M/iTfl QA Ml! CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall ,^, CORNELL UNIVEflSTTY LIBRAR- 924 050 927 163 •— ' •-- - ^ tftnn -O-J Qm... ■ JUU JllkiU^— 4ni.idi o lontf UctT 7 lijuj P4A^ HW-* Dpi" ^0Vt ^^^«3- f^'te-r =^y flflETm r HhtOB^— TT'*'-* — W •Aisor "> M- PRINTED IN U. 3. A. (SJ NO. 23233 Cornell University Library The original of tiiis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924050927163 A TREATISE ON OCTONIONS. EonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ffilasgoSn : 263, Akgtle Street. Ecipjifl: F. A. BEOCKHAUS. ia«fa Sotfi: THE MACMILLAN COMPANY. BamfiaE : E. SEYMOUR HALE. OCTONIONS A DEVELOPMENT OP CLIFFOED'S BI-QUATEBNIONS ALEX. M^'AULAY, M.A. PBOPESSOR OP MATHEMATICS AND PHYSICS IN THE UNIVERSITY OF TASMANIA. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1898 All Eights reserved. 't^c>k.$y A^ l^Vfe S'l Cambrilige : PRINTED By J. AND C. P. CLAY, AT THE nNITERSITY PRESS. In compliance with current copyright law, LBS Archival Products produced this replacement volume on paper that meets the ANSI Standard Z39.48-1984 to replace the irreparably deteriorated original. 1988 PREFACE. I OWE a great debt of gratitude to an old pupil for the results of a casual conversation I had with him some six or seven years ago. On that occasion Mr P. k M. Parker discoursed of rotors and motors in such wise that it seemed to his tutor high time to rub the dust from the volume of Clifford's Mathematical Papers lying on the shelves ; for otherwise the tutor and pupil bade fair to change places. Many days of most interesting work and thought have been the sequel of that talk within the walls of Ormond College. The treatment below of what Clifford called Bi-quaternions runs on two sharply-defined lines. Quaternions and the Aus- dehnungslehre have both been pressed into the service, and the help from them has led to very different kinds of development. Neither devehipment could, in my opinion, be well spared. The first seems to be allied to metrical geometry and the second to descriptive. At any rate I do not see how, in few words, better to describe the essential characteristics of the two. For more precise ideas the reader must study the subject itself. So far as the present treatise is concerned, these developments took place in two periods. I had done what I could on the quaternion model, but being dissatisfied because so many questions which presented themselves were thereby but im- perfectly answered, I put the work aside. Meanwhile I had been led by Sir Robert Ball's Theori/ of Screws (Dublin, 1876) VI PREFACE. to study the Ausdehnungslehre (1862), and was delighted to find that the gaps could apparently be filled from this source. On taking the subject up again it was found that this surmise was correct. Perhaps the most striking fact that has come to light in the investigation is one that appeared almost at the outset, and one which mainly induced me to proceed. I mean the fact that every quaternion formula except such as involve V admits of an octonion interpretation— a geometrical interpretation much more general than that which it was primarily meant to have. There is matter for refiection in that the founders of Quaternions, while they were busying themselves only with vector and quaternion conceptions, were, all the time, unknown to themselves, establishing motor and octonion truths. There is a corresponding though less striking fact connected with the second method of development. The statements of Quaternions were always intended to have but one meaning — of course with many variations of form when put into words — but the Ausdehnungslehre was intended to be a general frame- work of symbols whose applications should be in many provinces of thought. It is therefore not surprising that there are geometrical interpretations which were not developed, even if seriously contemplated, by Grassmann. Such are the applications of the Ausdehnungslehre below. I cannot believe that Grass- mann contemplated such applications of his calculus if only because apparently he never conceived of a magnitude other than zero whose "numerical value," in his own technical sense, was zero. It may be asked why, in this treatise, I start de novo, instead of taking all that Clifford has done for granted. The reasons are (1) the desirability of making the treatise self-contained; (2) the fact that Clifford uses a method dependent on the properties of non-Euclidean space, whereas I regard the subject as referring PREFACE. Vll to Euclidean space ; and (3) I do not altogether understand all of Clifford's arguments. The treatise suffers in form, somewhat, from the fact that it was not, in the making, meant to be a book but a " paper," as will be directly explained. If I had from the beginning con- templated the book form, or if when the treatise became destined to take that form, I had thought myself justified, or indeed had had the courage, to recast the whole appropriately, it would have been at least doubled in length, without probably any material mathematical amplification. Not only is it too condensed, where the argument is fairly covered, but many steps of reasoning are left out which the reader will require patience to supply, as I myself have found in reading the proof-sheets. I can but apologise to the reader for these rather irritating defects. There is a defect wholly unconnected with this, which qualified critics may help to remedy. I refer to the terminology. I have found myself compelled to invent quite a little vocabulary ; and if ever there was an author in such an uncomfortable position whose ignorance of dead and other languages was more profound than the writer's, I pity him. About the term " Octonion" I shall speak directly. The three groups of terms augmenter, tensor, additor, pitch ; and twister, versor, translator ; and velocity motor, force motor, momentum motor ; and their congeners I am (pending criticism) content with. The group of terms referring to linear motor functions of motors; general function, commutative function, pencil function, energy function ; are passable. The terms corawr*^, converter, axial quaternion (or axial) and some less frequently used seem to me like unwilling conscripts begging at any price for substitutes. The term variation as used in the treatise is objectionable on the ground that it clashes with the technical meaning of the same term in Algebra. If I had thought that any serious inconvenience would result I should have used some such term as replacement, but I thought variation better. [It must be Till PREFACE. remembered that in the technical use of this treatise the term variation is always qualified by some adjective such as combina- torial or circular.'] I have in the treatise itself tried to justify my deviation from Clifford's usage of vector and quaternion (replaced below by lator and axial), but I have given no reasons for the serious step of changing the name of the whole subject from Bi-quatemions to Octonions. The following reasons seemed to form sufficient justification. (1) I think it desirable to have a name for what Hamilton has called Bi-quaternions. For these there could scarcely be a better name. (2) I wish to imply that quaternions are not particular kinds of octonions but only very similar to such particular kinds. (3) Octonions like quaternions treat space impartially. By this I mean that they do not depend in any way on an arbitrarily chosen system of axes or arbitrary origin. But one of the two quaternions implied by Clifford's term does so depend on an arbitrary origin. This to me appears an absolute bar to the propriety of his term. If Clifford, in choosing his term, wished to emphasise his indebtedness to the inventor of Quaternions, this is scarcely a reason for one who merely follows Clifford to copy him in this respect, if there are intrinsic objections. Most of the methods and some of the results which follow are to some extent, I believe, novel. But I fear that many references to the work of others which ought to occur are wanting. The treatise was written at a distance from all mathematical libraries. I believe I should have been able to improve it in many respects if I had been able to consult the many authorities cited by Sir Robert Ball in his Theory of Screws. Explicit references to treatises on Quaternions are almost wholly omitted, as the reader must be assumed familiar with this subject. The references to the Theory of Screws and to the Ausdehnungs- lehre are copious. Except from these subjects and treatises I have, as far as I know, received no aid from the work of others. PREFACE. ix The treatise was communicated to the Royal Society on 28 Nov. 1895 and read on 12 Dec. The Council of the Royal Society, considering its nature to be more that of a book than a "paper," offered to aid its publication in book form by making (from a fund voted by the Treasury for such purposes) a sub- stantial pecuniary contribution to the expenses. The Cambridge Press Syndicate has borne all the other expenses. But for these aids, the treatise could not have been published. The deep gratitude I feel to both these bodies, and beg here to express, will be appreciated by all who have desired to present their reflections to the many sympathetically minded though difficult of access. For obvious geographical reasons, and for others scarcely less obvious, I was wholly unable to act for. myself in the arrangements which were terminated as above described. I am probably ignorant of some who disinterestedly did this work for me, but I know a large share was done by Dr Ferrers and Sir Robert Ball. My warmest thanks are due to them for this labour of kindness. To Prof. Forsyth I am indebted for the valuable suggestion among others that the treatise should be preceded by a short sketch of the main argument. There are probably many errors remaining in the book. That there are not more is due to the kindness of Mr P. a M. Parker of St John's College, Cambridge, and Mr G. H. A. Wilson of Clare College, Cambridge, in carefully revising the proof-sheets, and to the great care exercised at the Cambridge University Press. ALEX. MCAULAY. Univebsitt of Tasmania, 1 August, 1898. DEFINITIONS AND EXPLANATIONS OF TERMS. SECT. PAGE 2. Formal quaternion . . .... .7 Formal scalar 7 Formal vector ... 7 Linear formal quaternion function of a formal quaternion . 8 „ „ vector „ „ „ vector . . 8 3. Reciprocal (^"i) of a formal quaternion 9 4. Primary and secondary systems of formal quaternions . . 11 Q 11 5. Vector 17 Quaternion . . 18 Quotient of a vector by a vector . . . 18 6. Lator . . 19 Eotor, axis of rotor . ...... 19 Motor, axis of motor . . . . , 19 Octonion, axis of octonion 19 Ordinary scalar 19 Convert 19 Definite axis, indefinite axis 20 Axial quaternion or axial 20 Quaternion corresponding to an axial .... .20 7. Rotor equation of a straight line 23 OP a rotor 24 8. Velocity motor . .26 Force motor 26 Momentum motor . 26 Impulse motor 26 9. Axial, rotor, lator, and converter, of an octonion ... 27 Auementer, tensor, additor, and pitch, of an octonion . . 28 . 9ft Scalar octonion ^° Twister, versor, and translator, of an octonion .... 26 XU DEFINITIONS AND EXl>LANATIONS OF TERMS. SECI. PAQE 9. Scalar octonion part, ordinary scalar, scalar convertor, and convert, of an octonion 28 Motor, rotor, and lator, of an octonion 29 Convertor 29 Positive and negative scalar octonions 31 Positive and negative scalar converters 31 10. Quasi-tensor of a lator 34 Quotient of %wo motors . . 35 11. Displacement of an octonion ....... 37 The displacement Q 39 12. Positive and negative distances between directed lines . . 43 13. Formal quaternion function of any number of octonions . . 46 li. i,j,k 50 5 50 Independent motors . . 51 Complex of order n 51 Reciprocal motors 52 Reciprocal complexes ........ 52 15. General linear motor function of a motor 54 Commutative function 54 [For energy function, tee § 30.] ^{Zy Z,); x{Zi,Z„Z^, Z,); ir{C,C) 55 Conjugate of a general function and of a commutative function, self- conjugate functions 56 Pencil of rotors, centre of the pencil 56 Pencil function, centre of the pencil function .... 56 17. cubic, <^i cubic 63 18. Axial motor 65 Completely independent axial motors 65 Single root, repeated root 66 19. Principal roots of the <^ cubic 79 21. i,j, k; i^,j„, ko; C, Co; p 85 V, Vo 85 dS, dX 86 Suffix notation for v 87 V^ 87 >" 88 22. Rotor spin 90 Lator spin ... 91 23. r,, p', dS', dX' 92 Xoi Xf) X 93 V', * 93 24. ff 96 DEFINITIONS AND EXPLANATIONS OF TERMS. xiii 6BCT. PAGE 25. Motor intensity, motor flux 97 V 97 Motor spin 99 Intensity flux, flux intensity .... . . 101 28. Combinatorial product iqq {J-lA^sA^iM, _Ml£2-43^4^6} 108 ^1, A2, As, At, Ai,, Ag 110 29. Simple, multiple, positive, and negative, combinatorial variation 114 Linear variation, circular variation, hyperbolic variation . .115 30. Energy function, partial and complete 119 Complex of (m-l)th order and second degree . . . .119 Motors conjugate with regard to a general self-conjugate . . 119 31. Conjugate variation ••....... 122 Positive, negative, and zero norms 123 Semi-conjugate complexes 129 Conjugate with reference to a given complex of a given included complex 130 Partial and complete sets of conjugate norms .... 134 32. or-i 136 34. Homogeneous linear scalar function of one motor and of two motors 148 Homogeneous quadratic scalar function of a motor . . .149 35. Complex corresponding to a root of the n.-tic . . . 157 /(^) 159 38. Commutative combinatorial product 166 Commutative combinatorial variation 166 Motors fully conjugate with regard to a commutative self-conjugate function 167 Norms (with reference to a commutative self-conjugate function) of types A, B, Aq, Bg, C 168 Axial complex 168 [Commutative] conjugate variation 169 Ellipsoidal function 170 40. Axis of a second order complex 177 42. Tetrahedron p, a 183 Notation 12 when 1 and 2 represent straight lines . . . 183 Double twist, half twist, double rotation, half rotation . . 185 Semi-revolution 186 43. Singular cases of second order complex 187 Pitch conic, conjugate pitch conic 189 Cylindroid 190 Screw, virtual coefficient 190 44. Pitch quadric, conjugate pitch quadrio ... . .192 XIV DEFINITIONS AND EXPLANATIONS OF TERMS. SECT. PAQE 46. Displacement motor 198 F, G, E, F„ G„ H, . 198 Principal motors of inertia, absolute principal motors of inertia 202 Poinsot ellipsoid .... 211 Contact-plane . 211 Polhode, polhode cone . . . . 211 Normal cone, normal cone curve • .213 Outsider, plane-resident, resident . . • 213 47. Principal normal cone curve, reciprocal or conjugate polhodes . 234 48. Velocity complex, momentum complex, reaction complex . 242 Kew definition of a • 244 Generalised velocity motor, generalised momentum motor . . 246 49. Generalised force motor . . . . • 247 Principal motors of inertia, principal motors of the potential, harmonic motor.-; . . 248 Reduced wrench ... . • • 251 Virtual force motor, virtual impulse motor . • 251 Ellipsoid of inertia, ellipsoid of the potential . 252 OCTONIONS. SKETCH OF THE ARGUMENT. The following sketch is given— at the suggestion of Dr Forsyth — to enable the reader more easily to follow the main argument. It will probably also serve for the purposes of a table of contents. The numbers at the beginning of the following paragi-aphs refer to the sections of the text. 2. The subject of Quaternions may be described as a peculiar symbolic method of treating Geometry. But the formulae which occur may possibly have interpretations other than those which originally led up to them. A large part of the present treatise is concerned with the development of such other interpretations. It is desirable therefore to separate the purely symbolic laws of Quaternions from their geometrical interpretations. This has been done below under the heading " Fundamental formal laws of Quaternions." These so-called laws are below divided into six groups. 3. To convince the reader that these laws do really contain all the fundamental elements of quaternion formulae, some of the principal general formulae of Quaternions are deduced from them. 4. It is next shown that, connected with any system of symbols which obey the laws mentioned, there is a second system. These are called the primary and secondary systems of Formal Quater- nions respectively. The two systems are coimected by means of six equations in which an additional symbol, fl, occurs. This last behaves exactly like a formal scalar of the primary system except that D.^ always = 0, i.e. it behaves in a sense like an infinitesimal 1^ M. O. 1 2 OCTONIONS. formal scalar of the primary system. Q denoting a formal quater- nion of the secondary system, and q and r formal quaternions of the primary system, the equations are Q = q + nr (1), KQ=K^-|-nKr (2), eQ = Sq + n.Sr (3), VQ=Vg + nVr (4), TQ = (l + nSrq-')Tq (5), UQ = (l + D.\/rq-')Uq (6). It is shown that if the primary system obeys the six fundamental groups of laws, so does the secondary system. It is therefore subsequently assumed that if every quaternion formula is true for the primary system it is also true for the secondary system. V is subsequently changed to M because the connotation of V does not hold in Octonions. 5, 6. These purely symbolic preliminary matters being dis- posed of, an octonion as a geometrical magnitude is described. Rotors and motors are defined in ways equivalent to Clifford's. An octonion is defined as a magnitude requiring for its specifi- cation a motor and two scalars ; of which one is called its ordinary scalar and the other its convert. The term "lator" is used for Clifford's "vector" and "axial quaternion" or "axial" for his "quaternion" wherever these tenns occur in his papers on Bi- quatemions. It is taken as a fundamental definition that axials through a given point and their included system of rotors through and ordinary scalars obey among themselves all the laws of the corresponding quaternions and their included system of vectors and scalars. Axials through are then taken as the primary system spoken of above and octonions as the secondary system. Any octonion Q can be specified uniquely by means of two axials through 0. Denoting these by q and r and assuming equations (1) to (6) above to hold (M replacing V) the geometrical meanings of KQ, MQ, etc. are determinate, though not obvious. The specification of Q in terms of q and r and conversely of q and r in terms of Q may be expressed as follows. — Q is completely determined by its motor, its ordinary scalar and its convert. The motor of Q again is determined by its axis and the magnitudes (including the senses parallel to the axis) of its rotor and lator, SKETCH OF THE ARGUMENT. Ordinary scalar of Q = 8q, convert = Sr, rotor perpendicular from on axis of Q = M . MrfA'^q, rotor through parallel and equal to rotor of Q = Mq, „ lator „ =M5S.M?'M-'^ q = (ordinary scalar of Q) + (rotor through parallel and\ equal to rotor of Q), | r = (convert of Q) + vsMq + X, where ct is the rotor per- pendicular from on the axis of Q and X is the rotor through parallel and equal to the lator of Q.. (7), (8). 7. Every quaternion formula may now be read as an octonion formula and will have smie geometrical interpretation in connection with octonions. It is not obvious that this interpretation will have any great utility. It is therefore shown that although a fixed point is used in the above definitions the geometrical meaning of any such formula is quite independent of 0. For instance (to take an example from a later part of the treatise) if A, B, G are three motors connected by the equation C = IAAB, the axis of C is the shortest distance between the axes of A and B, its rotor bears to the rotors of A and B the same relation as the vector VayS does to the vectors a and /3 in Quaternions, and its pitch is the sum of d cot 6 and the pitches of A and B where d is the distance and 6 the angle between A and B, d being reckoned positive or negative, according as the shortest twist which will bring the rotor of either ^ or J5 into coincidence, both as to axis and sense, with the rotor of the other is a right-handed or left-handed one. 8 — 15. The geometrical interpretations of octonion formulae, thus shown to be independent of an arbitrary origin, are examined in considerable detail. 16 — 19. A classification of linear motor functions of motors is made. The analogue of the corresponding quaternion function, which is not the most general type of the octonion function, is called a commutative function. Many of its properties are ex- amined in detail. In particular the self-conjugate commutative function is examined. 20. The differentiation of all the octonion functions symbolised in the treatise is examined. 1—2 4 OCTONIONS. 21 — 26. Octonions are adapted for use in physical questions. They are in this respect very similar to quaternions. 