vector function, ^, 
 depends only upon the distribution of matter about the (temporary) 
 origin. 
 
 If OT be the vector of the centre of inertia, <r the vector of m 
 
 with respect to it, we have 
 
 jO = in- + 0-, 
 
 where (§ 31 (e)) Xma- = ; 
 
 and therefore Mh" = - tm {(Va^Y + ( Vaaf} 
 = -M(VcL^y-MSa^,oL. 
 Here <j>^ is the unique value of which corresponds to the 
 distribution of matter relative to the centre of inertia. The 
 equation last written gives the well-known relation between the 
 moment of inertia about any line, and that about a parallel line 
 through the centre of inertia. 
 
 Hence, to find the principal axes of inertia at any point (the 
 origin, whose vector from the centre of inertia is - •nr), note that h ■ 
 is to be made max., min., or max.-min., with the condition 
 
 a' = -l. 
 Thus we have So.' (ot Facr + ^^a) = 0, 
 
 <Sfa'a = 0; 
 * Proc. R. S. E., 1862—3. 
 
304 QUATERNIONS. \_3^7- 
 
 therefore ^^a + -57 Fm-ot =pa. = h^a. (by operating by 8 .a). 
 
 Hence (0, — h^ — ot") a = — ■nr/Satu- (1). 
 
 determines the values of a, W being found from the cubic 
 
 >Sfi;r (</),- A' --=rV^ = -l (2)- 
 
 Now the normal to 
 
 /S<r(0,-/i^-^T'<^ = -l (^)' 
 
 at the point o- is parallel to 
 
 But (3) passes through --nr, by (2), and there the normal is 
 
 which, by (1), is parallel to one of the required values of a. (3) is, 
 of course (§ 288), one of the surfaces confocal with the ellipsoid 
 
 Thus we prove Binet's theorem that the principal axes at any 
 point are normals to the three surfaces of the second degree, confocal 
 with the central ellipsoid, which pass through that point. 
 
 EXAMPLES TO CHAPTEE XI. 
 
 1. Form, from kinematical principles, the equation of the 
 cycloid ; and employ it to prove the well-known elementary 
 properties of the arc, tangent, radius of curvature, and evolute, 
 of the curve. 
 
 2. Interpret, kinematically, the equation 
 
 p = aU(^t-p), 
 where /3 is a given vector, and a a given scalar. 
 
 Shew that it represents a plane curve; and give it in an 
 integrated form independent of t. 
 
 3. If we write ■aT = ^t — p, 
 the equation in (2) becomes 
 
 /3 — OT = a ?7-s7. 
 
 Interpret this kinematically ; and find an integral. 
 
 What is the nature of the step we have taken in transforming 
 from the equation of (2) to that of the present question ? 
 
EXAMPLES TO CHAPTEE XI. 305 
 
 4. The motion of a point in a plane being given, refer it to 
 
 (a) Fixed rectangular vectors in the plane. 
 
 (b) Rectangular vectors in the plane, revolving uniformly 
 about a fixed point. 
 
 (c) Vectors, in the plane, revolving with diiferent, but uni- 
 form, angular velocities. 
 
 (d) The vector radius of a fixed circle, drawn to the point of 
 contact of a tangent from the moving point. 
 
 In each case translate the result into Cartesian coordinates. 
 
 5. Any point of a line of given length, whose extremities move 
 in fixed lines in a given plane, describes an ellipse. 
 
 Shew how to find the centre, and axes, of this ellipse ; and 
 the angular velocity, about the centre of the ellipse, of the tracing 
 point when the describing line rotates uniformly. 
 
 Transform this construction so as to shew that the ellipse is a 
 hypotrochoid. 
 
 When the fixed lines are not in one plane, what is the locus ? 
 
 6. A point. A, moves uniformly round one circular section of 
 a cone; find the angular velocity of the point, a, in which the 
 generating line passing through A meets a subcontrary section, 
 about the centre of that section. 
 
 7. Solve, generally, the problem of finding the path by which 
 a point will pass in the least time from one given point to another, 
 the speed at the point of space whose vector is p being expressed 
 by the given scalar function 
 
 fp- 
 Take also the following particular cases : — 
 
 (a) fp = a while Sap > 1, 
 fp = b while Sap < 1. 
 (6) fp = TSap. 
 (c) fp = - p\ (Tait, Trans. B. S. E., 1865.) 
 
 8. If, in the preceding question, fp be such a function of Tp 
 that any one swiftest path is a circle, every other such path is a 
 circle, and all paths diverging from one point converge accurately 
 in another. (Maxwell, Camb. and Dub. Math. Journal, IX. p. 9.) 
 
 T. Q. I. 20 
 
306 QUATERNIONS. 
 
 9. Interpret, as results of the composition of successive conical 
 rotations, the apparent truisms 
 
 a K t S y ^ ^ 
 
 and — a ~ o Z~ ^^ 
 
 K t a 7 p a 
 
 (Hamilton, Lectures, p. 334.) 
 
 10. Interpret, in the same way, the quaternion operators 
 
 5 = (Se-^)^(er)*(?n*. 
 
 11. Find the axis and angle of rotation by which one given 
 rectangular set of unit-vectors a, ^, y is changed into another 
 given set a^,0^,y^. 
 
 12. Shew that, if ^p = p+ Vep, 
 
 the linear and vector operator (f> denotes rotation about the vector 
 e, together with uniform expansion in all directions perpendicular 
 to it. 
 
 Prove this also by forming the operator which produces the 
 expansion without the rotation, and that producing the rotation 
 without the expansion ; and finding their joint effect. 
 
 13. Express by quaternions the motion of a side of one right 
 cone rolling uniformly upon another which is fixed, the vertices of 
 the two being coincident. 
 
 14. Given the simultaneous angular velocities of a body about 
 the principal axes through its centre of inertia, find the position 
 of these axes in space at any assigned instant. 
 
 15. Find the linear and vector function, and also the quater- 
 nion operator, by which we may pass, in any simple crystal of the 
 cubical system, from the normal to one given face to that to 
 another. How can we use them to distinguish a series of faces 
 belonging to the same zone ? 
 
 16. Classify the simple forms of the cubical system by the 
 properties of the linear and vector function, or of the quaternion 
 operator, mentioned in (16) above. 
 
EXAMPLES TO CHAPTER XI. 307 
 
 17. Find the vector normal of a face which truncates symme- 
 trically the edge formed by the intersection of two given faces. 
 
 18. Find the normals of a pair of faces symmetrically truncat- 
 ing the given edge. 
 
 19. Find the normal of a face which is equally inclined to 
 three given faces. 
 
 20. Shew that the rhombic dodecahedron may be derived from 
 the cube, or from the octahedron, by truncation of the edges. 
 
 21. Find the form whose faces replace, symmetrically, the 
 edges of the rhombic dodecahedron. 
 
 22. Shew how the two kinds of hemihedral forms are indi- 
 cated by the qiiaternion expressions. 
 
 23. Shew that the cube may be produced by truncating the 
 edges of the regular tetrahedron. If an octahedron be cut from a 
 cube, and cubes from its tetrahedra, all by truncation of edges, the 
 two latter cubes coincide. 
 
 24. Point out the modifications in the auxiliary vector func- 
 tion required in passing to the pyramidal and prismatic systems 
 respectively. 
 
 25. In the rhombohedral system the auxiliary quaternion 
 operator assumes a singularly simple form. Give this form, and 
 point out the results indicated by it. 
 
 26. Shew that if the hodograph be a circle, and the accelera- 
 tion be directed to a fixed point ; the orbit must be a conic section, 
 which is limited to being a circle if the acceleration follow any 
 other law than that of gravity. 
 
 27. In the hodograph corresponding to acceleration f{I>) 
 directed towards a fixed centre, the curvature is inversely as 
 
 28. If two circular hodographs, having a common chord, which 
 passes through, or tends towards, a common centre of force, be cut 
 by any two common orthogonals, the sum of the two times of 
 hodographically describing the two intercepted arcs (small or large) 
 will be the same for the two hodographs. (Hamilton, Elements, 
 p. 725.) 
 
 20—2 
 
308 QUATERNIONS, 
 
 29. Employ the last theorem to prove, after Lambert, that the 
 time of describing any arc of an elliptic orbit may be expressed in 
 terms of the chord of the arc and the extreme radii vectores. 
 
 30. If 5 ( ) q'^ be the operator which turns one set of rect- 
 angular unit- vectors a, /8, 7 into another set a^, ^^.y^, shew that 
 there are three equations of the form 
 
 81. If a ray, a, fall on a fine, polished, wire 7, shew that on 
 reflection it forms the surface 
 
 p^iSayy^a^iSypT, 
 a right cone. 
 
 32. Find the path of a point, and the mannef of its descrip- 
 tion, when 
 
 p = (p-a)-'-ip-\-ay\ 
 
 33. In the first problem of § 336 shew that 
 
 ^q'^q = — Vt;, or V . uq'^ = 0. 
 
 Also that (Vvf = - 2V V or 4m~V''m* = 0. 
 
 Again, shew that there are three equations of the form 
 
 dv „ .„„ , „ dv 
 ax ax 
 
 From these last deduce, by a semi-Cartesian process, the result 
 
 u = e'-=CjTp\ 
 as in the text. 
 
 34. Give the exact solution of 
 
 «j = « - /Sfo)V . 0-. (§ 385.) 
 
 [Note that we may assume, a- being given, 
 
 da- = (f)dp, 
 
 where the constituents of ^ are known functions of p. Thus we 
 have what is wanted for the problem above : — viz. 
 
 — ScoV . a- = <f)co ; 
 
 with various other important results, such as 
 
 <f>'o> = — V/SftKT, Vor = ^i(j)i, focj 
 
CHAPTER XII. 
 
 PHYSICAL APPLICATIONS. 
 
 388. We propose to conclude the work by giving a few in- 
 stances of the ready applicability of quaternions to questions of 
 mathematical physics, upon which, even more than on the Geo- 
 metrical or Kinematical applications, the real usefulness of the 
 Calculus must mainly depend — except, of course, in the eyes of 
 that section of mathematicians for whom Transversals and Anhar- 
 monic Pencils, &c. have a to us incomprehensible charm. Of course 
 we cannot attempt to give examples in all branches of physics, nor 
 even to carry very far our investigations in any one branch : this 
 Chapter is not intended to teach Physics, but merely to shew by 
 a few examples how expressly and naturally quaternions seem to 
 be fitted for attacking the problems it presents. 
 
 We commence with a few general theorems in Dynamics — the 
 formation of the equations of equilibrium and motion of a rigid 
 system, some properties of the central axis, and the motion of a 
 solid about its centre of inertia. The student may profitably 
 compare, with the processes in the text, those adopted by Hamilton 
 in his Elements (Book III., Chap. III., Section 8). 
 
 A. Statics of a Rigid System. 
 
 389. When any forces act on a rigid body, the force j8 at the 
 point whose vector is a, &c., then, if the body be slightly displaced, 
 so that a becomes a + Sa, the whole work done against the forces is 
 
 S/Sf/SSa. 
 
310 QUATERNIONS. [390- 
 
 This must vanish if the forces are such as to maintain equilibrium. 
 Hence the condition of equilibrium of a rigid body is 
 
 For a displacement of translation Sa is any constant vector, hence 
 
 S/3 = (1). 
 
 For a rotation-displacement, we have by § 371, e being the axis, 
 and Te being indefinitely small, 
 
 ga=F6a, 
 and ^S.^Vea = 'Z8.eVa^ = 8.eX(rcc^) = 0, 
 whatever be e, hence S . Fa/3 = (2). 
 
 These equations, (1) and (2), are equivalent to the ordinary six 
 equations of equilibrium. 
 
 390. In general, for any set of forces, let 
 
 2/8 = /3„ 
 2.Fay8 = a„ 
 
 it is required to find the points for which the couple a, has its axis 
 coincident with the resultant force /3j. Let j be the vector of such 
 a point. 
 
 Then for it the axis of the couple is 
 
 2.F(a-7);S = a,-F7/3,, 
 and by condition «yS, = a, — VJ^^. 
 
 Operate by >S/S, ; therefore 
 
 ^/3/ = /SaA,- 
 and Vj0, = a, - ^'^ 8u,^, = - /3, Fa,/3,-', 
 
 or y^Va,^-' + y^^, 
 
 a straight line (tbe Central Axis) parallel to the resultant force. 
 
 [If the resultant force and couple be replaced by an equivalent 
 in the form of two forces, /S at a, and /3' at a', we have 
 
 /3-l-/8' = /3^, Fa^ + Fa'y8' = a,. 
 
 The volume of the tetrahedron whose opposite edges are /8, /8' 
 (acting as above stated) is as /S . l3'V(a - a) /3. But 
 
 8 . /3'a/3 = ;S;S'a„ ;Sf . ^W^ = - S/Sa^, 
 
 so that the volume is as 8a^ {/3 + /3') = 8a^0^, a constant whatever 
 pair of equivalent forces be taken.] 
 
393-] PHYSICAL APPLICATIONS. 311 
 
 391. To find the points about which the couple is least. 
 Here !r(aj — F7/3j) = minimum. 
 
 Therefore 8.{cl,- Vj^,) ri3,ry' = 0, 
 
 where y is any vector whatever. It is useless to try 7' = /S^, but 
 we may put it in succession equal to a, and to Va^/3^. Thus 
 
 8.yV.l3Ja,l3, = 0, 
 
 and ( VcL,$,y - /3,^/Sf . yVa^^, = 0. 
 
 Hence 7 = a; Fa,/3i + y^^, 
 
 and by operating with S. Fa,/Sj, we get 
 
 or 7 = Fa,/3.-^ + 2//3,, 
 
 the same locus as in last section. 
 
 392. The couple vanishes if 
 
 a,-F7/3, = 0. 
 
 This necessitates Sa^^^ = 0, 
 
 or the force must be in the plane of the couple. If this be the 
 case, 
 
 still the central axis. 
 
 To assign the values of forces f, ^j, to act at e, e^, and be 
 equivalent to the given system. 
 
 ? + f, = ^.> 
 
 Hence Fe| + Fe, (/3. - f ) = a,, 
 
 and ? = (6 - 6,)-' (a, - Ve,^,) + x(6- e,). 
 
 Similarly for f^. The indefinite terms may be omitted, as they 
 must evidently be equal and opposite. In fact they are any equal 
 and opposite forces whatever acting in the line joining the given 
 points. 
 
 393. If a system of parallel forces act on a rigid body, say 
 
 iCyS at a, &c. 
 they have the single resultant ;SX {00), at a, such that 
 
312 QUATERNIONS. [393- 
 
 Hence, whatever be the common direction of the forces, the 
 resultant passes through 
 
 _ _ S (ica) 
 
 If S (x) = 0, the resultant is simply the couple 
 
 By the help of these expressions for systems of parallel forces 
 we can easily proceed to the case of forces generally. 
 
 Thus if any system of forces, ^, act at points, a, of a rigid body; 
 and if i, j, & be a system of rectangular unit vectors such that 
 
 the resultant force is 
 
 lU actmg at ^^^^, or ^^^^ 
 
 as we may write it. Take this as origin, then <^k = 0. 
 The resultant couple, m the same way, is 
 
 or V{i<l>i +j<fyj). 
 
 Now <l)i, (f}j, (j}k are invariants, in the sense that they retain 
 the same values however the forces and (with them) the system 
 i, j, k be made to rotate : provided they preserve their mutual 
 inclinations, and the forces their points of application. For the 
 as are constant, and quantities of the form S^i, 8^j, or 8^k are 
 not altered by the rotation. 
 
 We may select the positions of i and j so that (f>i and 4>j 
 shall be perpendicular to one another. For this requires only 
 
 /S . (f>i <j)j = 0, or 8 . i<l>'4'j = 0. 
 But (§ 381) (f)'(j) is a self-conjugate function; and, by our change of 
 origin, k is parallel to one of its chief vectors. The desired result 
 is secured if we take i, j as the two others. 
 
 With these preliminaries we may easily prove Minding's 
 Theorem : — 
 
 If a system of forces, applied at given points of a rigid body, 
 have their directions changed in any way consistent with the 
 preservation of their mutual inclinations, they have in an infinite 
 number of positions a single force as resultant. The lines of 
 action of all such single forces intersect each of two curves fixed 
 in space. 
 
393-] PHYSICAL APPLICATIONS. 313 
 
 The condition for the resultant's being a single force in the 
 line whose vector is p is 
 
 bVkp=V(i(l>i+j(j>j), 
 
 which may be written as 
 
 bp = xk —j4>i + i(f>j. 
 
 That the two last terms, together, form a vector is seen by- 
 operating on the former equation by S.k; for we thus have 
 
 Sj(f)i — Si^j = 0. 
 
 We may write these equations for convenience as 
 
 bp = a;k-ja + i^ (1), 
 
 8jci-Si^ = (2). 
 
 [The student must carefully observe that a and yS are now 
 used in a sense totally different from that in which they first 
 appeared, but for which they are no longer required. If this 
 should puzzle him, he may change a into 7, and /S into S, in the last 
 two equations and throughout the remainder of this section.] 
 
 Our object now must be to express i and j in terms of the single 
 variable k, which is afterwards to be eliminated for the final 
 result. 
 
 From (2) we find at once 
 
 yi^Vk^ + kVka I 
 
 yj^-Vka + kVk^ ^'^^' 
 
 whence we easily arrive at either of the following 
 
 or -y' = a' + ^' + (Skaf + (Sk^f + 28 . ka^) ^^■ 
 
 Substituting for i and j in (1) their values (3), we have 
 _ ybp = — zk — aSak — ^8^k — a/3 
 
 = (■as- — z) k — a^ (5), 
 
 where w, which is now used for a linear and vector function, is 
 defined by the equation 
 
 ■57P = - a8ap - /3/S/8/9. 
 
 Obviously w (ayS) = 0, 
 
 so that (-ST - z)-^ (a/S) = - - a/3. 
 
 Thus -yh(zT-zyp = k+'^ (6). 
 
 z 
 
314 QUATERNIONS. 
 
 [394- 
 
 Multiply together the respective members of (5) and (6), and take 
 the scalar, and we have ^ 
 
 fb'Sp (^ -zrp = z- (Sakf - {Sm' - 2Ska^ + ^ , 
 
 or, by (4), =f + z + u' + ^'+^- 
 
 " z 
 
 which, for ^ = - a\ ot z = - /3', gives as the required curves the 
 focal conies of the system 
 
 Sp{^-z)-'p = b-\ 
 
 394." The preceding investigation was based on the properties 
 of a system of parallel forces ; and thus has a somewhat composite, 
 semi-Cartesian, character. 
 
 That which follows is much more purely quaternionic. It is 
 taken from the Trans. R S. E. 1880. 
 
 When any number of forces act on a rigid system ; /3j at the 
 point Hj, /S^ at a^, &c., their resultant consists of the single force 
 
 acting at the origin, and the couple 
 
 K = -tV^OL (1). 
 
 If these can be reduced to a single force, the equation of the 
 line in which that force acts is evidently 
 
 YBp = tV^OL (2). 
 
 Now suppose the system of forces to turn about, preserving 
 their magnitudes, their points of application, and their mutual 
 inclinations, and let us find the fixed curves in space, each of 
 which is intersected by the line (2) in every one of the infinite 
 number of its positions. 
 
 Operating on (2) by V. jS, it becomes 
 
 pB' - 08Bp = t (a/Sf/3^ - 0Sci0) = 0- (fy'B 
 with the notation of Chap. V. Now, however the forces may 
 turn, <f>^ = ta8l30 
 
 is an absolute constant ; for each scalar factor as S0^0 is unaltered 
 by rotation. « Let us therefore change the origin, i.e. the value of 
 each a, so as to make 
 
 Sa-Sf/3^ = (/)3 = (3). 
 
395-] PHYSICAL APPLICATIONS. 315 
 
 This shews that /S is one of the three principal vectors of <ji, and 
 we see in consequence that (f) may be expressed in the form 
 
 ry'Syi ) + S'SBi ); 
 
 where 7 and S are unit vectors, forming with ^ a rectangular 
 system. They may obviously be so chosen that 7' and 8' shall be 
 at right angles to one another, but these (though constants) are 
 not necessarily unit vectors. 
 
 Equation (2) is now 
 
 bri3p = Vyy + rSB' (2'), 
 
 where b is the tensor, and /3 the versor, of 3. 
 
 The condition that the force shall lie in the plane of the couple 
 is, of course, included in this, and is found by operating by /S . /3. 
 
 Thus S(By'-yS')==0 (4). 
 
 395. We have here all the data of the problem, and solutions 
 can only differ from one another in the mode of attacking (2') 
 and (4). 
 
 Writing (4) in the form 
 
 ,Sf7(S'+Fj87') = 0, 
 we have at once ty = F/88' + /SF/87', ■> ,^,. 
 
 whence t8 = - V^y' + ^V^B' J 
 
 where t is an undetermined scalar. 
 
 By means of these we may put (2') in the form 
 
 btV^p = - F. /3 (F7'8' + y'Sy'^ + S'/SfS',8) 
 = -V.^(Vy'S'-i^fi) 
 where w = — 7'^7' ( ) — 8'SS' ( ). 
 
 Let the tensors of 7' and S' be e^, e^ respectively, and let l3' be a 
 unit vector perpendicular to them, then we may write 
 
 6«p = 35/3 - e.e^/S' + OTj8 (5). 
 
 Operating by (w + a;)"', and noting that 
 
 57^' = 0, 
 
 we have 6< (,^7 + «;)> = |S - ^" /3' (5'). 
 
 Taking the scalar of the product of (5) and (5') we have 
 
 b'fSp (^ + xyp = -(x^- e,e,^y + S^i^T^. 
 sc 
 
316 QUATERNIONS. [SQ^- 
 
 But by (4') we have 
 
 f = 8j3^^ + e,' + e,'-2e,e,S/3^' (6), 
 
 so that, finally, 
 
 b'Spi'HT + x) p=l--i ^ {')■ 
 
 Equation (7), in which f is given by (6) in terms of /3, is true 
 for every point of every single resultant. But we get an immense 
 simplification by assuming for x either of the particular values 
 -e^ or -e/. For then the right-hand side of (7) is reduced 
 to unity, and the equation represents one or other of the focal 
 conies of the system of confocal surfaces 
 
 Sp (■07 - hy^p = p , 
 a point of each of which must therefore lie on the line (6). 
 
 396. A singular form, in which Minding's Theorem can be 
 expressed, appears at once from equation (2'). For that equation 
 is obviously the condition that the linear and vector function 
 
 -bpS^i ) + 'y'8'y( ) + S'8B{ ) 
 
 shall denote a pure strain. 
 
 Hence the following problem : — GHven a set of rectangular unit 
 vectors, which may take any initial position : let two of them, after 
 a homogeneous strain, become given vectors at right angles to one 
 another, find what the third must become that the strain may be 
 pure. The locus of the extremity of the third is, for every initial 
 position, one of the single resultants of Minding's system ; and 
 therefore passes through each of the fixed conies. 
 
 Thus we see another very remarkable analogy between strains 
 and couples, which is in fact suggested at once by the general 
 expression for the impure part of a linear and vector function. 
 
 397. The scalar t, which was introduced in equations (4'), is 
 shewn by (6) to be a function of /3 alone. In this connection it 
 is interesting to study the surface of the fourth degree 
 
 Sti^t - (e/ + O T^ - 2e,e^TT80'T = 1, 
 where t = - /8. 
 
 V 
 
 But this may be left as an exercise. 
 
 Another form of t (by 4') is Syy + 8BS'. 
 
399-] PHYSICAL APPLICATIONS. 317 
 
 Meanwhile (6) shews that for any assumed value of ^S there 
 are but two corresponding Minding lines. If, on the other hand, 
 p be given there are in general four values of /3. 
 
 398. For variety, and with the view of further exploring this 
 very interesting question, we may take a different mode of 
 attacking equations (4) and (2'), which contain the whole matter. 
 In what follows 6 will be merged in p, so that the scale of the 
 result will be altered. 
 
 Operating by V. j8 we transform (2') into 
 
 p + ^8^p = -(ySry'^ + BSS'^) (2"). 
 
 Squaring both sides we have 
 
 p'' + S'^p = S^^l3 (8). 
 
 Since ;3 is a unit vector, this may be taken as the equation of a 
 cyclic cone; and every central axis through the point p lies upon it. 
 For we have not yet taken account of (4), which is the condition 
 that there shall be no couple. 
 
 To introduce (4), operate on (2") hy S.y and by 8 . S'. We 
 thus have, by a doiible employment of (4), 
 
 Sy'p + 8y'^8^p = Sy^m 
 
 8S'p + 88'^8^p = 8S^l3\ '^^''• 
 
 Next, multiplying (8) by )S/3ot/S, and adding to it the squares of 
 (9), we have 
 
 p''8/3'^^ - 28^p8l3^p- 8p^p = - 8^^'^ (10). 
 
 This is a second cyclic cone, intersecting (8) in the four directions 
 /3. Of course it is obvious that (8) and (10) are unaltered by the 
 substitution o{ p +y^ for p. 
 
 If we look on /3 as given, while p is to be found, (8) is the 
 equation of a right cylinder, and (10) that of a central surface of 
 the second degree. 
 
 399. A curious transformation of these equations may be made 
 by assuming p^ to be any other point on one of the Minding lines 
 represented by (8) and (10). Introducing the factor —jS''( = 1) in 
 the terms where /8 does not appear, and then putting throughout 
 
 0\\P.-P (11) 
 
 (8) becomes - p\' + 8'pp, = 8(p,- p)^{p,- p) (8'). 
 
 As this is symmetrical in p, p^, we should obtain only the same 
 
318 QUATERNIONS. [4OO. 
 
 result by putting p^ for p in (8), and substituting again for as 
 before. 
 