27. The quaternion analogy, fruitful as it is, proving — at any rate in my hands — nevertheless insufficient to furnish methods for answering various interesting questions that octonions present, recourse is had to Grassmann's Ausdehnungslehre. It seemed especially desirable to identify if possible Grassmann's meaning of " normal " with Sir Robert Ball's meaning of " reciprocal " as applied to screws. With Grassmann's own geometrical treatment of motors as quantities of the second order this is found to be impossible. But by treating motors as quantities of the first order the identification can be made. But it turns out that real motors with negative pitch must be regarded as imaginary quan- tities of the first order ; and there are motors, viz. all rotoi's and all lators, which cannot in a calculus of octonions be regarded as zero, which nevertheless have in Grassmann's sense zero numerical value. Hence while his methods or extensions of them are applicable his theorems have to be carefully revised for our purposes. 28. Products and combinatorial products are defined in ways which though different in form from Grassmann's definitions yet harmonise with his. Two particular kinds of combinatorial pro- ducts (of 5 and 6 motors respectively, the former product being a motor reciprocal to each of the five and the latter an ordinary scalar) directly suggested by some of Grassmann's products are introduced and used very frequently in subsequent parts of the treatise. 29. Grassmann's linear variation and circular variation are generalised to a new type called combinatorial variation. The uses, and they are many, that are subsequently made of this are all very similar to Grassmann's uses of his two species. The property of a combinatorial variation on which much of its usefulness depends is that a combinatorial product is unaltered by the variation. 30 — 34. By means of the methods thus introduced many properties of the general self-conjugate linear motor function, ■sr, of a motor are examined. They may all be expressed as properties of complexes of the second degree which are generally of arbitrary order. Putting ot = 1 we get also many properties of reciprocal SKETCH OF THE ARGUMENT. 5 motors and reciprocal complexes of the first degree. A particular form of ■ST, called an energy-function, which is of primary import- ance in the consideration of the motion of a rigid body proves to have, comparatively, very simple properties. 35, 36. Passing to the general function which is not necessarily self-conjugate Grassmann's methods are further utilised. 37. The bearing of these results on the general self-conjugate and on commutative functions is examined. It is found that several properties of the general self-conjugate that might, by analogy with the corresponding quaternion case, be expected to exist are non-existent. 38. The commutative self-conjugate is returned to and several additional properties are proved by a method analogous to that used for the general self-conjugate. 39. Similarly the general function is returned to and treated by a method analogous to that used for the commutative function earlier in the treatise. 40. Combinatorial variation is used to establish some miscel- laneous results especially in connection with second and third order complexes of the first degree. 41. The simplest forms of complexes of the first degree of all orders are established. 42 — 49. Several applications of Octonions are made. These are all suggested by Sir Robert Ball's ' Theory of Screws.' CHAPTER I. FORMAL QUATERNIONS. 1. Quaternion formulae susceptible of new geometrical interpretations. In the course of the development of our subject it will become apparent that every quaternion formula which in- volves any combination of symbols representing quaternions, scalars, vectors, linear quaternion functions of quaternions, linear vector functions of vectors, conjugate and self-conjugate ditto (i.e. the two separable cfy's) and the symbols i, j, k, K, 6, T, U, V, f is susceptible of a geometrical interpretation quite different from the ordinary one, an interpretation in the subject of Octonions. This interpretation like the ordinary one treats the different parts of and dii'ections in space with perfect impartiality. Each symbol in the new interpretation involves just double the number of ordinary scalars that are involved in the old. Thus the symbol correspond- ing to a scalar involves two ordinary scalars, a vector six, a quaternion eight, a linear vector function of a vector eighteen, a self-conjugate ditto twelve, i four and so on. I have not been able to include in this interpretation formulae involving V though two quite distinct octonion analogues of V are used below. It will be remembered by the readers of Clifford's Mathematical Papers that he continually speaks of the dual interpi'etation of certain quaternion equations. Below I shall give reasons for regarding this dual interpretation as in reality but a dual aspect of a single interpretation. But accepting Clifford's meaning of duality for the moment we must now add that there is a second dual interpretation also to be given to such equations. Whether or not in the future other symmetrical geometrical interpretations of quaternion formulae will come to light cannot of course be said, but there seems no reason for believing the contrary. § 2] FORMAL QUATERNIONS. 7 Since our interpi-etations of Quaternion fonnuke are to be different from the ordinary ones it is desirable to investigate the purely symbolic laws underlying those formulae. 2. Fundamental formal laws of Quaternions. Just as in ordinary Algebra there are certain fundamental formal laws on which the whole subject may be based quite apart from the applications to arithmetic quantity, so there are certain funda- mental formal laws of quaternion symbols (i.e. the sixteen symbols just enumerated) from which all quaternion formulae can be derived, quite apart from their geometrical interpretation. We must collect these here because we wish to consider interpretations of the formulae different from the ordinary ones. The subject when thus considered apart from its geometrical interpretation will be called Formal Quaternions. In the following few statements (§§ 2, 3) about formal quater- nions Hamilton's conventions as to notation are adopted though this course has not been found advisable in the paper generally. The conventions referred to are that formal vectors shall be denoted by small Greek letters, the linear function by , formal quaternions by p, q, r, scalars by x, y, &c. The following are the fundamental laws : — (1) Every formal quaternion q can be expressed as the sum of two parts called its formal vector part and its formal scalar part, denoted by \/q and Sq respectively. A formal scalar means a formal quaternion whose formal vector part is zero and a formal vector one whose formal scalar part is zero. A particular case of a formal scalar is an ordinary scalar. (2) All the fundamental laws of ordinary Algebra except the commutative law for multiplication are true of formal quater- nions. (3) Formal scalars and in particular ordinary scalars are commutative with formal quaternions (so that formal scalars obey among themselves all the fundamental laws of Algebra). (4) Of the five symbols Kg', Sg, Vg, Tq, Viq, the two Sq and Tq are formal scalars and the five satisfy the following con- ditions : — 8 OCTONIONS. [§ 2 (a) q = Sq + Wq = Tq.Uq, (b) T[Jq = l, T(qr) = TqTr, (c) Kq=Sq-Wq = (Tqyq-\ (d) X, y being any formal scalars and a, /S, any formal vectors, x + y and xy are also formal scalars and a + /3 and xa formal vectors. [For instance q\(^q =(Tqy= formal scalar, a^ = — aKa = - (Ta)= = formal scalar.] a;"' has a unique meaning, (x-y = ± x. (.5) a and /3 two formal vectors can be found such that Ta, TVo/3 and T (aVa/3) are all ordinary scalars not zero. (6) (j>q = Xaqh (where a and h are given formal quaternions and q an arbitrary one) is defined as the linear formal quaternion function of a formal quaternion ; and (^p = 2Vap6 (where a and h are as before and p is an arbitrary formal vector) is defined as the linear formal vector function of a formal vector. [In Octonions which are formal quaternions we shall, however, have moi'e general forms than these for linear functions.] It may be remarked that conditions (1) to (4) are alone sufficient for all formulae that do not involve , i, j, k and that if (6) be added all formulae not involving i, j, k are true. (1) to (.5) are sufficient for all formulae not involving <^. 3. Some general formulae deduced fi-om the f\inda- mental formal laws. It would involve too long a digression to attempt to give a satisfactory proof that these contain all the fundamental formal laws of quaternions. I am not sure that in the above the laws have been reduced to their simplest form. But though the above statements may to a certain extent be redundant, the following deductions from them of the chief general formulae of Quaternions will, I think, serve to convince the reader that they are at any rate sufficient. We will then prove first that Va;S7 = aS/87 - /3S7a + ySa/3 (from which may be at once deduced in the ordinary way that pSa/Sy = aS/dyp + . . . = V/SySap +...); secondly that three formal vectors i, j, k can be found such that jk = i, ki =j, ij — k, i^ =j'' = k^ = ijk= — 1 ; § 3] FORMAL QUATERNIONS. 9 and thirdly that if fi be defined by the equation ^ (&, ?,) =^{ii) + y!r (j, j) + ylr (k, k), &by and so to any number of pairs of f 's (yfr and x being formal quater- nion functions linear in each of their constituents) then where w is any formal vector and (p any linear formal vector function of a formal vector. [Note that from the last can be deduced the cubic (with formal scalar coefficients) satisfied by cj) just as in the case of Quaternions — Utility of Quaternions inPhysics, p. 17, foot-note. And again, from the cubic in this form can be deduced the usual forms and in particular the various forms for <^~'. This will be shown when we come to treat of the correspond- ing part of Octonions.J In proving these three general formulae we shall incidentally pi'ove several other general formulae well known in Quaternions. Notice first from (4) (c) that since it is assumed [(4) {d)'\ that when a; is a formal scalar «~' has a unique meaning, it follows that q~^ where g' is a formal quaternion also has a unique meaning. Next, notice that Ka; = x, and therefore that (Txf = x\() = (Tg)-^ Kqr=KrKq. From the second of these we have K {qiq.... qn) = Kg^K^n^i ... Kq^. 10 OCTONIONS. [§ 3 Again if a be a formal vector Ka = Sa - Va = - o, and therefore K (aitts ... a,J = (- l)"a„a,i_i ...a,. From this since q + Kq = 2Sq, q-Kq = 2\/q, 2S(ai...aJ = ai...a„+(-l)»a„...ai = (-l)''2S(a„...aO. 2V(a, ... a,0 = a, ... a„ - (- lfa„ ... a, = -{- l)»2V(a„ ... a,). In particular a/8 + /3a = 2Sa/S = 2Sy9a, Va/3 + V^a = 0. Thus Va (/S7 + 7/3) = 2aS/S7, V(ay3 + /3a)7 = 27Sa/3, V(a7./8 + /3.a7) = 2/3S7a. Adding the first two and subtracting the third of these we get WoLjSj = aS/37 - /3S7a + 7Sa/S. Next choose a and /8 as in (5) § 2 and put VayS = a', cLVal3 = a" = aa'. Thus by the equation just established, a'a" = a'aa' = — aa'-, a"a = aa'a = — a'aP, aa =a". Hence Ua, Ua', Ua" are three formal vectors which being put respectively equal to i, j, k give jk = i, ki =j, ij = k, i- =j- = k^ = ijk = — 1. Now notice that by the equations established if p be any formal vector p = - iSip -jSjp - kSkp = -^S^p, VrVp?=-pr^+?Spr=2p. If (f>(o be a linear formal vector function of a formal vector to the equation inSX/MV = S^X(f)fj,^v, for the formal scalar m where \, p,, v are three formal vectors gives a value for m which is independent of the particular values of X, p, V. For by the definition ^p = 'SiVapb, (j> is commutative with formal scalars [Perhaps it should be explicitly stated that § 4] FORMAL QUATERNIONS. H formal scalars are commutative with S and V. For x being a formal scalar S {xq) = S (xSq + xMq) = xSq. Similarly for V and K. Similarly also S, V and K are distributive i.e. S (g- + r-) = Sg' + Sr, &C.J. Hence as can be easily seen, the meaning of m is unaltered by changing X into xX + yfi + zv where X, y, z are any three formal scalars. The proposition follows at once since, as we have seen, any formal vector can be expressed as the sum of (formal scalar) multiples of three formal vectors (say With the meaning of m thus obtained we have in particular But by repeated applications of the equations we have Si;,U3U.X^ = 6, Eliminating m we have 4. Primary and secondary systems of formal quater- nions. We shall now assume that if laws (1) to (6) § 2 are satisfied by a set of formal quaternions all the formulae (always excepting such as involve V) of ordinary quaternions will be true of this formal set. We proceed to show that -connected with any system (denoted by q, r, &c.) of formal quaternions is another system (denoted by Q, R, &c.). The former system we shall call the primary system and the latter the secondary system. Let n be a symbol which behaves exactly like a formal scalar of the primary system except that fl^ always = 0. [fl is what by Clifford is denoted by w. We use H merely because it is a symbol whose want for other purposes is not so severely felt as that of CO. For many purposes the memory is greatly assisted by noticing that il behaves like a constant infinitesimal scalar of the primary system.] Further let il have the property that if q + D,r = 0, then q = r = 0. [From this it follows that if q + D,r = q' + fir', then q = q' and r = r'.] 12 OCTONIONS. [§ 4 Define as followa : Q = g + nr (1), KQ=K^ + nKr (2). SQ = Sq + nSr (3), VQ=Vg + nVr (4), TQ = (l + nSrg-i)Tg (5), UQ = {l + nVrq-^)Uq (6). It has been said that CI behaves lilte a formal scalar infinitesimal of the primary system. It may here be noticed that in an extended sense this statement has a bearing on these definitions. First it must be carefully noted however that fl is not such an infinitesimal. If it were such a scalar however He would be an arbitrary in- finitesimal formal quaternion of the primary system which might be put = dq. The definitions (1) to (6) would then give Q = q + dq, KQ=Kq + dKq, SQ = Sq + dSq, WQ=Wq + d\q, TQ = Tq + dJq, UQ = Uq + dUq. These relations might be made the basis of the proof of most that is required in the present section. But the most important use of them is to show as it were why the secondary system is a system of formal quaternions if the primary system is so. Note that if equations (1) to (6) hold and lead to the secondary system obeying laws (1) to (6) of § 2, a formal scalar of the secondary system must be of the form x -I- Cly and a formal vector of the form a + H/S where x, y are formal scalars and a, /S formal vectors of the primary system. Further note that since (§ 3 above) g~' has a unique meaning so has Q""^ For putting we have (g + fir) (^i + Hrj) = 1, or S'^i = 1, Sn + rq^ = 0, i.e. q^ = q-^^ r^ = — q~^rq~'^. Hence Q-^ = 5-1 - flg-^j-^ (7). [It may be noticed that from this if Q = fir, Q~' = floo or (let us say) Q~' is unintelligible. A clear geometrical reason for this will appear in the case of Octonions. It is of course due to the fact that n^ = 0.] § 4] FORMAL QUATERNIONS. 13 With these definitions : — If the primary system q, r, cLx. is a system of formal quaternions, so also is the secondary system Q, R, &c. To prove this we must assume (1) to (6) § 2 above to hold for the primary system and deduce that they hold for the secondary system. (1) is obviously true of the secondary system. Since fl behaves like a formal scalar of the primary system, and since formal scalars are commutative with formal quaternions in the primary system it follows that any ordinary algebraic law that holds for the primary system holds also for the secondary system. Hence (2) is true for the secondary system. [That is P + Q=Q + P, P+(Q+E) = (P + Q) + R, P.QR = PQ.R, {P + Q){R + S) = PR + PS + QR + QS.] (3) also follows for the secondary system from the similarity of the behaviour of Q, to that of a formal scalar of the primary system, [ft is not a formal scalar of the primary system but it is of the secondary system viz. when q = 0,r = l in eq. (1) above.] To prove (4) first notice that TQ and SQ are each of the form a' + fly and are therefore formal scalars. The equation (4) (a) Q=SQ+\/Q is obvious. To prove the second equation (4) (a) Q = TQUQ, we have to prove that 9 + fir = (1 + nSrq-') Tq (1 + flWrq-') \Jq. Noticing that by definition D, is commutative with every formal quaternion of the primary system and that n- = 0, the expression on the right = TqUq + n (Srq-' . TqUq + ^rq-' . TqUq) = q + nrq-^ -9 = 1 + ^''• To prove (4) (6) we have first to show that TUQ= 1. Now if UQ = q' + nr' we have by equation (5) TUQ = {l+nSr'q'-').Tq'. But by equation (6) q = Uq, r = Vrq-' Uq. Hence TUQ = 1 + nS (Vrg-' . \Jq . Uq-') = 1. 14 OCTONIONS. [§ 4 Next we have to show that T (QR) = TQTR or T (5 + Or) .T(q' + Xlr') = T . (g + fir) (q + fir'), where q, r, q', r' are any formal quaternions of the primary system. Thus by equation (5) we have to show that (1 + nSrq-') Tg . (1 + fiSr'g'-') Tg' = T (gg' + 11 [qr' + rg']) ^[l + aS.{qr+rq'){qqr'U{qq'). The left of this equation is TgTg' {1 + nS (rg-^ + r'g'->)), and the right is T{qq'){l + ilS{rq-' + r'q'-% so that the equation is true. The equation (4) (c) KQ = SQ-VQ is obvious. To establish the equation KQ= (TQ)- Q~' it is necessary to show that Kq + fiKr = {(1 + fiSrg-^) TglH?~' - nq-'rq-'). Remembering that Kq .q = (Tqf the right becomes (1 + 2n6rq-') Kq (1 - Hrq-') = Kq + nKq (- rq-' + 2Srq-') = Kq + nKqK (rq-') = Kg + flK {rq-' . q) = Kg + flKr. The statements of (4) {d) except those referring to x~^ and {x-Y' are obvious from the similarity of fl to a formal scalar of the primary system. That Q~' where Q is any octonion has a unique meaning has already been proved. It remains to prove that (A'^)* = ± X when X is a formal scalar of the secondary system. Y^ where F is such a formal scalar means of course a formal scalar whose square = T. Let Y = x + ny, Yi = x' + ny'. Thus x + (ly={x' + D,y'y = x'^ + D.2x'y'. Hence x = ± x^, y' = ± \x~^y, or V,* + % = ± V« ('l + fi ^) (8). § 4] FORMAL QUATERNIONS. 15 [This is obvious from the similarity of D. to an infinitesimal scalar.] Hence j(a; + ilyy}i = («= + 2nxy)i =±a;(l + n^^ = ±{x + ily), i.e. {X')i = ±X. (5) is obviously true for the secondarj^ system if true for the primary for r may be put zero. (6) consists of definitions only. CHAPTER II. OCTONIONS AS FORMAL QUATERNIONS. 5. The meanings of certain words. The following tabular comparison of the meanings of certahi terms used in the present treatise with the corresponding terms used by Clifford and Sir Robert Ball will probably prove convenient to the reader. Clifford. Ball. Present Treatise. Bi-quaternion Octonion Motor Motor Rotor Rotor Vector Lator Quaternion Axial quaternion or axial Twist Twist Screw Screw, unit-motor Twist about a screw Velocity motor Wrench on screw Force motor Impulsive wrench Momentum Motor Pitch Pitch Pitch With regard to the terms lator and aodal quaternion it should be remarked that these are equivalent to Clifford's vector and quaternion only when the latter occur in his papers on Bi-Quater- nions. Where he uses vector and quaternion strictly in Hamilton's sense the present writer would do the same. On p. 182 of Clifford's Mathematical Papers he says : — " The name vector may be conveniently associated with a velocity of translation, as the simplest type of the quantity denoted by it. In analogy with this, I propose to use the name rotor (short for rotator) to mean a quantity having magnitude, direction, and position, of which the simplest type is a velocity of rotation^ about § 5] OCTONIONS AS FORMAL QUATERNIONS. 