 From (10) we obtain the corresponding symmetrical result 
 ip' - Spp,) Sp^^p, + ip,' - Spp,) Sp^p = - 8pp,S {p, -p)^ (p, - p) 
 
 -S(p,-p)^'(p,-p) (10'). 
 
 These equations become very much simplified if we assume p and p^ 
 to lie respectively in any two conjugate planes ; specially in the 
 planes of the focal conies, so that SB'p = 0, and Sy'p^ = 0. 
 
 For if the planes be conjugate we have 
 
 ^P^Pi = o> 
 Sp^'p^ = 0, 
 
 and if, besides, they be those of the focal conies, 
 8pp, = -S^'pS^P„ 
 Spw^p = e^Sp'STp, &c., 
 and the equations are 
 
 -p'p^^+S'pp.^Sp^^p. + Sp^p, (8"), 
 
 and p'Spi'urp^ + p^^Sp'orp = - Sp^tff'p^- SpisT^'p (10"). 
 
 From these we have at once the equations of the two Minding 
 curves in a variety of different ways. Thus, for instance, let 
 
 p,=pB' 
 
 and eliminate p between the equations. We get the focal conic in 
 the plane of /3', 7'. In this way we see that Minding lines pass 
 through each point of each of the two curves; and by a similar 
 process that every line joining two points, one on the one curve, 
 the other on the other, is a Minding line. 
 
 400. Another process is more instructive. Note that, by the 
 equations of condition above, we have 
 
 Then our equations become 
 
 Sp^pSp.'arp, p,' + e,' „ p'+e/ „ „„ _ n 
 
 and (p« + e,') 8p,^p, + (p,' + e,') Sp^p = 0. 
 
 If we eliminate p^ or p^^ from these equations, the resultant 
 obviously becomes divisible by Sp'orp or Sp^'syp^, and we at once 
 obtain the equation of one of the focal conies. 
 
403- ] PHYSICAL APPLICATIONS. 319 
 
 401. In passing it may be well to notice that equation (10) 
 may be written in the simpler form 
 
 S . p/S/JW^ + Sp'urp = S^ht'^. 
 
 Also it is easy to see that if we put 
 
 = pSl3p-(w + p')/3 
 
 we have (8) in the form 8/36 = 0, 
 
 and by the help of this (10) becomes 
 
 ff'=Sp'BTp. 
 
 This gives another elegant mode of attacking the problem. 
 
 402. Another valuable transformation of (2") is obtained by 
 considering the linear and vector function, ;^ suppose, by which /3, 
 7, S are derived from the system /3', Uy, US'. For then we have 
 obviously 
 
 P = «'X/3' + X^*X^ (2'"). 
 
 This represents any central axis, and the corresponding form of the 
 Minding condition is 
 
 5f.7'x^"*S' = *Sf.S'xt3-"V (4"). 
 
 Most of the preceding formulae may be looked upon as results of 
 the elimination of the function % from these equations. This forms 
 probably the most important feature of such investigations, so far 
 at least as the quaternion calculus is concerned. 
 
 403. It is evident from (2'") that the vector-perpendicular 
 from the origin on the central axis parallel to %/3' is expressed by 
 
 ■^ = X^^X^'- 
 But there is an infinite number of values of jj; for which Ut is a 
 given versor. Hence the problem; — to find the maximum and 
 minimum values of Tt, when Ut is given — i.e. to find the surface 
 hounding the region which is filled with the feet of perpendiculars on 
 central axes. 
 
 We have Tr' = -8. x^'^X^'> 
 
 = TtS.x^'Ut. 
 
 Hence = 8. ^I^'^^X^'. 
 
 = S.xl3'Ur. 
 But as 2)3' is constant = 8. x^'x^'. 
 
320 QUATERNIONS. [404- 
 
 These three equations give at sight 
 
 where u, u' are unknown scalars. Operate by S . xP' and we have 
 
 - T\-u=0, 
 so that 8t (ot + t')"' t = 0. 
 
 This differs from the equation of Fresnel's wave-surface only in 
 having ra- + t*" instead of w + t"", and denotes therefore the reciprocal 
 of that surface. In the statical problem, however, we have w/S' = 0, 
 and thus the corresponding wave-surface has zero for one of its 
 parameters. [See § 435.] 
 
 [If this restriction be not imposed, the locus of the point 
 
 where </> is now any given linear and vector function whatever, 
 will be found, by a process precisely similar to that just given, to be 
 
 8.{t- </.'/8') (<^> + t'')-' (t - <^'/3') = 0, 
 where <\)' is the conjugate of ^.] 
 
 B. Kinetics of a Rigid System. 
 
 404. For the motion of a rigid system, we have of course 
 
 2<Sf(ma-/3)Sa = 0, 
 
 by the general equation of Lagrange. 
 
 Suppose the displacements Sa to con-espond to a mere transla- 
 tion, then Sa is any constant vector, hence 
 
 2 (ma - /S) = 0, 
 
 or, if Kj be the vector of the centre of inertia, and therefore 
 
 ttjlSm = 2ma, 
 
 we have at once cCjSm - 2/3 = 0, 
 
 and the centre of inertia moves as if the whole mass were 
 concentrated in it, and acted upon by all the applied forces. 
 
 405. Again, let the displacements Sa correspond to a rotation 
 about an axis e, passing through the origin, then 
 
 Sa = Vea, 
 
 it being assumed that Te is indefinitely small. 
 
406.J PHYSICAL APPLICATIONS. 321 
 
 Hence IS.eVa (md - ^8) = 0, 
 
 for all values of e, and therefore 
 
 S.Fa(ma-/3) = 0, 
 
 which contains the three remaining ordinary equations of motion. 
 
 Transfer the origin to the centre of inertia, i.e. put a = a, + ot, 
 then our equation becomes 
 
 2F (a, + tn-) (ma, + mii - /3) = ; 
 
 or, since Smtn- = 0, 
 
 S Fot (m* -I3)+ Fa, (a,2m - 1^) = 0. 
 
 But (§ 404) dy,m - S/3 = 0, 
 
 hence our equation is simply 
 
 SFt3-(mw-/3) = 0. 
 
 Now S Fsr/3 is the couple, about the centre of inertia, produced 
 by the applied forces ; call it f, then 
 
 2mFOT^ = ? (1). 
 
 406. Integrating once, 
 
 l,mV^^ = y + j^dt (2). 
 
 Again, as the motion considered is relative to the centre of 
 inertia, it must be of the nature of rotation about some axis, in 
 general variable. Let e denote at once the direction of, and the 
 angular velocity about, this axis. Then, evidently, 
 
 •dr = VevT. 
 Hence, the last equation may be written 
 2mor Few = 7 + J^dt. 
 Operating hy S.e, we get 
 
 tm{re^Y = Sey + SeJ^dt (3). 
 
 But, by operating directly by 2jS€dt upon the equation (1), we 
 
 get 
 
 Xm{ret^y = -h' + 2J8e^dt (4). 
 
 Equations (2) and (4) contain the usual four integrals of the first 
 order, h being here an arbitrary constant, whose value depends 
 upon the initial kinetic energy of the system. By § 387 we see 
 how the principal moments of inertia are involved in the left 
 member. 
 
 T. Q. I. 21 
 
322 QUATERNIONS. [4°7- 
 
 407. When no forces act on the body, we have ^=0, and 
 
 therefore 
 
 Smtn-Fetn- = 7 (5), 
 
 tm^' = Xm{V6i:TY = -h' (6). 
 
 and, from (5) and (6), Sey^-h" (7). 
 
 One interpretation of (6) is, that the kinetic energy of rotation 
 remains unchanged : another is, that the vector e terminates in an 
 ellipsoid whose centre is the origin, and which therefore assigns 
 the angular velocity when the direction of the axis is given ; 
 (7) shews that the extremity of the instantaneous axis is always 
 in a plane fixed in space. 
 
 Also, by (5), (7) is the equation of the tangent plane to (6) at 
 the extremity of the vector e. Hence the ellipsoid (6) rolls on the 
 plane (7). 
 
 From (5) and (6), we have at once, as an equation which e 
 must satisfy, 
 
 y't.m{ Vet^f = - h' (2 . mw Ve^f. 
 
 This belongs to a cone of the second degree fixed in the body. 
 Thus all the results of Poinsot regarding the motion of a rigid body 
 under the action of no forces, the centre of inertia being fixed, are 
 deduced almost intuitively : and the only difficulties to be met 
 with in more complex properties of such motion are those of 
 integration, which are inherent to the subject, and appear what- 
 ever analytical method is employed. (Hamilton, Proc. R. I. A. 
 1848.) 
 
 If we write (5) as 
 
 +"'e = 7 (5), 
 
 the special notation ^ indicating that this linear and vector 
 function is related to the principal axes of the body, and not to 
 lines fixed in space, and consider the ellipsoid (of which e is a 
 semidiameter) 
 
 Se^-'e = -h' (6), 
 
 we may write the equation of a confocal ellipsoid (also fixed in the 
 body) as 
 
 sp{^+prp=-h' (8). 
 
 Any tangent plane to this is 
 
 S.a{^+p)''p = -h' (9). 
 
408.] PHYSICAL APPLICATIONS. 323 
 
 If this plane be perpendicular to 7, we may write 
 
 so that, by (5) 
 
 p =x{e+py) (10). 
 
 The plane (9) intercepts on 7 a quantity h'/xTy, which is constant 
 by (8) and (10). 
 
 The vector velocity of the point p is 
 
 Vep = px Vey = —p Vyp 
 
 (by two applications of (10)). Hence the point of contact, p, 
 revolves about 7 with angular velocity —pTy. That is, if the 
 plane (9) be rough, and can turn about 7 as an axis, the ellipsoid (8) 
 instead of sliding upon it, will make it rotate with uniform angular 
 velocity. This is a very simple mode of obtaining one of Sylvester's 
 remarkable results. (Phil. Trans. 1866.) 
 
 408. For a more formal treatment of the problem of the 
 rotation of a rigid body, we may proceed as follows : — 
 
 Let a be the initial position of ra-, q the quaternion by which 
 the body can be at one step transferred from its initial position to 
 its position at time t. Then 
 
 ^ = qaq'^ 
 and Hamilton's equation (5) of last section becomes 
 
 S . mqaq'^ V. eqctq'^ = 7, 
 or S . mq {aS . aq'^ eq — q'^ eqa'] q~^ = 7. 
 
 The vector 7 is now written for 7 + J^dt of § 406, as f is required 
 for a new purpose. Thus 7 represents the resultant moment of 
 momentum, and will be constant only if there is no applied couple. 
 
 Let (^/3 = 2 . TO (ajSap — a^p) (1), 
 
 where <f) (compare § 387) is a self-conjugate linear and vector 
 fimction, whose constituent vectors are fixed in the body in its 
 initial position. Then the previous equation may be written 
 
 q(f> (q-' eq) q-' = y, 
 or <l>(q'^eq) = q-'yq. 
 
 For simplicity let us write 
 
 q''yq = u 
 
 Then Hamilton's dynamical equation becomes simply 
 
 <^'7 = ? (3). 
 
 21—2 
 
324 QUATERNIONS. [409- 
 
 409. It is easy to see what the new vectors r; and ^ represent. 
 For we may write (2) in the form 
 
 ^=7:;i (2'); 
 
 from which it is obvious that t] is that vector in the initial position 
 of the body which, at time t, becomes the instantaneous axis in the 
 moving body. When no forces act, 7 is constant, and ^ is the 
 initial position of the vector which, at time t, is perpendicular to 
 the invariable plane. 
 
 410. The complete statement of the problem is contained in 
 equations (2), (3) above, and (4) of | 372* Writing them again, 
 we have 
 
 qv = ^ (4), 
 
 yq = q^ (2), 
 
 </»? = ? (3). 
 
 We have only to eliminate f and r), and we get 
 
 2q = qi>-'{q-'yq) (5), 
 
 in which q is now the only unbnown ; 7, if variable, being supposed 
 given in terms of q and t. 
 
 It is hardly conceivable that any simpler, or more easily 
 interpretable, expression for the motion of a rigid body can be 
 presented until symbols are devised far more comprehensive in 
 their meaning than any we yet have. 
 
 411. Before entering into considerations as to the integration 
 of this equation, we may investigate some other consequences of 
 the group of equations in § 410. Thus, for instance, differentiating 
 (2), we have 
 
 yq+yq = q^+qt 
 
 and, eliminating q by means of (4), 
 
 yqr] + 2'Yq = qv^+2qt 
 
 * To these it is unnecessary to add 
 
 Tq = constant, 
 
 as this constancy of Tq is proved by the form of (4). For, liad Tq been variable, 
 there must have been a quaternion in the place of the vector 17. In fact, 
 
 f^(Tq)'' = 2!^.qKq = {Tq)^Sri = 0. 
 
412.] PHYSICAL APPLICATIONS. 325 
 
 whence, eliminating 7 by the help of (2), 
 
 which gives, in the case when no forces act, the forms 
 
 r='f^rrr (6), 
 
 and (as ^=<^ri) 
 
 ^rj = - V .7)^7) (7). 
 
 To each of these the term q~^<yq must be added on the right, if 
 forces act. 
 
 412. It is now desirable to examine the formation of the func- 
 tion (^. By its definition § 408, (1), we have 
 
 (^/3 = S . m {aSap — a^p), 
 
 = — S.maFa/9. 
 
 Hence -Sp<^p = t .m{TVap)\ 
 
 so that — Spj>p is the moment of inertia of the body about the 
 vector p, multiplied by the square of the tensor of p. Compare 
 § 387. Thus the equation 
 
 Spc^p = — K\ 
 
 evidently belongs to an ellipsoid, of which the radii-vectores are 
 inversely as the square roots of the moments of inertia about them; 
 so that, if i,j, k be taken as unit-vectors in the directions of its 
 axes respectively, we have 
 
 Sjj>j = -B\ (8), 
 
 8k4>k = -G] 
 
 A, B, G, being the principal moments of inertia. Consequently 
 
 (j>p = -{AiSip + Bj8jp + Ck8kp} (9). 
 
 Thus the equation (7) for r/ breaks up, if we put 
 
 into the three following scalar equations 
 
 A(b^ + (G-B)(o^^w, = 0\ 
 Bcb,+ {A-G)co^co^ = 0\, 
 G(b^+(B-A)(o^o}, = 0\ 
 
 which are the same as those of Euler. Only, it is to be understood 
 that the equations just written are not primarily to be considered 
 as equations of rotation. They rather express, with reference to 
 
326 QUATERNIONS. [413- 
 
 fixed axes in the initial position of the body, the motion of the 
 extremity, w,, w^, Wg, of the vector corresponding to the instan- 
 taneous axis in the moving body. If, however, we consider «,, w^, 
 ojg as standing for their values in terms of w, x, y, ^ (§ 416 below), 
 or any other coordinates employed to refer the body to fixed axes, 
 they are the equations of motion. 
 
 Similar remarks apply to the equation which determines ^, for 
 if we put 
 
 (6) may be reduced to three scalar equations of the form 
 
 '^ \G B 
 
 „ ,OT„1i7„= 0. 
 
 413. Euler's equations in their usual form are easily deduced 
 from what precedes. For, let 
 
 whatever be p ; that is, let c|> represent with reference to the moving 
 principal axes what ^ represents with reference to the principal 
 axes in the initial position of the body, and we have 
 
 = qtq-' =qV{^4>-'0q-' ' 
 
 = -qV(r)(t>v)q~' 
 
 = —V. qrj^Tjq'^ 
 
 = -V.qvq-\4>{q-'eq)q-' 
 
 = -F.6,|,e, 
 which is the required expression. 
 
 But perhaps the simplest mode of obtaining this equation is to 
 start with Hamilton's unintegrated equation, which for the case 
 of no forces is simply 
 
 % . mV'ST'ii = 0. 
 
 But from tir = Few 
 
 we deduce ■n-=Fetir+ Feisr 
 
 = ■nre''' — eSe^ + Feot, 
 so that S . m ( VewSe^iy — ezr" + vrSevr) = 0. 
 
 If we look at equation (1), and remember that <}> differs from </> 
 simply in having ct substituted for a, we see that this may be 
 written 
 
 Ve^e + (|>6 = 0, 
 
41 5- J PHYSICAL APPLICATIONS. 327 
 
 the equation before obtained. The first mode of arriving at it has 
 been given because it leads to an interesting set of transformations, 
 for which reason we append other two. 
 
 By (2) 
 
 7 = qKi \ 
 
 therefore = qq~\ q^q~^ + q^q''- - q^q~'qq~\ 
 
 or qiq-' = 2V.y Vqq-' 
 
 = F76. 
 
 But, by the beginning of this section, and by (5) of § 407, this 
 is again the equation lately proved. 
 
 Perhaps, however, the following is neater. It occurs in Hamil- 
 ton's Elements. 
 
 By (5) of § 407 <|,e = y. 
 
 Hence (j)e = — <|>e = — 2 . m (tb- Vew + •sr Fe^) 
 
 = — S . nH'iySe'sr 
 
 = — F. eS . m'SjSem 
 
 = -Fe<j.e. 
 
 414. However they are obtained, such equations as those of 
 § 412 were shewn long ago by Euler to be integrable as follows. 
 
 Putting 2/a>j(»2«3di = s, 
 
 we have Aw^ = AQ.^ + {B-G)s, 
 
 with other two equations of the same form. Hence 
 
 ds 
 
 2dt = 
 
 /„, B-G\i/^, G-A\i(^, A-B\i' 
 
 so that t is known in terms of s by an elliptic integral. Thus, 
 finally, r] or ^ may be expressed in terms of t ; and in some of the 
 succeeding investigations for q we shall suppose this to have, been 
 done. It is with this integration, or an equivalent one, that most 
 writers on the farther development of the subject have commenced 
 their investigations. 
 
 415. By § 406, y is evidently the vector moment of momen- 
 tum of the rigid body ; and the kinetic energy is 
 
 — "I S . m'6j'' = — h Sey. 
 
 But Sey = S . q'^eqq'^yq = Sr]^, 
 
328 QUATERNIONS. [4 1 6. 
 
 SO that when no forces act 
 
 But, by (2), we have also 
 
 T^=Ty, or T<f>v = Ty, 
 
 so that we have, for the equations of the cones described in the 
 initial position of the body by i? and f, that is, for the cones de- 
 scribed in the moving body by the instantaneous axis and by the 
 perpendicular to the invariable plane, 
 
 This is on the supposition that 7 and h are constants. If forces act, 
 these quantities are functions of t, and the equations of the cones 
 then described in the body must be found by eliminating t between 
 the respective equations. The final results to which such a process 
 will lead must, of course, depend entirely upon the way in which t 
 is involved in these equations, and therefore no general statement 
 on the subject can be made. 
 
 416. Recurring to our equations for the determination oiq, and 
 taking first the case of no forces, we see that, if we assume r] to 
 have been found (as in § 414) by means of elliptic integrals, we 
 have to solve the equation 
 
 qrj = 2q*, 
 
 * To get an idea of the nature of this equation, let us integrate it on the suppo- 
 sition that J) is a constant vector. By differentiation and substitution, we get 
 
 Hence q = Q^0OB-^ t + Q^ sin -^ t. 
 
 Substituting in the given equation we have 
 
 T„ ^ - Qi sin ^ t + §2 cos ^ t^ = (q^ cos ''^ t + Q„ sin ^^ «) ^. 
 Hence Tr,.Q.,= Q^n, 
 
 -Tn.Q, = Q^V, 
 which are virtually the same equation, and thus 
 
 g = (9i ^cos ~ t+Pijsin-^M 
 
 tTr, 
 
 And the interpretation of j ( ) g-i will obviously then be a rotation about tj 
 through the angle tTij, together with any other arbitrary rotation whatever. Thus 
 any position whatever may be taken as the initial one of the body, and Q^ ( ) Q^-' 
 brings it to its required position at time 8 = 0. 
 
4 1 7-] PHYSICAL APPLICATIONS. 329 
 
 that is, we have to integrate a system of four other differential 
 equations harder than the first. 
 
 Putting, as in § 412, 
 
 where «,, co^, w^ are supposed to be known functions of i, and 
 
 g = w + ix +jy + kz, 
 
 ,, . , . \ J. dw dx dy dz 
 
 this system is s ft* = tit = -i^ = -fI = ^ . 
 
 •^ 2 W X Y Z' 
 
 where W = - m^x - m^ — w^z, 
 
 X = co^w + co^y — Q}^, 
 
 Y = a)^w + w^z — (Bg«, 
 
 Z= Q},w + (o^x - (o,y; 
 
 or, as suggested by Cayley to bring out the skew symmetry, 
 
 X — w^y — (n^z + ft)jW, 
 
 F = — (i)^x . + (o^z + &)„w, 
 
 ^ = a^x — ai^y . + co^w, 
 
 W = -o}^x- w^y -w^z . . 
 
 Here, of course, one integral is 
 
 w^ + x' +y^ + z^ = constant. 
 
 It may suffice thus to have alluded to a possible mode of 
 solution, which, except for very simple values of t), involves very 
 great difficulties. The quaternion solution, when rj is of constant 
 length and revolves uniformly in a right cone, will be given 
 later. 
 
 417. If, on the other hand, we eliminate 77, we have to 
 integrate 
 
 q^-' (q-'yq) = 2q, 
 
 so that one integration theoretically suffices. But, in consequence 
 of the present imperfect development of the quaternion calculus, 
 the only known method of effecting this is to reduce the quaternion 
 equation to a set of four ordinary differential equations of the first 
 order. It may be interesting to form these equations. 
 
 Put q = w + ix +jy + kz, 
 
 j = ia +jb + kc, 
 
330 QUATERNIONS. [417- 
 
 then, by ordinary quaternion multiplication, we easily reduce the 
 given equation to the following set : 
 
 dt _d'w _dx _dy _ dz 
 2~W~X~'Y~'Z' 
 
 where 
 
 W = -x^-y^-z(S, or Z= . yOL-M + w^, 
 
 X= w^^-y(&-zm Y=-x<B . +z^ + w^, 
 
 Y= w^ + z^-x(& Z= x^-y^ . +w€, 
 
 Z= wQt+xm-ym W = -x^-y^-z(!L . , 
 
 and 
 
 a = ^ [a (w" - a;' - 2/" - z") + ^(c{ax + hy + cz) + 2w {hz - cy)], 
 23 = 4 [6 (w" - «" - 2/' - ^') + ^y(ax + by+ cz) + 2w (ex - az)], 
 
 GD = ^ [c [vf- af-y^- z^) + 2z{ax + by+ cz) + 2w (ay- bx)]. 
 
 W, X, Y, Z are thus homogeneous functions of w, x, y, z of the third 
 degree. 
 
 Perhaps the simplest way of obtaining these equations is to 
 translate the group of § 410 into w, x, y, z at once, instead of 
 using the equation from which f and t? are eliminated. 
 
 We thus see that 
 
 77 = tm+j23 + A;eD. 
 One obvious integral of these equations ought to be 
 w^ + x^ + y'^ + z^ = constant, 
 which has been assumed all along. In fact, we see at once that 
 
 wW + xX-\-yY+zZ=0 
 identically, which leads to the above integral. 
 
 These equations appear to be worthy of attention, partly 
 because of the homogeneity of the denominators W, X, Y, Z, but 
 particularly as they afford (what does not appear to have been 
 sought) the means of solving this celebrated problem at one step, 
 that is, without the previous integration of Euler's equations 
 (§ 412). 
 
 A set of equations identical with these, but not in a homo- 
 geneous form (being expressed, in fact, in terms of k, X, fi, v of 
 § 375, instead of w, x, y, z), is given by Cay ley (Gamb. and Bvb. 
 
4 1 9-] PHYSICAL APPLICATIONS. 331 
 
 Math. Journal, vol. i. 1846), and completely integrated (in the 
 sense of being reduced to quadratures) by assuming Euler's 
 equations to have been previously integrated. (Compare § 416.) 
 
 Cayley's method may be even more easily applied to the above 
 equations than to his own ; and I therefore leave this part of the 
 development to the reader, who will at once see (as in § 416) that 
 ^, 23, ® correspond to m^, o^, a), of the ij type, § 412. 
 
 418. It may be well to notice, in connection with the formulae 
 for direction cosines in § 375 above, that we may write 
 
 ^ = j[a (w^+ x^ -y" - z') + 26 {xy + wz) + 2c {xz - wy)\ 
 33 = -n [2a {xy - wz) + b {y/ -x^ + y^- z^) +'2c {yz + wx)\ 
 
 CD = ^[2a {xz + wy) + 26 {yz — wx) +c {w' -x' — y'' + z'')]. 
 
 These expressions may be considerably simplified by the usual 
 assumption, that one of the fixed unit-vectors {i suppose) is 
 perpendicular to the invariable plane, which amounts to assigning 
 definitely the initial position of one line in the body ; and which 
 gives the relations 
 
 6 = 0, c = 0. 
 
 419. When forces act, 7 is variable, and the quantities a, 6, c 
 will in general involve all the variables w, x, y, z, t, so that the 
 equations of last section become much more complicated. The 
 type, however, remains the same if 7 involves t only ; if it involve 
 q we must differentiate the equation, put in the form 
 
 7 = 2q^ {q-'q) q-\ 
 
 and we thus easily obtain the differential equation of the second 
 order 
 
 i/r = 4F. ^<^ {q-'q) q-' + ^qfj, { V. q-'q) q-' ; 
 
 if we recollect that, because q'^'q is a vector, we have 
 
 S.q-'q^{q-'qf. 
 
 Though the above formula is remarkably simple, it must, in the 
 present state of the development of quaternions, be looked on as 
 intractable, except in certain very particular cases. 
 