17 a certain axis. A rotor will be geometrically represented by a length proportional to its magnitude measured upon its axis in a certain sense. The rotor AB will be identical with CD if they are in the same straight line, of the same length, and in the same sense ; i.e. a vector may move anywise parallel to itself, but a rotor only in its own line." This meaning of rotor as also his meaning (explained later in the same treatise) of motor I propose to adopt, but not so his term vector. There is no objection generally to use vector to mean anything which has the same geometrical significance as Hamilton's vector. But when, as in the present treatise, we are using symbols to represent such geometrical quantities and when those symbols do not obey all the laws of Hamilton's symbols for vectors, and when further we frequently have to refer to Hamilton's symbols and their laws, it is necessary for clearness to use the term vector strictly in Hamilton's sense. Now Clifford does not use the term in quite that sense. Hamilton's vectors and Clifford's vectors have the same geometrical significance and obey the same laws of addition but they do not obey the same laws of multiplication. I propose then to use the term lator for Clifford's vector. Thus lators and vectors are only distinguishable by their laws of multiplication. Exactly similar remarks apply to Clifford's use in his papers on Bi-Quaternions of the term quaternion. His quaternion is a particular kind of octonion and consistently with his use of qaatemion he ought to call his rotor a vector ; i.e. his quaternion has like his rotor a definite axis fixed in space and not like Hamilton's quaternion fixed merely in direction. For Clifford's quaternion I therefore substitute axial quaternion. This will generally be contracted to axial. It will probably aid towards an understanding of the methods below if I here comment on another idea of Clifford's. He frequently speaks of the dual interpretation to be placed on certain quaternion equations. This dual interpretation exists and has been of great aid in the development of Quaternions. But I think that simplicity in our fundamental ideas is gained, and some doubtful metaphysics is avoided, when it is shown that the dual interpretation is but a dual aspect of a single fact. To show this in the case of Quaternions is not difficult. Defining a vector as anything which (1) requires for its specification M. O. 18 OCTONIONS. [§ 5 and is completely specified by a direction and magnitude, (2) obeys certain assigned laws of addition, and (3) obeys certain laws of multiplication assigned later; we may define a quaternion as anything which (1) requires for its specification and is com- pletely specified by a vector and a scalar, (2) obeys certain assigned laws of addition (depending on those of vectors), and (3) obeys certain assigned laws of multiplication. Among the other assigned laws is that the quaternion is the sum of the vector and scalar, and that the sum of any number of component quaternions is the quaternion whose vector part is the sum of the vector parts of the components and whose scalar part is the sum of the scalar parts of the components. Thus a vector is a particular case of a quaternion and a scalar is another particular case, and the laws of addition of quaternions harmonise with those of vectors and scalars. The laws of multiplication of quaternions likewise harmonise with those of scalars (by which of course it is not meant that the laws of quater- nion multiplication are the same as those of scalar, but only that when quaternions degenerate into scalars, the quaternion laws degenerate into the scalar laws) and the laws of multiplication of vectors are now determinate by reason of vectors being particular quaternions. From the assigned laws of multiplication of quaternions it can be shown that the product qr of two quaternions in the particular case when r reduces to a vector perpendicular to the vector part of q is another vector also perpendicular to q and definitely related to r. In this sense q is the quotient obtained by dividing the last vector by the vector r and may be looked upon as an operator which converts r into the other vector. We thus get the dual interpretation with one fundamental conception of a quaternion. Of course historically this is not the order in which the quaternion conceptions emerged, but there is no reason why after we have ourselves reached by a tangled route a desired goal we should not point out to others a smoother way. So with octonions. They may be defined as operators on motors just as quaternions may be defined as operators on vectors, and as in the case of quaternions we may later go back and enlarge our primitive conception of them as operators. But I think it is simpler to define them otherwise and show that from the definitions § 6] OCTONIONS AS FORMAL QUATERNIONS. 19 QA where Q is an octonion and A a motor (a particular case of an octonion) which intersects Q at right angles is itself a motor which intersects Q at right angles. Moreover just as the con- ception of a quaternion as a geometrical magnitude requires first the conception of a vector as such a magnitude, so the conception of an octonion as a geometrical magnitude requires first the conception of a motor as such a magnitude. 6. Fundamental conceptions and laws of Octonions. The definitions about to be given are complete only when taken as a whole. In other words the definitions of Lator, Rotor, «Src. are not complete till we have implied by the definition of the multiplication and addition of octonions what is meant by the multiplication and addition of lators, rotors, «fec. A lator is a quantity which requires for its specification and is cmnpletely specified by a direction and an ordinary magnitude (scalar). A rotor is a quantity which requires for its specification and is completely specified by a direction, an indefinitely long straight line parallel to that direction and an ordinary magnitude. The line is called the aads of the rotor. A motor is a quantity which requires for its specification and is completely specified by a rotor and a lator which are parallel to one another. The axis of the rotor is called the aads of the motor. [A particular case of a motor is a rotor and another particular case is a lator, and in the latter case the motor has no definite axis fixed in space, though of course any line (or the direction) parallel to the lator may be called the axis. Such an axis may be called an indefinite axis. The restriction that the rotor and lator should be parallel will be removed later. In this case the axis of the motor is not the axis of the rotor, though, as will appear, it is parallel to it.J An octonion is a quantity which requires for its specification and is completely specified by a motor and two scala^-s of which one is called its ordinary scalar and the other its convert. The aads of the motor is called the axis of the octonion. [A particular case of an octonion is a motor and another particular case is a scalar, which is the ordinary scalar mentioned and not the convert. A third particular case is a ' scalar octonion ' which involves both 2—2 20 OCTONIONS [§ 6 the scalars of the definition. Of course if the motor have no definite axis, i.e. if it be a lator or if the motor be zero, the octonion has no definite axis or none at all respectively.] An axial quaternion (or more shortly " an axial ") is an octonion of which both the lator and convert are zero. [Thus it has a definite axis except when it reduces to an ordinary scalar. It will be observed that a quaternion has an indefinite axis except when it reduces to an ordinary scalar. If the scalar of the axial be zero the latter reduces to a rotor, so that rotors and ordinary scalars are particular forms of axials.] It will thus be seen that if be a given point a system of axials through (i.e. with axes passing through 0), rotors through and ordinary scalars are specified — when once is fixed — in precisely the same way as quaternions, vectors and ordinary scalars. Indeed just as the vectors and ordinary scalars are included in the system of quaternions, so the rotors through and ordinary scalars are included in the system of axials through 0. If q stand for an axial through and 5' for the quaternion whose vector part is parallel and equal to the rotor specifying q and whose scalar part is equal to the scalar specifying q, q may be conveniently called the quaternion corresponding to q. Thus to every axial in space there is one corresponding quaternion. To a quaternion corresponds an infinite number of axials however, with parallel axes. But if we limit ourselves to the axials through an assigned point 0, there is but one axial corresponding to a given quaternion. A system of axials through a given point (including rotors through and ordinary scalars) obey among themselves all the laws of the corresponding quaternions (and their included vectors and ordinary scalars). This is our starting-point for the definitions of the multiplication and addition of octonions. A definite point is used in the definitions. Thus the definitions of multiplication and addition do not apparently treat space impartially. [For instance starting with we render definite the meaning of products and sums of any octonions in space. As particular cases we get the meanings of products and sums of axials through some second point P. So far as the definitions themselves go it is not evident that axials through F would obey among themselves all the laws of the § 6] OCTONIONS AS FORMAL QUATERNIONS. 21 corresponding quaternions. As a matter of fact they do, but this must be proved.] The real meanings of the definitions however apart from their form are independent of the point used ; i.e. if Q and R be two octonions, although is used in the definitions of QR arid Q + R, these latter octonions are dependent solely on Q and R and not on 0. Since when we limit ourselves to axials through we are to all intents and purposes dealing with ordinary quaternions we use q, r... for axials through ; p, a... for rotors through 0; x,y... for ordinary scalars. In other words we adopt Hamilton's con- ventions as to notation, using Hamilton's vector symbols for their corresponding rotors through 0, his quaternion symbols for axials through 0, and his scalar symbols for ordinary scalars. Whenever then we find it convenient as we sometimes shall to thus limit ourselves to a definite axial system we adopt the Theory of Quaternions in toto — in geometrical applications as well as notation. The V that has hitherto been used will now be changed to M and read ' motor part of The symbolic properties of M are precisely those of the quaternion V, but the connotation of the initial letter of vector does not hold in the present theory. Consider now how an octonion Q can be supposed specified. It is determined when its axis, its rotor and lator parts and its two scalars are known. It is therefore completely specified by m-, 0), X, X, y, where ts, a> and X are rotors through and x and y are ordinary scalars ; m being the rotor perpendicular on the axis, w the rotor parallel and equal to the rotor of (the motor of) Q,X the rotor parallel and equal to the lator of (the motor oi) Q, x the ordinary scalar of Q and y the convert. It will thus be seen that w and \ are coaxial being both parallel to the axis of Q, and tu- intersects each of these three parallels perpendicularly. Instead of ■sr, w and X we may use w and a where a is the rotor through defined + X = o- + Meo) = Mr + MeMg'. Hence Q' = ^ + Xlr + OMeM^ = g + fir + OMeM (? + Or) = Q+fiMeMQ, which is eq. (1). 24 OCTONIONS. [§ 7 Since 2MeMQ = eQ-Qe and since (1 + iOe)-! = 1 - ^Oe, eq. (1) may be put Q' = {l + ^ne)Q(l+ine)-' (2). Now any octonion Q has the same space relations with a point P as Q with its axis shifted a distance equal and parallel to PO has with 0. Otherwise worded — any octonion viewed from P has the same aspect as the same octonion with its axis shifted a distance equal and parallel to PO viewed from 0. Let OP = p (3), where OP denotes, as it will throughout this treatise, the rotor OP- It will now be seen that the fact that QR and Q+ R are octonions independent of follows from the obvious truths (1 - iQp) {QR) (1 - k^p)-' = {(1 - h^p) Q (1 - i"p)-'l {(i-inp)ij(i-inpn, (1 - \^p) (Q +R)(i- |np)-> = (1 - ^np) Q (1 - jHp)- + {i-^[ip)R(i-^np)-\ for put into words the first of these equations asserts that if Q and R be two octonions, then defining multiplication by means of 0, QR shifted through a distance equal and parallel to PO is the same as Q sitnilarly shifted multiplied into R similarly shifted ; i.e. defining multiplication by means of P instead of the octonion now denoted by QR is the same as before. Similar remarks apply to the second of the equations. The statement in § 1 above is now established, that there is a geometrical interpretation of every quaternion formula (not involving V) which treats the directions and regions of space impartially but which is quite different from the ordinary one. We have not yet found the principles on which that interpretation must be made. This is to be done by examining the geometrical meaning of the simpler combinations of octonions and especially motors. 8. Motors in Mechanics. Velocity-motor, force-motor, momentum-motor, impulse-motor. In § 4 it was assumed that if q+D,r=:q' + Clr', where q, r, q', r' belong to the primary system, then q = q' and § 8] OCTONIONS AS FORMAL QUATERNIONS. 25 r = r' without exception. It must now be remembered that the primary system (axials) that we have used to conduct to the secondary system (octonions) consists of a system of axials through one definite point 0. In this case the assumption is justified and this is all that is required for the results based on the assumption. But we may have g + O;- = 5' + Q.r' when q and r are axials through and q' and r' are axials through some other point P- In this case q' and r' do not belong to the primary system and we have not q= q' , r =^ r in general. Still it is geometrically evident that q' and r' are unique determinate functions of q, r and p where p stands for OP. Now 5 + Or = (^ + HMpMg) + O {{r - MpMq) -f- D,MpM (r - pMq)}. But q and r — MpNlq pass through 0. Hence by equation (1) § 7 q+riMpMq and r-tApfAq+nMpM{r-pMq) pass through P. Hence q' = q + nMpMq, r' = r- MpMq + OMpM (r - pMq). . .(1). We may notice in passing an important result. We see that q is q with its axis shifted to pass through P. From this we easily deduce that if q, q are axials and Q. Q' octonions such that q + nQ = q' + HQ' the quaternions corresponding to q and q' are identical, i.e. q' is q with its axis shifted. This is quite easily proved directly from the definitions of q and r in § 6. From (1) it follows that Sq = Sq', 6r = Sr, i.e. it is only that part oi q+ llr which is a motor that has its form changed. This is otherwise obvious. Supposing q + D,r = a motor = « + Ho- = 00' + fla-', where w, a are rotors through and w', ' and its moment of momentum about P is equal and parallel to cr'. A motor that possesses this property with reference to the system of matter we shall call the momentum- motor of the system. The line which is the locus of points about which the moment of momentum is parallel to the momentum is the axis of the motor. The momentum is equal and parallel to the rotor, and for these points the moment of momentum is equal and parallel to the lator. The motion is always such that it could be instantaneously produced from rest by a system of impulses. This system is related to a motor in the same way as a system of forces is related to the force-motor. The motor to which it is so related is the momentum-motor. When we are actually consider- ing such a system of impulses we shall sometimes call the momen- tum-motor the impulse-motor of the system. An impulse-motor is what Sir Robert Ball calls an impulsive wrench on a screw. § 9] OCTONIONS AS FORMAL QUATERNIONS. 27 It is now evident that the result of mechanically superposing two velocity, force, momentum, or impulse motors A and B is to obtain what in octonions is denoted by ^ + jB. Thus a distinct mental picture is formed of the sum of two motors, and indeed of the sum of two octonions since both scalars are added in the ordinary algebraic way. We shall later obtain a similar mental picture of the product of two octonions based on the geometrical interpretation of the operator Q ( ) Q~^ where Q is an octonion. Before attempting to do this it is best to introduce some new terms and symbols. 9. Various decompositions of an octonion into simpler elements. We have seen that the position of our original point of reference does not alter the geometrical meaning of the various functions of an octonion that have been introduced. Take then on the axis of Q and put Q = ?(3+ii'-g (1)> where qq and Tq are axials through 0. From the original definition (§ 6) of g' + D,r we see that qq and Tq are both coaxial with Q and are independent of the position of on the axis of Q. By the definitions indeed qq = rotor of Q + ordinary scalar of Q, Org = lator of Q + O x convert of Q. Thus qg and Vq are axials coaxial with Q which are definite func- tions of Q. qq is called the axial of Q ; flvq is called the convertor of Q. [When it is necessary to refer to rq by name it may be called the convertor-axial, but this is not a good name. It is sometimes convenient to refer to the octonion Hr^gQ-' by a name. It may be called the pitch-translation convertor, but this is very clumsy. Similarly rq^q"^ would be called the pitch-translation axial.] Since in (1) qq and Vq are coaxial we see that coaxial octonions like coaxial quaternions obey the commutative law of multiplication as well as the rest of the fundamental laws of ordinary algebra. By equation (5) § 4 above we have TQ = TqQ(l+^Srqqq-'). Put T,Q = Tqq, T,Q = 1 + a8rqqq-\ tQ = Srqqq-^) TQ = T,QT,Q = T,Q (1 f OtQ) J ' " ' ^ ^- 28 OCTONIONS. [§ 9 Here it is to be noticed that (just as MQ is not in octonions called the vector part of Q because though MQ behaves symbolically like Hamilton's vectors, yet the connotation of ' vector part ' does not apply here) we do not in Octonions call TQ the tensor of Q though it is TQ which is the exact symbolic analogue of the quater- nion tensor. TQ, T^Q, T^Q and XQ will be called the augmenter, the tensor, the additor and the pitch of Q respectively. [Aug- menter may be regarded as an English contraction of the Latin tensor-additor.] It will be observed that the tensor and the pitch are ordinary scalars, whereas the augmenter and the additor are not in general but are of the form x + Hy, where x and y are ordinary scalars. An expression of the form x + D,y will in the future be called a scalar octonion. Again, by equation (6) § 4 above Put U,Q=Ugg, U,Q = 1 + nM7•g9e"^ uQ = ^r^qQ-\ ,„. UQ =u,Q.aQ = u,Q(i + nuQ) j---^''^- Here again although UQ is the exact symbolic analogue of the quaternion versor it is not in Octonions called the versor. UQ, UiQ and U.2Q will be called the twister, the versor and the translator of Q. [Twister may be regarded as an English contraction of the Latin versor-translator.] It will be observed that the versor is an axial, the corresponding quaternion being a versor. The twister and the translator are not in general axials. [When it is necessary to refer to uQ by name it may be called the translation-rotor or simply the translation, but both names are bad.] Again, SQ = SqQ + nSrQ. Put S,Q = SqQ, S,Q = nSrQ, sQ = Srg) SQ = S,Q + S,Q = S,Q + nsQ J ^^^• SQ, SiQ, S.jQ, sQ will be called the scalar-octonion part, the ordinary scalar, the scalar-convertor and the convert of Q respec- tively. Again, W\Q = MqQ + nMrQ. Put M,Q = Mq^, M,Q = nMrg, mQ = MrJ MQ =M,Q + M,Q = M,Q + nmQ j ^ >' § 9] OCTONIONS AS FORMAL QUATERNIONS. 29 MQ, M,Q and M^Q will be called the motor, the rotur and the lator of Q respectively. It will be observed that the rotor being a rotor is an axial, but the motor and the lator are not axials. [When it is necessary to refer to mQ by name it may be called the lator-rotor, but the name is bad.] From these equations we have 9e = T,QU,Q = S,Q + M,Q (6), r-c = sQ+mQ, Dr^j = S,Q + M,Q (7), rQqQ-' = tQ + uQ, 1 + nr^q^-^ = T,QU,Q (8), from which it follows that sQ+mQ = (tQ + uQ) T,QU,Q = (tQ + uQ) (S^Q + M,Q). . .(9). Some analogous symbols might be added such as K^Q, K^Q, kQ, but it is best to symbolise only the absolutely necessary elements of Q. It will be observed that of the symbols which are contained in the following list, each in the second line is always an axial, whereas each in the first line is in general not an axial, and so on. Hrg, KQ, SQ, MQ, TQ, UQ, S^Q, M,Q, T^Q, U„Q (not axials), ?g, rQ, S,Q,M,Q,T,Q,UiQ, sQ, mQ, tQ, uQ (axials), M,Q, mQ, uQ (rotors), SjQ, TiQ, sQ, tQ, (ordinary scalars) Q,rQ , S,Q,M,Q,T,Q-l,[J,Q-l (convertors), M,Q, U2Q-l(lators). The only term in this list that has not been defined is convertor. Any octonion of the form D,r where r is an axial, or what is the same D,Q where Q is an octonion, is called a convertor. Thus in particular all lators and scalar-convertors such as MjQ, S^Q and T2Q — 1 are convertors. Observe that a convert is an ordinary scalar; thus the convert of a; + fly is not the scalar- convertor Cly but the ordinary scalar y. If Q is a convertor qg is zero and Tq not zero. We may sup- pose \JqQ = 1, TqQ = 0. In this case TQ = SQ = S,Q = nsQ, T,Q = 0, U,Q = 1, tQ = uQ=oo, and no intelligible meaning can be attached to T^Q, UQ, U2Q. 30 OCTONIONS. [§9 It may be observed that the octonion Q in the convertor a = nQ may without altering the meaning of R have (1) its axis translated arbitrarily since nQ = D,(Q + nMpMQ), .and (2) its convertor part altered arbitrarily since n{q + D,r) = D,q. To assist in remembering the relations between many of the symbols I give two figures. In these q and r are written instead of q(j and vq. If the relations between the tensor, vector part, c-fe- D [AC = TrlT^Q, BA=tMQ, CF=tSQ, CE=tQ = tTQ. ABC=FCD= Aq, AGD= ir, ACF= Lrq-'^\ Fig. 2. scalar part and angle of a quaternion be remembered, and also the fact that for two coaxial quaternions q and r, /. (rq~') = Z.r — Zq, § 9] OCTONIONS AS FORMAL QUATERNIONS. 