332 QUATERNIONS. [420. 
 
 420. Auother mode of attacking the problem, at first sight 
 entirely different from that in § 408, but in reality identical with 
 it, is to seek the linear and vector function which expresses the 
 Homogeneous Strain which the body must undergo to pass from its 
 initial position to its position at time t. 
 
 Let OT = ^a, 
 
 a being (as in § 408) the initial position of a vector of the body, 
 OT its position at time t. In this case % is a linear and vector 
 function. (See § 376.) 
 
 Then, obviously, we have, tn-j being the vector of some other 
 point, which had initially the value a,, 
 
 /Sotct, = S . %«%«! = Saa^, 
 (a particular case of which is 
 
 T'ST = Txa=Ta) 
 and Ftn-OTj = V. x'^X'^i ~ xY^-^'-v 
 
 These are necessary properties of the strain-function X' depending 
 on the fact that in the present application the system is rigid. 
 
 421. The kinematical equation 
 
 •ir =l^ecr 
 becomes xp'- = ^- ^X" 
 
 (the function ^ being formed from x by the differentiation of its 
 constituents with respect to t). 
 
 Hamilton's kinetic equation 
 
 2 . tn'S! FeOT = ly, 
 becomes S . mxpi V. e^cc = y- 
 
 This may be written 
 
 S . m (%a/S . exa. — ea^) = 7, 
 or 2 . m (a»Sf . a^'e - %"'e . a') = x''y. 
 
 where x is the conjugate of x- 
 
 But, because S . x^X^i = 'S'aai, 
 
 we have <Saa, = S . o-xx^v 
 
 whatever be a and a,, so that 
 
 X = %"' 
 Hence X . m {aS . oLx^e — x^e . a') = x'^Jt 
 or, by §408 <Ax"'e = %'V 
 
424-] PHYSICAL APPLICATIONS. 333 
 
 422. Thus we have, as the analogues of the equations in 
 §§ 408, 409, 
 
 %''« = n, 
 
 x"V = ?■. 
 
 and the former result %a = V. e;y;a 
 
 becomes pj^a = V. %^%a = ^Vrja. 
 
 This is our equation to determine t^, »? being supposed known. 
 To find 7] we may remark that 
 
 and ?=%"V 
 
 But xx'''^ = «> 
 
 so that XX~'« + XX'' a = 0- 
 
 Hence ?=-%"'%a;"V 
 
 = -v-vx"j-nv = v.^ct>-% 
 
 or ^fj = — Vrjcf)!]. 
 
 These are the equations we obtained before. Having found r) 
 from the last we have to find ^ from the condition 
 
 %"'%« = V-na. 
 
 423. We might, however, have eliminated t; so as to obtain 
 an equation containing ^ alone, and corresponding to that of 
 § 410. For this purpose we have 
 
 so that, finally, x"'%« = ^- 0"'x"Va, 
 
 or %^a=l^- %"'«</>" VV, 
 
 which may easily be formed from the preceding equation by 
 
 putting x^a for a, and attending to the value of %"' given in last 
 section. 
 
 424. We have given this process, though really a disguised 
 form of that in §§ 408, 410, and though the final equations to 
 which it leads are not quite so easily attacked in the way of 
 integration as those there arrived at, mainly to shew how free a 
 use we can make of symbolic functional operators in quaternions 
 
334 QUATERNIONS. [42 5- 
 
 without risk of error. It would be very interesting, however, 
 to have the problem worked out afresh from this point of view by 
 the help of the old analytical methods : as several new forms of 
 long-known equations, and some useful transformations, would 
 certainly be obtained. 
 
 425. As a verification, let us now try to pass from the final 
 equation, in % alone, of § 423 to that of § 410 in q alone. 
 
 We have, obviously, 
 
 OT = qixq^ = %a, 
 
 which gives the relation between q and %. 
 
 [It shews, for instance, that, as 
 
 S.^X^ = S.oix% 
 
 while S.l3xa = S. ^qaq'' = S . aq-'^q, 
 
 we have X'/3 = q'^^q, 
 
 and therefore that xx'^ = 1 i'f^^l) ?' = ^' 
 
 or X — X^' ^^ above.] 
 
 Differentiating, we have 
 
 qaq-"- - qaq'^ qq~' = x«- 
 
 Hence X'X*" — q^qa — aq^q 
 
 = 2V.V{q-^q)a. 
 
 Also (/>"' x"V = <^~' (q'^yq), 
 
 so that the equation of § 423 becomes 
 
 2V.V(q-'q)a=V.cf>-'{q-'yq)a, 
 
 or, as a may have any value whatever, 
 
 2V.q-^q = <t>-'{q-'ryq), 
 
 which, if we put Tq = constant 
 
 as was originally assumed, may be written 
 
 2q = q4>-' (q-'yq), 
 as in §410. 
 
 G. Special Kinetic Problems. 
 
 426. To form the equation for Precession and Nutation. Let 
 o- be the vector, from the centre of inertia of the earth, to a particle 
 m of its mass: and let p be the vector of the disturbing body, whose 
 mass is M. The vector-couple produced is evidently 
 
426. j PHYSICAL APPLICATIONS. 335 
 
 ■Ml.m 
 
 Vcyp 
 
 rip -a) 
 
 = MX 
 
 mVcrp 1 
 
 •^> / 2Sap TV 
 
 V^ + T'p ^ r, 
 
 rr 
 
 no farther term.s being necessary, since -^ is always small in the 
 
 actual cases presented in nature. But, because (t is measured from 
 the centre of inertia, 
 
 2 . mo- = 0. 
 
 Also, as in § 408, ^p = S . m (aSap — a-^p). 
 
 Thus the vector-couple required is 
 
 y^ y-p<Pp- 
 
 Referred to coordinates moving with the body, becomes ^ as in 
 § 413, and § 413 gives 
 
 ^e = y = SMJ^dt. 
 
 Simplifying the value of <|> by assuming that the earth has two 
 principal axes of equal moment of inertia, we have 
 
 Be -{A- B) aSae = vector-constant +3M(A- B) f p^ dt. 
 This gives 8ae = const. = 12, 
 
 whence e = — Oa -I- ad, 
 
 so that, finally, 
 
 BVa:d-An,a = ^{A-B) VapSap. 
 
 The most striking peculiarity of this equation is that the form 
 of the solution is entirely changed, not modified as in ordinary cases 
 of disturbed motion, according to the nature of the value of p. 
 
 Thus, when the right-hand side vanishes, we have an equation 
 which, in the case of the earth, would represent the rolling of a 
 cone fixed in the earth on one fixed in space, the angles of both 
 being exceedingly small. 
 
336 QUATERNIONS. [42 7- 
 
 If p be finite, but constant, we have a case nearly the same as 
 that of a top, the axis on the whole revolving conically about p. 
 
 But if we assume the expression 
 
 p=r(j cos mt + k sin mt), 
 
 (which represents a circular orbit described with uniform speed,) 
 a revolves on the whole conically about the vector i, perpendicular 
 to the plane in which p lies. (§§ 408—426, Trans. R. S. E., 
 1868—9.) 
 
 427. To form the equation of motion of a simple pendulum, 
 taking account of the earth's rotation. Let a be the vector (from 
 the earth's centre) of the point of suspension, X its inclination to 
 the plane of the equator, a the earth's radius drawn to that point; 
 and let the unit- vectors i,j, k be fixed in space, so that i is parallel 
 to the earth's axis of rotation ; then, if to be the angular velocity 
 of that rotation 
 
 a — a[i sin X + (j cos oot + k sin mt) cos X] (1). 
 
 This gives d = aw {—j sin cot + k cos (ot) cos X 
 
 = (oVia (2). 
 
 Similarly 'd = w Via = — w^ (a — ai sin X) (3). 
 
 428. Let p be the vector of the bob m referred to the point of 
 suspension, R the tension of the string, then if Hj be the direction 
 of pure gravity 
 
 m{a + p) = — mg Ua^ — RUp (4), 
 
 which may be written 
 
 Vpd+Vpp^^Va,p (5). 
 
 To this must be added, since r (the length of the string) is 
 constant, 
 
 Tp = r (6), 
 
 and the equations of motion are complete. 
 
 429. These two equations (5) and (6) contain every possible case 
 of the motion, from the most infinitesimal oscillations to the most 
 rapid rotation about the point of suspension, so that it is necessary 
 to adapt different processes for their solution in different cases. 
 We take here only the ordinary Foucault case, to the degree of 
 approximation usually given. 
 
430. ] PHYSICAL APPLICATIONS. 337 
 
 430. Here we neglect terms involving w^ Thus we write 
 
 a = 0, 
 
 and we write a for a^, as the difference depends upon the ellipticity 
 of the earth. Also, attending to this, we have 
 
 p = -\ + ^ (7)^ 
 
 where (by (6)) >Sfaii7 = (8), 
 
 and terms of the order -^^ are neglected. 
 With (7), (5) becomes 
 
 Faro- = - Fa-CT ; 
 
 a a 
 
 so that, if we write - = ri^ (9), 
 
 we have Va (ot + nV) = (10). 
 
 Now, the two vectors ai — a sin X and Via 
 
 have, as is easily seen, equal tensors ; the first is parallel to the line 
 drawn horizontally northwards from the point of suspension, the 
 second horizontally eastwards. 
 
 Let, therefore, sr = x{ai — asin X) + yVia. (11), 
 
 which (x and y being very small) is consistent with (6). 
 
 From this we have (employing (2) and (3), and omitting ui^) 
 
 ■s! = x {ai — a sin X) + ^ Via. — xw sin X Via — yco(a — ai sin X), 
 TS! = x(ai — a sin X) + yVia — 2xco sin X Via — 2y(o (a — ai sin X). 
 
 With this (10) becomes 
 Va [x (ai — a sin X) -I- y Via — 2xco sin XFia — 2y(o (a — ai sin X) 
 
 + n'x (ai — a sin X) + r^yVioC^ = 0, 
 or, if we note that V.aVia = a(ai — a sin X), 
 {—X — 2jfoa sin X - in'x) aVia + (y- 2xw sin X +n^y)a(ai—aB\n\) = 0. 
 
 This gives at once x + n?x + 2(oy sin X = 0] ,.. „, 
 
 y + n^y — 2a)x sin X = OJ 
 
 which are the equations usually obtained; and of which the solution 
 is as follows : — 
 
 If we transform to a set of axes revolving in the horizontal plane 
 at the point of suspension, the direction of motion being from the 
 T. Q. I. 22 
 
338 QUATERNIONS. [431- 
 
 positive (northward) axis of x to the positive (eastward) axis of y, 
 with angular velocity O, so that 
 
 x = ^ cos O^ — 7? sin Oil ,-. „-. 
 
 2/ = ^ sin 0< + ■)? cosfiij 
 
 and omit the terms in O^ and in cbO (a process justified by the 
 results, see equation (15)), we have 
 
 (^ + n^^) cos D,t — (rj + n'r)) sin fit — 2y{D, — w sin X.) = ] .... 
 
 i^ + ii'^) sin nt+{ij + nS) cos IW + 2*(0 - &> sin A,) = 1 " " 
 
 So that, if we put fl = w sin X (15), 
 
 we have simply | + n^f = 01 
 
 7? + nr] = OJ 
 
 the usual equations of elliptic motion about a centre of force in the 
 centre of the ellipse. {Proc. R. S. E., 18G9.) 
 
 D. Geometrical and Physical Optics. 
 
 4.31. To construct a reflecting surface from which rays, emitted 
 from a point, shall after reflection diverge uniformly, hut horizontally. 
 
 Using the ordinary property of a reflecting surface, we easily 
 obtain the equation 
 
 S.dp{^-±^)\^0. 
 
 P 
 
 By Hamilton's grand Theory of Systems of Rays, we at once 
 write down the second form 
 
 Tp-T{^ + ci Vap) = constant. 
 
 The connection between these is easily shewn thus. Let -sr and 
 T be any two vectors whose tensors are equal, then 
 
 /T^y ^ 1 + 2^r-' + {-^T-y 
 
 = 2i;rT-' (1 + Svtt") (Chapter III Ex. 2), 
 whence, to a scalar factor prhs, we have 
 
 Hence, putting 'st= U{^ + aVap) and t= Up, we have from the 
 first equation above 
 
 S.dp [Up + U(^ + aVap)] = 0. 
 
432.] PHYSICAL APPLICATIONS. 339 
 
 But d(^ + aVap) = aVadp = — dp — aSa.dp, 
 
 and S.a{0 + aVap)==O, 
 
 so that we have finally 
 
 S.dpUp-S.d(0 + aVap) U(l3 + aVap) = 0, 
 
 which is the differential of the second equation above. A curious 
 particular case is a parabolic cylinder, as may be easily seen 
 geometrically. The general surface has a parabolic section in the 
 plane of a, /8 ; and a hyperbolic section in the plane of /3, a/8. 
 
 It is easy to see that this is but a single case of a large class of 
 integrable scalar functions, whose general type is 
 
 S.dp(^-^]''p = 0, 
 
 p 
 
 the equation of the reflecting surface ; while 
 
 8{a-p)da- = 
 
 is the equation of the surface of the reflected wave : the integral 
 of the former being, by the help of the latter, at once obtained in 
 the form 
 
 Tp±T(a-p) = constant*. 
 
 432. We next take Fresnel's Theory of Double Refraction, but 
 merely for the purpose of shewing how quaternions simplify the 
 processes required, and in no way to discuss the plausibility of the 
 physical assumptions. 
 
 Let ii3- be the vector displacement of a portion of the ether, 
 with the condition 
 
 ^=^ = -1 (1), 
 
 the force of restitution, on Fresnel's assumption, is 
 
 t {aSSiw + H'jSj-BT + c'kSk'ST) = t^'sr, 
 
 using the notation of Chapter V. Here the function <j) is the 
 negative of that of Chapter IX. (the force of restitution and the 
 displacement being on the whole towards opposite parts), and it is 
 clearly self-conjugate, a', b\ c" are optical constants depending on 
 the crystalline medium, and on the wave-length of the light, and 
 may be considered as given. 
 
 Fresnel's second assumption is that the ether is incompressible, 
 or that vibrations normal to a wave front are inadmissible. If, 
 
 * Proc. R. S. E., 1870-71. 
 
 22—2 
 
340 QUATERNIONS. [433- 
 
 then, a be the unit normal to a plane wave in the crystal, we have 
 of course 
 
 a'' = -l (2), 
 
 and /SttOT = (3) ; 
 
 but, and in addition, we have 
 
 w~^V-u7<f>T!7 II a, 
 
 or S.avT^ = Q (4). 
 
 This equation (4) is the embodiment of Fresnel's second assump- 
 tion, but it may evidently be read as meaning, the vormal to the 
 front, the direction of vibration, and that of the force of restitution 
 are in one plane. 
 
 433. Equations (3) and (4), if satisfied by ct, are also satisfied 
 by CTa, so that the plane (3) intersects the cone (4) in two lines 
 at right angles to each other. That is, for any given wave front 
 there are two directions of vibration, and they are perpendicular to 
 each other. 
 
 434. The square of the normal speed of propagation of a 
 plane wave is proportional to the ratio of the resolved part of the 
 force of restitution in the direction of vibration, to the amount of 
 displacement, hence 
 
 Hence Fresnel's Wave-surface is the envelop of the plane 
 
 Sap = y*SiOT(^CT (5), 
 
 with the conditions tr-^ = — 1 (1), 
 
 «'=-! (2), 
 
 Saw = (3), 
 
 S . aiB-^OT = (4). 
 
 Formidable as this problem appears, it is easy enough. From (3) 
 and (4) we get at once, 
 
 x-a7= V . aVacjizy. 
 
 Hence, operating by (S . ot, 
 
 — a; = — S'!!T(f>'ST = — v^. 
 
 Therefore (^ + v') ot = - aSa^-sr, 
 
 and S.a{<p + v"-pa==0 (6). 
 
435-] PHYSICAL APPLICATIONS. 341 
 
 In passing, we may remark that this equation gives the normal 
 speeds of the two rays whose fronts are perpendicular to a.. In 
 Cartesian coordinates it is the well-known equation 
 
 l^_ m' n^ 
 
 „'=_«-' + 6'= _t,^ + ^^37- "• 
 
 By this elimination of ct, our equations are reduced to 
 
 ;S.a((/. + «Va = (6), 
 
 v = -Sap (5), 
 
 a^ = -l (2). 
 
 They give at once, by § 326, 
 
 (^ + vya + vpSa. (<j> + vy'oL = ha. 
 Operating by /S . a we have 
 
 v'SoL(<l) + vYoc = h. 
 Substituting for h, and remarking that 
 
 Sa (</) + vTa. = -ri<l) + vy'a, 
 because <^ is self-conjugate, we have 
 
 This gives at once, by rearrangement, 
 
 ?; (<^ + !)=)-'« = (</>- /'TV- 
 
 TT / . 2\-l VOL — P 
 
 Hence i^-p) P = -J^^- 
 
 Operating by S.p on this equation we have 
 
 Sp{<i>-pTp — l (7), 
 
 which is the required equation. 
 
 [It will be a good exercise for the student to translate the last 
 ten formulae into Cartesian coordinates. He will thus reproduce 
 almost exactly the steps by which Archibald Smith* first arrived 
 at a simple and symmetrical mode of effecting the elimination. 
 Yet, as we shall presently see, the above process is far from being 
 the shortest and easiest to which quaternions conduct us. J 
 
 435. The Cartesian form of the equation (7) is not the usual 
 one. It is, of course, 
 
 x' y' / 
 
 a'-r' 
 
 ^V-r'^c'-r' 
 
 * Cambridge Phil. Trans., 1835. 
 
342 QUATERNIONS. [436- 
 
 But write (7) in the form 
 
 and we have the usual expression 
 
 aV Vy c'z' „ 
 
 The last-written quaternion equation can also be put into either 
 of the new forms 
 
 or T{p-''-4>-y^p = o. 
 
 436. By applying the results of §§ 183, 184 we may introduce 
 a multitude of new forms. We must confine ourselves to the most 
 simple ; but the student may easily investigate others by a process 
 precisely similar to that which follows. 
 
 Writing the equation of the wave as 
 
 where we have 9 — ~ p'\ 
 
 we see that it may be changed to 
 
 _;S/, ((/)-' + /i)-v = o, 
 if mSp(f)p = ghp' = — h. 
 
 Thus the new form is 
 
 8p (<!>-' -mSp4,prp = (1). 
 
 Here m = -5T5-2 , 8p4>p = aV + &y + cV, 
 
 and the equation of the wave in Cartesian coordinates is, putting 
 
 x' f z" 
 
 h'c' - r 
 
 437. By means of equation (1) of last section we may easily 
 prove Pliicker's Theorem : — 
 
 The Wave-Surface is its own reciprocal with respect to the 
 ellipsoid whose equation is 
 
 ,K 1 
 
 sprp = -, ■ 
 
438.] PHYSICAL APPLICATIONS. 343 
 
 The equation of the plane of contact of tangents to this surface 
 from the point whose vector is p is 
 
 8^A^p = -^. 
 
 The reciprocal of this plane, with respect to the unit-sphere 
 about the origin, has therefore a vector a where 
 
 a = \/tn . <^ p. 
 
 Hence p = -; — (f>~"cr, 
 
 and when this is substituted in the equation of the wave we have 
 for the reciprocal (with respect to the unit-sphere) of the reciprocal 
 of the wave with respect to the above ellipsoid, 
 
 S.aU-^- S(7cj}-'a\ ' o- = 0. 
 
 This differs from the equation (1) of last section solely in having 
 ^~' instead of <f), and (consistently with this) 1/m instead of m. 
 Hence it represents the index-surface. The required reciprocal 
 of the wave with reference to the ellipsoid is therefore the wave 
 itself 
 
 438. Hamilton has given a remarkably simple investigation 
 of the form of the equation of the wave-surface, in his Elements, 
 p. 736, which the reader may consult with advantage. The 
 following is essentially the same, but several steps of the process, 
 which a skilled analyst would not require to write down, are 
 retained for the benefit of the learner. 
 
 Let Sp,p = -1 (1) 
 
 be the equation of any tangent plane to the wave, i.e. of any wave- 
 front. Then p, is the vector of wave-slowness, and the normal 
 velocity of propagation is therefore 1/Tp. Hence, if ot be the vector 
 direction of displacement, pT'iiT is the effective component of the 
 force of restitution. Hence, </)ot denoting the whole force of 
 
 restitution, we have 
 
 <pTff — p'^-sr II p, 
 
 or in- II ((^ — p'^y^p, 
 
 and, as ts- is in the plane of the wave-front, 
 
 Spts = 0, 
 
 or Sp{(f>-p-Tp^O (2). 
 
344 QUATEKNIONS. [439- 
 
 This is, in reality, equation (6) of § 4.34 It appears here, 
 however, as the equation of the Index-Surface, the polar reciprocal 
 of the -wave with respect to a unit-sphere about the origin. Of 
 course the optical part of the problem is now solved, all that 
 remains being the geometrical process of § 328. 
 
 439. Equation (2) of last section may be at once transformed, 
 by the process of § 435, into 
 
 Let us employ an auxiliary vector 
 
 whence /:t = (/a" — <^~')t (1). 
 
 The equation now becomes 
 
 'S/.T = 1 (2), 
 
 or, by(l), /^V-St^-V = 1 (3). 
 
 Differentiating (3), subtract its half from the result obtained by 
 operating with ;S. t on the differential of (1). The remainder is 
 
 T^ Sfidfi - SrdiJL = 0. 
 
 But we have also (§ 328) 
 
 Spdfi, = 0, 
 
 and therefore, since dfi has an infinite number of values, 
 
 Xp — flT^ — T, 
 
 where a; is a scalar. 
 
 This equation, with (2), shews that 
 
 Stp = (4). 
 
 Hence, operating on it by »Si . p, we have by (1) of last section 
 
 and therefore p"^ = — ^ -j- t~'. 
 
 This gives p~^ = A'" — '^~'- 
 
 Substituting from these equations in (1) above, it becomes 
 
 r = (<!>-' -p-Tp"- 
 Finally, we have for the required equation, by (4), 
 
 8p-'{r-p-Vp-"=0, 
 or, by a transformation already employed, 
 
 Sp(cj,-pVp = 'i. 
 
441.] 
 
 PHYSICAL APPLICATIONS. 
 
 345 
 
 440. It may assist the student in the practice of qiiaternion 
 analysis, which is our main object, if we give a few of these 
 investigations by a somewhat varied process. 
 
 Thus, in § 432, let us write as in § 180, 
 
 aSSisr + VjSJTs + cVcSkts = X'S/j/st + fju'Sx'^sT -p'lir. 
 
 We have, by the same processes as in § 432, 
 
 S . isoiK'8fjf-n7 + 8 . OTa/i'/SW = 0. 
 
 This may be written, so far as the generating lines we require are 
 concerned, 
 
 8 . ■S7aV . X'sTfl' =0 = 8. ■stoK'tst/m, \ 
 since wa is a vector. 
 Or we may write [ 
 
 8 . jjJV . tn-X'OTa = = 8 . jjI'ss'K'ma.] 
 
 Equations (1) denote two cones of the second order which pass 
 through the intersections of (3) and (4) of § 432. Hence their 
 intersections are the directions of vibration. 
 
 441. By (1) we have 
 
 >S . wXWa/t' = 0. 
 
 Hence otX'ot, a, (j! are coplanar ; and, as cr is perpendicular to a, 
 it is equally inclined to Vk'ol and F/i'a. 
 
 For, if L, M, A be the projections of X', fi, a on the unit 
 sphere, BG the great circle whose 
 pole is. .4, we are to find for the 
 projections of the values of ot on 
 the sphere points P and P', such 
 that if LP be produced till 
 
 PQ = LP, 
 
 Q may lie on the great circle AM. 
 Hence, evidently, 
 
 CP = PB, 
 
 and GT' = JPB; 
 
 which proves the proposition, since 
 
 the projections of V\'a and Vfi'cc on the sphere are points b and 
 
 c in BC, distant by quadrants from C and B respectively. 
 
346 QUATERNIONS. [442- 
 
 442. Or thus, S^a = 0, 
 
 S . v^V . aX'^sTfi' = 0, 
 
 therefore x-sr = V.aV. aK'-arfju, 
 
 = -V. X'-sJii -aSaV. XV/i'. 
 
 Hence {SX'fi' - «) ot = (V + a^S^aX') Sfi'^sr + (fi' + aSa/u,') S'XW. 
 
 Operate by S . X', and we have 
 
 {x + SX'aSfi'a) SXV = [X'^'a' - 8\'a] S/i'zr 
 
 = S/x'^rVX'a. 
 Hence by symmetry, 
 
 oX 'UT b/M ■ar 
 
 or 
 
 8x\ 
 
 
 and as S^ira = 0, 
 
 ^ = C/'(CrFX'a± UVfjIoi). 
 
 443. The optical interpretation of the common result of the 
 last two sections is that the planes of polarization of the two rays 
 whose wave-fronts are parallel, bisect the angles contained by planes 
 passing through the normal to the wave-front and the vectors (optic 
 axes) X', fi. 
 
 444. As in § 434, the normal speed is given by 
 
 v^ = S^4>-!iT = 2/SX'CT/SfA<,'o7 -p'm' 
 S" . X'^'a 
 
 P '^'(T + S).VX'aV/j:oL- 
 
 [This transformation, effected by means of the value of ot in 
 § 442, is left to the reader.] 
 
 Hence, if v^, v^ be the velocities of the two waves whose 
 normal is a, 
 
 «: sin X'a sin /jb'a. 
 