31 it will be seen that tig. 1 is an immediate deduction from equations (6). (7), (8). To prove the relations implied by fig. 2, put z g = 6, Zr= + HMtM . {1 + n (;; +p')] qa. Hence QA is obtained from A by (1) multiplying A's tensor (T]4=Tw) by the tensor (T,Q=Tg) of Q, keeping the pitch unaltered ; (2) turning the motor round Q through the angle of Q ; M. o. 3 34 OCTONIONS. [§ 10 (3) adding to the motor's pitch (t^ =p') the pitch (tQ = p) of Q; (4) translating the motor through a distance equal and parallel to the translation-rotor (uQ = t) of Q. These four commutative operations are performed by multiply- ing the motor A successively by (1) the tensor TiQ, (2) the versor UiQ, (3) the additor TjQ, (4) the translator UjQ. Operations (1) and (2) are performed simultaneously by multiplying the motor by the aadal (g' = TiQUiQ) of Q, and (3) and (4) are performed by multiplying by T0QU2Q. Operations (1) and (3) are performed simultaneously by multiplying by the augmenter (TQ) of Q, and operations (2) and (4) by multiplying by the twister (UQ) of Q. The reasons for the names adopted are now evident. If A have no definite axis, i.e. is a lator Ho- where o- is a rotor and Q have a definite axis, the above phraseology requires further elucidation. We have seen (§ 9) that a may have any position in space that is convenient. We may therefore suppose it to inter- sect Q. It is still supposed to be perpendicular to Q. Thus QA={q + D,r) ilcr = Qqa: If we call the tensor of the rotor a- the quasi-tensor of the lator n and e. The conventions as to the signs of d and e are as in § 12. By eq. (13) § 12 we have tSABG = tS (MAB) C = iMAB + tC-e tan 4,. Hence by eq. (8) § 12 tSABG = iA + tB + tG + dcote-etsin(f> (1). Since SABC=SBGA = SGAB the following deduction may be made from this:— 7/1, 2, 3 be three straight lines; d„ d^, d, the distances and Bu 6»„ 6, the angles between the pairs 23, 31, 12; e„ e„ Bs the distances and , = d^ cot 6^ - e^ tan 4>r, = d^ cot 6, - e, tan <^3- • • (2)- 48 OCTONIONS. [§ 13 Before finding tMABC, note that if Q, R, ... be any number of octonions T,{QR...) = T,QT,R..., t(QR...) = tQ + tR+ (3). For two octonions Q and R these are at once deduced from the equation T(QR) = TQTR by putting TQ = TiQ (1 + fltQ) and similarly for R and QR. The extension to any number of octonions is obvious. Next note that from the equation we get by equating the ordinary scalar and the convertor parts [remembering that SQ = (1 + fitSQ) S,Q, MQ = (1 + mMQMQ, TQ = (1 + atQ) T,^], SrQ - M,'Q = T,^ Q, tSQ . S,=Q -tMQ.M,^Q = tQT.'Q. . .(4). For our purposes the last is put more conveniently in the form '"«' '^^v:ts.r (^)- Now put Q = ABC. Byeq. (3) tQ = t^ + t5 + tC, and by quaternion interpretation T,Q = T,AT,BT,C, S,Q = - T,AT,BT^C sin cos . Hence by equations (1) and (5) tMABC =tA+tB + W /cotg-etan0 cot^6'tan2(^ + cot2^ + tan2^ ^ ''■ Since MABC = MCBA a result similar to eq. (2) may be written down. By means of eq. (2) however it easily reduces, to the well-known and easily proved equation sin ^1 cos i = sin 02 cos (f>2 = sin 0^ cos ^3. tM (MAB) C may be found by a process exactly similar to that used in establishing eq. (1). Thus by eq. (8) § 12 tM (MAB) C= tfAAB + W + e cot , or tM (MAB) C = tA + tB + tC + dcote + e cot (7). § 14] OCTONIONS AS FORMAL QUATERNIONS. 49 From the geometrical interpretation of Sij4£C just mentioned we see that in order that S^ABC may not be zero it is necessary and sufficient that not one of the motors A,B,C shall be a lator, and that they shall not all be parallel to one plane. Since [eq. (7) § 4] S-^ABC = Sr^ABG- S^ABCSc^ABC=Sc^ABG{\ - D.tS ABC), we see that the conditions just mentioned are the necessary and sufficient conditions that S~'^ABG should have a definite in- telligible meaning. Hence the equations ESABC = ASBCE + BSCAE + CSABE ) = MBCSAE + MCASBE + MABSCe]'"^ ^' serve to express any motor E in terms of motors coaxial with A, B, C OT with MBC, MCA, MAB when A, B, C satisfy the conditions mentioned. By putting 8-^^50= Sr'ABC(l - ntS ABC), SBCE = S,BCE (1 + mSBGE), SAE = S,AE (1 + ntS AE), &c. it will be seen that when E is expressed in terms of (ordinary scalar) multiples oi A, B, G, ilA, CIB, flO, the A component is AS.BCES.-'ABC, and the D,A component is nAS,BCESr'ABC (tSBCE - tSABC), which may be transformed by eq. (1) to a fairly simple form. Similarly when expressed in terms of MBC, Q,MBC, &c., the MBC component is MBC6,AESr'ABC, and the QMBC component is Q.MBCS,AESr'ABC(tSAE-tSABC), which may be transformed by eq. (13) § 12 and eq. (1) of the present section. 14. Miscellaneous remarks. We collect here chiefly for future reference some miscellaneous statements most of which are almost obvious. S, Si, S2, s are all distributive, i.e. S(Q + R) = SQ + SR, S,(Q + It)=S,Q + S,R, S,(Q + R)=S,Q + SJi, s(Q+R) = sQ + sR (1). M is distributive (and also K) but M„ M^ and m are not. 4 M. o. 50 OCTONIONS. [§14 In 1 13 we saw that t(QR) = tQ + tR The similar equation in U, viz. U (QR) = UQU-K is true, but those in Uj, U2 and u are not true. We have seen that D, and therefore every scalar octonion is commutative with each of the symbols K, S, M. It is not commuta- tive with Si, S2, s, Ml, M2, m, T, Ti, T2, t, U, Ui, Us, u. It may be remarked that a positive scalar octonion is commutative with T, just as in Quaternions a positive scalar is. The following relations are obvious snQ = 8iQ, S,nQ = n8,Q, SifiQ = 0, nS,Q = 0...(2). From these we have SQ = snQ + nsQ (3), so that all the S's can be expressed by means of s and O. A similar remark is not true of the M's; because mflQ is indeterminate. Suppose i, j, k are three mutually perpendicular intersecting unit rotors. Then A being any motor A = -iSiA-jSjA-kSkA (4), or by eq. (3) A =—is. iliA —js . D,jA — ks . HkA — HisiA — ^jsjA — Q,kskA . . .(5). Let A be an independent variable motor given by A =xi + yj + zk + im + mD,j + n£lk (6), where x, I, ... are ordinary scalars. Then ^ is defined by the equation ^ = idjdl +jd/dm + kd/dn + nid/dx + Djd/dy + nkd/dz. . .(7). This gives sdA'i[ = -d (8). Hence if A=x'i'+... + I'm' + ... where i', f, k' are any other set of mutually perpendicular inter- secting unit rotors i'dldl' + ... + nid/dx'+ ... = -i's . D.i"^ - ... - ni'si"^ -...=% so that ^ is an invariant, i.e. is independent in meaning of the i, j, k used in defining it. § 14] OCTONIONS AS FORMAL QUATERNIONS. 51 The following are easy deductions from equations (4) and (5):— A = B where A and B are motors _ (1) if SpA = SpB where p is an arbitrary rotor through a given point and a fortiori when p is an arbitrary rotor or motor; (2) if s^^ = sEB when E is an arbitrary motor through a given point and a fortiori when E is an arbitrary motor ; (3) if spA = SpB when p is an arbitrary rotor. The last of these statements may be deduced from the first of the following two : — (4) If spA = where p is an arbitrary rotor through a given point, ^ is a rotor through the same point. (5) If SpoA = where p„ is an arbitrary later, A is a later. s . J. 2 is an expression that frequently occurs below. If J. be a later, J.^ = and therefore sA^ = 0. If ^4 be not a later it can be put in the form (1 + [It A) M^A. Hence A' = (l + 2ntA)MM (9). In particular s^^ = 2t J.Mi=^ = - 2t^Ti=J. (10). [This can be easily generalised to s.T^Q = 2tQT,=Q (11), but we shall not have any use for this more general form.] It follows that the necessary and sufficient condition to ensure that sA" = is that A shall be either a later or a rotor. n motors A-^, A^.-.An are said to be independent when no relation of the form x^A-^ + . . . + «n.4„ = %xA = (where Xi...x„ are ordinary scalars not all zero) holds between them. When they are independent all motors of the form XxA are said to form a complex of order n. The complex will generally be called the complex Ai, Az-.-An, and when they are not independent the complex (though then of lower order than n), containing them, will be called the complex A^, A^...Ar. Since [eq. (5)] any motor can be expressed in terms of six particular motors the complex of highest order is the sixth and all motors in space belong to it. li A-,...Ae\)e any six independent motors, any motor in space can be expressed in the form XiA-^ + ... +XsAs. This can be easily deduced from eq. (5). 4—2 52 OCTONIONS. [§14 Two motors A^ and A.^ are said to be reciprocal when sA^A^ = 0. Thus every lator and every rotor, but no other motor is self- reciprocal. The n motors Ai...An are said to be co-reciprocal when every pair of them is a reciprocal pair. It is obvious that Ai is then reciprocal to every motor of the complex A2...An. More generally every motor of the complex Ai...Ar is reciprocal to every motor of the complex Ar+i...An- When two complexes are so related they are said to be reciprocal complexes. Of six independent co-reciprocal motors not one can be self- reciprocal, for if it were it would be reciprocal to the complex of all six, i.e. to every motor in space. But this by statement (2) above requires that the motor should vanish. Thus of six inde- pendent co-reciprocal motors not one is a lator or a rotor. If A, B, C are motors the necessary and sufficient condition to ensure that SABG = is that either (1) two independent motors of the complex A, B, C are lators or (2) XA + YB + ZC = 0, where X, Y, Z are scalar octonions whose ordinary scalar parts are not all zero. Before proving this, note the two following consequences of it. A necessary condition is that XA + YB + ZG= where X, Y, Z are scalar octonions not all zero. For if xA + yB -f ^C is a lator D,xA + ^yB + D.zC = Q. A sufficient condition is that XA -\- YB + ZC=0, where X, Y, Z are scalar octonions whose ordinary scalar parts are not all zero. This is merely a part of the general enunciation. The condition is sufficient. If XA + YB + ZG = where SjX is not zero, A=~YX-^B-ZX-^G= Y'B + Z'G where Y', Z' are finite scalar octonions. Hence SABC= will be called a linear motor function of a motor, or more frequently a general function when (pE is such that whatever be the motor value of U, E is a motor, and that E+<}>F=(E + F) (1), whatever be the motor values of E and F. will be called a commutative linear motor function of a motor, or more shortly a commutative function, when ^JS" = M iQ,ER, + Q,ER, + ...) = tMQER (2), where E is an arbitrary motor and Qi, R^, Q^, R^- ■ ■ given octonions. In Quaternions two such definitions would be precisely equi- valent, but it is not so in Octonions. A commutative function is commutative with il and therefore with any scalar octonion, whence its name. The general function (except when it degenerates into a commutative function) is not thus commutative. It will be seen that when octonions are regarded as formal quaternions, the linear formal vector function of a formal vector [(6) § 2 above] is the commutative and not the general function. We proceed to show that the commutative function involves eighteen scalars and the general function thirty-six. Let A, B, G be any three constant motors such that SiABG is not zero. Since the commutative E=(4>ASBCE + (f)BSCAE+ (l>CSABE)S-'ABC ...(S). Hence when the three motors ^A, B, E is altered for some value of E. Hence eighteen and only eighteen scalars are required to specify a commutative function. § 15] OCTONIONS AS FORMAL QUATERNIONS. 55 Next let he a. general function. A, B, C being as before, we have seen (§ 13) that any motor E can be expressed in the form E = xA + yB + zG + mA ■irmD.B + nD.G (4), where x, y, z, I, m, n are ordinary scalars. Although is not commutative with scalar octonions in general, yet by its definition it is commutative with ordinary scalars. Hence <}>E = x(l)A+yC+l^(nA) + m^(aB) + n(P(nG)...(5). Hence 4> is now completely given by six motors 4>A ... ^> S) being an octonion function linear in each of its constituents, Z^ is defined by X{Z,,Z„Z„Z,) = x{%,A„%,A,) (7), and so to any number of pairs of Z's. If, as in equation (6), there is only one pair the suffix may be dropped. f is defined as in the case (§ 3) of formal quaternions, i, j, k now standing for three mutually perpendicular intersecting unit rotors. In discussing formal quaternions we only considered what is now denoted by the commutative function. But the definition of f will be supposed to hold good with the more general meanings of -v/f and % now contemplated. Z is an invariant (§ 14). f is an invariant with regard to the directions of i, j, k but not with regard to their point of intersection, unless we restrict ourselves to commutative functions. When this restriction is not made, then, 5' must be regarded as a function of the point of intersection just mentioned. It will be noticed that by equations (4) and (5) of § 14, E=-!;SE^=-ZsEZ (8). 56 OCTONIONS. [§15 The conjugate (f)' of ^ is defined by either of the equivalent equations sE<}>F=sF'E (9), for all motor values of E and F or ,f>'E=-ZsE4>Z (10). In the case of the commutative function these may [eq. (3) § 14] be replaced by SE'E (11), 4>'E = -tSE4,^ (12). If the function is said to be self-conjugate. In this case sEF = %F^E. There are fifteen such independent scalar equations that must be satisfied in order that the general may be self-conjugate, because the number of such independent equa- tions is the number of pairs that can be chosen out of six independent motors. Hence a general self-conjugate function involves twenty-one scalars. Equation (11) is equivalent to two scalar equations. Three independent equations of the form SE(f>F= SF(j)E must be satisfied by a commutative self-conjugate function. Such a function there- fore involves twelve scalars. Rotors which pass through one point may be said to form a pencil of rotors. If a commutative function is such that it reduces every rotor of such a pencil to a rotor of the same pencil it will be called a pencil function, and the point through which the rotors pass will be called the centre of the function or of the pencil according to convenience. Since every motor in space can be put in the form 0) + ilcr where o) and a- belong to the pencil and since D, is com- mutative with a commutative function, the present function is known for every motor in space when it is known for every rotor of the pencil. So long as we restrict ourselves to rotors of the pencil, the pencil function has exactly the same properties as a linear vector function of a vector. Besides the three scalars required to specify the centre, a pencil function thus involves nine other scalars, and a self-conjugate pencil function six. A pencil function is therefore not the most general form of a com- mutative function. § 15] OOTONIONS AS FORMAL QUATERNIONS. 57 We now proceed to express various functions (general, com- mutative, pencil, self-conjugate) in terms of each other. Any general function can he expressed in the form ^E = %E + UE (13), where T^ is a commutative function and H is given in terms of another commutative function Tj by the equations n« = o, un,co = %co (14), where w is an arbitrary rotor of an assigned pencil. For taking i, j, k as unit perpendicular rotors belonging to the pencil, •I' can be expressed in the form ^E = - (AsEm + A'sEi + BsEaj + B'sEj + GsE£lk + C'sEk) = - 2 [ASEi + (A' - nA) sEi} = r,E+UE, where Tj is the commutative function — SJ.S( )i and Tl is the function —l,(A' — D,A)s()i. Here we clearly have 11 to = and nfio) = T2C1) where Tj is the commutative function given by r,E = --Z(A'-nA)SEi. It will be observed that Tj involves eighteen scalars and Tj eighteen so that the full number thirty-six of $ is accounted for. Hence when the centre of the pencil is given Tj and Tj are unique. It does not appear from the above that Tj and Tj are self- conjugate when is. It will appear directly that in general they are not (though the two self-conjugates Tj and T would involve twenty-four scalars and the self-conjugate only twenty- one). The centre of the pencil however can be so chosen in general that they are self-conjugate. The general function O can always be put in the form Oft) = !&) -1- Xi023, 4>i are four pencil functions with a common assigned centre and w is an arbitrary rotor through the centre. For ^(o is a linear function of w which can be expressed as the sum of two linear functions, the first being a rotor (!&)) through the assigned centre and the second a lator (0,E = - IXSEi, ^,E = - Sa'S^i, <^4^= -S\'S^i.] This determination is also unique since each of the four pencil functions involves nine scalars and $ thirty-six. Comparing equations (13), (14), (15) we see that Ti=' with an assigned common centre and where a> is an arbitrary rotor through the centre. To prove this first suppose '^ to have the form of O in equation (15) and use the equation S (o) + Ho-) ^ (o)' + n,a-') = S (w' + ila') ^ (co + [la-), where co, a, a , a' are rotors through the assigned centre. Putting a and a' zero and leaving co and a>' arbitrary, we find that 2 is self-conjugate. Similarly putting w and tu' zero we find that ^ is self-conjugate. Finally, putting o) and a' zero we find that yfr^^ and i/tj involve six scalars each and is unique. When of equation (13) is self-conjugate and equal to the present '^ we see from equation (16) that % = <(> + fiA/Ti, r, = f, + n{'-^ of (f>. Thus in general Tj and Tj are not self-conjugate. (p of equation (17) can in general be made self-conjugate by suitably choosing the centre. We see that this is probably the case by noticing that in the choice of the point there are three disposable scalars. <}> is self-conjugate if SiWi^w^ = SiW^'^o), for any two rotors through the centre. If the given centre does not possess this § 16] OCTONIONS AS FORMAL QUATERNIONS. 59 property suppose if possible that P does where OP = p. Any two rotors through P may be expressed [eq. (1) § 7] as Wi + OMpw^ and Wa + flMpcoj where Wi and ©a are rotors through 0. Now Sj (tBi + nMpWi) ^ (o), + nMpoJa) = S (ft)i(^a)2 + i/rjWi . pw.,). Hence the required condition is that Swi (^ — (^') Wa + S (&)ipi/r2«2 — o)2pyfr^0)j) = 0, or ScBaWiMf^f = - SwsWiMf M/j-faf = Sa^Wi (i/ra + Sfilr^O p ; P satisfies the required conditions then if (t, + S?,|r,Op = M?<^r (19), which can generally but not invariably be satisfied. When (ji is self-conjugate Tj and Tj are also self-conjugate as appears by equation (18). A commutative function T can always he put in the form <^i -I- Ii<^2 where i and (p^ are self- conjugate. This is obvious from what has gone before. It may be noticed that if a commutative function T reduces every motor it acts on to a lator it is of the form D.2- Hence it reduces every lator it acts on to zero and may be said only to act on rotors. Moreover it reduces equal parallel rotors to the same lator. Its properties are precisely similar to those of the linear vector function of a vector corresponding to the pencil function ^2- I shall postpone the more detailed consideration of general functions to a later part of the treatise, as at present I wish to limit the discussion as far as convenient to those properties of octonions which are most immediately connected with the fact that octonions are formal quaternions. 16. Properties of the commutative function derived ftom Formal Quaternions. Confining our attention then to commutative functions we will in this section put down certain results, without proof, that flow fi:om the fact that octonions are formal quaternions. 60 OCTONIONS. [§16 <^^ = - B8AE - B'SA'E - B"SA"E = - 1.BSAE. . .(1) is a perfectly general form for this case, even when the number of terms is only three. A, A', A" may be assumed such that S-^AA'A" is not zero. [If S^AA'A" is zero equation (1) defines a commutative function but not one of the most general form. All commutative functions are of the form given in (1) when A, A', A" are arbi- trarily chosen as long as S^^AA'A" is not zero.] Put ' = 24> (2), ^ is self-conjugate and is called the self-conjugate part of ^=2H (3). Then 4>E=4>E + MHE, ^'E=^E-MHE (4). ■\lr(E, F) being a commutative linear motor function in both the motors E and F, ^(?. <^r)=t ('?> (5). [Proved at once by putting (^'f = — fiSf^^i. Note that putting <^ = Q ( ) Q~^ it follows that f is an invariant when associated only with commutative functions.] When