 That is, the difference of the squares of the speeds of the two 
 waves varies as the product of the sines of the angles between the 
 normal to the wave-front and the optic axes (X', /u,'). 
 
447-J PHYSICAL APPLICATIONS. 347 
 
 445. We have, obviously, 
 
 (T'-S'). V\'aViJi'a=T'V. FX'aF/^'a = ,Sf^ xy'a. 
 Hence v'=p+(T±S). Vx'a Vfi'a. 
 
 The equation of the index surface, for which 
 
 is therefore 1 = - p'p^ + {T ± S) . Vx'p Vfji'p. 
 
 This will, of course, become the equation of the reciprocal of the 
 index-surface, i.e. the wave-surface, if we put for the function <f) 
 its reciprocal : i.e. if in the values of X', fi, p' we put 1/a, 1/6, 1/c 
 for a, b, c respectively. We have then, and indeed it might have 
 been deduced even more simply as a transformation of § 434 (7), 
 
 ] = -pp' +{T±8). rxprp.p, 
 
 as another form of the equation of Fresnel's wave. 
 
 If we employ the i, k transformation of § 128, this may be 
 written, as the student may easily prove, in the form 
 
 (k' - ij = S'(l-k)p + {TVip + TVkp)\ 
 
 446. We may now, in furtherance of our object, which is to 
 give varied examples of quaternions, not complete treatment of any 
 one subject, proceed to deduce some of the properties of the wave- 
 surface from the different forms of its equation which we have 
 given. 
 
 447. Fresnel's construction of the wave by points. 
 
 From § 290 (4) we see at once that the lengths of the principal 
 semidiameters of the central section of the ellipsoid 
 
 by the plane Sap = 0, 
 
 are determined by the equation 
 
 8.a(^-'-p-Ya = 0. 
 
 If these lengths be laid off along a, the central perpendicular to the 
 cutting plane, their extremities lie on a surface for which a = Up, 
 and Tp has values determined hy the equation. 
 
 Hence the equation of the locus is 
 as in §§ 434, 439. 
 
348 QUATERNIONS. [448- 
 
 Of course the index-surface is derived from the reciprocal ellip- 
 soid 
 
 Sp(t>p = 1 
 
 by the same construction. 
 
 448. Again, in the equation 
 
 1 = ^pp' +{T±S). VXp Vfip, 
 suppose 7\p = 0, or F/ip = 0, 
 
 we obviously have 
 
 p= ± —r or p = ±—r- 
 
 and there are therefore four singular points. 
 
 To find the nature of the surface near these points put 
 
 UX 
 
 where Tot is very small, and reject terms above the first order in 
 2W. The equation of the wave becomes, in the neighbourhood of 
 the singular point, 
 
 2^/SXot + 8.vjV. WK/m =±T. FXct VXfi, 
 
 which belongs to a cone of the second order. 
 
 449. From the similarity of its equation to that of the wave, it 
 is obvious that the index-surface also has four conical cusps. As 
 an infinite number of tangent planes can be drawn at such a point, 
 the reciprocal surface must be capable of being touched by a plane 
 at an infinite number of points ; so that the wave-surface has four 
 tangent planes which touch it along ridges. 
 
 To find their form, let us employ the last form of equation of 
 the wave in § 445. If we put 
 
 TVip = TVKp (1), 
 
 we have the equation of a cone of the second degree. It meets the 
 wave at its intersections with the planes 
 
 S{i-K)p=±{H:'~i') (2). 
 
 Now the wave-surface is touched by these planes, because we 
 cannot have the quantity on the first side of this equation greater 
 in absolute magnitude than that on the second, so long as p 
 satisfies the equation of the wave. 
 
45I-] PHYSICAL APPLICATIONS. 349 
 
 That the curves of contact are circles appears at once from (1) 
 and (2), for they give in combination 
 
 p^ = :^8{,+ K)p (3), 
 
 the equations of two spheres on which the curves in question are 
 situated. 
 
 The diameter of this circular ridge is 
 
 TV.{. + .) U{,. - «) = ^^^ = \ V(a= - 60 (6^ - O- 
 
 [Simple as these processes are, the student will find on trial 
 that the equation 
 
 Sp{4r'-p-r^p = o, 
 
 gives the results quite as simply. For we have only to examine 
 the cases in which —p~^ has the value of one of the roots of the 
 symbolical cubic in <^~\ In the present case Tp = b is the only 
 one which requires to be studied.] 
 
 450. By I 438, we see that the auxiliary vector of the succeed- 
 ing section, viz. 
 
 is parallel to the direction of the force of restitution, ^sr. Hence, 
 as Hamilton has shewn, the equation of the wave, in the form 
 
 Srp = 0, 
 
 (4) of § 439, indicates that the direction of the force of restitution is 
 perpendicular to the ray. 
 
 Again, as for any one versor of a vector of the wave there are 
 two values of the tensor, which are found from the equation 
 
 s. Up {<}>-' ~p-^y'Up=o, 
 
 we see by § 447 that the lines of vibration for a given plane front 
 are parallel to the axes of any section of the ellipsoid 
 
 S.p<}>-'p = l 
 
 made by a plane parallel to the front ; or to the tangents to the lines 
 of curvature at a point where the tangent plane is parallel to the 
 wave-front. 
 
 451. Again, a curve which is drawn on the wave-surface so as 
 to touch at each point the corresponding line of vibration has 
 
 <pdp\\ict,-^-p-Tp- 
 
350 QUATERNIONS. [452- 
 
 Hence S4)pdp = 0, or Sp4>p = C, 
 
 so that such curves are the intersections of the wave with a series 
 of ellipsoids concentric with it. 
 
 452. For curves cutting at right angles the lines of vibration we 
 have 
 
 dpwvpcp-'icf^-'-p-'rp 
 
 \\Vp(4>-pVp. 
 
 Hence Spdp = 0, or Tp = C, 
 
 so that the curves in question lie on concentric spheres. 
 
 They are also spherical conies, because where 
 
 Tp = G 
 the equation of the wave becomes 
 
 s.p{r'+o'Vp=f>. 
 
 the equation of a cyclic cone, whose vertex is at the common centre 
 of the sphere and the wave-surface, and which cuts them in their 
 curve of intersection. (§§ 432 — 452, Quarterly Math. Journal, 1859.) 
 The student may profitably compare, with the preceding investi- 
 gations, the (generally) very different processes which Hamilton 
 (in his Elements) applies to this problem. 
 
 E. Electrodynamics. 
 
 453. As another example we take the case of the action of 
 electric currents on one another or on magnets ; and the mutual 
 action of permanent magnets. 
 
 A comparison between the processes we employ and those of 
 Ampere {Thiorie des PMnomenes Electrodynamiques) will at once 
 shew how much is gained in simplicity and directness by the use 
 of quaternions. 
 
 The same gain in simplicity will be noticed in the investiga- 
 tions of the mutual effects of permanent magnets, where the 
 resultant forces and couples are at once introduced in their most 
 natural and direct forms. 
 
 454. Ampere's experimental laws may be stated as follows : 
 
 I. Equal and opposite currents in the same conductor produce 
 equal and opposite effects on other conductors. 
 
455-1 PHYSICAL APPLICATIONS, 351 
 
 II. The effect of a conductor bent or twisted in any manner is 
 equivalent to that of a straight one, provided that the two are 
 traversed by equal currents, and the former nearly coincides with 
 the latter. 
 
 III. No closed circuit can set in motion an element of a 
 circular conductor about an axis through the centre of the circle 
 and perpendicular to its plane. 
 
 IV. In similar systems traversed by equal currents the forces 
 are equal. 
 
 To these we add the assumption that the action between two 
 elements of currents is in the straight line joining them. [In a 
 later section (§ 473) other assumptions will be made in place of 
 this.] We also take for granted that the effect of any element of 
 a current on another is directly as the product of the strengths of 
 the currents, and of the lengths of the elements. 
 
 455. Let there be two closed currents whose strengths are 
 a and a^ ; let a', a, be elements of these, a being the vector 
 joining their middle points. Then the effect of a' on a, must, 
 when resolved along a^, be a complete differential with respect 
 to a (i.e. with respect to the three independent variables involved 
 in a), since the total resolved effect of the closed circuit of which 
 a' is an element is zero by III. 
 
 Also by I, II, the effect is a function of To., Saa.', Saa^, and 
 
 /Sa'Op since these are sufiScient to resolve a and a, into elements 
 
 parallel and perpendicular to each other and to a. Hence the 
 
 mutual effect is 
 
 aaJJaf{Ta., Saa', (Saa,, Sa'aJ, 
 
 and the resolved effect parallel to a^ is 
 
 aa^SUa^Uaf. 
 
 Also, that action and reaction may be equal in absolute magnitude, 
 / must be symmetrical in Saa' and Saa^. Again, a (as differential 
 of a) can enter only to the first power, and must appear in each 
 term of/ 
 
 Hence /= ^^Sfa'cc, + BSaa'Saa^. 
 
 But, by IV, this must be independent of the dimensions of the 
 system. Hence A is of — 2 and B oi —4\ dimensions in Ta. 
 Therefore 
 
 yp- [ASaa^Sa'a^ +BSaa'S^aa^} 
 
352 QUATERNIONS. [456- 
 
 is a complete differential, with respect to a, if da = a'. Let 
 
 where (7 is a constant depending on the units employed, therefore 
 
 or ^""^ Ta" 
 
 and the resolved effect 
 
 = g^ d ^y5 = Gaa, y^, (- aW«. + iSaa'Saa,) 
 
 = Caa^ p^, {S . Faa'Faa, + ^SacL'Saa,). 
 
 The factor in brackets is evidently proportional in the ordinary 
 
 notation to 
 
 sin 6 sin 6' cos to — | cos ^ cos 6'. 
 
 456. Thus the whole force is 
 
 Caa^a , S^aa^ _ Gaa^a , S'^oux 
 2^ Ttf^ " 2^^ ' Tot ' 
 
 as we should expect, d^a being = a^. [This may easily be trans- 
 formed into 
 
 _2Caa,Ua . 
 
 (Taf '^ ' 
 
 which is the quaternion expression for Ampere's well-known form.] 
 
 457. The whole effect on a.^ of the closed circuit, of which a' is 
 an element, is therefore 
 
 Gaa^ { OL ^ (Saa.J' 
 
 
 
 between proper limits. As the integrated part is the same at both 
 limits, the effect is 
 
 -^F«A where ^=j-^=j-—, 
 and depends on the form of the closed circuit. 
 
45 8- J PHYSICAL APPLICATIONS. 353 
 
 458. This vector /S, which is of great importance in the whole 
 theory of the effects of closed or indefinitely extended circuits, cor- 
 responds to the line which is called by Ampere " directrice de 
 I'action dlectrodynamique." It has a definite value at each point 
 of space, independent of the existence of any other current. 
 
 Consider the circuit a polygon whose sides are indefinitely 
 small ; join its angular points with any assumed point, erect at 
 the latter, perpendicular to the plane of each elementary triangle 
 so formed, a vector whose length is co/r, where o) is the vertical 
 angle of the triangle and r the length of one of the containing 
 sides ; the sum of such vectors is the " directrice " at the assumed 
 point. 
 
 [We may anticipate here so far as to give another expression 
 for this important vector, in terms of processes to be explained 
 later. 
 
 We have, by the formula (for a closed curve) of § 497 below, 
 
 ^ = jvP^,=fvdpV^^=jjdsV.V(Uv\f)vl-^, 
 
 (where ds is an element of any surface bounded by the circuit, Uv 
 its unit normal) 
 
 = Ijds Uv^' Y~^ II ^^'^^^^ Y 
 
 Tp JJ— Tp- 
 
 But the last integral is obviously the whole spherical angle, O 
 suppose, subtended by the circuit at the origin, and (unless Tp = 0) 
 we have (§ 145) 
 
 ^^i = «- 
 Hence, generally, 
 
 Thus ft may be considered as representing a potential, for which 
 /3 is the corresponding force. 
 
 This is a " many- valued " function, altering by 4?^ whenever we 
 pass through a surface closing the circuit. For if a be the vector 
 of a closed curve, the work done against ,3 during the circuit is 
 
 jS^da- = - j Sdtr'^a = /dft. 
 
 The last term is zero if the curve is not linked with the 
 circuit, but increases by ± iir for each linkage with the circuit.] 
 T. Q. L 23 
 
354 QUATERNIONS. [459- 
 
 459. The mere form of the result of § 457 shews at once that 
 if the element a, be turned about its middle point, the direction of the 
 resultant action is confined to the plane whose normal is /3. 
 
 Suppose that the element a, is forced to remain perpendicular 
 to some given vector S, we have 
 
 and the whole action in its plane of motion is proportional to 
 
 TV.SVa,^. 
 
 But F.SFa,/3 = -a.fif/8S. 
 
 Hence the action is evidently constant for all possible positions 
 of a, ; or 
 
 The effect of any system of closed currents on an element of a 
 conductor which is restricted to a given plane is (in that plane) 
 independent of the direction of the element. 
 
 460. Let the closed current be plane and very small. Let e 
 (where ^e = 1) be its normal, and let 7 be the vector of any point 
 within it (as the centre of inertia of its area) ; the middle point of 
 Hj being the origin of vectors. 
 
 Let a = 7 + p ; therefore a' = p', 
 
 and g_rFaa'_/ -F(7 + p)p - 
 
 and ^-JTa' "j Tiy + pf 
 
 = A,/f(7 + ,),'{i + ^^^^} 
 
 to a sufficient approximation. 
 
 Now (between limits) JVpp' = 2Ae, 
 where A is the area of the closed circuit. 
 Also generally 
 
 JVyp'Syp = J (SypVyp + yV. yJVpp) 
 — (between limits) AyVye. 
 
 Hence for this case 
 
 - TyX^~W)' 
 
463-J PHYSICAL APPLICATIONS. 355 
 
 461. If, instead of one small plane closed current, there be a 
 series of such, of equal area, disposed regularly in a tubular form, 
 let X be the distance between two consecutive currents measured 
 along the axis of the tube ; then, putting 7' = xe, we have for the 
 whole effect of such a set of currents on a 
 
 2x '^•"'JVry+ Ty' 1 
 
 Try' ' Ty' J 
 
 GAaa^ Fa,7 r -^ \ 
 
 = — s — rn s (between proper limits). 
 
 If the axis of the tubular arrangement be a closed curve this will 
 evidently vanish. Hence a closed solenoid exerts no influence on an 
 element of a conductor. The same is evidently true if the solenoid 
 he indefinite in both directions. 
 
 If the axis extend to infinity in one direction, and 7,, be the 
 vector of the other extremity, the effect is 
 
 GAaa^ ^«,7o 
 
 and is therefore perpendicular to the element and to the line joining 
 it with the extremity of the solenoid. It is evidently inversely as 
 T7/ and directly as the sine of the angle contained between the 
 direction of the element and that of the line joining the latter with 
 the extremity of the solenoid. It is also inversely as x, and there- 
 fore directly as the number of currents in a wnit of the axis of the 
 solenoid. 
 
 462. To find the effect of the whole circuit whose element is 
 a, on the extremity of the solenoid, we must change the sign of 
 the above and put a, = 70' ; therefore the effect is 
 
 CAaa , [V%j„ 
 2x ] Tyi ' 
 
 an integral of the species considered in § 458, whose value is easily 
 assigned in particular cases. 
 
 463. Suppose the conductor to be straight, and indefinitely 
 extended in both directions. 
 
 Let hO be the vector perpendicular to it from the extremity of 
 the solenoid, and let the conductor be || 1?, where Td = Tr) = 1. 
 Therefore yg = h6 + yr/ (where y is a scalar), 
 
 1^7o'7o = %' ^V^' 
 
 23—2 
 
356 
 
 QUATERNIONS. 
 
 [464- 
 
 and the integral in § 462 is 
 The whole effect is therefore 
 
 and is thus perpendicular to the plane passing through the conductor 
 and the extremity of the solenoid, and varies inversely as the distance 
 of the latter from the conductor. 
 
 This is exactly the observed effect of an indefinite straight 
 current on a magnetic pole, or particle of free magnetism. 
 
 464. Suppose the conductor to be circular, and the pole nearly 
 in its axis. [This is not a proper subject for Quaternions.] 
 
 Let EPD be the conductor, AB its axis, and G the pole ; BG 
 perpendicular to AB, and small in comparison with AE = h the 
 radius of the circle. 
 
 Let ABhea^i, BG=bk, AP = h(jx + ky) 
 
 where ^l = MzJ^^P= M 0. 
 
 y] (smj [sinj 
 
 CP=:y = a^i + bk — h (jx + ky). 
 
 Then 
 
 And the effect on C 
 
 /■ 
 
 Vyy' 
 
 J (a^' + J}' + h'-2bhyf ' 
 where the integral extends to the whole circuit. 
 
465-] PHYSICAL APPLICATIONS. 
 
 357 
 
 465. Suppose in particular G to be one pole of a small magnet 
 or solenoid GG' whose length is 21, and whose middle point is at G 
 and distant a from the centre of the conductor. 
 
 Let /.GGB = A. Then evidently 
 
 a^ = a + l cos A, 
 
 & = Z sin A. 
 Also the effect on G becomes, if a,^ + }>' +h^ = A'', 
 
 ■77/17 ^^'* ^^i^^'' 15 K'bS 
 
 A' \ A'^ A' ^2 A 
 since for the whole circuit 
 
 J9'xy"' = 0. 
 
 If we suppose the centre of the magnet fixed, the vector axis 
 of the couple produced by the action of the current on C is 
 
 IV. (i cos A + ^ sin A) I -^ 
 
 -rrhH sin A . f„ 36'' 15 hV Sa,b cos A 
 
 °^ A' -^'f A''^ 2 A' A'sinA 
 
 If A, &c. be now developed in powers of l, this at once becomes 
 
 -n-A'isinA.L 6aicosA 15aTcos'A Sl^ 
 :^ 2 ^-— + 
 
 (a'+ hy " I a^ + K' (a' + hj a' + h' 
 
 Si'sin'A ISAVsin'^A 
 
 a' + h' 
 
 ■ + 
 
 ISAVsin'A (a + Zcos A)Zcos A ' 5aZcosA\ 
 T(^+A7~ ¥Th' V'' a' + h' ) 
 
 Putting — Z for I and changing the sign of the whole to get that 
 for pole G', we have for the vector axis of the complete couple 
 
 iTrmdnA.C J\W- h')(4<-5sm''A) 1 
 
 which is almost exactly proportional to sin A, if 2a be nearly 
 equal to h and I be small. (See Ex. 15 at end of Chapter.) 
 
 On this depends a modification of the tangent galvanometer. 
 (Bravais, Ann. de Ghimie, xxxviii. 309.) 
 
358 QUATERNIONS. [466. 
 
 466. As before, the effect of an indefinite solenoid on a, is 
 GAaa^ Va^y 
 
 Now suppose a, to be an element of a small plane circuit, S the 
 vector of the centre of inertia of its area, the pole of the solenoid 
 being origin. 
 
 Let 7 = 8 + p, then a^ = p. 
 
 The whole effect is therefore 
 
 GAaa, [ V{l^■p)p 
 
 GAAaa, I 3S/SSe 
 
 where A^ and e^ are, for the new circuit, what A and e were for 
 the former. 
 
 Let the new circuit also belong to an indefinite solenoid, and 
 let S„ be the vector joining the poles of the two solenoids. Then 
 the mutual effect is 
 
 2xx^ }\TB'^ TW J 
 
 _GAAfla, \_ JIB, 
 - 2xx, (TB^f^^iTB^y 
 
 which is exactly the mutual effect of two magnetic poles. Two 
 finite solenoids, therefore, act on each other exactly as two magnets, 
 and the pole of an indefinite solenoid acts as a particle of free 
 magnetism. 
 
 467. The mutual attraction of two indefinitely small plane 
 closed circuits, whose normals are e and 6j, may evidently be 
 deduced by twice differentiating the expression UB/TB^ for the 
 mutual action of the poles of two indefinite solenoids, making dB 
 in one differentiation || e and in the other || e,. 
 
 But it may also be calculated directly by a process which will 
 give us in addition the couple impressed on one of the circuits by 
 the other, supposing for simplicity the first to be circular. [In 
 the sketch we are supposed to be looking at the faces turned 
 towards one another.] 
 
467-] PHYSICAL APPLICATIONS. 359 
 
 Let A and B be the centres of inertia of the areas of A and B, 
 
 e and e, vectors normal to their planes, a any vector radius of B 
 AB = ^. 
 
 Then whole effect on o-', by §§ 457, 460, 
 
 A 
 
 V(T'\e + 
 
 8{/3 + a)S(0 + <T)e 
 
 3Va'l3Sa€ „ Va'aS/3e] 
 
 + - 
 
 • + 3 
 
 W ' " T/3-' y 
 But between proper limits, 
 
 JVa'r,Sda = -A,V.r}Vd6^, 
 
 for generally jVa'rjSdcr =-\{ VrjaSda +V.r)V. 6 jVaa'). 
 
 Hence, after a reduction or two, we find that the whole force 
 exerted by A on the centre of inertia of the area of B 
 
 QC 
 
 
 This, as already observed, may be at once found by twice 
 
 US 
 differentiating ^^^ . In the same way the vector moment, due to 
 
 A, about the centre of inertia of B, 
 
 AA^( SV^e^\ 
 
 These expressions for the whole force of one small magnet on 
 the centre of inertia of another, and the couple about the latter, 
 seem to be the simplest that can be given. It is easy to deduce 
 
360 QUATERNIONS. [468. 
 
 from them the ordinary forms. For instance, the whole resultant 
 couple on the second magnet 
 
 may easily be shewn to coincide with that given by Ellis (Camb. 
 Math. Journal, iv. 95), though it seems to lose in simplicity and 
 capability of interpretation by such modifications. 
 
 468. The above formulae shew that the whole force exerted 
 by one small magnet M, on the centre of inertia of another m^ 
 consists of four terms which are, in order, 
 
 1st. In the line joining the magnets, and proportional to the 
 cosine of their mutual inclination. 
 
 2nd. In the same line, and proportional to five times the product 
 of the cosines of their respective inclinations to this line. 
 
 3rd and 4th. Parallel to \yX and proportional to the cosine of 
 
 the inclination of \ [to the joining line. 
 
 All these forces are, in addition, inversely as the fourth power 
 of the distance between the magnets. 
 
 For the couples about the centre of inertia of m we have 
 
 1st. A couple whose axis is perpendicular to each magnet, and 
 which is as the sine of their mutual inclination. 
 
 2nd. A couple whose axis is perpendicular to m and to the line 
 joining the magnets, and whose moment is as three times the product 
 of the sine of the inclination of m, and the cosine of the inclination 
 of M, to the joining line. 
 
 In addition these couples vary inversely as the third power of 
 the distance between the magnets. 
 
 [These results afford a good example of what has been called 
 the internal nature of the methods of quaternions, reducing, as 
 they do at once, the forces and couples to others independent of 
 any lines of reference, other than those necessarily belonging to 
 the system under consideration. To shew their ready applicability, 
 let us take a Theorem due to Gauss.] 
 
47 1-] PHYSICAL APPLICATIONS. 361 
 
 469. If two small magnets he at right angles to each other, the 
 moment of rotation of the first is approximately twice as great when 
 the axis of the second passes through the centre of the first, as when 
 the axis of the first passes through the centre of the second. 
 
 In the first case 
 
 ell/Sle.; 
 
 therefore moment = 
 
 c 9r" 
 
 In the second 
 
 6.ll/3±6; 
 
 therefore moment = 
 
 C 
 -=^3 Tee^. Hence the theorem. 
 
 470. Again, we may easily reproduce the results of § 467, if 
 for the two small circuits we suppose two small magnets perpen- 
 dicular to their planes to be substituted. /3 is then the vector 
 joining the middle points of these magnets, and by changing the 
 tensors we may take 2e and 26, as the vector lengths of the 
 magnets. 
 
 Hence evidently the mutual effect 
 oc p(/S + e-e,)- j^(/3-e-6.)+y,(^-6 + e.)-y,(/3 + e + 6,), 
 which is easily reducible to 
 
 as before, if smaller terms be omitted. 
 
 If we operate with V. e, on the two first terms of the unreduced 
 expression, and take the difference between this result and the 
 same with the sign of e, changed, we have the whole vector axis of 
 the couple on the magnet 26,, which is therefore, as before, seen to 
 be proportional to 
 
 ^/ 3F6./3>g^6 \ 
 
 T/S'V ' T^ )' 
 
 471. Let F (7) be the potential of any system upon a unit 
 particle at the extrenoity of 7. Then we have 
 
 8vd-/ = 0, 
 where i' is a vector normal giving the force in direction and 
 magnitude (§ 148). 
 
362 ■ QUATERNIONS. [472. 
 
 Now by § 460 we have for the vector force exerted by a small 
 plane closed circuit on a particle of free magnetism the expression 
 
 lyV 
 
 . + »!) 
 
 merging in A the factors depending on the strength of the current 
 and the strength of magnetism of the particle. 
 
 Hence the potential is 
 
 ASey 
 
 area of circuit projected perpendicular to 7 
 
 °^ 2y ' 
 
 oc spherical opening subtended by circuit. 
 
 The constant is omitted in the integration, as the potential must 
 evidently vanish for infinite values of Tiy. 
 
 By means of Ampere's idea of breaking up a finite circuit into 
 an indefinite number of indefinitely small ones, it is evident that 
 the above result may be at once ex- 
 tended to the case of such a finite closed 
 circuit. 
 