2 are commutative functions, then (f>i = 2. And if the same equation hold when ^^ is a perfectly arbitrary self- conjugate commutative function, then ^i = ^.,. The equation proved in § 3 above for formal quaternions ^s?,r,r3s^ - M"<^^ + M'^ - M)E = Q (8), where 6ilf = S^.U^HKM^Hz (9) 2if'=-sMr,r2Mri<^r2 ao), iif"=-s?A', B" = ASAE - A"SA"E]. Hence by equations (12), (13), (14) MSAA'A" = Scj>A4>A'(l>A" .^....^....„...^ (24), M'SAA'A" = S (A4>A'A + A"4>A^A') . . .(25), M"SAA'A" = 6 (A'A"4>A + A"AA' + AA'A') (26). 62 OCTONIONS. [§ 16 Here A, A', A" are any three motors whatever. Equation (24), it will be noticed, is the original definition (§ 2 above) of M. From the last three equations X'-M"X' + M'X-M = S(X - )!' (X - is changed to (j)'. Hence the ^' cubic is the same as the cubic [eq. (8)]. It follows that (f> may be changed to ' in equations (24) to (27). Since in equation (24) A is an arbitrary motor we have (§ 14) MMA'A" = 4>'M(pA'cf>A" (28). Similarly MMA'A" = 'A'^'A" (29). Here again it must be remembered that A' and A" are arbitrary motors. These results are exactly of the same form as corresponding quaternion results. We have therefore treated them very briefly. As the geometrical interpretations of octonion formulae have to be made independently of quaternion forms we must now enter into more detail. 17. The ] and ^2 are pencil functions with any assigned common centre. It is now important to observe that the form of ^1 is independent of the position of this centre. By this is meant that if a, /3, ... be given inde- pendent rotors and p an arbitrary rotor through the centre and a, /3', ..., p' be the parallel equal rotors through the centre 0'; and if ^ = ^1 + n2 = (/)/ + n^j' where ip = — SySSap, we shall have ^/p' = — S/S'Sa'p'. A=(0+ flo- = O)' + Off', (f)A = GJi + flo"! = «/ + n<7i', where A is any motor, where a, a-, toj, o-j are rotors through and where «', a', w/, o-/ are rotors through 0'. Then q),=^i&), a)i=i'co'; § 17] OCTONIONS AS FORMAL QUATERNIONS. 63 and Wi and w/ are equal and parallel, being the rotor of iM/3( )-MpA ), the form of ^2 to that of ^, + ^,Mp( )-Mp^,( ), and €2 becomes changed to the rotor through 0' equal and parallel to 62 — M/sei. On account of the similarity of ! to a linear vector function of a vector we see that its cubic must have ordinary scalar coefficients. In accordance with equations (9), (10), (11) § 16 these are — Sfc^if, &c. Substituting 0i + O^, for in those equations we see by equating ordinary scalar and converter parts that the cubic satisfied by (f)^ is (j>,'-S,M".^^ + S,M'.i cubic. One of the roots of the (pi cubic is always real when as we assume '' + M' -M = {-X) i4>-X') {-X")...{2), where X, X ', X" are scalar octonions ? Putting X = ^ + %, X' = x' + ny', X" = x" + nf (3), where x, y, &c. are ordinary scalars and expanding the right of (2) .(4). 64 OCTONIONS. [§17 we find by equating the ordinary scalar part and the converter part of each coefficient that xx'x" = S,M x'x" + x"x + xx' = SiM' x + x' + x" = S,M" yx'x" + y'al'x + y"xx' = %M y {x' + x") + y' (x" + x) + y" (x + x) = sM' y + y'+y" = sM" Hence x, x', x" are the roots of the (f>i cubic, x will be always assumed real. Supposing these roots determined we have _ n-n'x + n"x' , _ n - n'x' + n V^ „ _n- n'x" + w' V" y~ {x-x){x-x"y y ~ {x'-x"){x'-xy y ~{x"-x){x"-x') (5), where n, n, n" have been put for sM, sM', sM". This shows that when x, x', x" are all unequal, X, X' and X" can all be determined. Further when x is real and x' and x" are imaginary (in which case they are all unequal) both X and {-X'){4>-X") are real. But when two of the <^] cubic roots are equal, say x' and x", there are in general no corresponding roots X', X" of the cubic; and when all three roots of the ! cubic are equal there is in general no root of the <^ cubic. If x' = x' and x is not equal to either it will be found by a similar process to the above that there is one root X = x-\- ^y of the (j) cubic, y being given by eq. (5). In this case and sometimes in others it is convenient to define the scalar octonions iV" and N' by the identity (l}'-M"(j>^ + M'^-M={(l>-X){. This, which is not a truism, appears from the identity (27) § 16. If two roots or three roots of the 0i cubic are equal it is not always true that there are no corresponding roots of the cubic may have a root equal to X + Cly the M of the cubic and the two corresponding to x are arbitrary to the extent that their convertor parts may have any values con- sistent with the sum having a given value. For we have seen that the third root x" of the 0i cubic has a root x" + D,y" of the <^ cubic corresponding to it. If a; + D,y is another root (4)-x- ny) (<}) - x" - ay") is a factor of {(f) - xY - M" {(j> - xf + M'(-x)- M. There must therefore be a third factor and this can only be — xf — D,{y + y') { — x) of (^ — x)- D,y and ( — x)-- riy' is unaltered if y and y' are changed arbitrarily so long as their sum remains unaltered. And in the product of ■ these and — «;; i.e. the M of the cubic. If three roots of the i cubic are equal to x ; in order that the (f) cubic may have a root the M of the - xy -nz(4>-x) + Viz' is the other {quadratic) factor of {(j) - x)' - M" { cubic may have two roots both the M and M' of the ^ — X cubic must be zero. There are then three roots of the cubic and these are arbitrary to the extent that their convertor parts may have any values consistent with their sum having a given value. The proof of this is exactly similar to the proof of the last propo- sition. 18. Geometrical properties of a commutative function. By discussion of the nature of the roots of the <^i cubic and of the ^ cubic we are able to deduce many important geometrical pro- perties of a commutative function. For the sake of brevity the following terms will be used : — An axial motor will mean a motor which is not a lator, i.e. a motor which has a definite axis. Two completely independent axial motors are two axial motors which are not parallel. [According to the definition of § 14 two coaxial motors and two parallel motors are in general independent. Hence the necessity for the term completely independent.] M. o. ^ 66 OCTONIONS. [§18 A single root of a cubic will mean a root which is not equal to any other root of the cubic. A repeated root will mean a root which is equal to some other root. The following conventions as to notation will be strictly- adhered to in this section. i, j, k will mean a set of mutually perpendicular intersecting unit rotors. A, B, A', E, F, E' &c. will stand for motors, a, ^, 7, a! &c. will stand for rotors. X, Y, Z, Y', Z' will stand for scalar octonions. x, y, a, h, p &c. will stand for ordinary scalars. For the sake of clearness it is convenient to arrange the most important assertions of this section in formal propositions. Prop. I. If X is a root of the (f) cubic corresponding to a single root of the E = (X + £ly) E but when there is not, and when the root of the 0i cubic is not zero Mi4>p and M^Xp are equal and parallel where p is any rotor parallel to a certain plane. When the repeated root of the ^1 cubic is zero <^p is a lator. [Note that the condition that S^AA'IABB' is not zero is equi- valent to the conditions that not one of the rotors M^A, MjA', Wi^B, Mi£', VA^AA', W^BB' is zero and the two last are not perpendicular ; i.e. A, A', B, B', MAA' and MBB' are all axial motors and the last two are not perpendicular.] Since X is a root qf the cubic we have by the identity (27) § 16 that SiX-)A(X-6)A'(X-4>)A" = (2), where A, A', A" are any three completely independent axial motors. Hence by § 14 either X, (Z - (/>) 2 + z, (X - ) 2" = 0, where the ordinary scalars of Xj , X^ and X3 are not all zero, or else two independent motors of the complex iX-)A, (X-4,)A', {X-)A" § 18] OCTONIONS AS FORMAL QUATERNIONS. 67 areJators._ In the first case we may take A" for the axial motor X^A + X^' + X^A', when we have ^A" = XA" (3). Now choose A, A', A", B, B', B" to depend on A, A', A", (-X)A, B' = (-X)A'. 5" = (-X)E = -BSAE-B'SA'E, which is the same in form as eq. (1). If now the root S^X of the (/>! cubic corresponding to the root X of the tp cubic is not repeated, one root but not two of the ((/>! - SjZ) cubic is zero, i.e. the S^M' of the ( - Z) cubic is not zero, i.e. S^MAA'MBB' is not zero by eq. (15) § 16. In the case when two independent motors of the complex (X — ) A, (X — -X)E = - BSE A - n^'SEa' - ^^''SEol' (6), where a' and a" are any rotors equal and parallel to MjJ.' and MyA". In this case we see by eq. (13) § 16 that the S^M' of the (j) — X cubic is zero and therefore SiZ is a repeated root of the <^i cubic. In this case we have by equation (5) that (f)A' = XA' + a lator. Hence if p is any rotor parallel to A', p is a lator). Similarly for any rotor cr parallel to A". This can be seen perhaps still more easily from eq. (6) from which we see that if p is any rotor perpendicular to A 4>p = Xp + a lator. That there is not always in the case of eq. (6) an axial motor E for which (0 - X - ily) E =0 is seen by taking a particular case. If we put — fly) E=0. 5—2 68 OCTONTONS. [§ 18 We may notice that equation (1) may be reduced to either of the forms {(f> - X) E = - BSjE - B'SkE (7), or (-X)E = -jSAE-kSA'E (8), for taking i along the shortest distance of A and A' of eq. (1) we have SAi = SA'i = 0, and therefore A = -jSjA - kSkA, A' = -jSjA' - kSkA'. Substituting these values in eq. (1) we get (4>-X)E = (BSjA + B'SjA') SjE + (BSkA + B'SkA') SkE, which is of the form (7). Similarly taking i along the shortest distance of B and B' of eq. (1) we get the form (8). In these forms the conditions that SjX may not be a repeated root of the ^i cubic become respectively SiiBB' not zero and SiiAA' not zero. Since {4,'-M"' + M'- M)E=0 where E is any motor, we see in particular that when a root of the ipi cubic is repeated three times there are three completely independent axial motors B, F, B" such that for any motor E = X^B + X^' + X^B", ((f>' — ...)E=0. Similarly when S^X is a single root of the ^i cubic we have now seen that there is one axial motor B such that ii E = XiB, (0 — X) E = 0. A particular result of the next pro- position is that when a root of the '-N'^-\-N)E = (9). This is not always true if X corresponds to a repeated root of the (pi cubic. § 18] OCTONIONS AS FORMAL QUATERNIONS. 69 When X corresponds to a single root of the ^i cubic eq. (1) is true with the condition that SiMAA'MBB' is not zero by Prop. I. By this condition it follows that MiBB' is not zero and therefore that B and B' are two completely independent axial motors. But since {^ - JSf'' - N'4> + ]Sr)(4>-X)A =0, (^'-JSf'^ + N)(4>- X)I' = 0, i.e. ((j>' - N''-N'4> + N)B =0, which proves the first part of the proposition. That this is not always true when X corresponds to a repeated root (even when we change ^^ — N'(p + iV to (/)^ - (N' -D,y)E = — jSEi — nkSEj — D,iSEk. The cubic is <^' = so that N', iV and X are all zero and it will be found that (^ + ny4>) E=-n{(k + yj) SEi +jSEk}, so that in this case [(/)» - {N' -ny)(l> + {N+ D.y (X-N')]]E = 0, when E is any axial motor parallel to _;' but is not true for any other axial motor value of E, whatever be the value of y. Prop. III. If ac, is a repeated root of the (pi cubic, either (y ■ B=(pi [(pk = Q§", tf,k' = Q{^"-yB")]. ^j=A' FiQ. 5. Since a; is a repeated root of the i cubic we have by § 17 that the S^M and S^M' of the -x)E = -BS{A+ yA") E-B'S{A' + zA") E - n^"Sa"E, § 18] OCTONIONS AS FORMAL QUATERNIONS. 71 where a" is any rotor parallel and equal to MiA". Since A, A' and A" are completely independent axial motors so are A + yA", -4' + zA" and a". Denoting the first two by A and A' respectively we have {^-x)E=^-B^AE-B'^A'E-a^"Sa:'E (15), where Si4J.'a" is not zero. Expressing now the condition that Si.¥" = we have by equation (13) § 16 Hence either MiJS5' is zero or M^^^' is not zero and ^AA' and M55' are perpendicular to one another. In the first case we may put B' = z'B + H/S' whereupon chang- ing A + z'A! to A we get {^-x)E = - BSAE - n^'SaE - n^'Sa"E. Here we may take i coaxial with A and express a' and a" in terms of i, j, k. Doing this and collecting the terms in SiE, SjE, SkE we get an equation of the form (-x)E = - BSiE - a^'SjE - n^"SkE, (where of course the meanings of B, /3' and 0" are in general different from their meanings in the last equation). In this form we notice that ( - x) (i + DMpi) = B. Hence if i' is any unit rotor parallel to i, ((f) — x) i has the constant value B. i may therefore when B has a definite axis be taken to intersect that axis. Lastly j and k may be so chosen that the plane of i,j is parallel to B. We thus get equations (10) and (11). [Eq. (11) is put in such a form that B may be a lator.] When MiBB' and Mi.4^' are not zero but S,MAA'MBB' = 0, first in eq. (15) express fla" in terms of HA, HA' and D,MAA' (which can always be done since M^AA' is not zero), and incor- porate in the first two terms of eq. (15) the terms resulting from the first two components of D,a". The form of eq. (15) is not thereby altered but we may now assume a" to be perpendicular to A and A'. Now put eq. (15) in the form ((f>-x)E=- BSA,E - B'SA.'E - D,^,"Sa"E, 72 OCTONIONS. [§18 where A, = A + y£l! cubic we have further that the S^M" of the cj) —x cubic is zero, so that by equation (14) § 16 the further condition in the case of eq. (15) is S,(AB + A'B') = 0. In the case of eq. (10) this gives SiBi = or SiFcos^ = 0. In the case of eq. (12) it gives S^B'j = or S^Y' = 0. In this case B' in fig. 5 must be drawn parallel to k. § 18] OCTONIONS AS FORMAL QUATERNIONS. 73 Prop. IV. In the case of eq. (10) if i' is any unit rotor parallel to i, ((/> — x) i' has the constant value B. If E is any axial motor perpendicular to i, fAi(pE is equal and parallel to xfA^E. The first statement has been already incidentally proved. The second follows at once from the fact, obvious from eq. (10), that when E is perpendicular to i, ((f>— x)E is a lator. Prop. V. In the case of equation (12) if j', k' are unit rotors parallel to j and k, intersecting i at a distance y from the point of intersection of i, j, k; ((f)— x)j' has the constant value B' and (E is equal and parallel to xM^E. We have / = (1 + %i) j = j + ayk, k' = (l+ nyi) k = k- nyj, from which the statements about / and k' follow. If £' is a motor parallel to k, (^ —x) E is a, lator, from which the statement about E follows. [We saw in making the transformation Ai = A+ y^a", &c., above that y and y' were to a certain extent arbitrary. It will be found that the present statements about / and k' depend on this. The present y is the former — y'. The statements about / and k' are represented graphically in fig. 5.] From this proposition we see that j and k may be supposed to intersect i at any point which is convenient, for instance either at the point of intersection of i and B or that of i and B'. In the case of X thrice repeated the last is most convenient for we then have B' = Z'k. Other conditions might be satisfied, for instance /9" can always be made perpendicular to B'. By a process exactly similar to the proof of Prop. V. we may prove : — Prop. VI. If i', j', k' he unit rotors parallel to i, j, k and intersecting B of eq. (11) at a distance r from the point of inter- section of i, j, k ; in the case of eq. (10), {(f> — x)i' has the constant value B, (^ — x)f has the constant value H^', and ( — x) k' has the value n (/3" + Br sin 6). Here one condition can be satisfied by properly choosing the point of intersection of i, j, k on B; for instance /S" can always be made perpendicular to B. Prop. VII. If there is no axial motor E for which -x)E = - YjSEi - il^'SEj - D.^"6Ek, 6,i0 = 0, SJ/8" = (16), 74 OCTONIONS. [§ 18 where S^Y, Sji/S" aiid SJc^ are not zero ; or (^-x)E = -(Yj + Zk) SEi - Z'kSEj - D.^"SEk (17), where SjF, S^Z' and Sji'/S" are not zero. Conversely if these con- ditions are satisfied there is no such axial motor. Note that in this proposition ! cubic are all equal. Calling each of them X, (j) will be of one of the forms (10) or (12). In the case of eq. (10) we have seen that when ir is a thrice repeated root of the ^i cubic SjFcos ^ = 0. If 8iF= 0, £ becomes a lator fl/3 and in this case (<^ — a;) ^ is a lator function of E and bears to the rotor part of E exactly the same geometrical relations that a linear vector function of a vector bears to the vector. Hence in this case there is always an axial motor E of the kind mentioned (and any motor parallel to E is also of the same kind). If SjF is not zero we have the first of equations (16). We have seen (Prop. VI.) that in this case we may suppose S^j^" = 0. If now there is an axial motor E of the kind mentioned it cannot have a component parallel to i and therefore we may assume that it has the rotor form E = yj + zk + np. [Since ! cubic SiY' = and by taking, as we have seen we may, the point of intersection of i, j, k on B this becomes T' = 0. We thus have eq. (17) with the conditions SjF and S^Z' not zero (since M^BB' is not zero). In this case we see that if E is a motor of the kind desired, it must be parallel to k so that we may put E=k + np = k + il,(^i+rij+^k). Thus (^-x)E=n \I3" + ? ( Fj + Zk) + rjZ'k], so that the lator on the right is parallel to k. With the given conditions this can clearly be satisfied by real values of ^ and rj (which may be zero) if and only if Sii/8" = 0. In the case of eq. (10) x + il^/ (where y is arbitrary) is a root of the cubic. In the case of eq. (11) a; + D,y is not always a root. For by equation (12) § 16 the If of (^ - a; is in the case of eq. (10) zero but is not generally zero in the case of eq. (12) [see end of § 17 above]. The following statements are not of sufficient importance to embody in formal propositions. Both equations (10) and (12) are of the form ( — a; cubic 76 OCTONIONS. [§ 18 M=-nSBB'^", M'=- {SkBB' + nS^" (iE -jB)}, M" = - [SiB + SjB' + nSJfc/3"). In the case of eq. (10) these give if = 0, if' = - nF {S/3' (- i sin 6 + j cos 6) + S0'k cos 6], M" = -{- Fcos ^ + fiS ( ;/3' + Jfc/8")}. Hence the cubic is (<^ - a;) {((^ - a;)^ + [- 7 cos (9 + OS ( j/3' + W)] { - x) + nF[S/3' (i sin (9 - j cos 6) - S^'k cos ^]) = 0, which can generally be put in the form ((j)-x){-x-Ycose + n tan d Si/3'i = 0. The axial motor (when SjFcos d is not zero) E for which {-x- Y cos e + ntan e Si^'} E = 0, is {(f) - x) {(p - X - a [S/9' (i tan 6 -j) - S^"k]] i. E is however written down more simply by noticing that if {-x)E = - BSE A - n^'SEa - D,^"SEct", then (-x)B = - BSAB - il^'Sa'B - fl/9"Sa"£, from which it is obvious that if we again operate by (-xfB=- {SAB + D.SAeS-^AB) { - x) B. Hence in our present case the desired axial motor E is {— x) B, i.e. £ = - BSiB - n/3'Sj5 - D,^'SkB. The splitting of the cubic into three linear factors fails generally when a; is a thrice repeated root, i.e. when ^ = „ • [a; is a thrice repeated root if SiF=0 but in this case the above linear factor form of the cubic does not fail.] If however (§ 17) M' is zero we know that there are three linear factors, ilf ' = gives in the present case S^'i = and then the above form of the cubic still holds and takes the simplified form ( -x) + nS {j^' + k^")\ = 0. §19] OCTONIONS AS FORMAL QUATERNIONS. 77 For eq. (12) we have M = -n{YZ'- Y'Z) Sir, M' = nS/S" (iZ +jZ' - kT), M" = Y'-aSk/3", so that the cubic is (<^ - xy +{(f>- xy (- F + nsk/3") + {(f>-oi;) nS/3" (iZ +jZ' -kY') + n(YZ'- Y'Z) Si0" = 0, which may when x is not thrice repeated be put in the form [^-a,-Y' + nZ'Y'-' S/3" ( jF' + iY)} {((^ - xy -(- x)nY'-' Sy8" (iYZ +jY'Z' - kT') - SlY'-^YZ' - Y'Z) S/3"i} = 0. The axial motor which is reduced to zero by the first factor is E = {^-x)B'- BnZ'Y'-^Si^" = Y'B' + D,Z' (/3" - BY'-'Si^"). li X is a thrice repeated root of the <^i cubic S^M" = or S] F' = and the above factorising of the cubic fails. In this case in order that it may be possible to find one linear factor, M must be zero (§ 17), i.e. Sii/3" = 0. In this case we may put ^" = b"j+c"k and the cubic is (

-x)+n (c"Y' - h"Z')} = o. In order that there may be three factors M' must be zero (§17), i.e. 8i (b"Z' - c"Y') = 0. In this case the cubic is (-x-(Y' + nc")}^0. These two factorisations are true when SJ.^" = and when Sii(S" = and S,(6"Z'- c"F') = respectively, whether or not SiF' = 0. If the last be true, however, we may write -Q,b"Z' instead of fl (c"F' - h"Z') in the first of the two cases. 19. The self-conjugate commutative flinction. We proceed to show that when is self-conjugate it can always be put in the form 4,E = - XiSiE - X'jSjE - X"kSkE (1). 78 OCTONIONS. [§ 19 [In § 38 below a proof of this is given which is independent of § 18. The reader who chooses can at once pass to that proof.] In the first place when is self-conjugate the cases of equa- tions (16) and (17) of § 18 do not occur. For by equations (3), (4) § 16, when is self-conjugate = Mf^f = M (ii +j<}>j + ki = X,i + Y,j + YA <}>j=Y:i + X,j+Y,k, (f>k=Y,'i+Y,'j + Xsk. Making self-conjugate we get F/ = Fi, Y^' = F^, Y^' = Yg, and making (jsi coaxial with i we get Y,= F3 = 0.] Now Z' can always be determined so that <^ (j + Z'k) is coaxial vfith j+Z'k. For ^ (j +^Z'k) = (Y + Z' W)j + (Z'Z -t- F) k, and this is coaxial with j ■\- Zk if Z'^W-^Z' (J-Z)- F=0 (2), which gives Z' = ^-W-^ [Z- F+ \(Z - Yy -)- 4Tf ^]i). By eq. (8) § 4 above this gives a real finite value for Z' except when Tf is a converter. If F" is a convertor eq. (2) is satisfied by Z' = W/(Y-Z) unless F- .^ is also a convertor. If both W and Y— Z are convertors, say D,w and D,y where w and y are ordinary scalars, eq. (2) is satisfied by Z' = iw-' {-y + [y^ + 4>w'']i], unless w = 0. But when w = 0, W=0 and both (j>j and is self-conjugate there are always three real roots of its cubic. In § 17 we saw that if any two, say SjZ and SjZ' of the ordinary scalar parts of X, X', X" were equal, the cubic had an infinite number of roots of the form X + fly. We will now show that if for any axial motor E E = YE, Y must have one of the values X, X' or X" of eq. (1). If is given even when the roots of the (f) cubic are indeterminate. When the roots are indeterminate X, X' and X" may be called the principal roots. It may be noticed in passing that by the beginning of § 17 above the fact that it is always possible to put a self-conjugate commutative function in the real form (1) involves the following quaternion theorem : — If ^i and tp^ are two self-conjugate linear vector functions of vectors it is always possible to determine a real vector p, in general uniquely so that the principal directions of ^1 and 02 + 4'i^P ()~ ^P4'i ( ) are the same. Quaternion analogy suggests the examination of the self- coniugate given by (f>E=MAEB (4). We have at once MAB = -MABSAB (5), {UA ± UB)= + TATB{UA ± UB) (6). 80 OCTONIONS. [§19 Hence the principal axes are the shortest distance of A and B and the two lines bisecting the shortest distance perpendicularly and bisecting the directions of A and B. The principal roots of the cubic are — SAB and + TATB, i.e. the cubic is {^^-A'B^)((f> + SAB) = (7). It is natural now to enquire whether every self-conjugate can be put in the form in the form (8), viz. when two but not three of the roots of the ^ cubic are equal and when the corresponding principal roots of the ^ cubic are unequal. If S^X = SjX' and these are not equal to S-^X", X-X' = TATB + SAB = TATB {l + S\^A UB). Hence T^AT^B + S^AB is zero or A and B are parallel. Hence by eq. (10) § 12 SU^U5= ± 1. The lower sign must be taken since Sj {X — X') = 0. Thus X — X' = or the transformation is impossible. Similarly the transformation is impossible when SiX = SiX" and these are not equal to S^X and X' and X" are unequal. § 20] OCTONIONS AS FORMAL QUATERNIONS. 