 472. Quaternions give a simple 
 method of deducing the well-known 
 property of the Magnetic Curves. 
 
 Let A, A' be two equal magnetic 
 poles, whose vector distance, 2a, is bi- 
 sected in 0, QQ' an indefinitely small 
 
 magnet whose length is 2/3', where p = OP. Then evidently, 
 taking moments, 
 
 V{p±a)_p_ _ V(p-a)p' 
 'T(p + ay - Tip-af ' 
 where the upper or lower sign is to be taken according as the poles 
 are like or unlike. 
 
 [This equation may also be obtained at once by differentiating 
 the equation of the equipotential surface, 
 
 1 1 _ , 
 
 T(p + a) + T{p-a)~''°''^^-' 
 and taking p' parallel to its normal (§ 148).] 
 
473'] PHYSICAL APPLICATIONS. 363 
 
 Operate by 8 . Vap, 
 Sap'(p + ar-^Sai^ + u)Sp(p + .) ^ ^ ^^^^^ ^.^^ _ ^^^ 
 
 or 8.aV(~^jU{p + a.)= ±[samewith -a], 
 
 i.e. Sad U{p + a) = ± Sad U(p - a), 
 
 Sa{U(p + a)+ U{p — a)} = const., 
 or cos Z OAP ± cos Z OA'P = const., 
 
 the property referred to. 
 
 If the poles be unequal, one of the terms to the left must be 
 multiplied by the ratio of their strengths. 
 
 (§§ 453—472, Quarterly Math. Journal, 1860.) 
 
 F. General Expressions for the Action between Linear 
 Elements. 
 
 473. The following general investigation of different possible 
 expressions for the mutual action between elements of linear 
 conductors is taken from Proc. R. S. E. 1873 — 4. 
 
 Ampere's data for closed currents are briefly given in § 454 
 above, and are here referred to as I, II, III, IV, respectively. 
 
 (a) First, let us investigate the expression for the force exerted 
 by one element on another. 
 
 Let a be the vector joining the elements a,, a', of two circuits ; 
 then, by I, II, the action of a, on a is linear in each of a^, a', 
 and may, therefore, be expressed as 
 
 where is a linear and vector function, into each of whose con- 
 stituents a^ enters linearly. 
 
 The resolved part of this along a' is 
 
 S. Ua'cpa', 
 and, by III, this must be a complete differential as regards the 
 circuit of which a, is an element. Hence, 
 
 4,a' = -{8. a,V) fa + Fa'^a^ , 
 where f and % are linear and vector functions whose constituents 
 involve a only. That this is the case follows from the fact that 
 (pa' is homogeneous and linear in each of a^, a. It farther 
 
364 QUATERNIONS. [473- 
 
 follows, from IV, that the part of 0a' which does not disappear 
 after integration round each of the closed circuits is of no dimen- 
 sions in ToL, To!, Tu^. Hence ;;^ is of — 2 dimensions in Ta, and 
 
 thus 
 
 _paSaoL^ qoL^ rVaa^ 
 
 where p, q, r are numbers. 
 Hence we have 
 
 . / o/ nx . / pVa'aSaa. qVa'a, rV.a'Vaa, 
 <f>a'=-8 (a,V) ^a' + ^ y^^ ' + =^y^' + j^ ■ 
 
 Change the sign of a in this, and interchange a' and a,, and we 
 get the action of a on a^. This, with a' and a^ again interchanged, 
 and the sign of the whole changed, should reproduce the original 
 expression — since the effect depends on the relative, not the abso- 
 lute, positions of a, a^, a'. This gives at once, 
 
 p = 0, q = 0, 
 and 
 
 4>a=-S(a,V)yjra + ^, — ', 
 
 with the condition that the first term changes its sign with a, and 
 thus that 
 
 fa' = a/Sfaa'F (Ta) -I- a'F(Ta.), 
 
 which, by change of F, may be written 
 
 = aS(a.'V)f(Ta) + a'F{Ta), 
 where / and F are any scalar functions whatever. 
 Hence 
 
 <f>a'=-S (a,V) [otS {aV)f{Ta) + aF (fa)] + ^^"j^J""' , 
 
 which is the general expression required. 
 
 (6) The simplest possible form for the action of one current- 
 element on another is, therefore, 
 
 , , rV. a'Faa, 
 Here it is to be observed that Ampere's directrice for the circuit 
 
 e 
 
 -I 
 
 Faa. 
 
 Ta' ' 
 the integral extending round the circuit ; so that, finally. 
 
473-] PHYSICA.L APPLICATIONS. 365 
 
 (c) We may obtain from the general expression above the 
 absolutely symmetrical form, 
 
 rF. a'aaj 
 
 if we assume 
 
 /(ra) = const., F{Ta) = ~. 
 
 Here the action of a! on a^ is parallel and equal to that of a^ on a!. 
 The forces, in fact, form a couple, for a is to be taken negatively 
 for the second — and their common direction is the vector drawn 
 to the corner a of a spherical triangle abc, whose sides ah, he, ca in 
 order are bisected by the extremities of the vectors Ua.', Ua, ?7a,. 
 Compare Hamilton's Lectures on Quaternions, §§ 223 — 227. 
 
 (d) To obtain Ampere's form for the effect of one element on 
 another write, in the general formula above, 
 
 /(ra) = ^, F{Ta)=0, 
 
 and we have 
 
 ^ </>«' = - 5a, V. 
 
 aSaa! 
 
 V. a'Vaa , 
 
 a^Saa.' aScc^a SaSaa'Saa^ V. a'Faa, 
 2a / ,„ , 3 
 
 "^Ta' 
 
 ; ( oc'Sa^a — ^ Saa'8aa^ j , 
 = — =-5 1 S . Faa' Faaj + ^ Saa'Saa^ 1 
 
 which are the usual forms. 
 
 (e) The remainder of the expression, containing the arbitrary 
 terms, is of course still of the form 
 
 - 8 (a,Vj [aS (a'V)/(ra) + oc'F {To}]. 
 
 In the ordinary notation this expresses a force whose com- 
 ponents are proportional to 
 
 (Note that, in this expression, r is the distance between the 
 
 elements.) 
 
 dF 
 (2) Parallel to a j- ■ 
 
366 QUATERNIONS. [473- 
 
 (3) Parallel to a, - ^ . 
 
 If we assume /= ^ = - Q, we obtain the result given by Clerk- 
 Maxwell {Electricity and Magnetism, § 525), which differs from the 
 above only because he assumes that the force exerted by one 
 element on another, when the first is parallel and the second per- 
 pendicular to the line joining them, is equal to that exerted when 
 the first is perpendicular and the second parallel to that line. 
 
 (/) What precedes is, of course, only a particular case of the 
 following interesting problem : — 
 
 Required the most general expression for the mutual action of 
 two rectilinear elements, each of which has dipolar symmetry in the 
 direction of its length, and which may be resolved and compounded 
 according to the usual kinematical law. 
 
 The data involved in this statement are equivalent to I and 
 II of Ampere's data above quoted. Hence, keeping the same 
 notation as in (a) above, the force exerted by a, on a' must be 
 expressible as 
 
 (f)a' 
 
 where <^ is a linear and vector function, whose constituents are 
 linear and homogeneous in a^ ; and, besides, involve only a. 
 
 By interchanging a, and a, and changing the sign of a, we get 
 the force exerted by a on a,. If in this we again interchange a, 
 and a, and change the sign of the whole, we must obviously repro- 
 duce <pa'. Hence we must have ^a' changing its sign with a, or 
 
 (f)a' = PaSa^a.' + QaSaa^Sm' + Ra^Saa! + Ra'Saa^, 
 where P, Q, R, R are functions of Ta only. 
 
 (g) The vector couple exerted by a, on a must obviously be 
 expressible in the form 
 
 V, a'cro,, 
 
 where nr is a new linear and vector function depending on a alone. 
 Hence its most general form is 
 
 CTa, = Pflj + QaSaa^, 
 
 where P and Q are functions of Ta only. The form of these func- 
 tions, whether in the expression for the force or for the couple, 
 depends on the special data for each particular case. Symmetry 
 shews that there is no term such as 
 
 RVaa,. 
 
473-] PHYSICAL APPLICATIONS. 367 
 
 Qi) As an example, let a, and a' be elements of solenoids or of 
 uniformly and linearly magnetised wires, it is obvious that, as a 
 closed solenoid or ring-magnet exerts no external action, 
 
 ^a! = - Say . yira'. 
 
 Thus we have introduced a different datum in place of Ampere's 
 No. III. But in the case of solenoids the Third Law of Newton 
 holds — hence 
 
 ^a' = 8a,VSa'V . ^a, 
 
 where % is a linear and vector function, and can therefore be of no 
 other form than 
 
 c^fiTa). 
 
 Now two solenoids, each extended to infinity in one direction, act 
 on one another like two magnetic poles, so that (this being our 
 equivalent for Ampere's datum No. TV.) 
 
 Hence the vector force exerted by one small magnet on another is 
 
 pSaySaV^,. 
 
 (i) For the couple exerted by one element of a solenoid, or of 
 a uniformly and longitudinally magnetised wire, on another, we 
 have of course the expression 
 
 V. a'roaj, 
 
 where m is some linear and vector function. 
 
 Here, in the first place, it is obvious that 
 
 for the couple vanishes for a closed circuit of which a^ is an 
 element, and the integral of otk, must be a linear and vector 
 function of a alone. It is easy to see that in this case 
 
 F{Ta) oc (Taf. 
 
 (j) If, again, a, be an element of a solenoid, and a' an element 
 of current, the force is 
 
 0a' = — Say . ■yfra', 
 where 
 
 i^a' = Pa' + QaSaa' + RVaa'. 
 
368 QUATERNIONS. [473- 
 
 But no portion of a solenoid can produce a force on an element of 
 current in the direction of the element, so that 
 
 </>a'=F.a'xa„ 
 whence P = 0, Q = 0, 
 
 and we have ^a' = - Sa^V (RVaa). 
 
 This must be of — 1 linear dimensions when we integrate for the 
 effect of one pole of a solenoid, so that 
 
 If the current be straight and infinite each way, its equation being 
 
 a = l3 + x'Y, 
 
 where Ty = l and S^y = 0, 
 
 we have, for the whole force exerted on it by the pole of a solenoid, 
 the expression 
 
 
 which agrees with known facts. 
 
 (k) Similarly, for the couple produced by an element of a 
 solenoid on an element of a current we have 
 
 Fa'crMj, 
 
 where •nra^ = — Say . yjra, 
 
 and it is easily seen that 
 
 ra 
 
 (I) In the case first treated, the couple exerted by one current- 
 element on another is, by (g), 
 
 V. aVa,, 
 
 where, of course, + ctHj are the vector forces applied at either end 
 of a.'. Hence the work done when a changes its direction is 
 
 — S. SaOTttj, 
 
 with the condition 8 . a'Ba' = 0. 
 
 So far, therefore, as change of direction of a' alone is concerned, 
 the mutual potential energy of the two elements is of the form 
 
 8 . cl'tttol, . 
 
473-] PHYSICAL APPLICATIONS. 369 
 
 This gives, by the expression for •sr in {g), the following value 
 
 P/Sfa'a, + QSaa'Saa,. 
 
 Hence, integrating round the circuit of which a^ is an element, we 
 have (§ 495 below) 
 
 f{P8a.\ + QSaa'Sa.a,)=JJds^S. Uv,V (Pa' + QolSolo!), 
 = !Jds,S.Uv,(^^-oLaqy 
 
 = ffds^S.UvJcLa'^, 
 
 F 
 where <E) = y- + Q. 
 
 Integrating this round the other circuit we have for the mutual 
 potential energy of the two, so far as it depends on the expression 
 
 above, the value 
 
 !!ds,8. UvJVaa'^ 
 
 = -fJds,S. UvJJds'V. ViUv'V) aO 
 
 = JJds^ JJds' \S . Up, Up' (2<I> + Ta*') + Soi Up' 8a Up, ~\ . 
 
 But, by Ampere's result, that two closed circuits act on one another 
 as two magnetic shells, it should be 
 
 jjdsjjds's.Uvys.Up-v^^ 
 
 = JJds, JJds' [S . Up, Uv' ^3 + ZSa Uv'Sa Uv, ^) . 
 Comparing, we have 
 
 — = ra<E>' 
 
 giving ^^~w r?' 
 
 which are consistent with one another, and which lead to 
 Hence, if we put 
 
 P' ^ \_ ■ 
 
 we get P = „ rr ■ 
 
 T. Q. 1. 24 
 
370 QUATEENIONS. [474- 
 
 and the mutual potential of two elements is of the form 
 
 a+n)^ + {l-n) y^3 ' , 
 
 which is the expression employed by Helmholtz in his paper Ueber 
 die Bewegungsgleichungen der Electricitdt, Grelle, 1870, p. /6. 
 
 G. Application of V to certain Physical Analogies. 
 
 474 The chief elementary results into which V enters, in 
 connection with displacements, are given in § 384 above. The 
 following are direct applications. 
 
 Thus, if a be the vector-displacement of that point of a homo- 
 geneous elastic solid whose vector is p, we have, p being the 
 consequent pressure produced, 
 
 V^ + VV = (1), 
 
 whence SBpVa = — SBpVp = Bp, a complete differential (2). 
 
 Also, generally, p = cSVa; 
 
 and if the solid be incompressible 
 
 /SVo- = (3). 
 
 Thomson has shewn (Camb. and Dub. Math. Journal, ii. p. 62), 
 that the forces produced by given distributions of matter, electricity, 
 magnetism, or galvanic currents, can be represented at every point 
 by displacements of such a solid producible by external forces. It 
 may be useful to give his analysis, with some additions, in a 
 quaternion form, to shew the insight gained by the simplicity of 
 the present method. 
 
 475. Thus, if SaBp = 8 ^ .. we may write each equal to 
 -SBpvl-^. 
 
 This gives a- = — V 7^- . 
 
 the vector-force exerted by one particle of matter or free electricity 
 on another. This value of <r evidently satisfies (2) and (3). 
 
 Again, if 8 . BpVc7 = B -^ , either is equal to 
 _^.8pV-^by(4)of§146. 
 
478.] PHYSICAL APPLICATIONS. 371 
 
 Here a particular case is 
 
 __Vap 
 
 which is the vector-force exerted by an element a of a current upon 
 a particle of magnetism at p. (§ 461.) 
 
 476. Also, by § 146 (3), 
 
 _ Vap __ ap^ — SpSap 
 
 and we see by §§ 460, 461 that this is the vector-force exerted by 
 a small plane current at the origin (its plane being perpendicular 
 to a) upon a magnetic particle, or pole of a solenoid, at p. This 
 expression, being a pure vector, denotes an elementary rotation 
 caused by the distortion of the solid, and it is evident that the 
 above value of a- satisfies the equations (2), (3), and the distortion 
 is therefore producible by external forces. Thus the effect of an 
 element of a current on a magnetic particle is expressed directly 
 by the displacement, while that of a small closed current or 
 magnet is represented by the vector-axis of the rotation caused by 
 the displacement. 
 
 477. Again, let 8BpV'a = Sp§. 
 
 It is evident that o- satisfies (2), and that the right-hand side of 
 the above equation may be written 
 
 -S.SpV^. 
 Hence a particular case is 
 
 and this satisfies (3) also. 
 
 Hence the corresponding displacement is producible by external 
 forces, and Vo- is the rotation axis of the element at p, and is seen 
 as before to represent the vector-force exerted on a particle of 
 magnetism at p by an element a of a current at the origin. 
 
 478. It is interesting to observe that a particular value of a- 
 in this case is 
 
 as may easily be proved by substitution. 
 
 24—2 
 
372 QUATERNIONS. [479- 
 
 » Sap 
 Again, if Sopa- = — o ^-5 , 
 
 _ Sap 
 we have evidently o- = V ^-j . 
 
 Now, as p^ is the potential of a small magnet a, at the origin, 
 
 on a particle of free magnetism at p, a- is the resultant magnetic 
 force, and represents also a possible distortion of the elastic solid 
 by external forces, since Vo- = W = 0, and thus (2) and (3) are 
 both satisfied. {Proc. R. S. E. 1862.) 
 
 H. Elementary Properties of V. 
 
 479. In the next succeeding sections we commence with a 
 form of definition of the operator V somewhat different from that 
 of Hamilton (§ 145), as we shall thus entirely avoid the use of 
 Cartesian coordinates. For this purpose we write 
 
 where a. is any unit-vector, the meaning of the right-hand operator 
 (neglecting its sign) being the rate of change of the function to 
 which it is applied per unit of length in the direction of the 
 vector a. If a be not a unit-vector we may treat it as a vector- 
 velocity, and then the right-hand operator means the rate of 
 change per unit of time due to the change of position. 
 
 Let a, /3, 7 be any rectangular system of unit-vectors, then by 
 a fundamental quaternion transformation 
 
 V = - aSaV - /3SI3V - ySjV = ad^. + ^dp + jdy, 
 
 which is identical with Hamilton's form so often given above. 
 {Lectures, § 620.) 
 
 480. This mode of viewing the subject enables us to see at 
 once that V is an Invariant, and that the effect of applying it to 
 any scalar function of the position of a point is to give its vector of 
 most rapid increase. Hence, when it is applied to a potential u, 
 we have the corresponding vector-force. From a velocity-potential 
 we obtain the velocity of the fluid element at p ; and from the 
 temperature of a conducting solid we obtain the temperature- 
 
481.] PHYSICAL APPLICATIONS. 373 
 
 gradient in the direction of the flux of heat. Finally, whatever 
 series of surfaces is represented by 
 
 the vector Vm is the normal at the point p, and its length is 
 inversely as the normal distance at that point between two con- 
 secutive surfaces of the series. . 
 
 Hence it is evident that 
 
 8 . dpVu = — du, 
 or, as it may be written, 
 
 -S.dpV = d; 
 
 the left-hand member therefore expresses total differentiation in 
 virtue of any arbitrary, but small, displacement dp. 
 
 These results have been already given above, but they were 
 not obtained in such a direct manner. 
 
 Many very curious and useful transformations may easily be 
 derived (see Ex. 34, Chap. XI.) from the assumption 
 
 da- = (f)dp, or <f)a = — SaV . cr, 
 
 where the constituents of are known functions of p. For 
 instance, if we write 
 
 _ . d . d , d 
 
 ^^^'T^+^dn+^M' 
 
 where cr = i^ + jrj + k^, 
 
 we find at once V = <f)V^, or V^ = 0'"'V ; 
 
 a formula which contains the whole basis of the theory of the 
 change of independent variables from x, y, z to f, 1), t„ or vice versa. 
 The reader may easily develop this application. Its primary 
 interest is, of course, purely mathematical : — but it has most 
 important uses in applied mathematics. Our limits, however, 
 do not permit us to reach the regions of its special physical 
 usefulness. 
 
 481. To interpret the operator V. aV, let us apply it to a 
 potential function u. Then we easily see that u may be taken 
 under the vector sign, and the expression 
 
 r(oLV)u=V.a.Vu 
 
 denotes the vector-couple due to the force at p about a point whose 
 relative vector is a. 
 
374 QUATERNIONS. [482. 
 
 Again, if cr be any vector function of p, we have by ordinary 
 quaternion operations 
 
 F(aV) .cr = 8. aVVa- + aSVo- - V,Sfa<7. 
 
 The meaning of the third term (in which it is of course understood 
 that V operates on a alone) is obvious from what precedes. The 
 other terms were explained in § 384. 
 
 J. Applications of V to Line-, Surface-, and Volume-Integrals. 
 
 482. In what follows we have constantly to deal with integrals 
 extended over a closed surface, compared with others taken through 
 the space enclosed by such a surface; or with integrals over a 
 limited surface, compared with others taken round its bounding 
 curve. The notation employed is as follows. If Q per unit of 
 length, of surface, or of volume, at the point p, Q being any 
 quaternion, be the quantity to be summed, these sums will be 
 denoted by 
 
 JJQds and JJJQd<;, 
 
 when comparing integrals over a closed surface with others through 
 the enclosed space ; and by 
 
 JJQds and JQTdp, 
 
 when comparing integrals over an unclosed surface with others 
 round its boundary. No ambiguity is likely to arise from the 
 double use of 
 
 JJQds, 
 
 for its meaning in any case will be obvious from the integral with 
 which it is compared. What follows is mainly from Tran^. R. S. E. 
 1869—70. See also Proc. R. 8. E. 1862—3. 
 
 483. We have shewn in § 384 that, if o- be the vector displace- 
 ment of a point originally situated at 
 
 p = ix-\-jy-irkz, 
 
 then (S'.Vo- 
 
 expresses the increase of density of aggregation of the points of 
 the system caused by the displacement. 
 
 484. Suppose, now, space to be uniformly filled with points, 
 and a closed surface S to be drawn, through which the points can 
 freely move when displaced. 
 
486.] PHYSICAL APPLICATIONS. 375 
 
 Then it is clear that the increase of number of points within 
 the space 2, caused by a displacement, may be obtained by either 
 of two processes — by taking account of the increase of density at 
 all points within 2, or by estimating the excess of those which 
 pass inwards through the surface over those which pass outwards. 
 These are the principles usually employed (for a mere element of 
 volume) in forming the so-called ' Equation of Continuity.' 
 
 Let V be the normal to S at the point p, drawn outwards, then 
 we have at once (by equating the two different expressions of the 
 same quantity above explained) the equation 
 
 which is our fundamental equation so long as we deal with triple 
 integrals. [It will be shewn later (| 500) that the corresponding 
 relation between the single and the double integral can be deduced 
 directly from this.] 
 
 As a first and very simple example of its use, let p be written 
 for cr. It becomes 
 
 -^m'i=!SSpUvds, 
 
 i.e. the volume of any closed space is the sum of the elements of 
 area of its surface, each multiplied by one-third of the perpendicular 
 from the origin on its plane. 
 
 485. Next, suppose o- to represent the vector force exerted 
 upon a unit particle at p (of ordinary matter, electricity, or 
 magnetism) by any distribution of attracting matter, electricity, 
 or magnetism partly outside, partly inside 2. Then, if P be the 
 potential at p, 
 
 and if r be the density of the attracting matter, &c., at p, 
 
 Va- = V''P = 47rr- 
 
 by Poisson's extension of Laplace's equation. 
 
 Substituting in the fundamental equation, we have 
 
 47r ///rd? = 47rif = ///S . VP Uvds, 
 
 where M denotes the whole quantity of matter, &c., inside 2. 
 This is a well-known theorem. 
 
 486. Let P and P, be any scalar functions of p, we can of 
 course find the distribution of matter, &c., requisite to make either 
 
376 QUATERNIONS. [.4^7- 
 
 of them the potential at p ; for, if the necessary densities be r and 
 r^ respectively, we have as before 
 
 V=P = 47rr, V'P, = 47rr,. 
 Now V(PVP,) = VPVP. + PVT,. 
 
 Hence, if in the formula of § 484 we put 
 
 we obtain 
 
 JJJS . VPVP.d, = - JJJPV'P.d, + If PS . VP. Uvds, 
 
 = - JJfPyPd, + fJP.S . VP Uvds, 
 
 which are the common forms of Green's Theorem. Sir W. Thomson's 
 extension of it follows at once from the same proof 
 
 487. If Pj be a many-valued function, but VP^ single-valued, 
 and if 2 be a multiply-connected* space, the above expressions 
 require a modification which was first shewn to be necessary by 
 Helmholtz, and first supplied by Thomson. For simplicity, suppose 
 X to be doubly-connected (as a ring or endless rod, whether knotted 
 or not). Then if it be cut through by a surface s, it will become 
 simply-connected, but the surface-integrals have to be increased 
 by terms depending upon the portions just added to the whole 
 surface. In the first form of Green's Theorem, just given, the 
 only term altered is the last : and it is obvious that if p^ be the 
 increase of P^ after a complete circuit of the ring, the portion to be 
 added to the right-hand side of the equation is 
 
 pJJS.VPUvds, 
 taken over the cutting surface only. A similar modification is 
 easily seen to be produced by each additional complexity in the 
 space X. 
 
 488. The immediate consequences of Green's theorem are 
 well known, so that I take only a few examples. 
 
 Let P and P^ be the potentials of one and the same distribution 
 of matter, and let none of it be within 2. Then we have 
 
 JJJ{VPyd<;=JJP8.VPUvds, 
 
 * Called by Helmholtz, after Eiemann, mehrfach zusammenMngend. In trans- 
 lating Helmlioltz'g paper (Phil. Mag. 1867) I used the above as an English 
 equivalent. Sir W. Thomson in his great paper on Vortex Motion {Trans. R. S. E. 
 1868) uses the expression "multiply-continuous." 
 
489.] PHYSICAL APPLICATIONS. 377 
 
 SO that if VP is zero all over the surface of 1,, it is zero all through 
 the interior, i.e., the potential is constant inside 2. If P be the 
 velocity-potential in the irrotational motion of an incompressible 
 fluid, this equation shews that there can be no such motion of the 
 fluid unless there is a normal motion at some part of the bounding 
 surface, so long at least as S is simply-connected. 
 Again, if S is an equipotential surface, 
 
 !jj(ypf d, = pjjs . vpuvds ^Pijjv'Pd, 
 
 by the fundamental theorem. But there is by hypothesis no 
 matter inside S, so this shews that the potential is constant 
 throughout the interior. Thus there can be no equipotential 
 surface, not including some of the attracting matter, within which 
 the potential can change. Thus it cannot have a maximum or 
 minimum value at points unoccupied by matter. 
 
 489. Again, in an isotropic body whose thermal conductivity 
 does not vary with temperature, 'the equation of heat-conduction is 
 
 dt c 
 where (for the moment) k and c represent as usual the conductivity, 
 and the water equivalent of unit volume. 
 
 The surface condition (assuming Newton's Law of Cooling) is 
 
 iiS.UvVv-hv^O. 
 Assuming, after Fourier, that a particular integral is 
 
 we have V^m — ^^ w = 0, 8 . UvVu — - u = 0. 
 
 k k 
 
 Let u^ be a particular integral of the first of these linear 
 differential equations. Substitute it for u in the second ; and we 
 obtain (with the aid of the equation of the bounding surface) a 
 scalar equation giving the admissible values of m. 
 
 Suppose the distribution of temperature when t = to be 
 given ; it may be expressed linearly in terms of the various values 
 of u, thus 
 
 W = Ut=n = ^K^^- 
 
 For if Mj, Mj be any two of these particular integrals, we have 
 by Green's Theorem, and the differential equations. 
 
 m 
 "k 
 
 fjjju.u^d^ - l^jju^u^ds = '^jfju.u.d, - ^jju^s. 
 
378 QUATERNIONS. [49°- 
 
 Hence, unless m^ = m^, we have 
 
 Thus we have ^^!Iful,d<; = JJJwujA<;. 
 
 A^ being thus found, we have generally 
 
 490. If, in the fundamental theorem, we suppose 
 
 which imposes the condition that 
 
 >SVo- = 0, 
 
 i.e., that the o- displacement is effected without condensation, it 
 becomes 
 
 SSS . Vt Uvds = ///-S VVfZ? = 0. 
 
 Suppose any closed curve to be traced on the surface S, dividing 
 it into two parts. This equation shews that the surface-integral is 
 the same for both parts, the difiference of sign being due to the 
 fact that the normal is drawn in opposite directions on the two 
 parts. Hence we see that, with the above limitation of the value 
 of a, the double integral is the same for all surfaces bounded by a 
 given closed curve. It must therefore be expressible by a single 
 integral taken round the curve. The value of this integral will 
 presently be determined (§ 495). 
 
 491. The theorem of § 485 may be written 
 
 JJJV'Pd^ = JJS UvVPds = JJS ( UvV) Pds. 
 From this we conclude at once that if 
 
 (which may, of course, represent any vector whatever) we have 
 
 JJJV'ad,=JJ8{UvV)ads, 
 
 or, if W = T, 
 
 JIIrd,^JJ8{UvV-')Tds. 
 
 This gives us the means of representing, by a surface-integral, a 
 vector-integral taken through a definite space. We have already 
 seen how to do the same for a scalar-integral — so that we can now 
 express in this way, subject, however, to an ambiguity presently 
 to be mentioned, the general integral 
 
 Ilkd,, 
 
493-] PHYSICAL APPLICATIONS. 379 
 
 where q is any quaternion whatever. It is evident that it is only 
 in certain classes of cases that we can expect a perfectly definite 
 expression of such a volume-integral in terms of a surface-integral. 
 
 492. In the above formula for a vector-integral there may 
 present itself an ambiguity introduced by the inverse operation 
 
 to which we must devote a few words. The assumption 
 
 is tantamount to saying that, if the constituents of a are the 
 potentials of certain distributions of matter, &c., those of t are the 
 corresponding densities each multiplied by 47r. 
 
 If, therefore, t be given throughout the space enclosed by 2, 
 or is given by this equation so far only as it depends upon the 
 distribution within S, and must be completed by an arbitrary vector 
 depending on three potentials of mutually independent distributions 
 exterior to %. 
 
 But, if cr be given, t is perfectly definite ; and as 
 
 Vcr = V-V, 
 
 the value of V"' is also completely defined. These remarks must 
 be carefully attended to in using the theorem above : since they 
 involve as particular cases of their application many curious 
 theorems in Fluid Motion, &c. 
 
 493. As a very special case, the equation 
 
 FVo- = 
 
 of course gives Vo- = m, a scalar. 
 
 Now, if V be the potential of a distribution whose density is u, we 
 have 
 
 Vv = 47rM. 
 
 We know that when u is assigned this equation gives one, and but 
 one, definite value for v. We have in fact, by the definition of a 
 potential. 
 
 '=111 Ti 
 
 where the integration (confined to Wj and /oj extends to all space 
 in which u differs from zero. 
 
380 QUATERNIONS. [494- 
 
 Thus there is no ambiguity in 
 
 and therefore o- = j- Vd 
 
 is also determinate. 
 
 494. This shews the nature of the arbitrary term which must 
 be introduced into the solution of the equation 
 
 rVa- = T. 
 
 To solve this equation is (§ 384) to find the displacement of any 
 one of a group of points when the consequent rotation is given. 
 
 Here ^fVT = 6^. VFV<7 = >SfVV = 0; 
 
 so that, omitting the arbitrary term (§ 493), we have 
 
 VV = Vt, 
 
 and each constituent of a- is, as above, determinate. Compare 
 §503. 
 
 Thomson* has put the solution in a form which may be written 
 
 a = yVTdp + Vu, 
 
 if we understand by /( ) dp integrating the term in da; as if y 
 and z were constants, &c. Bearing this in mind, we have as 
 verification, 
 
 F^ff = JS Vi { ^Ti + /F J dpi 
 
 = i{2. + 2/|d. + 2/dp^4;} 
 = J{3t+/c^/3/SVt} = t. 
 
 495. We now come to relations between the results of integra- 
 tion extended over a nou-closed surface and round its boundary. 
 
 Let o" be any vector function of the position of a point. The 
 line-integral whose value we seek as a fundamental theorem is 
 
 where r is the vector of any point in a small closed curve, drawn 
 from a point within it, and in its plane. 
 
 * Electrostatics and Magnetism, § 521, or Phil. Trans., 1852. 
 
495- J PHYSICAL APPLICATIONS. 381 
 
 Let o-j be the value of o- at the origin of t, then 
 o- = o-„ - /S (tV) o-„, 
 so that J8(rdT =JS.{(7,-S (tV) o-J dr. 
 
 But /dr = 0, 
 
 because the curve is closed ; and (antd, § 467) we have generally 
 
 JSrVSa-^dT = i/SfV {rSaoT - <7„ IVrdr). 
 Here the integrated part vanishes for a closed circuit, and 
 
 lJVTdT = dsUv, 
 where ds is the area of the small closed curve, and Up is a unit- 
 vector perpendicular to its plane. Hence 
 
 fSa,dT = S.Va,Uv.ds. 
 
 Now, any finite portion of a surface may be broken up into small 
 elements such as we have just treated, and the sign only of the 
 integral along each portion of a bounding curve is changed when 
 we go round it in the opposite direction. Hence, just as Ampere 
 did with electric currents, substituting for a finite closed circuit 
 a network of an infinite number of infinitely small ones, in each 
 contiguous pair of which the common boundary is described by 
 equal currents in opposite directions, we have for a finite unclosed 
 surface 
 
 J8(rdp=jjS.VaUv.ds. 
 
 There is no difficulty in extending this result to cases in which 
 the bounding curve consists of detached ovals, or possesses multiple 
 points. 
 
 This theorem seems to have been first given by Stokes (Smith's 
 Prize JEJccam. 1854), in the form 
 
 f(adx + ^dy + <ydz) 
 
 =*'K|-f)-(£-£)-(i-| 
 
 It solves the problem suggested by the result of § 490 above. It 
 will be shewn, however, in a later section that the equation above, 
 though apparently quite different from that of § 484, is merely a 
 particular case of it. 
 
 [If we recur to the case of an infinitely small area, it is clear 
 that 
 
 JJ8.V<TUvds 
 is a maximum when 
 
 V. UvWo- = 0. 
 
382 QUATEENIONS. [496- 
 
 Hence VVa- is, at every point, perpendicular to a small area 
 for whicli 
 
 JSadp 
 is a maximum.] 
 
 496. If a represent the vector force acting on a particle of 
 matter at p, ~8.a-dp represents the work done by it while the 
 particle is displaced along dp, so that the single integral 
 
 J8<rdp 
 
 of last section, taken with a negative sign, represents the work 
 done during a complete cycle. When this integral vanishes it is 
 evident that, if the path be divided into any two parts, the work 
 spent during the particle's motion through one part is equal to 
 that gained in the other. Hence the system of forces must be 
 conservative, i.e., must do the same amount of work for all paths 
 having the same extremities. 
 
 But the equivalent double integral must also vanish. Hence a 
 conservative system is such that 
 
 ffdsS.VaUv = 0, 
 
 whatever be the form of the finite portion of surface of which ds 
 is an element. Hence, as Vo- has a fixed value at each point of 
 space, while Uv may be altered at will, we must have 
 
 FVa- = 0, 
 
 or Vo" = scalar. 
 
 If we call X, Y, Z the component forces parallel to rectangular 
 axes, this extremely simple equation is equivalent to the well- 
 known conditions 
 
 ^-— = —-^ = — _^-0 
 dy dx ' dz dy ' dx dz ~ ' 
 
 Returning to the quaternion form, as far less complex, we see 
 that 
 
 Vo- = scalar = iirr, suppose, 
 
 implies that a = VP, 
 
 where P is a scalar such that 
 
 V"P = 47rr; 
 
 that is, P is the potential of a distribution of matter, magnetism, 
 or statical electricity, of volume-density r. 
 
498.] PHYSICAL APPLICATIONS. 383 
 
 Hence, for a non-closed path, under conservative forces, 
 -J8.<Tdp = -JS.VPdp = -J8(dpy)P 
 = Jda,P=JdP = P,-P„ 
 depending solely on the values of P at the extremities of the path. 
 
 497. A vector theorem, which is of great use, and which cor- 
 responds to the scalar theorem of § 491, may easily be obtained. 
 Thus, with the notation already employed, 
 
 /F.^dT=/FK->Sf(TV)cr„}d7, 
 
 = -JS(TV)r.a,dr. 
 
 Now F ( F. V F. Tdr) a, = -8 (tV) F. a.dr - 8 {drV) Ftc7„, 
 
 and d{8irV)V<T,T} = S (rV) F. cr^c^r + »S (drV) Fcr„T. 
 
 Subtracting, and omitting the term which is the same at both 
 
 limits, we have 
 
 JV. <TdT = - F. {VUv^) a.ds. 
 
 Extended as above to any closed curve, this takes at once the form 
 
 jr.adp = -JJdsV.{rUvV)a. 
 
 Of course, in many cases of the attempted representation of a 
 quaternion surface-integral by another taken round its bounding 
 curve, we are met by ambiguities as in the case of the space- 
 integral, § 492 : but their origin, both analytically and physically, 
 is in general obvious. 
 
 498. The following short investigation gives, in a complete 
 form, the kernel of the whole of this part of the subject. But 
 |§ 495 — 7 have still some interest of their own. 
 
 If P be any scalar function of p, we have (by the process of 
 
 5 495, above) 
 
 . JPdr=J{P,-8{rV)P,]dT 
 
 = -J8.TVP,.dr. 
 
 But r.VV.rdr = dT8.rV-TS.dTV, 
 
 and d (t8tV) = drS .tV + t8. drV. 
 
 These give 
 
 JPdr = - i {r/SfrV - F. JF(t(^t) V} P„ = dsV. Uv^P,. 
 
 Hence, for a closed curve of any form, we have 
 
 !Pdp=HdsV.Uv^P, 
 
 from which the theorems of §§ 495, 497 may easily be deduced. 
 
384 QUATEENIONS. [499- 
 
 Multiply into any constant vector, and we have, by adding 
 
 three such results 
 
 jdpa = Jfdsrl UvV) a. [See § 497.] 
 
 Hence at once (by adding together the corresponding members 
 of the two last equations, and putting 
 
 fdpq=JJdsViUvV)q, 
 
 where q is any quaternion whatever. 
 
 [For the reason why we have no corresponding formula, with S 
 instead of V in the right-hand member, see remark in [ ] in 
 § 505.] 
 
 499. Commencing afresh with the fundamental integral 
 
 JfJSVad, = JJ8<7Uvds, 
 
 put a- = u^, 
 
 and we have ////S/SVmcZ? = JfuS^ Uvds ; 
 
 from which at once fffVud<; = ffuUvds (1), 
 
 or mrd^=JfUv.Tds (2), 
 
 which gives 
 
 ///FV F<7T(is = rjJUvr<7Tds = JJirSUva - aSUvr) ds. 
 
 Equation (2) gives a remarkable expression for the surface of a 
 space in terms of a volume-integral. For take 
 
 T=Uv= UVP, 
 
 where P = const. 
 
 is the scalar equation of the closed bounding surface. Then 
 
 - Jfds = JJUv Uvds = ///V UVPd^. 
 
 (Note that this implies 
 
 fJIV^UVPd, = 0, 
 which in itself is remarkable.) 
 
 Thus the surface of an ellipsoid 
 
 Sp4>p = G, 
 is mUcf^pd^, 
 
 the integration being carried on throughout the enclosed space. 
 (Compare § 485.) 
 
500.] PHYSICAL APPLICATIONS. 385 
 
 Again, in (2), putting u^ for t, and taking the scalar, we have 
 jflSrVu^ + m,SVt) d^ = JJu^SrUrds, 
 
 whence fJJ{S (rV) a + <tSS7t\ d<; = JJaSr Uvds (3). 
 
 The sum of (1) and (2) gives, for any quaternion, 
 
 Hnqd, = HUv.qds. 
 
 The final formulae in this, and in the preceding, section give 
 expressions in terms of surface integrals for the volume, and the 
 line, integrals of a quaternion. The latter is perfectly general, but 
 (for a reason pointed out in § 492) the former is definite only when 
 the quaternion has the form Vq. 
 
 500. The fundamental form of the Volume and Surface 
 Integral is (as in § 499 (1)) 
 
 j^jVud'i^SJUvuds. 
 
 Apply it to a space consisting of a very thin transverse slice of 
 a cylinder. Let t be the thickness of the slice, A the area of one 
 end, and a a unit-vector perpendicular to the plane of the end. 
 The above equation gives at once 
 
 V (aV) u.tA=tiVaUv. udl, 
 
 where dl is the length of an element of the bounding curve of the 
 section, and the only values of Uv left are parallel to the plane of 
 the section and normal to the bounding curve. If we now put p 
 as the vector of a point in that curve, it is plain that 
 
 V.aUv=Udp, dl = Tdp, 
 
 and the expression becomes 
 
 V{aV)u.A =Judp. 
 
 By juxtaposition of an infinite number of these infinitely small 
 directed elements, a. (now to be called Uv) being the normal vector 
 of the area A (now to be called ds), we have at once 
 
 JJV {UvV) uds = Judp, 
 
 which is the fundamental form of the Surface and Line Integral, 
 as given in | 498. Hence, as stated in § 484, these relations are 
 not independent. 
 
 In fact, as the first of these expressions can be derived at once 
 from the ordinary equation of "continuity," so the second is merely 
 the particular case corresponding to displacements confined to a 
 T. Q. I. 25 
 
386 QUATERNIONS. [SOI- 
 
 given surface. It is left to the student to obtain it, simply and 
 directly, (in the form of § 495) from this consideration. 
 
 [Note. A remark of some importance must be made here. It 
 may be asked :— Why not adopt for the proof of the fundamental 
 theorems of the present subject the obvious Newtonian process (as 
 applied, for instance, in Thomson and Tait's Natural Philosophy, 
 § 194, or in Clerk-Maxwell's Electricity, § 591)? 
 
 The reply is that, while one great object of the present work is 
 (as far as possible) to banish artifice, and to shew the "perfect 
 naturalness of Quaternions," the chief merit of the beautiful 
 process alluded to is that it forms one of the most intensely 
 artificial applications of an essentially artificial system. Cartesian 
 and Semi-Cartesian methods may be compared to a primitive 
 telegraphic code, in which the different signals are assigned to 
 the various letters at hap-hazard; Quaternions to the natural 
 system, in which the simplest signals are reserved for the most 
 frequently recurring letters. In the former system some one word, 
 or even sentence, may occasionally be more simply expressed than 
 in the latter : — though there can be no doubt as to which system 
 is to be preferred. But, even were it not so, the methods we have 
 adopted in the present case give a truly marvellous insight into 
 the real meaning and " inner nature '' of the formulae obtained.] 
 
 501. As another example of the important results derived 
 from the simple formulae of § 499, take the following, viz.: — 
 
 //F. V (o- Uv) Tds = JJaSr Uvds - fJUvSaTcls, 
 
 where by (3) and (1) of that section we see that the right-hand 
 member may be written 
 
 = /// {S (tV) <7 -f- aS^T - VSar] d? 
 
 = -ffIV.V{Va)Td': (4). 
 
 In this expression the student must remark that V operates on t 
 as well as on a. Had it operated on t alone, we should have 
 inverted the order of V and a, and changed the sign of the whole ; 
 or we might have had recourse to the notation of the end of § 133. 
 This, and similar formulae, are easily applied to find the 
 potential and the vector-force due to various distributions of 
 magnetism. To. shew how they are to be introduced, we briefly 
 sketch the mode of expressing the potential of a distribution. 
 
503-] PHYSICAL APPLICATIONS. 387 
 
 K. Application of the V Integrals to Magnetic <&c. Problems. 
 
 502. Let o- be the vector expressing the direction and intensity 
 of magnetisation, per unit of volume, at the element cZ?. Then if 
 the magnet be placed in a field of magnetic force whose potential 
 is u, we have for its potential energy 
 
 E = -JJJSaVud<i 
 
 = JJfu8V<,d,-fJ!SV(ua)d'; 
 
 = ///mS Vo-d? - jJuSa Uvds. 
 
 This shews at once that the magnetism may be resolved into a 
 volume-density SVa, and a surface-density — Sa Up. Hence, for 
 a solenoid al distribution, 
 
 /SVo- = 0. 
 
 What Thomson has called a lamellar distribution (Phil. Trans. 
 1852), obviously requires that 
 
 Scrdp 
 be integrable without a factor ; i.e., that 
 
 FVo- = 0. 
 A complex lamellar distribution requires that the same expression 
 be integrable by the aid of a factor. If this be u, we have at once 
 
 FV (mo-) = 0, 
 or S.aVa- = 0. 
 
 But we easily see that (4) of § 501 may be written 
 - fJV. ( Va Up) rds = - JJJV. rVVad^; - JJJV. a-Vrd^ + JJJSctV . rd^. 
 
 Now. if ^ = ^0' 
 
 where r is the distance between any external point and the element 
 d<i, the last term on the right is the vector-force exerted by the 
 magnet on a unit-pole placed at the point. The second term on 
 the right vanishes by Laplace's equation, and the first vanishes as 
 above if the distribution of magnetism be lamellar, thus giving 
 Thomson's result in the form of a surface integral. 
 
 503. As another instance, let a be the vector of magnetic 
 induction, /3 the vector potential, at any point. Then we know, 
 
 physically, that 
 
 JS^dp=JI8Upc(,ds. 
 
 25—2 
 
388 QUATERNIONS. [504- 
 
 But, by the theorem of § 496, we have 
 
 Since the boundary and the enclosed surface may be any what- 
 ever, we must have 
 
 a=FV/3; 
 
 and, as a consequence, 
 
 SVOL = 0. 
 
 Hence V/3 = a + «, 
 
 = Va + V-'u + Vg, 
 where 5 is a quaternion satisfying the equations 
 V^^ = 0, SVq = 0. 
 To interpret the other terms, let 
 V'u = a-, 
 so that VVa- = ; 
 
 and ff = Vw, 
 
 where w is a scalar such that 
 
 V'v = u. 
 
 Thus V is the potential of a distribution it/47r, and can there- 
 fore be found without ambiguity when u is given. And of course 
 
 V'ti = Vv 
 is also found without ambiguity. 
 
 Again, as SVa = 0, 
 
 we have Va = 7, suppose. 
 
 Hence, for any assumed value of 7, we have 
 V-'« = V'V, 
 
 so that this term of the above value of ^ is also found without 
 ambiguity. 
 
 The auxiliary quaternion q depends upon potentials of arbitrary 
 distributions wholly outside the space to which the investigation 
 may be limited. [Compare this section with § 494.] 
 
 504. An application may be made of similar transformations 
 to Ampere's Directrice de Vaction ^lectrodynamique, which, § 458 
 above, is the vector-integral 
 
 Vpdp 
 Tp' ' 
 
 I 
 
505-J PHYSICAL APPLICATIONS. 389 
 
 where dp is an element of a closed circuit, and the integration 
 extends round the circuit. This may be written 
 
 -jvdpvl, 
 
 so that its value as a surface integral is 
 
 JJS ( UvV) V i rf,9 - jjUvV' ~ ds. 
 
 Of this the last term vanishes, unless the origin is in, or infinitely 
 near to, the surface over which the double integration extends. 
 The value of the first term is seen (by what precedes) to be the 
 vector-force due to uniform normal magnetisation of the same 
 surface. Thus we see the reason why the Directrice can be 
 expressed in terms of the spherical opening of the circuit, as in 
 §§ 459, 471. 
 
 505. The following result is obviously but one of an extensive 
 class of useful transformations. Since 
 
 we obtain at once from § 484 
 
 a curious expression for the gravitation potential of a homogeneous 
 body in terms of a surface-integral. 
 
 [The right-hand member may be written as 
 
 JJSUvVTpds ; 
 
 and an examination into its nature shews us why we ought not to 
 expect to have a general expression for 
 
 fJSUv'^Pds 
 
 in terms of a line-integral. It will be excellent practice for the 
 student to. make this examination himself. Of course, a more 
 general method presents itself in finding the volume-integral 
 which is equivalent to the last written surface-integral extended 
 to the surface of a closed space.] 
 
 From this, by differentiation with respect to a, after putting 
 p + a for p, or by expanding in ascending powers of Ta (both of 
 which tacitly assume that the origin is external to the space 
 
390 QUATERNIONS. [S^^. 
 
 integrated throtigh, i.e., that Tp nowhere vanishes within the 
 limits), we have 
 
 and this, again, involves 
 
 Jlw-IIP''"^' 
 
 506. The interpretation of these, and of more complex formulae 
 of a similar kind, leads to many curious theorems in attraction and 
 in potentials. Thus, from (1) of § 499, we have 
 
 ///B^-///f^*=//f^ ''•• 
 
 which gives the attraction of a mass of density t in terms of the 
 potentials of volume distributions and surface distributions. Putting 
 
 this becomes 
 
 [ff Va-d'i _ rCf Up . ad's _ /■[ Uv . ads 
 J J] Tp jj] Tp' -jj ^^■ 
 
 By putting a = p, and taking the scalar, we recover a formula 
 given above ; and by taking the vector we have 
 
 VfJUvUpds = 0. 
 
 This may be easily verified from the formula 
 
 fPdp=VJJUv.VPds, 
 
 by remembering that ^Tp= Up. 
 
 Again if, in the fundamental integral, we put 
 
 a = t Up, 
 
 we 
 
 -- ///^'*-.///|-//^.«.... 
 
 [It is curious how closely, in fact to a numerical factor of one 
 term pres, this equation resembles what we should get by operating 
 on (1) hy S.p, and supposing that we could put p under the signs 
 of integration.] 
 
50 8.] PHYSICAL APPLICATIONS. 391 
 
 L. Application of V to the Stress-Function. 
 
 507. As another application, let us consider briefly the Stress- 
 function in an elastic solid. 
 
 At any point of a strained body let X be the vector stress per 
 unit of area perpendicular to i, fi and v the same for planes per- 
 pendicular to j and h respectively. 
 
 Then, by considering an indefinitely small tetrahedron, we 
 have for the stress per unit of area perpendicular to a unit-vector 
 o) the expression 
 
 \Si(o + fJ'Sja) + vSJca = — (^m, 
 
 so that the stress across any plane is represented by a linear and 
 vector function of the unit normal to the plane. 
 
 But if we consider the equilibrium, as regards rotation, of an 
 infinitely small parallelepiped whose edges are parallel to i, j, k 
 respectively, we have (supposing there are no molecular couples) 
 
 V{i\ +JIJ, + kv) = 0, 
 
 or S Vi^i = 0, 
 
 or, supposing V to apply to p alone. 
 
 This shews (§ 185) that in the present case ^ is self -conjugate, and 
 thus involves not nine distinct constants but only six. 
 
 508. Consider next the equilibrium, as regards translation, of 
 any portion of the solid filling a simply-connected closed space. 
 Let u be the potential of the external forces. Then the condition 
 is obviously 
 
 !S4>{Uv)ds+HSd,-^u = o, 
 
 where v is the normal vector of the element of surface ds. Here 
 the double integral extends over the whole boundary of the closed 
 space, and the triple integral throughout the whole interior. 
 
 To reduce this to a form to which the method of § 485 is 
 directly applicable, operate by ;S. a where a. is any constant vector 
 whatever, and we have 
 
 ///Sf . <^a Uvds -t- JJJdsSaVu = 
 
392 QUATERNIONS. [SoS. 
 
 by taking advantage of the self-conjugateness of 0. This may be 
 written (by transforming the surface-integral into a volume- 
 integral) 
 
 and, as the limits of integration may be any whatever, 
 
 S.V(pa + 8aVu = (1). 
 
 This is the required equation, the indeterminateness of a rendering 
 it equivalent to three scalar conditions. 
 
 There are various modes of expressing this without the a. 
 Thus, if A be used for V when the constituents of (f> are considered, 
 we may write 
 
 Vm = SVA. (/>/). 
 
 It is easy to see that the right-hand member may be put in 
 either of the equivalent forms 
 
 V8.A4ip or S.AcjN.p. 
 In integrating this expression through a given space, we must 
 remark that V and p are merely temporary symbols of construction, 
 and therefore are not to be looked on as variables in the integral. 
 
 Instead of transforming the surface-integral, we might have 
 begun by transforming the volume-integral. Thus the first equa- 
 tion of this section gives 
 
 JJ{(f> + u) Uvds = 0. 
 
 From this we have at once 
 
 //,S. TJv{<^ + u)oLds=0. 
 
 Thus, by the result of § 490, whatever be a we have 
 
 which is the condition obtained by the former process. 
 