81 In all other cases (j> can be put in the form (8). It will be found that - XiSEi - X'jSEj - X"kSEk = -M (Zi + Z'k) E (Zi - Z'lc) + YE. . .(10), if 2Z^ = X-X', 2Z'^ = X'-X", 2Y=X + X" (11), and these equations give real finite values for Z, Z\ Y when S,Z, SiZ' and S,Z" are all different and in descending order If - Xi^Ei - Xj^Ej - X'kSEk = -D,M (zi + z'k)E(zi-z'k) + YE.. .(12), if 2nz' = X-X', 2nz'' = X'-X", 2Y=X + X"...(13), and these equations give real finite values for z, z, Y when sA", %X' and %X" are in descending order. Lastly - Xi^Ei - X'jSEj - X'kSEk = \{X'-X)iEi + J {X' + X)E...{1^). After the general linear function has been considered below several other theorems relating to commutative fuuctions and especially self-conjugate commutative functions will be enunciated. For the present we leave this part of the subject. 20. Differentiation of octonion functions. We have not in the above considered all the different kinds of octonion func- tions that are even immediately suggested by quaternion analogy. For instance, we have not entered into interpretations of Q", where both Q and R are octonions, or of log Q, or of the simpler concep- tion J.", where A is any motor and n any scalar not necessarily positive and integral. In this section we propose to provide materials for writing down the diff'erential of any combination of octonions which can be made of such functions as have been considered. In this section explicit references will not be made to § 9 above. The following are obvious dSQ = SdQ, dS,Q = SM, dS,Q = S4Q, dsQ = sdQ...(l). M. o. 6 82 OCTONIONS. [| 20 Also from the fact that octonions are formal quaternions, dMQ = MdQ (2), dKQ = KdQ (3), dUQIUQ = M.dQ/Q (4), dTQITQ = S.dQIQ (5). Putting in the last TQ = T^Q (1 + HtQ) we obtain dJ,Q/T,Q = S,.dQIQ (6), dtQ=^s.dQIQ, dT,Q = S,.dQ/Q (7). Since dM,Q = ndmQ, dU,Q = nduQ (8), it only remains to find the differentials of MjQ, mQ, U^Q and uQ. From equation (2), Put now MdQIMQ = dR (9), so that dR is the octonion which, when viewed as an operator, changes the motor MQ into the motor MdQ and therefore inter- sects both Q and dQ perpendicularly. Thus dMQ = SdR . MQ + MdR . MQ, and therefore Q + dQ=Q-+ (SdQ + SdR . MQ) + MdR . MQ] = {l+^dR){Q + d'Q)(l + ^dR)-' j ^' where d'Q = SdQ + SdR .MQ (11). Thus Q + dQ is obtained from Q by first adding the coaxial increment d'Q and then displacing Q + d'Q as a rigid body. Denote the increments of the various functions of Q, such as MjQ, qQ due to the increment d'Q of Q, by the symbol d'. Since all these increments are coaxial with Q we have I 20] OCTONIONS AS FORMAL QUATERNIONS. 83 d'M,Q = M4'Q = S4R.M,Q (12), d'q^ =S,d'Q + M,d'Q = S4Q + S4R.MiQ ...(13), dV,Q = Md'q^q^-Kyj,Q I = Tr^ QM,Q (- S4Q + S,QS,dR) U.Qj ' '^ '' [by putting q^-' = Tr'Q^qQ = Tr'Q (S,Q - M,Q)] d'mQ = md'Q = SjcZJ? . mQ + sdiJ . M,Q (15), d\ = sc^'Q + md'Q = sc^Q + S,dR . mQ + sc^E . M,Q...(1C), d' (tQ + uQ) = d' (r^^g-O = 9(2-' ((^V<2 - TQqg-'d'qQ) [since qq,rQ, d'qq and dV^ are all coaxial] = Tr^Q (S,Q - M,Q) {dV,j - (tQ + uQ) (^'^^j, or d' (tQ + uQ) = Tr'Q (SiQ - M,Q) (scZQ + S,(^E . mQ + sdRM,Q-{\Q + \iQ){S4Q + S4RM,q)] (17). Hence utilising equation (10), dM,Q = ^(dRM,Q-M,QdR) + M,QSM (18), dmQ = i (dRmQ - mQdR) + mQS.dR + M.QsdR (19), d\J,Q = WR\J,Q-\J,QdR) + Tr'QM,QS,(dRSQ-dQ).U,Qi20), duQ =^(dRuQ-uQdR) + Tr'QM . (S,Q - M,Q) {sdQ + mQSM + M,QsdR - (tQ + uQ) {S4Q + M,QS4R)} = ^{dRviQ-\iQdR) + Ml . Q-' {sdQ + mQSdR + MQsdR - (tQ + uQ) {SdQ + MQSdR)] dqq =^(dRqQ-qQdR) + S4Q+M,QS4R (22), drg = i {dRvQ - VQdR) + SdQ + mQS^dR + M,QsdR (23). Equations (18), (19), (22) and (23) may also be written dM,Q = {M + S,)dR.M,Q (24), dmQ=(M + S,)dR.mQ + M,QsdR (25), 6—2 ■(21), 84 OCTONIONS. [§ 20 dqQ = (M + S,)dRM,Q + S4Q (26), dr^ = (M + SO dR.mQ + M.QsdR + sdQ (27). The differential of Q", where n is any scalar constant, can be written down from the corresponding quaternion case (Proc. R. S. E. 1888-89, p. 201), d.Qr = n(^-^dQ+l{W-'Q-nq^-m-'Qr){Q'dQ-dQ.Q^)\ = 7iQ"-idQ+ (MQ« - wQ«-iMQ) MM-^QMc^Q Also if ^ be a motor d . A-^ = nA'^SA-^dA +IAA'^ MA-MA (29), and therefore de^ = Ae^SA-'dA ^lAe^ MA-'dA (30). CHAPTER HI. ADAPTATION TO PHYSICAL APPLICATIONS. 21. Meaning^ and properties of V in Octonions. We have for the sake of more readily utilising quaternion analogy altered the geometrical significance of many symbols which occur in Quaternions. For the same reason we now propose to take the same liberty with V. It is first necessary to be precise in the meanings we shall in our physical applications attach to the symbols i, j, k and ^. Let be some fixed point and P a variable point, i, j, k will be supposed to be three mutually perpendicular unit rotors inter- secting in P, and their directions will be supposed fixed. The values of i, j, k at 0, which will be called the origin, will be denoted by t'o, jo, ^o- Thus i, j, k are not constant rotors but i„, j„, h are. On the other hand i, j, k are independent in mean- ing of an arbitrary origin, whereas i^, jo, K are not. The rotor OF will be denoted by p = i^x +joy + koZ = ix+jy + kz (1). Thus X, y, z have their ordinary Cartesian meanings. The symbolic rotor V is defined by the equation SJ = vdldx+jdldy + kdldz (2), and V„ will denote what may be called the value of V at the origin, i.e. iod/dx+ .... Thus V is a symbolic rotor which in so far as it is a rotor is a function of the position of a point and is inde- pendent in meaning of an arbitrary origin; whereas V„ is a syrabolic rotor which in so far as it is a rotor is a constant depend- ing in meaning on the selection of an arbitrary origin. ? will still be defined by the equation 86 OCTONIONS. [§ 21 where "^{A, B) is any octonion function of two motors A and B which is linear in each, ^i will be used for the value of ^ at the origin. The rotor element of a curve will be denoted by d\ and the rotor element of a surface by dt; that is to say, if P and Q are two near points on a curve d\ = PQ and if P, Q, R are three, not coUinear, near points on a surface d2 will denote the rotor iMPy . PR which passes through P, is normal to the surface at P and equal in tensor to the element PQR of surface. [More strictly, i.e. taking account of the third order of small quantities, dl, should be defined as the rotor equal and parallel to ^MPQ.PR which passes through the centroid of the triangle PQR, i.e. 6dt = M (PQQR + QRRP + RPPQ).] The usual conventions as to positive directions will be adopted when the boundary of a surface is compared with the surface and when the boundary of a volume is compared with the volume ; i.e. in the first case d\ will be in the direction of positive rotation round a proximate dS, and in the second case dX will point away from the volume bounded. With these definitions we have [eq. (1) § 7 above] i = io + nMpio, V=V„+ilM,cV„,| ^=^o+^Mp?o, dX = dp + nMpdp\ *■ ''■ Thus dp might for some purposes be appropriately denoted by rfXo- It will be observed that the properties of our present *0) Jot «''0) Pi V(|, ^0, are practically identical with those of the corresponding qua- ternion symbols i, j, k, p, V, f, but it is obviously inappropriate in Octonions, on account of the arbitrary origin, to denote them by the latter symbols. With the present notation SdXV = SdpV,= -d (.5). There is a peculiarity in the use of the present V which must be carefully attended to. It is well illustrated by the statement § 21] ADAPTATION TO PHYSICAL APPLICATIONS. 87 that SV (MV^), where E is any motor function of the position of a point, is not in general zero. At first sight this statement appears absurd, for V is a motor operator and hence apparently SV (MV^) = SV (V^") = 0. The last statement (SV(V£') = 0) is not in general true because the present V unlike the quaternion V is a function of the position of a point and therefore is itself subject to space differen- tiations. Thus in the expression SV (V^) the left V has two operands, viz. E and the right V. We cannot then treat V as a mere rotor in the same way as we may treat the quaternion V as a mere vector, but we can do something very similar. To enable us to symbolise the process here referred to we must indicate the operator and its operand in a rather peculiar way. Since a V may be either an operator or an operand, or as with the second V of SV (V^) simultaneously the one and the other, it is necessary to indicate what is its character by a system of suffixes. We shall indicate, when necessary, an operator by a capital letter suffix and its operand by the corresponding small letter suffix. Thus SV(V£:) = S(V^ + V^)V„5£'6, in which Vj, and V^^ are operators both having E,, for operand and where V^ is an operator having V^^ for operand. Thus SV (V^) = SVj^j^aEi + S'^^sEb Here the suffixes might be removed if the usual convention be adopted that when not otherwise indicated the operator of a V is the single symbol immediately succeeding it. Thus SV(MV£') = SV(V£') = SVVi; (6), which is not a truism as may be illustrated by the more general equation SV(VQ) = SVVQ + V^^SQ„ (7), where Q is any octonion function of the position of a point. It will be noticed that V^ has not been used in the above. Adopting the ordinary convention by which for instance 88 OCTONIONS. [§ 21 V» should be defined by the equation V=Q = V(VQ) (8). With this meaning V= is not in general a scalar operator, though the square of a rotor is an ordinary scalar. ^-^Qa or more generally V"^Qa where w is a positive integer, is rather a cumbrous expression for the thing signified. It is easy to see that V%Q<, = V„»Q[?ieven] V%Qa = VV„«-^Q[7iodd]| ^^^' but this alternative method of indication is if anything still more objectionable. I shall therefore use the notation defined by V%Q. = >"Q (10). We have not yet found the meaning of the expression SVV.S or SVjV aB^b- Let i/r have the meaning just now given to it. Required the meaning of where V„ has some operand which however it is not necessary here to indicate. Although V cannot be treated as a constant rotor, V,, can. Thus by eq. (4) the only variable part of V is flM/3V„. Thus = ^{^„ + nMp^„ nMr„v„), or ^(V^, V,)=t(r, OM^V) (11), where V on the right has the same operand as V^, on the left. As a particular result we have V2 = (v^ + v^) v<,5 = orMf V + 1>^ or V==-2nV + t>2 (12). Hence SV^E = -2nSVE (13). 22. Line-surface integral and surface-volume integral. To lead up to the octonion theorems corresponding to the well- known quaternion integration theorems, first notice that for a closed curve and a closed surface respectively we have jd\ = 2njjd-^ (1), Jld^=0 (2). § 22] ADAPTATION TO PHYSICAL APPLICATIONS. 89 The truth of these are easily seen by their physical meanings. The first asserts that the system of forces which is represented by the sides of any closed polygon taken in order is equivalent to twice the system of couples represented by the area of the polygon — i.e. what would in Quaternions be called the vector area and what here must be called the lator area. The second asserts that the system of forces represented by a uniform hydrostatic pressure on a closed surface is in equilibrium. It is easy to furnish octonion proofs of both statements. It is only necessary to prove equation (1) for a triangle and equation (2) for a tetrahedron, since a finite surface or volume can be split up into a series of elementary triangles or tetrahedra. Let the triangle be OPQ. Then FO + OQ is the rotor through equal and parallel to PQ, i.e. PO + OQ is obtained by translating PQ along the rotor PO or [eq. (1) § 7] PO+OQ = PQ + DMPOPQ, or PQ + QO + OP = VMOPPQ, which proves the property for the triangle. Next let pi, p-i... be the rotor perpendiculars from a fixed point on a series of rotors equal and parallel to mA, m^i^... where \ is a unit rotor through 0. The sum of the rotors is by eq. (1) § 7 above 2m (1 + rip) io = (1 + ^p) i^m, where p is defined by the equation p'S.m = l,mp. Interpreting this sum by aid of equation (1) § 7 it follows that a rotor plane area is a rotor whose tensor is the area and which passes normally to the area through its centroid. If then a, /9, 7 be three of the rotor edges of a tetrahedron all starting from one angular point, the rotor areas of the faces are - iM/S7 - ^OM (/3 + 7) M/37, -^Mya- iOM (7 + a) M7a, - ^Ma/S - inM (a + yS) Ma/S, and iM (^y + 7a + a/S) + ^HM (a + /3 + 7) M (^7 + 7a + a^X the sum of which is easily seen to be zero (since MaM^ry + M^M7a + MjMa^ = 0). 90 OCTONIONS. [§22 Let now (f) be any linear octonion function of an octonion. Then Jcj^dX = fj{2ct> (nd%) + JMdXV^] (3), //.^ that the vector ^/V^ has with the coordinates of the linear vector function (/>'. The last statement is seen at once from the equation ^^a^ A = ^<» coupled with the similarity of lators to vectors, and of a linear lator function of a rotor to a linear vector function of a vector. To prove that r = M^(^^ we have merely to prove that (^aV^ — flM^<^f is a rotor through P. Now by equation (4), [since, as it is easy to prove, -i/r (4, V^) = 11^(11^1 ^^', ^)]. Now Si^dl, is an ordinary scalar. It follows that si {4>a'^A — ^^a^A — ilMf^^" is a rotor 92 OCTONIONS. [§ 23 through P. Putting ^ = ^M ( ) o- the above statements about the spin follow. 23. Strain. In treating of strain it is convenient to utilise the arbitrary origin 0, though as might be expected many of the results are independent of it. Let Q be a point near to P (fig. 6) and let OP = p, OQ = p + dp, PQ = dX (1). Suppose in a general displacement that P moves to P' and Q to Q'. Put PP'=V, QQ' = V + dv (2), so that 7j and r] + dr) are the rotor displacements of P and Q. Also put 6P' = p', OQ' = p' + dp', TQ' = d\' (3). Let Oq and PQ be equal and parallel, and again let Oq', PQ" and P'Q' be equal and parallel. Mmmmmmm, Fig. 6. Thus Oq = dp, PQ = d\ = dp + DMpdp •(^X Oq' = dp', PQ" = dp' + nMpdp' FQ' = d\' = dp' + DMp'dp' = PQ' + QM-qPQ w\ ■■■<''■ § 23] ADAPTATION TO PHYSICAL APPLICATIONS. 93 V = p' - p + nMpp' (6), dp' = -Sd\V .p' (7), d\' = - {SdW . p' + D.Mp' (SdV^ . p')} (8). A being any motor define ^ix %p, % by the first of each of the following three sets of equations — XoA^-SA-^ .p', dp' = xodX (9), XpA = XoA + nMpx«A, PW^Xpd^ (10). XA = XoA + DMp'x^A = Xp-^ + ^^VXp^' dX'=x'i^---(^'^)- The rest of the equations are easily seen to follow from the defining equations. From equations (10) and (11) it is obvious that x ^nd Xp ^-re independent in meaning of the origin. In fact XpA = A + nMrjA - SAV .r, (12), which may be proved from the above results or in a manner not involving the origin as follows. — Expressing the fact PQ + QQ' + Q'P' + P'P = twice the lator area of the quadrilateral PQQ'F', we get dr)+d\- d\' = flM (d\ + d\') ■q, whence putting d\' = %dX = Xp^^ + ^^VXp^^ ^'^^ di] = — Sd\V . -q we get Xpd'>^ = d\ + DMrjdX — ScZXV . tj, which proves eq. (12) for A = any rotor dX through P. Operat- ing by n and assuming that Xp i^ a commutative function eq. (12) is true for A =any lator. Adding these results eq. (12) is true for A = any motor. From the second of equations (10) it follows that Xp is a pencil function (§ 15 above) with centre P. Hence by Tait's Quaternions, 3rd ed., § 381, we may put XpA=q^A Xa "^® have by equations (13), (14), (16) r = XX = Xp'Xp = Xo'Xo = '^^S ( ) V^Spa'pb = ^. . .(18), say. From eq. (12) and eq. (18) '^ may be expressed in terms of r] and its derivatives. Eq. (12) may be written XpA = - (1 + ifiT?) S^ (V^ + f) . (,;„ - iiMnva + 0(1 + h^v)-' (19). Utilising eq. (12) directly From either eq. (19) or eq. (20) we have ^A = (V^ + ?0 SA (V^ + ?,) S (,,„ - nMvva + ?0 (Vi-nMvVi + Q (21). In expanding the expressions on the right of equations (19) and (21) it is understood that any term in which a f occurs without the corresponding ^, is put zero. It will be noticed from equations (18) and (21) that ^ is the same function of V^ + ?i , Va- ^MriTja + ?i , ^B+ ?2, Vb- ^Mrjrjj, + f^, as it is of ^A! Pa, ^s, pi- In the present case the expression depending on the arbitrary origin is considerably simpler than the other. In the case of small strain we may put %p=1+(Xp). Xi''=l + (%p)'. where (%p) and (%p)' are small and conjugate to one another. Thus ^ = %p'%p=1+(Xp) + (Xp)', or from eq. (12) {^-\)A = 2{yir-\)A=-^^SAr),-y^,SAV^...{22), so that in the case of small strain the expression independent of the arbitrary origin is quite simple. § 24] ADAPTATION TO PHYSICAL APPLICATIONS. 95 24. Strain (continued). We propose now to consider the straining of a volume element, the straining of a surface element, and differentiation with regard to the coordinates of strain. These are all now so similar to the corresponding quaternion considerations that they have been marked off by being put in a separate section. The m [§ 3] or the M [eq. (9) § 16] of % has the usual physical significance, viz. the ordinary scalar ratio of the strained element of volume at P' to the corresponding unstrained element at P. This is seen at once by the original definition [§ 3] of m for the \, fi, V of that definition are ordinary motors. Taking them to be three elementary rotors through the point P, — SXfiv will be the element of volume at P (contained by a parallelepiped of which X, /x, V are adjacent edges) and — S)(\')(^fj,)^v will be the correspond- ing element at P'. Change X into dX and Mfiv into dl,, 'xX into dx' and M^f^X" into dX' so that dX, dX' have the meanings hitherto given and d'Z, d%' are the unstrained and strained rotor values of an element of surface. Thus mSdXdt = SdX'dX' = SdXx'dX'. Since dX is an arbitrary rotor through P it follows [statement (1) of § 14 above] that md'Z = xdl,' or d2' = mx'-ic^2 (1), exactly as in Quaternions, though this result has a more general meaning than the quaternion one. In connection with this last equation it may be noticed that we have as in Quaternions [§ 3 above and Utility of Quat. in Phys. eq.(6e)§3a] =-4Mxr,%?.sd2r.r2 = - i (1 + fi^) M {r)a - i^MrjVa + ^i) (Vb - ^^^h + ?=) • (1 - ^r,) xSciS(V^ + ?0(VB + ?.)i (2), which may also be put into a number of other forms. In many important physical applications a pencil function with a definite centre is an independent variable. Differentiation with regard to the nine independent scalars involved may be 96 OCTONIONS. [§ 24 treated in a manner practically identical with the same treatment in Quaternions [Utility, §§ 1, 5]. Thus for instance the potential energy of strain may be considered a function of '^ and it might be required to find the stress in terms of such a function. This will involve differentiation of the type mentioned. Let i, j, k have the usual meanings with regard to the centre of ^. Then the six ordinary scalars a, b, c, e, f, g, where ■^^ = - eiSiA -fjSjA - gkSkA - 1 {a (jSkA + kSjA) + h {kSiA + iSkA) + c {iSjA +jSiA)] (3), are independent variables and completely specify ^. Define (7 (or ' is a rotor through P'. Hence E' is a motor through P' whose pitch is p. Similarly when P is a motor through P. From these statements and equations (1) and (2) it follows til at x'~' and m"'x reduce any rotor through P to a rotor through P'. This is otherwise obvious from the equations XA =xp^ + AMt^xp^, X'^A = Xp'"'^ + fiM7;xp'-'^...(3), and from remembering that Xf i^ a pencil function with centre P and that r) = PP". If V have the same meaning with regard to the strained space as V has with regard to the unstrained space Sd\V=Sd\'V' (4) [eq. (5) § 21], so that V is a symbolic rotor intensity. If E^.Ei are intensities ME^Ei is a flux. If E is an intensity M^E + 2nE is a flux. This statement it will be noticed is not quite identical with the corresponding quaternion statement. It might be thought 7 M. O. 98 OCTONIONS. [§ 25 that it was inconsistent with the two preceding statements. The explanation is that E in MV£ is not in all respects analogous to E^ in M^i^2- To be more precise, ^^^'y^'^^a bears the same relation to MV^ as F' to F (and this is all that can be deduced from the facts that V and E are intensities and MEiE^ is a flux) whereas MV^' or MV {x'~''E) does not because of the differentia- tions of x'~^- The reason that the same sort of thing does not occur in Quaternions is that the quaternion expression is identically zero, whereas the octonion expression is not. The proof of these statements is obtained by noticing that in the quaternion case X-'dp' = dp=-Sdp'V'.p, and therefore ;)^'-i = — V/S ( ) pi , whereas in the octonion case X~^dk' = dX=dp + ilMpdp, = - SdX'V . p + nMpx-'d\', and therefore x~' = " ^a'^ ( )Pa+ ^x'-'M ( ) p. From this it does not follow that MV ^'xa~^E = but by eq. (12) §21 MV^'Xa'-'E = n {IV^'SEpa + M . V^'x„'->M^p„) (5). We can however deduce a result of a different form fron. (5), a result independent of the arbitrary origin, from the facts that V and E are intensities and ME^E^ and MVE ■+ 2ilE are fluxes. For from these we have M V (x'-'E) + 2D,x'-'E = m-'x (MV^ + 2nE), and MVyx'~'Ea = wr^lAVE, from which by subtraction MV^'X;-'^ = 212 {m-'x - X-') E (6). By a similar process or by observing the reciprocal relations between E' and E we obtain MV^;V;,'^' = 2n(mx--x')^' 0). These two results are not independent. The second can easily be obtained from the first by substituting for E and V in terms of E' and V. § 25] ADAPTATION TO PHYSICAL APPLICATIONS. 