 As a verification, it may perhaps be well to shew that from this 
 equation we can get the condition of equilibrium, as regards 
 rotation, of a simply-connected portion of the body, which can be 
 written by inspection as 
 
 //F. p<^ ( Uv) ds + JJJ VpVud, = 0. 
 
 This is easily done as follows : (1) gives 
 
 if, and only if, a satisfy the condition 
 
 S.<j>{V)a = 0. 
 
509-J PHYSICAL APPLICATIONS. 393 
 
 Now this condition is satisfied hy 
 
 a = Vap 
 
 where a is Any constant vector. For 
 
 S. ^ (V) Vap = -S.aV^ (V) p = S . aF V,^p, = 0, 
 
 in consequence of the self-conjugateness of ^. Hence 
 
 ///c^9 {8 . V4> Vap + 8 . apVu) = 0, 
 
 or JJds8 .ap(f>Uv+ JJJd'i8 . apVu = 0. 
 
 Multiplying by a, and adding the results obtained by making a in 
 succession each of three rectangular vectors, we obtain the required 
 equation. 
 
 509. To find the stress-function in terms of the displacement at 
 each point of an isotropic solid, when the resulting strain is small, 
 we may conveniently apply the approximate method of § 384. As 
 the displacement is supposed to be continuous, the strain in the 
 immediate neighbourhood of any point may be treated as homo- 
 geneous. Thus, round each point, there is one series of rectangular 
 parallelepipeds, each of which remains rectangular after the strain. 
 Let a, /3, 7 ; a^, /S^, 7, ; be unit vectors parallel to their edges 
 before, and after, the strain respectively ; and let e^, e^, e^ be the 
 elongations of unit edges parallel to these lines. We shall not 
 have occasion to determine these quantities, as they will be 
 eliminated after having served to form the requisite equations. 
 
 Since the solid is isotropic and homogeneous, the stress is 
 perpendicular to each face in the strained parallelepipeds ; and 
 its amount (per unit area) can be expressed as 
 
 P, = 2ne, + (c - f?i) Se, &c (1), 
 
 where n and c are, respectively, the rigidity and the resistance to 
 compression. 
 
 Next, as in § 384, let a be the displacement at p. The strain- 
 function is 
 
 yfrzy = 07 — (St«rV. a 
 
 so that at once 
 
 te=-8^a; (2), 
 
 and, ii q { ) ?~' be the operator which turns a into a,, &c., we 
 
 have 
 
 qnsq-' = -ST+^VV{s7<T)i!T (3). 
 
394 QUATERNIONS. [SOQ- 
 
 Thus, if (f> be the stress-function, we have (as in § 507) 
 
 ^a) = - 2 . P,a,Sa,« (4). 
 
 But i/rw = - 2 . (1 + e,) a,/Sa&), 
 
 so that ■x/r'o) = - 2 . (1 + ej) aSa^o}, 
 
 and qyfr'coq''- = - 2.(1 +61) a^Sa^a (5). 
 
 By the help of (1), (2), and (5), (4) becomes 
 
 <f)a) = 2nq^p''o)q'^ — 2n(o — (c — 1«) mSVa ; 
 and, to the degree of approximation employed, (3) shews that this 
 may be written 
 
 <^ay = -n (ScoV. a + VSwa) - (c - f w) coSVa (6), 
 
 which is the required expression, the function <f> being obviously 
 self- conjugate. 
 
 As an example of its use, suppose the strain to be a uniform 
 dilatation. Here 
 
 (T = ep, 
 
 and (j)(o = 2neQ) + 3 (c — §n) ea> = Sceco ; 
 
 denoting traction See, uniform in all directions. If e be negative, 
 there is uniform condensation, and the stress is simply hydrostatic 
 pressure. 
 
 Again, let a- = — eaSap, 
 
 which denotes uniform extension in one direction, unaccompanied 
 by transversal displacement. We have (a being a unit vector) 
 
 ^(B = — 2nea8caa + (c — fw) eco. 
 
 Thus along a there is traction 
 
 {c + ^n)e, 
 
 but in all directions perpendicular to a there is also traction 
 
 (c-fw)e. 
 
 Finally, take the displacement 
 
 cr = — eaS^p. 
 
 It gives (pco = — ne (aSa^ + ^Saa) - (c — |w) ecoSa.^. 
 
 This displacement gives a simple shear if the unit vectors a and /3 
 are at right angles to one another, and then 
 
 <f>(o = — ne (ajSftj/S -I- ^Scoa), 
 
 which agrees with the well-known results. In particular, it shews 
 that the stress is wholly tangential on planes perpendicular either 
 
5'I-] PHYSICAL APPLICATIONS. 395 
 
 to a or to /3 ; and wholly normal on planes equally inclined to them 
 and perpendicular to their plane. The symmetry shews that the 
 stress will not be affected by interchange of the unit vectors, 
 a and ^, in the expression for the displacement. 
 
 olO. The work done by the stress on any simply connected 
 portion of the solid is obviously 
 
 because ^ ( Uv) is the vector force overcome per unit of area on 
 the element ds. [The displacement at any moment may be written 
 xa-; and, as the stress is always proportional to the strain, the 
 factor xdx has to be integrated from to l.J This is easily trans- 
 formed to 
 
 W=^SHS.V^cyd^. 
 
 511. We may easily obtain the general expression for the 
 work corresponding to a strain in any elastic solid. The physical 
 principles on which we proceed are those explained in Appendix 
 G to Thomson and Tait's Natural Philosophy. The mode in 
 which they are introduced, however, is entirely different; and a 
 comparison will shew the superiority of the Quaternion notation, 
 alike in compactness and in intelligibility and suggestiveness. 
 
 If the strain, due to the displacement a, viz. 
 
 ■>JrT = T — /SfrV . a 
 
 be a mere rotation, in which case of course no work is stored up 
 by the stress, we have at once 
 
 8 . -\|ra)i|rT = Scot 
 
 for all values of co and t. We may write this as 
 
 ;Si . « (^yjr'ylr — 1) T = Sm-^T = 0, 
 
 where % is (§ 380) a self-conjugate linear and vector function, whose 
 complete value is 
 
 Xr=- StV . a - VSto- + ^.StVSo-o-^. 
 The last term of this may, in many cases, be neglected. 
 
 When the strain is very small, the work (per unit volume) 
 must thus obviously be a homogeneous function, of the second 
 degree, of the various independent values of the expression 
 
 S(OXT. 
 
396 QU ATERN IONS. [5^2. 
 
 On account of the self-conjugateness of x there are but six 
 such values : — viz. 
 
 Sixi ^hJ' %^. ^JXJ' ^JX^' ^^X^- 
 Their homogeneous products of the second degree are therefore 
 twenty-one in number, and this is the number of elastic coefficients 
 which must appear in the general expression for the work. In 
 the most general form of the problem these coefficients are to be 
 regarded as given functions of p. 
 
 At and near any one point of the body, however, we may take 
 i,j, k as the chief vectors of ;^ at that point, and then the work 
 for a small element is expressible in terms of the six homogeneous 
 products, of the second degree, of the three quantities 
 
 ^^X^' SJXJ' ^^X^- 
 This statement will of course extend to a portion of the body of 
 any size if (whether isotropic or not) it be homogeneous and 
 homogeneously strained. From this follow at once all the 
 elementary properties of homogeneous stress. 
 
 M. The Hydrokinetic Equations. 
 
 512. As another application, let us form the hydrokinetic 
 equations, on the hypothesis that a perfect fluid is not a molecular 
 assemblage but a continuous medium. 
 
 Let a be the vector-velocity of a very small part of the fluid at 
 p ; e the density there, taken to be a function of the pressure, p, 
 alone ; i.e. supposing that the fluid is homogeneous when the 
 pressure is the same throughout ; P the potential energy of unit 
 mass at the point p. 
 
 The equation of " continuity" is to be found by expressing the 
 fact that the increase of mass in a small fixed space is equal 
 to the excess of the fluid which has entered over that which 
 has escaped. If we take the volume of this space as unit, the 
 condition is 
 
 de 
 
 j^ = \jSUv(ea)ds = SV(ea) (1). 
 
 We may put this, if we please, in the form 
 de de „ _ „_ 
 
 a"r^-'^"^-^ = ^^^- (2). 
 
 where 9 expresses total differentiation, or, in other words, that we 
 follow a definite portion of the fluid in its motion. 
 
5 1 3-] PHYSICAL APPLICATIONS. 397 
 
 The expression might at once have been written in the form 
 (2) from the comparison of the results of two different methods of 
 representing the rate of increase of density of a small portion of 
 the fluid as it moves along. Both forms reduce to 
 
 8Va- = 0, 
 
 when there is no change of density (§ 384). 
 
 Similarly, for the rate of increase of the whole momentum 
 within the fixed unit space, we have 
 
 where the meanings of the first two terms are obvious, and the 
 third is the excess of momentum of the fluid which enters, over 
 that of the fluid which leaves, the unit space. 
 
 The value of the double integral is, by § 499 (3), 
 
 o-SV (eo-) + e-Sfo-V .a = a^ + eSaV . <t, by (1). 
 
 Thus we have, for the equation of motion 
 or, finally 
 
 ot at 
 
 This, in its turn, might have been even more easily obtained by 
 dealing with a small definite portion of the fluid. 
 
 It is necessary to observe that in what precedes we have 
 tacitly assumed that a is continuous throughout the part of the 
 fluid to which the investigation applies : — i.e. that there is neither 
 rupture nor finite sliding. 
 
 513. There are many ways of dealing with the equation (3) 
 of fluid motion. We select a few of those which, while of historic 
 interest, best illustrate quaternion methods. 
 
 We may write (3) as 
 Now we have always 
 
398 QUATEBNIONS . [S I 3 • 
 
 Hence, if the motion be irrotational, so that (§ 384) 
 
 VVa = 0, 
 the equation becomes 
 
 But, if w be the velocity-potential, 
 
 o- = Vw ; 
 and we obtain (by substituting this in the first term, and operating 
 on the whole by S . dp) the common form 
 
 d~ + ^d.v' = -dQ, 
 
 where ^v" (= — ^a^) is the kinetic energy of unit mass of the fluid. 
 
 If the fluid be incompressible, we have Laplace's equation 
 
 for w, viz. 
 
 V% = SVo- = 0. 
 
 When there is no velocity-potential, we may adopt Helmholtz' 
 method. But first note the following quaternion transformation 
 {Proc. R. S. E. 1869—70) 
 
 [The expression on the right has many remarkable forms, the 
 finding of which we leave, as an exercise, to the student. For our 
 present purpose it is sufficient to know that its vector part is 
 
 - S . VjO-jV . o- + Wa- . /S Vo-.] 
 This premised, operate on (3) by V. V, and we have 
 
 Hence at once, if the fluid be compressible, 
 
 iFVo- = -/S.V,o-V.o-+FVo-.SVo-= F.VK.o-FVo-. 
 at ' ' 
 
 But if the fluid be incompressible 
 
 |FV<7 = -S.V,C7,V.(r. 
 
 Either form shews that when the vector-rotation vanishes, its 
 rate of change also vanishes. In other words, those elements of 
 the fluid, which were originally devoid of rotation, remain so during 
 the motion. 
 
SH]- physical applications. 399 
 
 Thomson's mode of dealing with (3) is to introduce the 
 integral 
 
 f^-TSadp (5) 
 
 J a 
 
 which he calls the " flow " along the arc from the point a to the 
 point p ; these being points which move with the fluid. 
 
 Operating on (3) with S . dp, we have as above 
 TT-Sadp — Sada- = dO.; 
 
 01 
 
 so that, integrating along any definite line in the fluid from a to 
 p, we have 
 
 dt 
 
 Q-l 
 
 5' 
 
 •(6) 
 
 which gives the rate at which the flow along that line increases, as 
 it swims along with the fluid. 
 
 If we integrate round a closed curve, the value of df/dt vanishes, 
 because Q is essentially a single-valued function. In this case the 
 quantity f is called the " circulation,'' and the result is stated in 
 the fortn that the circulation round any definite path in the fiuid 
 retains a constant value. 
 
 Since the circulation is expressed by the complete integral 
 
 -jSadp 
 
 it can also be expressed by the corresponding double integral 
 
 -JJ8. UvVads, 
 
 so that it is only when there is at least one vortex-filament passing 
 through the closed circuit that the circulation can have a finite 
 value. 
 
 N. Use of V in connection with Taylor's Theorem. 
 514. Since the algebraic operator 
 
 6 dx, 
 
 when applied to any function of x, simply changes x into x + h, it 
 is obvious that if o- be a vector not acted on by 
 
 — _ . d . d , d 
 dx '' dy dz' 
 
 we have e'^^^fip) =f{p + a). 
 
400 QUATERNIONS. [515- 
 
 whatever function / may be. From this it is easy to deduce 
 Taylor's theorem in one important quaternion form. 
 
 If A bear to the constituents of a the same relation as V bears 
 to those of p, and if / and F be any two functions which satisfy 
 the commutative law in multiplication, this theorem takes the 
 curious form 
 
 of which a particular case is (in Cartesian symbols) 
 
 ^f{x)F{y)=f{x+^^F{y)=F{^y+^^f{x). 
 
 The modifications which the general expression undergoes, when 
 / and F are not commutative, are easily seen. 
 
 If one of these be an inverse function, such as, for instance, 
 may occur in the solution of a linear differential equation, these 
 theorems of course do not give the arbitrary part of the integral, 
 but they often materially aid in the determination of the rest. 
 
 One of the chief uses of operators such as »SaV, and various 
 scalar functions of them, is to derive from IjTp the various orders 
 of Spherical Harmonics. This, however, is a very simple matter. 
 
 515. But there are among them results which appear startling 
 from the excessively free use made of the separation of symbols. 
 Of these one is quite sufficient to shew their general nature. 
 
 Let P be any scalar function of p. It is required to find the 
 difference between the value of P at p, and its mean value 
 throughout a very small sphere, of radius r and volume v, which has 
 the extremity of p as centre. This, of course, can be answered at 
 once from the formula of § 485. But the somewhat prolix method 
 we are going to adopt is given for its own sake as a singular piece 
 of analysis, not for the sake of the problem. 
 
 From what is said above, it is easy to see that we have the 
 following expression for the required result : — 
 
 l///(e-v_l)P,, 
 
 where o- is the vector joining the centre of the sphere with the 
 element of volume d<s, and the integration (which relates to o- 
 and d<i alone) extends through the whole volume of the sphere. 
 
5 1 6.] PHYSICAL APPLICATIONS. 401 
 
 Expanding the exponential, we may write this expression in the 
 form 
 
 higher terms being omitted on account of the smallness of r, the 
 limit of To-. 
 
 Now, symmetry shews at once that 
 
 IIH, = o. 
 
 Also, whatever constant vector be denoted by a, 
 
 /// {SaaY d, = - aV// {8a- Uaf d,. 
 
 Since the integration extends throughout a sphere, it is obvious 
 that the integral on the right is half of what we may call the 
 moment of inertia of the volume about a diameter. Hence 
 
 /// 
 
 {8aUoLfd^ = '"^. 
 
 If we now write V for a, as the integration does not refer to V, 
 we have by the foregoing results (neglecting higher powers of r) 
 
 lJjJ(e-..v_i)P,, = _!L;v^P, 
 
 which is the expression given by Clerk-Maxwell *. Although, for 
 simplicity, P has here been supposed a scalar, it is obvious that in 
 the result above it may at once be written as a quaternion. 
 
 516. As another illustration, let us apply this process to the 
 
 finding of the potential of a surface-distribution. If p be the 
 
 vector of the element ds, where the surface density is fp, the 
 
 potential at a is 
 
 SJdsfp.FT{p-<T), 
 
 F being the potential function, which may have any form whatever. 
 
 By the preceding, § 514, this may be transformed into 
 
 SJdsfp.e^'^'FTp; 
 
 or, far more conveniently for the integration, into 
 
 JSdsfp.e'<-^FT<y, 
 
 * London Math. Soc. Proc, vol. iii, no. 34, 1871. 
 T. Q. L 26 
 
402 QUATERNIONS . L 5 ^ 7 • 
 
 where A depends on the constituents of a in the same manner as 
 V depends on those of p. 
 
 A still farther simplification may be introduced by using a 
 vector o-„, which is finally to be made zero, along with its corre- 
 aponding operator A„, for the above expression then becomes 
 
 where p appears in a comparatively manageable form. It is obvious 
 that, so far, our formulae might be made applicable to any dis- 
 tribution. We now restrict them to a superficial one. 
 
 517. Integration of this last /orm can always be easily effected 
 in the case of a surface of revolution, the origin being a point in 
 the axis. For the expression, so far as the integration is concerned, 
 can in that case be exhibited as a single integral 
 
 
 ■<7 
 
 da; fee e"*, 
 p 
 
 where F may be any scalar function, and x depends on the cosine of 
 the inclination of p to the axis. And 
 
 I da5Fa;e'"'= F(t-) . 
 
 As the interpretation of the general results is a little trouble- 
 some, let us take the case of a spherical shell, the origin being the 
 centre and the density unity, which, while simple, sufficiently 
 illustrates the proposed mode of treating the subject. 
 
 We easily see that in the above simple case, a being any 
 constant vector whatever, and a being the radius of the sphere, 
 
 JJds e^"" = 2'n-a r^e'T'^dx = ^ (e"^" - e""^"). 
 
 J -a ■'-^ 
 
 Now, it appears that we. are at liberty to treat A as a has just 
 been treated. It is necessary, therefore, to find the effects of such 
 operators as TA, e"^^, &c., which seem to be novel, upon a scalar 
 function of Ta ; or '^, as we may for the present call it. 
 
 Now (TAfF=-A'F = F" + ~, 
 
 whence it is easy to guess at a particular form of TA. To make 
 sure that it is the only one, assume 
 
5 1 8. J PHYSICAL APPLICATIONS. 403 
 
 where f and f are scalar functions of '^ to be found. This gives 
 
 = i'F" + (ff ' + f f + ff ) F' + (if + r) F. 
 Comparing, we have 
 
 f = 1, 
 
 ff' + ff+ff=|; 
 
 ff' + f'' = 0. 
 From the first, f=±l, 
 
 whence the second gives f = + =, ; 
 
 the signs of f and f being alike. The third is satisfied identically. 
 
 That is, finally, ±TA = ^f^ + ^. 
 
 Also, an easy induction shews that 
 
 Hence we have at once 
 
 caTA. 
 
 
 + &c. 
 
 by the help of which we easily arrive at the well-known results. 
 This we leave to the student*. 
 
 0. Applications of V in connection with Galctdus of Variations. 
 
 518. We conclude with a few elementary examples of the use 
 of V in connection with the Calculus of Variations. These depend, 
 for the most part, on the simple relation 
 
 SQ = -/Sf8pV.Q. 
 Let us first consider the expression 
 A=JQTdp, 
 where Tdp is an element of a finite arc along which the inte- 
 gration extends, and the quaternion Q is a function of p, generally 
 a scalar. 
 
 * Proc. E. 8. E., 1871-2. 
 
 26—2 
 
404 QUATERNIONS. [SIQ- 
 
 To shew the nature of the enquiry, note that if Q be the 
 speed of a unit particle, A will be what is called the Action. If 
 Q be the potential energy per unit length of a chain, A is the 
 total potential energy. Such quantities are known to assume 
 minimum, or at least "stationary", values in various physical 
 processes. 
 
 We have for the variation of the above quantity- 
 
 BA = J [BQTdp + QBTdp) 
 
 =JiBQTdp-QS.UdpdBp) 
 
 = - [QSUdpBp] + / {BQTdp + S.Bpd (Q Udp)], 
 
 where the portion in square brackets refers to the limits only, and 
 gives the terminal conditions. The remaining portion may easily 
 be put in the form 
 
 8SBp{d{QUdp)-'^Q.Tdp]. 
 
 If the curve is to be determined by the condition that the varia- 
 tion of A shall vanish, we must have, as Bp may have any direction, 
 
 d{QUdp)-'^Q.Tdp = Q, 
 
 or, with the notation of Chap. X, 
 
 This simple equation shews that (when Q is a scalar) 
 
 (1) The osculating plane of the sought curve contains the 
 vector VQ. 
 
 (2) The curvature at any point is inversely as Q, and directly 
 as the component of VQ parallel to the radius of absolute curvature. 
 
 519. As a first application, suppose A to represent the Action 
 of a unit particle moving freely under a system of forces which 
 have a potential ; so that 
 
 Q = Tp, 
 
 and p^ = 2{P-H), 
 
 where P is the potential, H the energy-constant. 
 These give TpVTp = ^VQ = - VP, 
 
 and Qp' = p, 
 
 so that the equation above becomes simply 
 
 p + VP = 0, 
 
52 2.] PHYSICAL APPLICATIONS. 405 
 
 which is obviously the ordinary equation of motion of a free 
 particle. 
 
 520. If we look to the superior limit only, the first expression 
 for 8 A becomes in the present case 
 
 - [TpSUdpBp] = - SpSp. 
 
 If we suppose a variation of the constant H, we get the following 
 term from the unintegrated part 
 
 tBH. 
 
 Hence we have at once Hamilton's equations of Varying Action in 
 the forms 
 
 VA=p 
 
 and ^Tff = t. 
 
 an 
 
 The first of these gives, by the help of the condition above, 
 
 {VAy = 2{P-H), 
 
 the well-known partial differential equation of the first order and 
 second degree. 
 
 521. To shew that, if A be any solution whatever of this 
 equation, the vector VJ. represents the velocity in a free path 
 capable of being described under the action of the given system of 
 forces, we have 
 
 = -£f(V.4.V)V^. 
 But ^.V^ = -fi'06V)VA 
 
 A comparison shews at once that the equality 
 
 VA^p 
 is consistent with each of these vector equations. 
 
 522. Again, if 9 refer to the constants only, 
 
 ^3 {VAf = S . ^AdVA = -dH 
 by the differential equation. 
 
 But we have also -^rf = t, 
 
 which gives ^^(dA) = -8{pV)dA=dH. 
 
406 QUATERNIONS. [523. 
 
 These two expressions for dH again agree in giving 
 
 and thus shew that the differential coefficients of A with regard to 
 the two constants of integration must, themselves, be constants. 
 We thus have the equations of two surfaces whose intersection 
 determines the path. 
 
 523. Let us suppose next that A represents the Time of 
 passage, so that the brachistochrone is required. Here we have 
 
 the other condition being as in § 519, and we have 
 
 which may be reduced to the symmetrical form 
 
 It is very instructive to compare this equation with that of the 
 free path as above, § 519 ; noting how the force VP is, as it 
 were, reflected on the tangent of the path (§ 105). This is the 
 well-known characteristic of such brachistochrones. 
 
 The appHcation of Hamilton's method may be easily made, as 
 in the preceding example. (Tait, Trans. R. 8. E., 18G5.) 
 
 524. As a particular case, let us suppose gravity to be the 
 only force, then 
 
 VP = «, 
 a constant vector, so that 
 
 The form of this equation suggests the assumption 
 
 /J"* = yS — pa. tan qt, 
 
 where p and q are scalar constants, and 
 
 /Sa/3 = 0. 
 Substituting, we get 
 
 -pq sec' qt+ {-^^~ p'oi^ tan' qt) = 0, 
 which gives pq = T'^ = p^T\. 
 
526.] PHYSICAL APPLICATIONS. 407 
 
 Now let ^^-1 a = 7 ; 
 
 this must be a unit-vector perpendicular to a. and /S, so that 
 
 '^" = cof^^(''°'?*-^'^"?^)' 
 
 whence p = cos qt (cos qt + y sin gi) jQ~' 
 
 (which may be verified at once by multiplication). 
 
 Finally, taking the origin so that the constant of integration 
 may vanish, we have 
 
 2/3/3 = t + ^ (sin 2qt — y cos 2qt), 
 Zq 
 
 which is obviously the equation of a cycloid referred to its vertex. 
 The tangent at the vertex is parallel to ^, and the axis of symmetry 
 to a. The equation, it should be noted, gives the law of descrip- 
 tion of the path. 
 
 525. In the case of a chain hanging under the action of given 
 forces we have, as the quantity whose line-integral is to be a 
 minimum, 
 
 where P is the potential, r the mass per unit-length. 
 
 Here we have also, of course, 
 
 !Tdp = l, 
 
 the length of the chain being given. 
 
 It is easy to see that this leads, by the method above, to the 
 equation 
 
 where m is a scalar multiplier. 
 
 526. As a simple case, suppose the chain to be uniform. 
 Then we may put ru for u, and divide by r. Suppose farther 
 that gravity is the only force, then 
 
 P = SoLp, VP = -a, 
 and ^{{8<xp + u)p']+a = 0. 
 
408 QUATEENIONS. [52 7- 
 
 Differentiating, and operating by Sp', we find 
 
 or, since p' is a unit-vector, 
 
 du 
 
 ds ' 
 which shews that u is constant, and may therefore be allowed for 
 by change of origin. 
 The curve lies obviously in a plane parallel to a, and its equation is 
 
 {SapY + a"" s" = const., 
 which is a well-known form of the equation of the common 
 catenary. 
 
 When the quantity Q of § 518 is a vector or a quaternion, we 
 have simply an equation (like that there given) for each of the 
 constituents. 
 
 527. Suppose P and the constituents of a- to be functions 
 which vanish at the bounding surface of a simply-connected space 
 X, or such at least that either P or the constituents vanish there, 
 the others (or other) not becoming infinite. 
 
 Then, by § 484, 
 
 JjJd^S . V(P<7) = // ds PSa- Uv = 0, 
 
 if the integrals be taken through and over S. 
 
 Thus /// d^ 8 . o-VP = - /// d? P/Sf Vo-. 
 