99 It is to be observed that since in equations (5), (6), (7), E and E' are perfectly arbitrary motors, these equations are not equations involving intensities, but differential identities satisfied by the strain function 'x_. Before going further we will deduce some more relations of the same type. From the symmetrical relations between V, p, •y^ and V, p, x~^ we can write down a similar equation to (5), viz. MV^Xa'E' = n (2V^S^ V + M . V^Xa'f^J^'pa) (8). Two other relations may be written down by expressing the equa- tion fdX = 20//dS in terms of the dashed letters and the equation fdX' = 2njjdl,' in terms of the undashed letters. They are Xa-'ME'V^' = 2n (m- x^E'-x-'E') (9), Xa^EV^ = 2D.{mx^-^E-xE) (10). Each of these may be deduced from the other. Eliminating the expressions from the right of equations (6) and (10) and expressing V^' in terms of V^^ and E in terms of E', we have XaMV^X/^' = (11). Similarly X^"^^ a X'^" ^ = ^ (12). Eq. (11) may be verified directly by expressing x i^i terms of Xa by eq. (11) § 23 and putting Xo = - S ( ) V . p' by eq. (9) § 23. Two other results may be obtained by expressing the equation //dS = in terms of the dashed letters and ^^d^' = in terms of the undashed letters. They are m»-'xa'v; = o (13), m.Xa'-'V^ = (14), each of which may be derived from the other. Of the nine equa- tions x'V = V, (6), (7), (9), (10), (11), (12), (13), (14) only four are independent. They may be taken as x'^' = ^. 0)> (H)' (!*)• ^^ we generalise the meaning of "spin" so as to call MVG'-|-2n(T the "spin" of the motor G, eq. (7) asserts that if E' is any constant motor the "spin" of xE' is the lator 2£lmx~'^ E' . Also the statement that MV^-|- 2n^ is a flux may be put:— the spin of an intensity is a flux. A curious property of some of these equations with regard to " dimensions " may be noticed [see Hamilton's Elements of Quaternions, § 347 (6) et seq.]. All the corresponding quaternion formulae are consistent with the assumption that x is of arbitrary 7—2 100 OCTONIONS. [§ 25 homogeneous dimensions but some of the present formulae are not. They are only consistent in the matter of dimensions if x ^^ assumed of zero order in dimensions. This need not surprise us since the original physical definition necessarily implied this. Equation (7) may be taken as a typical example. If x ^^ °^ n dimensions, m is of Sn, and therefore mx~^ is of 2n while x' ^ of n. There is a way other than that of regarding x as of zero order in dimensions of avoiding the anomaly. We may assume x to be of arbitrary homogeneous dimensions, but then we must not assume that JdX' = 2D,JJdl' where fl has the same meaning as in the equation JdK = 2D,JJdX. Instead we may put fd\' = 2n'JJdX, where D,' is similar to il when it is in combination with itself and ordinary scalars but is of different dimensions from 12. The relation between the dimensions of fl and D,' is the same as the relation between the dimensions of the O and fi' of Phil. Trans. 1892 A, p. 691. Indeed [ibid. eq. (9)] eq. (7) above takes the con- sistent form and the right would be identically zero if the relations between fl and Of were precisely those of that paper. This apparent anomaly in " dimensions " is ultimately due to the fact that fl, though a constant octonion, is not of order zero in dimensions, but of the same dimensions as the reciprocal of a length. If p' be assumed of arbitrary homogeneous dimensions different from p, 7} cannot be of homogeneous dimensions and therefore x cannot. This may- be illustrated by taking p'=cp, where c is a constant ordinary scalar, when the particular forms assumed by all the above equations can easily be written down. In the expressions for 77 and %, c"' will be found to occur so that if c be not of zero dimensions 7; and x cannot be assumed homogeneous in dimensions. We now return to the consideration of fluxes and intensities. If E and F be an intensity and flux respectively SEFd^=SE'F'd^', § 26] ADAPTATION TO PHYSICAL APPLICATIONS. 101 where d? and d<;' are corresponding unstrained and strained elements of volume. A somewhat interesting result may be deduced from this. Suppose a, a^ are two arbitrary rotor intensities passing thnjugh F and t, t, are two arbitrary rotor fluxes passing through F- Define the motors 0, G', H, H' by the equations O^a + ^T, G' = a' + fir' (15), H=T^ + D,(T„ H' = T,'+ila,' (16). If we wish to give G and H names, G may be called an intensity- flux and H a flux-intensity. By the result just given we have 6d^ is the force due to stress exerted on any region at the rotor element dl, of its boundary. The whole force motor exerted on the region by the rest of matter is by eq. (4) § 22, and therefore the force motor exerted on the element d? is ^a^^d?. We saw in § 22 that the motor ^ through the point considered and a lator ClM^^^, and that the rotor bore to the stress function . We see then that as in quaternions the effect of a stress on the element is the force <^aV^— HM^c^f through the element and the couple OMf^f . In particular the necessary and sufficient condition that there should be no molecular couple due to stress is that (p should be self-conjugate. And further if (/> is any self-conjugate pencil function which is a function of the position of a point and has the point for centre, <^o^^ is a rotor through the point. It is now evident that the discussion of the connection between stress and strain can be carried on by means of Octonions on lines practically identical with the quaternion ones. CHAPTER IV. MOTORS AS MAGNITUDES OF THE FIKST ORDER IN THE AUSDEHN UNGSLEHRE. 27. Can reciprocal motors be regarded as normal magnitudes of the first order ? Just as, hitherto, we have freely drawn on quaternion theorems, we shall now utilise parts of Grassmann's Ausdehnungslehre. The references will always be to the 1862 (Berlin) edition. [These references are equally to " Hermann Grassmanns gesammelte Mathematische und Physika- lische Werke. Ersten Bandes Zweiter Theil : Die Ausdehnungs- lehre von 1862." Leipzig, 1896. Both §§ and pages of the 1862 edition are indicated in the reprint.] References will also be frequently made to Sir Robert Ball's Theory of Screws, 1876 (Dublin). I shall for the future refer to these two treatises as ' Ausd! and ' Screws ' respectively. In a note on p. 182 of Screws the author seems to imply that in the Aiosd., if two motors A and B are " normal " in Grassmann's sense they would be "reciprocal" in Ball's sense, i.e. sAB would be zero. Now as far as Grassmann's own geometrical applications are concerned this is not the case. Grassmann says (Ausd. § 330) " Fiir die innere Multiplikation nehme ich als urspriingliche Einheiten im Raume stets drei zu einander senkrechte und gleich lange Strecken, (e^, e.^, e^." Thus he deliberately refuses to apply his theorems concerning inner multiplication and therefore in particular concerning normals to the more general system of primitive units E, gj, e^, 63, where E is a simple point which he introduces in § 216. And the reason for his alteration is not far to seek. Taking the gross complex (Hauptgebiet) as that of E, e^, e,, e-,, we see by § 89 that the com- plements of [Ee^], [Ee.^, [Ee^], [e^^], [eaej, [e.e^} would be [e^ej, 104 OCTONIONS. [§ 27 [6361], [6162], [j^Ci], [Ee^], [^63] respectively. Now from the geome- trical interpretations which Grassmann gives in § 239 et seq. we see that with our notation we must identify [-fc'ei], [-Ee^], [^63] with i, j, k respectively and [6263], [eaej, [eiSj] with fli, D,j, D,k respectively, where i, j, k are the unit rotors through E parallel to 61, e.2, e, respectively. Heuce i and Hi are complementary. But a definite point is required for the specification of the system i, j, k and no such point for Hi, Dj, D.k. Hence complementariness in this meaning does not treat space impartially. This indeed is pointed out by Grassmann himself [note to § 337 of Ausd.]. It does not follow from this however that all processes such as that of inner multiplication which are based on the idea of com- plements are similarly unsymmetrical. This is well illustrated by Grassmann's theorems concerning planimetric and stereometric multiplication (§ 288 et seq.). As a matter of fact — sAB is what Grassmann would call the stereometric product of the two motors A and B ; but this has nothing to do with " normals." Let us examine then the meaning of inner multiplication of motors. Denoting {Amd. § 89) the complement by a vertical line (so that \i = Q,i, \D,i = i, &c.) the inner product of two motors A and B is defined as [A \ B], and by § 142 we have [i\i] = i, [i\j] = o, [i\ni]=o, [ni\ni] = i, &c. Now the value of si' is zero and the value of siD,i is not zero. Hence the inner product of two motors cannot be identified with + sAB and in particular two motors are not in general " normal " if they are " reciprocal," and conversely. [This of course does not prove that the inner multiplication of motors treats space unsym- metrically, but this is not hard to show since the inner product of Xii + x^j + xjc -V liCli + mSlj + riiSlk, and x^i + y^j ■'r zJc-\- l^i + rn^Dj + 7up,k, would {Ausd. § 143) be x^x^ + y^y^^-ZiZi + lyli+mjm^ + n^ni, from which it can be deduced that the meaning of {A \ 5] is not inde- pendent of the point of intersection of i, j, k. As an example notice that the inner square {Ausd. § 145) \{i-\-xilj) | (i -f- 0;%)] of the unit rotor i -\- D,j is not unity but 1 -t- a?^ This want of symmetry follows from the fact that in Grass- maim's own geometrical interpretations motors are magnitudes of the second order. But there is no obvious reason why they should not be treated as magnitudes of the first order. § 27] MOTOES AS MAGNITUDES OF THE FIRST ORDER. 105 Let us see if by so treating them we can make Grassmann's " normal " mean the same as Ball's " reciprocal." If E and F are two of the primitive units this would necessitate that sEF= 0. If we are to identify - sAB with Grassmann's [A | B] we must further make sE' = sF' = - 1. Hence by eq. (10) § 14 above 2tET,'E=l (1). This shows that a real motor with negative pitch cannot be a primitive unit. But if the tensor T,E" of a real motor E" with negative pitch tE" be so chosen that 2tE"T^'E" = -1, then E' = vE", where v = V(- 1), is such that sE" = - 1. We are thus led to make trial of the following coreciprocal motors as primitive units — E= (i + ni)/^2, F= (y+flj)/V2, G= (ifc + nifc)/v2| ,^^ E'=v (i - ni)/V2, F' = v (j - %-)/V2, G' = v(Jc- f2yfc)/V2j " ' ■^^>- Here sE^ = sE'^= sF' = si?"= = sG^ =sG'^ = -l] sEE' = sEF = sEF' = sE'F = sE'F' = . . . = j • ' •^^)- It is now easy to see that if A^, A^ be any two motors [A,\A,] = [A,\A,] = -sA,A, (4), for we may put A, = e,E + e,'E' +f,F +f,'F + g,G + g,'G' (5), A, = e,E + e^'E' +f,F +f,'F + g,Q + g^O' (6), where e^ &c. are all ordinary scalars. Thus [Ausd. § 143] - s^,^ , = e,e, f e/e/ +//, +//; + g,g, + g^g^ = [A,\A,] = [A,\A,] (7). We are now in a position to utilise Grassmann's geueral theorems, but in doing so certain errors easily fallen into must be carefully avoided. The necessity for this caution is due to two connected facts : — (1) a real motor with negative pitch is not a real extensive magnitude of the first order but a simple imaginary one ; (2) there are real motors which in our calculus we cannot regard as zero but whose numerical values {^Avsd. § 151] in Grass- mann's sense are zero. These last are rotors and lators. With regard to (2) it is to be noted that Grassmann always assumes that a magnitude is zero if its numerical value is zero. In applying Grassmann's theorems then we have carefully to observe what parts of his proofs depend on (1) the assumed reality 106 OCTONIONS. [§ 28 of his magnitudes, and (2) the assumption that magnitudes with zero numerical values are themselves zero. 28. Some combinatorial products. If ^i, A.^...An are any n magnitudes (octonions, scalars, &c.) and if (/> (A^, A^ ... An) is such a function of them that if any one, say A^, is expressed in the form B^ + G^ we always have ^{A^,B^ + G^,A^...An) = (/> (.4,, 5„ A,...A„) + <}> (A„ C„ As ... An), ^ (J.1, A^... An) would by Grassmann be called a product of the magnitudes -4i ... J.„ {Ausd. § 44]. If further ^ is of such a form that when any two of the magnitudes change places <^ becomes changed to its own negative, the prodtict is called a combinatorial product [Ausd. § 55. These are not Grassmann's definitions, but are most convenient for our definitions]. Thus if Qi, Q2, Qi are octonions and A, B, C, D, E motors, Q1Q2Q3 and MABSCDE are products, but not in general combinatorial products. Again MAB, M,AB, SCDE, sCDE are combinatorial products. Again if (^ is a general linear motor function of a motor, sCB - sDcf>AsC(l>B is a product, but not a combinatorial product in general, of ^, B, C and i) ; it is however a combinatorial product of A and B and again of G and D. There are thus many kinds of combinatorial products of magnitudes of any assigned kind. When the magnitudes are motors and their number is either five or six there is a particular meaning which can -be attached to the combinatorial products which will make them symbolically harmonise with Grassmann's " Produkt in Bezug auf ein Hauptgebiet " [Ausd. § 94 et seq.] and which he denotes by [AiA^aA^As] and [AiA^AsAtA^Ae]. The ideas we attach to the latter are precisely the same as Grassmann's, and those that we attach to the former are essentially the same. In the first place Grassmann puts the product of the primitive units in an assigned order equal to the scalar unity, i.e. in our case [§ 27 above] [EE'FF'QO'^ = \ (1). (2). § 28] MOTORS AS MAGNITUDES OF THE FIRST ORDER For all values of r let A, = BrE + er'E' +f^F+f;F' + g,0 + gJG'\ = Xri + yrj + z)c + Ir^i + rrir^j + ririlk ]"" Thus [eq. (2) § 27] Xr = (er + ve/)/V2, Ir = (e, - ue/)/V2, .v, = (/, + v//)/V2, &c.. . .(3), Br = {xr + ir)/v'2, e/ = u (?, - a;,)/V2, &c (4). Grassmann then proves in § 63 that ei e/ /, // g, g^ 107 .(5). Bi 64 fi fi Qi g* [It is sufficient for our purposes to take this as the definition of [^i.-.^ej and to notice that by the definition [Ai...Ae] is a combinatorial product and that eq. (1) follows from this definition.] The rules for combinatorial multiplication are given in the Ausd. § 52 e< seq. The most important for our purpose is that to any factor of a combinatorial product we may add a multiple of any other factor without altering the value of the product. Thus 8 [EE'FF'GG'] = -v [(i + ni) (i - ni) (j + nj) lj-nj)(k+nk)(k-nk)] = 8v [i . m .j . Qj . k . D,k], so that [i .j .k. D,i.D,j .nk] = v Hence [Avsd. § 63] «i 2/i 2i h nh % x^ 2/2 z, k m^ n.2 Xi 2/3 ^3 k 1^h W3 Xi TJi Zi h m^ rii lAiA^AsAfAfA^j — V aJ5 2/» z. L m. n. Xe 3/6 Ze le m^ n^ .(6). .(7). Equations (5) and (7) apparently depend for meaning on the arbitrary point of intersection of i, j, k. That in reality they do not so depend could easily be directly proved from eq. (7). It is unnecessary to give the proof, as the fact follows iacidentally firom a result we shall prove at the end of this section. 108 OCTONIONS. [§28 In Grassmann's mode of expression [AiA^A^A^As] would be the complex (Gebiet) of -4i, A2, A,, A^, A^, [§ 14 above] associated with a numerical coefficient. Similarly | A would be a complex of the fifth order all of whose motors were reciprocal to the motor A. This complex and its numerical coefficient are completely specified by A and therefore in our calculus it is more convenient to regard \A as meaning, not the complex in question, but, the motor A itself This of course is exactly parallel to the quaternion process of identifying Grassmann's line-vectors ("Strecken") with his surface-vectors (products of two " Strecken ") and calling them both vectors. This leads at once to the only meaning consistent with the Ausd. that we can give to [^j.-.^ls]. For put [^,..A] = 5, where .B is a motor. Then if j4o be any motor whatever - zAJ^ = -%A,\B = lA, II £] = - lA^I {Ausd. § 93) — — [j4o^i^2-^3-'^-i-^5] ^0 ^0 5'o 9l = s4o 65 65 ... gt, EE ... G fii e/ ... 5-1 «1 2/1 65 6s = vsA X, y, ... n m nj nk i j k «i yi ^1 ii Wi «i 2/6 z. L ?)i, «, Since Aq is arbitrary it follows from § 14 above that EE' ...G' Q.i nj nk i j k Bi ei ... gi ^ _ ^ «, 2/1 z^ li m-, n^ 6565'... gs os^ 2/5 Zs Is m^ W5 [4,..^J = . ...(8). [For our purposes these equations may be taken as definitions. I thought it desirable to show what connection [j4i...^b] and [Ai ... Ae] had with the Ausd.] Very often the imaginary v recurring in equations (7) and (8) is an inconvenience. We therefore define as follows {A,...An] = -v[A,...A„] (9), §28] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 109 for the values 5 and 6 of n ; and in the definitions about to be given of {^i ... J.„} for values from 1 to 4 of n the same equation may be supposed to hold. Thus {A...A,} = ^2 2/2 •"g Vi Tie .(10), Ui...A)=- From these .(11). ■(12), .(14). .(15). {i.j.k.ni.Q.j.nk} = l (13), {j.k. m . nj . nk] = - Hi, {k.i .m. nj . nk] = - nj {i . j . m . nj .nk} = - dk {i .j ■ k . nj . nk} = i [i ■ j . k . nk . m] =j [i .j ■ k . m . nj] = k {A-i...As} maybe regarded as a linear motor function of A^; also as a motor function of A^, A^ linear in each; and so on. Thus [A^... A,} = ^,A, = , (A„ A,) = (^, (^3, A„ A,) \ = (j), (A^, A„ A„ A,) = ^0 (A„ A„ As, A„ A,)] With these meanings of <^i . . . ^4 we may put from m = 1 to n = 4, 4>n={A,...An] (16), and <^o = {l} (17). Thus [A^... An] is for all values of n from to 6 a combinatorial product of .4], A2...An. It is to be noticed that with the meanings of i/r and Z given in equation (6) § 15 above, ^|r(Z,Z) = [-^|r(A„{A^,...A,}) + ■^}r(A„{A,A,...A,])-...\ + Vr(A,{A^...A))]/{^A...A}l^ ''' where A^, Az-.-A^ are any six independent motors. For if we change A^ into any other motor XjAi + 0SiA2+ ■..+XeAs (x^ not I (19). 110 OCTONIONS. [§ 28 zero) independent of A^, A^.-.A^ the expression on the right is unaltered. Similarly for the other motors involved. We may therefore change A-^...A^ to i, j, k, m, ilj, ilk, whereupon the right becomes yjr (i, Hi) + yjr (D,i, i)+ ... or yjr (Z, Z). Since [eq. (8) § 15 above] E = -ZsEZ it follows from eq. (18) that E{A,...A,} = A,{EA,...A,} + A,{A,EA,...A,] + ... + A,{A,...A,E} = {A^As...Ae]sEAi-{A^A3...Ae}sEA^ + -{A,A,...A,}sEA, Define A^, A„...Ae hy the equations J- ^ _ { A^As-.-A^} J _ {Aj^As...As} -^ _ [A^Aj-.-A ^ .^^. [AiAi...A^ \A-^A2...A^ {.4 1.4.2... .^sj Eq. (19) then gives E = - A.sEA, - A,sEA, - ... - A,sEA,] = -A,sEA,-A^sEA^- ... -A.sEA^] ^ ^' Also by eq. (19) E [A,. ..A,} =A {EA,...A,}+A, {A,EA,...A,} + ... Comparing this with the last equation we see by eq. (12) and § 14, that _ \A^s...A s] . _ { AiA.i...A e} ^'- {ZA...2e)' ""'-{AJ,...!,}' ^ ^' so that the relations between the two sets AyAz.-.Ae and A1A2 ...Ae are symmetrical. Again by eq. (1 2) sJ.]Zi = sA^i — s^a^s = 8^44^4 = zAJl^ = 8.46^^0 = - l...(23). Again since {A ^A^A^ . . .via} = 0, s J ,^2 = 0. Thus S^iZj = S^iZs = SJ.1Z4 = S^jZa = ... =0 (24). j4i,J.2...^6 have been assumed independent. [Otherwise {.4.1 ... ^e} = 0.] A-^, A^... A^ are also independent since any motor [eq. (21)] can be expressed in terms of them. In § 14 we stated what was meant by reciprocal complexes. By equations (24) the complex AnJ^-y...A^ is reciprocal to the complex A^... -4„. Moreover no motor which does not belong to the former complex is reciprocal to the latter. For any motor § 28] MOTORS AS MAGNITUDES OF THE FIRST ORDER. Ill 6 _ can by eq. (21) be put in the form ^=2a;-4, and expressing that this is reciprocal to each of the motors A^, A2...An we get by equations (23)^ and (24) x, = x,= ...=x,,= 0, so that E belongs to the complex An+i-.-Ag. Hence to ^very complex of order n there is a reciprocal complex of order 6 - ?i ai^ no motor not belonging to the latter is reciprocal to the former. In particular to every complex of order five there is one (and ordinary scalar multiples of it) and only one motor reciprocal. "We have just seen that A-,...A^ are independent. Hence the six motors which are reciprocal to each set of five out of six given independent motors are themselves independent. More generally if (n) be a complex of order n and (6 - n) an independent complex of order (6 — n), then if (6 — n) is the complex reciprocal to (n), and (n) the complex reciprocal to (6 — n), (7i) and (6 - n) are independent complexes. For (n) may be taken as Ai...An and (6-?i) as An+i...Ae. Then (n) is A^...!,, and (6-n) is An+i ■■• Ag. If a motor is reciprocal to five out of six independent motors it is not reciprocal to the sixth. For sA^A^ is not zero. These are well-known results or easily deducible from such results, but they serve the double purpose of exemplifjdng the present methods and of showing the physical meaning of the con- nections between the two sets of motors Ai...Ae and A^...Ae. [I am not sure that the second and third italicised statements are known.] If ^1, A^... Ag are six independent coreciprocal motors and E=x,A^+... +XsAe, we have by operating by sA^ ( ), sEA^ = x^sAi". Hence E = A:,sEAJsAi' + A^sEA^/sAj' + ... + A,sEA,lsAs'.. .