 By the help of this expression we may easily prove a very 
 remarkable proposition of Thomson {Cam. and Dub. Math. Journal, 
 Jan. 1848, or Reprint of Papers on Electrostatics, § 206). 
 
 To shew that there is one, and but one, solution of the equation 
 
 (SV(e'Vw) = 47rr 
 
 where r vanishes at an infinite distance, and e is any real scalar 
 whatever, continuous or discontinuous. 
 
 Let V be the potential of a distribution of density r, so that 
 
 Vv = i-jrr, 
 and consider the integral 
 
 ■Q^-jjjd.(eVu-]vv)\ 
 
52 7-] PHYSICAL APPLICATIONS. 409 
 
 That Q may be a minimum as depending on the value of u (which 
 is obviously possible since it cannot be negative, and since it may 
 have any positive value, however large, if only greater than this 
 minimum) we must have 
 
 = JSQ = - JJJd<;8 . {e'Vu - Vv) VSu 
 
 By the lemma given above this may be written 
 
 =///d? BuSV (e'Vw - Vv), 
 
 =Jffd<! Su {8V (e'Vu) - 4>7rr}. 
 
 Thus any value of u which satisfies the given equation is such as to 
 make Q a minimum. 
 
 But there is only one value of u which makes Q a minimum ; 
 for, let Q, be the value of Q when 
 
 is substituted for this value of u, and we have 
 
 The middle term of this expression may, by the proposition at the 
 beginning of this section, be written 
 
 2fJJd^w{8V{e'Vu) - 4>7rr}, 
 
 and therefore vanishes. The last term is es.sentially positive. Thus 
 if Mj anywhere differ from u (except, of course, by a constant 
 quantity) it cannot make Q a minimum: and therefore m is a 
 unique solution. 
 
 MISCELLANEOUS EXAMPLES. 
 
 1. The expression 
 
 denotes a vector. What vector ? 
 
 2. If two surfaces intersect along a common line of curvature, 
 they meet at a constant angle. 
 
410 QUATERNIONS. 
 
 3. By the help of the quaternion formulae of rotation, translate 
 into a new form the solution (given in § 248) of the problem of 
 inscribing in a sphere a closed polygon the directions of whose sides 
 are given. 
 
 4. Find the point, the sum of whose distances from any 
 number of given points is a minimum. 
 
 If p be the sought point, a^, a^, &c. the given points : shew 
 that 
 
 'Z.U(p-a) = 0. 
 
 Give a dynamical illustration of this solution. 
 
 (Proc. R. 8. E. 1866-7.) 
 
 5. Shew that 
 
 8. Val3yV^yaVya& = 'iSa^ySa08^y8y(x. 
 
 6. Express, in terms of the masses, and geocentric vectors of 
 the sun and moon, the sun's vector disturbing force on the moon, 
 and expand it to terms of the second order; pointing out the 
 magnitudes and directions of the separate components. 
 
 (Hamilton, Lectures, p. 615.) 
 
 7. If g' = r*, shew that 
 
 2dq= 2dr^ = ^(dr + Kqdrq-') 8q-' = ^(dr + q'^drKq) 8q-' 
 
 = {drq + Kqdr) g~' {q + Kq)'"^ = {drq + Kqdr) {r + Tr)'^ 
 
 _ dr+ Uq-'drUq'^ _ drUq+Uq-'dr _ q'\ Uqdr + dr Uq") 
 ~ Tq{Uq+Uq-')~ q{Uq+UY^ Uq+Uq-' 
 
 _ q'\qdr+ Trdrq'') _drUq+ Uq'^dr _ drKq'^ + q^dr 
 Tq{Uq+ Uq-') ~ ~Tq (iVUr)' ~ 1 + Ur 
 
 = \dr + V. Vdr ^q\q~' = ldr-V (vdr J?"')}?"' 
 
 dr ( drV \ dr f dr V _,\ 
 
 = drq-' + V(Vq-\Vdr) (l + Jg"'") : 
 
 and give geometrical interpretations of these varied expressions 
 for the same quantity. (Ibid. p. 628.) 
 
 8. Shew that the equation of motion of a homogeneous solid 
 of revolution about a point in its axis, which is not its centre 
 of gravity, is 
 
 BVpp-AD,p = Vpy, 
 where O is a constant. (Trans. R. 8. E., 1869.) 
 
MISCELLANEOUS EXAMPLES. 411 
 
 9. Find the point P, such that, if J.,, A^, &c. be any fixed 
 points in space, and m^, m^, &c. positive numerical quantities, 
 X.mAP shall be a minimum. 
 
 Shew that a closed (gauche) polygon can be constructed whose 
 sides are parallel to PA^, PA^, &c. while their lengths are as m,, m^, 
 &c., respectively. 
 
 If T% . mAP is to be a minimum, what is the result ? 
 
 10. Form the quaternion condition that the lines joining the 
 middle points of the sides of a closed polygon (plane or gauche) 
 may form a similar polygon. 
 
 When this condition is satisfied, find the quaternion operator 
 which must be applied to the second polygon to make it similarly 
 situated with the first. 
 
 11. Solve the equations in linear and vector functions ; it 
 being given, while ^ and ■)(^ are to be found : 
 
 (a) </>' = ^, 
 (/3) 4> + f = ^, 
 
 12. Put the equation of a Minding line (§ 394) directly into 
 the normal form for a line passing through each of two fixed 
 curves : — 
 
 p = x<f)t + (1 — a;) yjru, 
 
 where ^ and yjr are definite vector functions of the arbitrary 
 scalars t, and u, respectively. 
 
 13. Shew that 
 
 /SVs7^.o-o-j and VV,/So-ct, 
 
 are different expressions for the same scalar, and give a number of 
 other forms. (§ 513.) How are the values of these quantities 
 respectively affected, if the V without suffix acts on o-^ as well 
 as on o"? 
 
 14. If (as in §511) 
 
 for all values of m and of t, shew by actual transformation, not by 
 the obvious geometrical reasoning, that we have also 
 
412 QUATERNIONS. 
 
 15. Shew that, to the third order in T^, the couple, due to 
 any closed circuit, on a small magnet, 2/S, whose centre is at the 
 origin, is proportional to 
 
 Simplify, by means of this formula, the quasi-Cartesian investiga- 
 tion of § 464. 
 
 16. Calculate the value of 
 
 /•TV 
 ./ Tp' 
 
 for a circular current, the origin being taken at any point. 
 
 17. Find an approximate formula for the potential, near the 
 centre of the field, when two equal, circular, currents have a 
 common axis, and are at a distance equal to their common 
 radius. (This is v. Helmholtz' Tangent Galvanometer.) 
 
 18. Deduce the various forms of Spherical Harmonics, and 
 their relations, from the results of the application of scalar 
 functions of SaV, and similar operators, to 1/Tp. 
 
 19. Integrate the differential equations : 
 
 (a) ^ + ^2 = ^' 
 
 
 where a and b are given quaternions, and and i|r given linear 
 and vector functions. (Tait, Proc. R. S. E., 1870-1.) 
 
 20. Derive (4) of § 92 directly from (3) of § 91. 
 
 21. Find the successive values of the continued fraction 
 
 (M)"' 
 
 where i and _;' have their quaternion significations, and x has the 
 values 1, 2, 3, &c. (Hamilton, Lectures, p. 645.) 
 
MISCELLANEOUS EXAMPLES. 413 
 
 22. If we have 
 
 ^^=t>T^' 
 
 where c is a given quaternion, find the successive values. 
 
 For what values of c does u become constant ? {Ibid. p. 652.) 
 
 23. Prove that the moment of hydrostatic pressures on the 
 faces of any polyhedron is zero, (a) when the fluid pressure is the 
 same throughout, (6) when it is due to any set of forces which 
 have a potential. 
 
 24. What vector is given, in terms of two known vectors, by 
 the relation 
 
 Shew that the origin lies on the circle which passes through the 
 extremities of these three vectors. 
 
 25. Tait, Trans, and Proc. R. S. E., 1870-3. 
 With the notation of §§ 484, 495, prove 
 
 {a) fff8{aV)Td<s=JfTSaUvds. 
 
 (b) If/Sf(pV)T = -nT, 
 
 (n + 3)JJfrd^ = -JjT8pUvds. 
 
 (c) With the additional restriction W = 0, 
 
 fJ8.Uv{2np + (n + S)p'V}.Tds = 0. 
 
 (d) Express the value of the last integral over a non-closed 
 surface by a line-integral. 
 
 (e) -JTdp=fJdsS.UvV<T, 
 if <r = Udp all round the curve. 
 
 (/■) For any portion of surface whose bounding edge lies 
 wholly on a sphere with the origin as centre 
 JfdsS.(UpUvV).a- = 0, 
 whatever be the vector a. 
 
 (g) JVdpV. a = Sjds { UvV^ -8{Uv'^)V} <r, 
 whatever be a. 
 
 26. Tait, Trans. R. 8. E., 1873. 
 
 Interpret the equation 
 . ■ , dcy — uqdpq'^, 
 
414 QUATERNIONS. 
 
 and shew that it leads to the following results 
 
 W = qVuq~^, 
 
 V. uq'' = 0, 
 
 V^ ui = 0. 
 
 Hence shew that the only sets of surfaces which, together, cut 
 space into cubes are planes and their electric images. 
 
 27. What problem has its conditions stated, in the following 
 
 six equations, from which ^, rj, ^ are to be determined as scalar 
 
 fiinctions of x, y, z, or of 
 
 p = ix +jy + kz'i. 
 
 , „ . d . d T d 
 
 where v=i^j- + ?-r+«^-r- 
 
 dx ■' dy dz 
 
 Shew that they give the farther equations 
 
 = v^^Tj = v'7)X=v^^^ = v^ ^7,1;: 
 
 Shew that (with a change of origin) the general solution of these 
 equations may be put in the form 
 
 where (/> is a self-conjugate linear and vector function, and f, tj, ^ 
 are to be found respectively from the three values of f at any 
 point by relations similar to those in Ex. 24 to Chapter X. 
 
 28. Shew that, if p be a planet's radius vector, the potential 
 P of masses external to the solar system introduces into the 
 equation of motion a term of the form, 8 {pV)yP. 
 
 Shew that this is a self-conjugate linear and vector ftinction 
 of p, and that it involves only five independent constants. 
 
 Supposing the undisturbed motion to be circular, find the chief 
 effects which this disturbance can produce. - 
 
 29. In § 430 above, we have the equations 
 Fa(^-f-nV) = 0, (Sfaw = 0, d = «Fia, Ta = l, 
 
 where w" is neglected. Shew that with the assumptions 
 
 q = i'^, a = q/3q~^, r = /3'', ■or — qrrr'^q'^ , , 
 we have ' 
 
 ,3 = 0, W = l, S^T^O, F/3(¥ + nV) = 0, 
 provided coSia — w, = 0. . Hence deduce the behaviour of the 
 
MISCELLANEOUS EXAMPLES. 415 
 
 Foucault pendulum without the x, y, and ^, r] transformations 
 in the text. 
 
 Apply analogous methods to the problems proposed at the end 
 of § 426 of the text. 
 
 30. Hamilton, Bishop Law's Premium Examination, 1862. 
 (a) If OABP be four points of space, whereof the three first 
 
 are given, and not coUinear ; if also 
 
 OA = a, OB = /8, ov = p; 
 and if, in the equation 
 
 a. a. 
 
 the characteristic of operation F be replaced by 8, the locus of p 
 is a plane. What plane ? 
 
 (6) In the same general equation, if F be replaced by V, 
 the locus is an indefinite right line. What line ? 
 
 (c) If F be changed to K, the locus of p is a point. What 
 point ? 
 
 (d) If F be made = JJ, the locus is an indefinite half-line, 
 or ray. What ray ? 
 
 (e) If F be replaced by T, the locus is a sphere. What 
 sphere ? 
 
 (/) If F be changed to TV, the locus is a cylinder of 
 revolution. What cylinder ? 
 
 (^r) If F be made TYTJ, the locus is a cone of revolution. 
 What cone ? 
 
 (A) If (SC/" be substituted for F, the locus is one sheet of 
 such a cone. Of what cone ? and which sheet ? 
 
 (i) If F be changed to VTJ, the locus is a pair of rays. 
 Which pair ? 
 
 31, Hamilton, Bishop Law's Premium Examination, 1863. 
 
 (a) The equation 
 
 Spp' + a' = 
 
 expresses that p and p are the vectors of two points p and p', 
 which are conjugate with respect to the sphere. 
 
 p' + «^ = 0; 
 or of which one is on the polar plane of the other. 
 
 (6) Prove by quaternions that if the right line pp', connect- 
 ing two such points, intersect the sphere, it is cut harmonically 
 thereby. 
 
416 QUATERNIONS. 
 
 (c) If p' be a given external point, the cone of tangents 
 drawn from it is represented by the equation, 
 
 (Vppy = a\p-py; 
 
 and the orthogonal cone, concentric with the sphere, by 
 
 {Sppy+ay = o. 
 
 (d) Prove and interpret the equation, 
 
 T(np-a) = T(p- no), if Tp = Ta. 
 
 (e) Transform and interpret the equation of the ellipsoid, 
 
 T {ip + pk) = K^ - i". 
 
 (/) The equation 
 
 (k' - ij = (l' + k') Spp' + 2SipKp' 
 
 expresses that p and p' are values of conjugate points, with respect 
 to the same ellipsoid. 
 
 (g) The equation of the ellipsoid may also be thus written, 
 
 Svp = 1, if («" - I'JV = (t - Kfp + 2t,SKp + 2KStp. 
 
 (h) The last equation gives also, 
 
 iK'-i,yv = (i,' + K')p + 2VipK. 
 
 (i) With the same signification of v, the differential equa- 
 tions of the ellipsoid and its reciprocal become 
 
 Svdp = 0, Spdv = 0. 
 
 (j) Eliminate p between the four scalar equations, 
 
 Sap = a, S^p = b, Syp = c, Sep = e. 
 
 32. Hamilton, Bishop Law's Premium Examination, 1864. 
 (a) Let A^B^, A.fi^, ...A^B^he any given system of posited 
 right lines, the 2n points being all given; and let their vector 
 sum, 
 
 AB = A,B, + A,B, + ...+A^B^, 
 
 be a line which does not vanish. Then a point H, and a scalar h, 
 can be determined, which shall satisfy the quaternion equation, 
 
 HA,.A,B, + ... + HA^.A^B^=^h.AB; 
 namely by assuming any origin 0, and writing, 
 
 j^^^OA A,B^+... 
 AA + - 
 
MISCBLLiANEOUS EXAMPLES. 417 
 
 (6) For any assumed point G, let 
 
 then this quaternion sum may be transformed as follows, 
 
 Qc=QH + GH.AB = {h + GH).AB; 
 and therefore its tensor is 
 
 TQc={h^ + ClPf.AB, 
 in which AB and GH denote lengths. 
 
 (c) The least value of this tensor TQc is obtained by 
 placing the point G at H; if then a quaternion be said to be a 
 minimum when its tensor is such, we may write 
 
 min. Qc=Qn=h.AB; 
 so that this minimum of Qc is a vector. 
 
 (d) The equation 
 
 TQc = c = any scalar constant > TQs 
 expresses that the locus of the variable point (7 is a spheric 
 surface, with its centre at the fixed point H, and with a radius r, 
 or GH, such that 
 
 r.AB={TQ'c- TQ'b)^ = {c^ - h' . AE'f ; 
 so that H, as being thus the common centre of a series of 
 concentric spheres, determined by the given system of right lines, 
 may be said to be the Gentral Point, or simply the Centre, of that 
 system. 
 
 (e) The equation 
 
 TVQc = Cj = any scalar constant > TQ^ 
 represents a right cylinder, of which the radius 
 
 divided by AB, and of which the axis of revolution is the line, 
 
 Vqc^QB = h.AB; 
 wherefore this last right line, as being the common axis of a series 
 of such right cylinders, may be called the Gentral Axis of the 
 system. 
 
 (/) The equation 
 
 SQ(, = Cj = any scalar constant 
 represents a plane ; and all such planes are parallel to the Gentral 
 Plane, of which the equation is 
 
 sqc=o. 
 
 T. Q. I. 27 
 
418 QUATERNIONS. 
 
 {g) Prove that the central axis intersects the central plane 
 perpendicularly, in the central point of the system. 
 
 Qi) When the n given vectors Afi^,...A,^B^ are parallel, 
 and are therefore proportional to n scalars, 6,, ... 6„, the scalar h 
 and the vector Qg vanish ; and the centre H is then determined 
 by the equation 
 
 h,.HA, + \.HA, + ... + K.HA^ = Q, 
 
 or by the expression, 
 
 where is again an arbitrary origin. 
 
 33. Hamilton, Bishop Law's Premium Examination, 1860. 
 
 (a) The normal at the end of the variable vector p, to the 
 surface of revolution of the sixth dimension, which is represented 
 by the equation 
 
 {p'-oLy = 21oi'{p-ar (a), 
 
 or by the system of the two equations, 
 
 p'~a' = U'd\ (p-ay = fa' (a'), 
 
 and the tangent to the meridian at that point, are respectively 
 parallel to the two vectors, 
 
 v = 2(p-a)-tp, 
 
 and T = 2 (1 - 2t) (p-a.) + fp ; 
 
 so that they intersect the axis a, in points of which the vectors 
 are, respectively, 
 
 2a 2(1 -2*) a 
 
 2—t' ^"""^ (_2-ty-2- 
 
 (b) If dp be in the same meridian plane a.s p, then 
 
 t{l-t){'k- 1) dp = Srdt, and 8 ^- = ~ . 
 
 dp 3 
 
 (c) Under the same condition, 
 
 {d) The vector of the centre of curvature of the meridian, 
 at the end of the vector p, is, therefore, 
 
 ^ \ dp) f^ 2 1-t 2{l-t) ■ 
 
MISCELLANEOUS EXAMPLES. 419 
 
 (e) The expressions in (a) give 
 
 v' = a¥ (1 - ty, t" = aV (1 - ty (i-t); 
 
 hence (a-py = ^a.r, and dp' = ^^df; 
 
 the radius of curvature of the meridian is, therefore, 
 
 and the length of an element of arc of that curve is 
 
 t ^i 
 
 ds^Tdp = STai^^~^j dt. 
 
 (/) The same expressions give 
 
 4>(rapy==-a.r(l-ty(i-t); 
 thus the auxiliary scalar t is confined between the limits and 4, 
 and we may write t = 2 vers 0, where ^ is a real angle, which varies 
 continuously from to 27r ; the recent expression for the element 
 of arc becomes, therefore, 
 
 ds = STa.tdd, 
 and gives by integration 
 
 s = 6Ta(0-sm0), 
 if the arc s be measured from the point, say F, for which p = a, 
 and which is common to all the meridians; and the total periphery 
 of any one such curve is = 127rTa. 
 
 (g) The value of a- gives 
 
 4 (a' - a') = SaH (4 - 1), 16 (Vaay = - a*f (4 - tf ; 
 
 if, then, we set aside the axis of revolution a, which is crossed by 
 all the normals to the surface (a), the surface of centres of 
 curvature which is touched by all those normals is represented by 
 the equation, 
 
 4,(o^-ay + 27a'(Va<Ty = (b). 
 
 (A) The point F is common to the two surfaces (a) and 
 (b), and is a singular point on each of them, being a triple point 
 on (a), and a double point on (b) ; there is also at it an infinitely 
 sharp cusp on (b), which tends to coincide with the axis a, but 
 a determined tangent plane to (a), which is perpendicular to that 
 axis, and to that cusp ; and the point, say F', of which the vector 
 = — a, is another and an exactly similar cusp on (b), but does not 
 belong to (a) . 
 
420 QUATERNIONS. 
 
 (i) Besides the three universally coincident intersections of 
 the surface (a), with any transversal, drawn through its triple 
 point F, in any given direction /3, there are always three other 
 real intersections, of which indeed one coincides with F if the 
 transversal be perpendicular to the axis, and for which the 
 following is a general formula : 
 
 p = Ta.[UcL + [2SU (a/3fYU^]. 
 
 (j) The point, say V, of which the vector is p = 2a, is a 
 double point of (a), near which that surface has a cusp, which 
 coincides nearly with its tangent cone at that point; and the 
 
 semi-angle of this cone is = ^ • 
 Auxiliary Equations : 
 
 (2Sp (p - a) = aV (3 + 1), 
 [2S(x{p-a) = d'f{S-t). 
 
 f Svp=--oi't{l-t){l-2t), 
 \2Sv(p-a.)=a'f(l-t). 
 
 ( Spr = a.V(l-t){4!-t), 
 
 \2S (p - a) T = a'f (1 - «) (4 - 1). 
 
 34. A homogeneous function of p, of the nth degree, is 
 changed into the same function of the constant vector, a, by the 
 operator 
 
 n\ ^ 
 
 35. If r be the cube root of the quaternion q, shew that 
 
 dr =p +(V.r'' + rV7^) Vq^ (rp —pr), 
 where p = ^r^dq. (Hamilton, Lectures, p. 629.) 
 
 36. Shew from § 509 that the term to be added to VQ of 
 § 512 (3), in consequence of viscosity, is proportional to 
 
 VV + |V>SfVo-. 
 
 37. Shew, from structural considerations, that 
 
 must be a linear and vector function of Fa/3. Also prove, directly, 
 
 that its value is 
 
 "^,Sf.(Fa/3)VV,.Fo-tr,. 
 
MISCELLANEOUS EXAMPLES. 421 
 
 38. The spherical opening subtended at a by the sphere 
 
 Tp=r 
 
 according as Ta = a ^r. 
 
 Hence shew, without further integration, that (with the same 
 conditions) 
 
 Ih 
 
 ds . 47rr^ 
 
 = 47rr, or 
 
 '^(p-a) ' a 
 
 ■ ■ whence, of course, 
 
 [ dsTJ(,p-a) _ _ Wf^ 
 
 Tip-oif 
 
 A.lso that 
 
 //- 
 
 k 
 
 ds 47rr iTrr' 
 
 or 
 
 39. Find also the values of 
 
 ff Uvds rr Uvds , , ,., 
 
 nTjp^y JJiT^^:r^.andsuch-hke, 
 
 the integration extending over the surface of the sphere Tp = r. 
 
 40. Shew that the potential at /3, due to mass m at a, is to 
 the potential at /S~', due to mass m' at a~\ as 1 : T^, provided 
 m' : m :: 1 : Ta. 
 
 Ih 
 
 Hence, by the results of 38 above, shew that, with a<r, 
 
 ds 47rr 47rr' 
 
 or 
 
 T(p-P)T{p- oif ~ {r' - a') T(l3-a)' a (r'-a') T{r'a'+^) 
 
 according as T^ 5 r, the limits of integration being as before. 
 
 Obtain these results directly from Green's Theorem (§ 486) 
 without employing the Electric Image transformation. 
 
 41. Shew that the quaternion 
 
 ILs ( CTW) rds - /TJo-V Vd? 
 
 (in the notation of § 482) is changed into its own conjugate by 
 interchange of a and r. Express its value (by §§ 133, 486) as a 
 single volume-integral. 
 
422 QUATERNIONS. 
 
 42. Prove that 
 
 jjj(c V't + KVa . V r) d? =jja- UvVrds, 
 and thence that 
 
 ff [(tV V - VV . 0-) cZ? = [[(t UvVa-KVT.Uv<7)ds. 
 
 43. Find the distribution of matter on a given closed surface 
 which will produce, in its interior, the same potential as does 
 a given distribution of matter outside it. 
 
 Hence shew that there is one, and only one, distribution of 
 matter over a surface, which will produce, at each point of it, any 
 arbitrarily assigned potential. 
 
 CAMBBIDGE : PKINTED EV C. 3. CLAY, M.A. AND SONS, , AT THE UNIVERSITY SRESB. 
 
By the same Author. 
 
 A Treatise on Natural Philosophy. By Sir W. Thomson, LL.D., 
 
 D.C.L., P.B.S., Professor of Natural Philosophy in the UniTersity of Glasgow, 
 and P. G. Tait, M.A., Professor of Natural Philosophy in the University of 
 Edinburgh. Parti. Demy 8vo. 16s. Part II. Demy 8vo. 18s. 
 
 Elements of Natural Philosophy. By Professors Sir W. Thomson 
 and P. G. Tait. Demy 8vo. 9s. 
 
 ILDitlan: C. J. CLAY and SONS, 
 
 CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 
 AVE MARIA LANE. 
 
 Lectures on some Recent Advances in Physical Science. 
 
 With Illustrations. Revised and enlarged, with the Lecture on Force delivered 
 before the British Association. By P. G. Tait, M.A. Third Edition. Crown 
 8vo. 9s. 
 
 Heat. With numerous Illustrations. By P. G. Tait, M.A. Crown 
 8vo. 6s. 
 
 A Treatise on Dynamics of a Particle. With numerous Examples. 
 
 By P. G. Tait, M.A., and the late W. J. Steele, B.A., Fellow of St Peter's 
 College, Cambridge. Sixth Edition, carefully Eevised. Crown 8vo. 12s. 
 
 The Unseen Universe ; or, Physical Speculations on a Future 
 
 state. By Professors Baleoub Stewakt and P. G. Tait. Fifteenth Edition. 
 Crown Svo. 68. 
 
 MACMILLAN AND CO. LONDON. 
 
 Light. By P. G. Tait, M.A., Sec. R.S.E. Second Edition. Revised 
 and Enlarged. Crown 8vo. 6s. 
 
 Properties of Matter. By P. G. Tait, M.A., Sec. E.S.E. Crown 
 8vo. 7s. 6d. 
 
 ADAM & CHARLES BLACK, EDINBUBGH.