(25). Comparing this with eq. (21) we have sEAJsA,^ = -sEA,. Hence in this case by § 14 ^,/s^,= = - A = {^2^3 ... A,}/{A,A,... A,} (26). A relation that we shall require later is the following {ABCaA'ilB'ilC'} = S,ABCS,A'B'C (27), 112 OCTONIONS. [§ 28 where A, B, C, A', B', C are any six motors. The right vanishes if and only if either A, B, G or A', B', C are not three completely independent axial motors. \i A, B, C are completely independent axial motors and A', B', C also are, the six motors on the left are independent and therefore the left expression does not vanish. If A', B', C are not completely independent axial motors the lators on the left are not independent and therefore the expression on the left vanishes. I{ A, B, C are not completely independent axial motors one motor at least of the complex A, B, G is a, lator, so that the six motors on the left are not independent (since four lators are never independent) and therefore the expression on the left vanishes. Hence if one of the two expressions of eq. (27) vanishes, the other does. If neither vanishes A, B, G are com- pletely independent axial motors and ilA', ilB', Q.G' are indepen- dent lators. Hence xA+yB + 2G+ inA' + mD,B' + nSlC when x is not zero is any axial motor completely independent of B and G. If this be substituted for A both expressions are altered in the ratio of x to 1. Similarly if D.A' be changed to any other lator xQ,A' + yi^B' -t- zVlC (x not zero) independent of £25' and nC, the expressions are both altered in the ratio of x to 1. It follows that {ABGn.A'nB'D,C}/S,ABGS,A'B'G' has the same value for any six motors A, B, G, A', B', G' of which the first three are completely independent axial motors and of which the last three are also. Changing A, B, G to i,j, k and A', B! , (7 also to i,2, k, we see by eq. (13) that this value is unity. Since S,ABGS,A'B'G' = sA {CMBG) S,A'B'G' = s . (D,A') MFG'S^ABC, we see by equations (27) and (12) and § 14 that {BCnA'nBnG'} = ntABCS.A'FG (28), and {ABGilB'n,C'] = - MB'GS.A BG+a, lator. If SiABG is not zero this lator may be put in the form n {xlABG + ylAC A ■\-zlAAB). Operating on the last equation by sA{ ) we get X = sAB'G" and similarly for y and z. Hence {ABGnB'D.G'] = - MB'G'8,ABG + n (MBGsAB'G' + fAGAsBB'G + MAB6CBG')...(29), when SiABG is not zero. This equation is also true when SiABG is zero, for then xA + yB + zG=a lator = nA', § 28] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 113 say, where one at least (say x) of the ordinary scalars x, y, z, is not zero. Operating on the last equation by flMC ( ) we get and similarly £mAB= - DMBC. X Thus \ABQ^B'^U\ = - X-' {BCnA'nB'nC} = - .r-i nMBCS.A'B'C [eq. (28)] = - .r-i Q.MBCS (xA +yB + zC) B'C = - fl IMBCsAB'C + ^MBCsBB'C' + - MBCsCB'C] X ' = - fl [lABCsAFC + MCAsBBV + MABsCB'C], which proves eq. (29) when S^ABC = 0. Eq. (29) may by eq. (8) § 13 be put in the form [ABC^B'^C] = MBV'S,ABC- MBGS.AB'C - fACAS.BB'C - MABS,CBO' (30). Putting in equations (29) and (30) B' = B, C' = C we get {ABGD.BnC} = MBC(-S, + S.^ABC (31). It will be observed that equations (28) to (31) are generalisa- tions of equations (13) and (14). Also eq. (28) is a pai'ticular ease of eq. (29) as we see by changing A of eq. (29) into ilA'. These results are rather more general than the following which are also instructive. By eq. (8) § 13 if ^ is any motor and A, B, C are three given completely independent axial motors E = ASEBCS-'ABC+... = A [sE(nBGS-'ABC)+nsE{BCS-'ABC)} + .... Hence if we take A, B, C, ilA, ilB, D.C for the A,,A....A,o{ equations (20) to (24) A = -nMBCS-'ABG, nA=-MBGS-'ABG (32), and similarly for B, D,B, G, ilG. [Notice that what was denoted in § 16 by -4 is here therefore denoted by ClA.] The second of M. o. 8 114 OCTONIONS. [§ 28 equations (32) easily leads to eq. (31) and the first to the particular case of eq. (28) obtained by putting B' = B, C = G. Since E = - AsEA - nAsEDA -BsEB - ..., we see that {ABCB'C} = {ABCn (AsB'nA + BsB'nB+ CsB'TlC) n(AsC'ilA + BsCiiB+ CsCnG)}. Equation (30) may now be utilised and we shall then have expressed {ABCB'C'] in a manner independent of any arbitrary origin. Similarly for [ABCA'B'C], The expressions are however too cumbrous to be of much use. 29. Combinatorial, linear, circular and hyperbolic variation. In § 71 oiAusd. Grassmann explains what he means by a simple and a multiple linear variation. In § 154 he explains what he means by a simple and a multiple and also a positive and negative circular variation. In § 391 he somewhat extends the latter. The modification that we thus get for real motors will be called a (simple or nniltiple. positive or negative) hyperbolic variation. These are all particular cases of a more general kind of varia- tion which will be called a combinatorial variation (simple or multiple, positive or negative). Any group of motors Aj ... Ap ... Ag ... An is said to be changed by simple combinatorial variation to the group A^ ... Ap ... Aq ... An (the only two motors changed being Ap and A^ if Ap=cAp -irsAA A^ = s'Ap + ca\ (1). c^-ss'=l J If further one of the motors Ap, Aq have its sign changed this combined with the former is said to constitute a simple negative combinatorial variation. [By the latter variation Ap, Aq are changed to Ap, Aq where Ap =cAp + sAq, Aq =- s'Ap-cAq, c^ — ss' = 1.] A series of combinatorial variations performed on the group are said to constitute a multiple combinatorial varia- tion. [So far as the fundamental property of combinatorial variation in connection with a combinatorial product is concerned (see § 29] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 115 below) we might define it more generally. The above definition is the most convenient for our purposes. The more general definition is given by the equations Ap = x^Ap + y^Aq, Aq = x^Ap + j/^Ag, x^y. - x^y^ = l.J When c = 1 (and therefore s or s' is zero) the variation is called a linear variation. When s' = -s the variation is called a circular variation. In this case we may put c = cos ^, s= -s' = sin (2). When s' = s the variation is called a hyperbolic variation. In this case we may put c = cosh0, .s = s' = sinh^ (3). From eq. (1) we have Ap = cAp - SAg, Ag=- S'Ap + cAg (4). Thus P' f" = f-f'-^l , (.5), Ap, Ag Ap , — Ag which in Grassmann's notation (Avsd. § 377) expresses tlie fact that Ap', - Ag' are the same linear functions of Ap, Ag that the latter are of the former. From this we see that if .4, B be changed by negative com- binatorial variation to A', B' A',B' A,B „, -a;b^a'7b' ^^^' that is to say that if we put A' = (j>A, B' = 05 ; then A = A', B = ^B'. This can be made the basis of the definition of combi- natorial variation. On account of the relation (6) some few properties of negative variation are simpler than the corresponding properties of positive variation. Since c^ — (— s) (— s')= 1 we see from eq. (4) that Ap, Ag are obtained from Ap', Ag by a simple positive combinatorial variation. And further, if the given variation {Ap, Ag into Ap, Ag) is linear, circular or hyperbolic, the derived variation {Ap, Ag into Ap, Ag) is linear, circular or hyperbolic respectively. In the cases of circular and hyperbolic variations the derived variation is obtained from the given one by changing the 6 of equations (2), (3) to — 6. 8—2 116 OCTONIONS. [§ 29 The fundamental property of a combinatorial variation, on which all its usefulness may be said to depend, is that : — Any combinatorial product of A^, A^.-.An is unaltered by combinatorial variation. In § 156 of Ausd. this is proved for the case of circular varia- tion applied to the magnitudes of a " normal system." It is just as easy to prove it in general. Let the product be denoted by {A,...Ap...Ag... An). Then (A, ... Ap' ... Ag' ... An) = {A, ... {cAp + sAg) ...(s'Ap + cAg) ... An) = (Ay... {{cAp + sAg)-- (s'Ap + cAg)} ... (s'Ap + CAg)... An) = {A^...^ ... {s'Ap + cAg)...An) c = iA,...^...cAg...A„) c = {Ai . . . Ap ... Ag ... An). [Here it has been assumed that c is not zero. The reader can easily supply the proof for the very simple case when c is zero.] The proposition is therefore true for a simple variation. It follows at once for a multiple variation. In particular if ^i, J. 2 be combinatorially varied to A/, A/, ma,a,=ma;a2' ; if A„ A„ A, to A,', A,', A,', SA,A,A, = SA,'A.:A,' ; if E, F to E', F' and ^, 5 to A', F, sE^AsFB-sF^AsEcj)B=sEA 'sF'4>B' -sF'^A'sE'4,B'\"-^ ^' where ^ is a general linear motor function of a motor. A par- ticular case of eq. (7) is extremely useful. Put E = A, F=B, E' = A', F' = B', and ^ = ts where cr is self-conjugate. Then zA'-ssA'sF'stB' - s^^'ctF = sA'^AsBisB - sU«-£. . .(8). A still further restricted case is obtained by putting -a — \ when we have sA'^&B'^-i^A'B' = %A-'%B:'-%-'AB (9). The equivalent of this last (for any two magnitudes of the first order) is proved in § 391 of Ausd. for the cases of circular and hyperbolic variation. Grassmann's use of the equation is analogous §29] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 117 to our use below of eq. (8). Needless to say the use below was suggested by his. In equations (7), (8), (9) we may put S or Sj for each s. CT having the meaning just given to it we have from eq. (1) SA'-utA' = c^SAvtA + 2csSAv7B + s'^SB^B] SB'vtB' = s'^SA^A + 2cs'8A^B+ c'^SBztB) ^ ^• from which we deduce that s'6A'^A'-sSB''^B' = s'6A^A-sSB'mB (11). Particular cases are (circular variation) SA':!TA'+SB'r^B'=SAvrA+SB-eTB...(12), (hyperbolic variation) SA'ztA' -SB'ztB' = SA'!!tA -S5orB...(13). Putting w= 1 we have in place of eq. (11) s'A''-sB'' = s'A'-sB' (14). Similar deductions may be made from the other equations. Again it may be noticed that if we put we have SA-srA + -, SB^B = 2b, , S^«r5 = b, s SA'ztA' + |,S£V£' = 2b/, SA'^B' = K I 0=c^ + ss', 8 = 2cs, S' = 2cs' I .(15), .(16), 6/ = Cb, + Sb,, K = S% + Gb,, G-'-SS'=l. so that 6/, 6/ are obtained from b,, b^ by a combinatorial variation. In particular, for circular variation, i (SA'^A' - SB'htB') ) = ^(SA^A-SB^B)cos2e + SAzTBsm2e I ...(17), SA'mB' = - i (S^t^4 - SBmB) sin 29 + SAnrB cos 26) and for hyperbolic variation ^(SA'^A'+SB't^yF) = ^ (SArsA + SBijtB) cosh 26 + SAn^B sinh 26 I ...(18). SA'^B' = i (SA^A + SB^B) sinh 26 + SAwB cosh 26 Since SQ = S,Q+nsQ, where Q is any octonion, we may in equations (10) to (18) write either S, or s in place of S. Suppose A„A^... Ae are six independent motors and A„ A.,...A^ have the meanings defined in § 28. A/, A/_...As wHl be defined in the enunciation about to be given. Let A/.A.! ...A^ 118 OCTONIONS. [§29 have the same relations with ^i, A.^... Ag that A^, A^ ...As have with ^1, A^... Ae. We proceed to show that : — If by a series of negative circular vuriations Ai,...Aa become transformed to A-^, ... A^, then A(, ... 4/ will be exactly the same linear functions of A^, ... A^, that A(, ... Ag are of A^, ... Ag] or in Grassmann's notation A^ , A.i, A -^3. -^i: -^6' -^6 -^ 1 , A2, A-^, At, A^, Ag It is sufficient to prove the theorem for a simple negative circular variation. Suppose then -0.3 =.0-3, A^ = A^, A^ = A^, Ag = Ag Ai =cAi+sA.i, A^'^-s'Ai- cA^, c^-ss'=l'i [We at first take the variation as any negative combinatorial variation as, though the above theorem does not then hold good, certain very simple formulae hold for the general simple negative combinatorial variation.] A combinatorial product which involves both Ai and A., has its sign changed, but is otherwise unaltered by this negative varia- tion. Hence •J / _ — \A^ A.J Aj A^ Ag ) _ [AiA^AjA-^g] _ -j [A.^A.jA:iA^A^Ag^ — ^A^A2A^AiA^Ag] and similarly for .4/, A^, A^. Again we have ■j,_ - [A^A.: ... A:\_ [{ s'A, + cA.:)A,... a,] j ,j ^^- {A,' A.'... A,'} - - {A,A, ...A,] -<'^>-*'^- J , _ {a m: • ■ ■ a: } _ {(cA, + sA,) A,.. . A,] J J '''-{AM.'-A,'}- -{AA...A~ -*^'-^^- Thus A,' = A^Al = A,M,':=A„A6' = Ae ] A,' = cA,-s'A,, A,' = sA,-cA„ c^- (- s)(-s') = 1] " which expresses the connections between A^', ... A^ and A^, ... Ag in the case of any simple negative combinatorial variation. We see that the former are obtained from the latter by a similar (linear^circular or hyperbolic) negative variation. Also by eq. (6) Aj = cAi — s'A^, An = sA^ — cAl express the converse relations. In the case of circular variation s' = — s. Hence in this case Al , ... Al are the same functions of .4i, ... Ag&s A^, ... A^ are of §30] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 119 Aj, ... Af). The theorem can be easily extended from simple to multiple variation. 30. Coi\jugacy with regard to a general self-conjugate function. Let w be a real general self-conjugate linear motor function of a motor. We are about to establish certain properties of CT. Two particular cases will frequently be considered. (1.) ss may be put equal to unity; conjugacy then reduces to reciprocity. (2) ST may be an energy function as we shall term it because of its intimate connection with the energy (kinetic and potential) of a rigid body. An energy function may be complete or partial. CT is a complete energy function when sE^E is negative and not zero for every motor E in space. It is a partial energy function when sEiffE is zero for some values of E and is negative for all other values. We shall usually write yjr for cr when we restrict it to being an energy function. When sEtsE = we shall find later that E belongs to what Sir Robert Ball (Screws, § 90) calls a complex of the fifth order and second degree. If E be restricted by this equation and further restricted by belonging to a complex (as defined in § 14 above) of the )ith order it would be said to belong to a complex of the (n — l)th order and second degree (Screws, § 158). The theorems we are about to establish can all be expressed as theorems relating to such a complex of the second degree. Two motors E and F are said to be conjugate with regard to ■w when sEvrF = 0. ?i motors are said to be conjugate when every pair of them is a conjugate pair. Defining a complex of order ji as in § 14 above, we proceed to show that: — In any complex of order n, n real independent conjugate motors can he found. Let J.1 ...An be any n independent motors of the complex. If any pair is not already a conjugate pair it can be made conju- gate by a simple circular variation. Let A^ and A, be not conjugate. Circularly vary them to A^, A^ so that A,' = Aicosd + A._sva.e, A^ = - Aisind + A,cose. 120 OCTONIONS. [| 30 Thus sAi'btA^' = sA^'srAijios 26 — (sAicrAi — sA.^'uyAi) sin 6 cos 6, so that 4/, J./ will be conjugate if and 6 can always be determined so as to satisfy this equation. When any such pair is thus made conjugate the product sA^s^AiSA^'ctA^ ... sJ„iir^„ is algebraically diminished. For the only factors of this product that are altered by the variation are the two involving the varied motors. And the product of these two factors (sAi^r^yA^ and S.42CT-J.2) is by eq. (8) § 29 diminished by the amount sMi'cr^a which by hypothesis is not zero and since ro- is real is not negative. There is an exception to this statement, viz. when one of the factors, not involving a varied motor, say sA^ii^As is zero; A^ is then self-conjugate. The product then remains zero. But unless A3 is conjugate not only to itself but to all the other motors it can be varied with one of them to which it is not conjugate, whereupon the product diminishes to less than zero by what has just been proved. If A^ is self-conjugate and conjugate to all the other n— 1 motors it is conjugate to the whole complex. In this case A3 does not require to be varied. We now see that the following statement is true : — If any pair of ^1 ... An is not conjugate we can by a circular variation make it conjugate and the product sA^vrAiSA^-^^A^... involving all the motors which are not conjugate to the whole complex thereby diminishes. This last product then has a minimum value, and this value is only attained when the 71 motors are conjugate. Hence ?i such real motors exist. Moreover if any one of such a set of conjugate motors is self-conjugate it is conjugate to the whole complex. There are in general an infinite number of such conjugate sets of n motors. It is easy to prove however that the complex con- sisting of the self-conjugate motors is a definite complex. In other words : — If A^ ... Ai, Bi+i ... Bn are one set of conjugate motors and Ai ...Ai„', B',n+i---Bn' are another set, none of the A's being § 30] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 121 self-conjugate hut all of the B's being self-conjugate, then the complex Bi+i ... Bn is the same as the complex B'm+i ■ ■ . B^. [Thus in particular m = Z.] For while every motor of the complex 5^+1 . . . 5„ is conjugate to the whole complex, no motor of the complex A^...Bn which does not belong to the complex Bi+i...Bn is conjugate to the whole complex. Suppose that x,A, + . . . + yi+, Bi+, + ...= l,xA + 2y5 is such a motor conjugate to the whole complex. Operating by sJ.iOT ( ), remembering that sA^-^A^ is not zero since A^ is not self- conjugate and that A^ is conjugate to A^... AiBi+i ... -B„, we see that «! = 0. Similarly x^= ...=xi = or the motor belongs to the B complex. It follows that every motor of the complex B'm+i . . . Bn belongs to the complex Bi+j ... Bn and every motor of the latter belongs to the former. Hence these complexes are identical. If we put w = 1 we obtain the following : — In every complex of order n, n real coreciprocal motors can he found and tJie complex consisting of lators and rotors in such a set is a definite one. [The reader should perhaps be cautioned against supposing that this means that there are no real self-reciprocal motors (lators and rotors) in the complex of the not self-reciprocal motors. The assertion only is that no such self-reciprocal motor can form one of a set of n coreciprocal motors.] If Ai ... Ai he a conjugate set of motors not one of which is self-conjugate they must he independent, and if B he conjugate to the whole complex it cannot helong to the complex A^ ... Ai. [Com- pare Ausd. § 157.] Suppose yB -1- {x-,A^ + ...+ xiAi) = 0. Operating by sAinr ( ) we obtain a;i = and similarly X2= ... = xi = 0. If J.1 ...Ai, Bi+i ...Bn have the meanings they had just now: — If E he a motor of the complex conjugate to each of the motors Ai, A.2 ■•■ Ap, it must helong to the complex Ap+i ... Ai, Bi+i ... Bn. [Compare Ausd. § 159.] Suppose E=lxA + '^yB. 122 OCTONIONS. [§ 30 Since E is conjugate to Ai we have by operating by sAt^s ( ), that x^ = 0. Similarly Xi= ... =Xp = Q, or E belongs to the complex Apj^-i ... Ai, Bi+i ... -D„. By putting ot- = 1, in both these statements we may read " reciprocal " instead of " conjugate." 31. Conjugate variation; positive, neg^ative and zero norms; semi-conjugate complexes. Suppose A^ and A^ are two motors conjugate with regard to cr. Suppose A/ and A^' are derived from A^, A„ by the combinatorial variation ^i' = c-4j + s^2, Ai = s'A-,-^cA.i, c=-ss' = l (1), with the condition s's^i=rj4i + ss42CTJ.2 = (2). Such a variation will be called a is -conjugate variation or, when there is no risk of ambiguity, simply a conjugate variation. [The meaning will be very slightly extended directly.] Since sAj-stA., = we have at once from equations (1) and (2) that when Ai and A^ are conjugate and are by a conjugate varia- tion transformed to J./, A^'; -4/ and -4./ are also conjugate, i.e. sA,'-stA^' = ...(3), and also s.4,Vj4i' = sJ.iotJ.,, sAJtitA^' = sA./stA., (4). When sAjvrAi = s^s^.^., a conjugate variation is a circular variation. When sA^tffA^ = — sA^srA-i a conjugate variation is a hyperbolic variation. When s.42or.4, = and sAiSsAi is not zero a conjugate variation is a linear variation. Note that the circular and hyperbolic variations here men- tioned are perfectly arbitrary variations of those types. The linear variation can only be of the type A^' = Ai+sA2, A.^^^A, where s is however arbitrary ; it must not be of the type .4/ = j1,, A.^ = A^ + s'A-,. According to equations (1) and (2) if s.4itn-Ji = $^2^.42 = a conjugate variation is any combinatorial variation. In this case however, viz. that of A^ and A., being both self-conjugate, it is convenient to regard conjugate variation as being of a more § 31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 1 23 arbitrary type than combinatorial variation according to the following definition: — If Ai and A^ are conjugate, independent and both self -conjugate, and A-l, Ai are any two independent motors of the complex A^, A..; the variation from A^, A^ to A^, A^ is said to he a conjugate variation. This may be put into symbols thus : A^ = xA^-iryA,„ A^' = x'A, + y'A.„ and xy' — x'y is not zero. If a series of conjugate variations be performed on a gi-oup of motors the whole variation may be called a multiple conjugate variation. Whatever motor A be we can by multiplying it by a real ordinary finite (and not zero) scalar, make sAx^A take one of the three values +1, — 1 or zero. For the sake of more clearly stating many of the results of this section it is convenient to define as follows : — A is called a positive norm if sA-mA = — 1 ; a negative norm if sAthA = + 1 ; and a zero norm if s^^^ = 0. Thus by § 30 any complex of order n can be expressed as a complex of n independent conjugate norms. Equations (3) and (4) it will be observed remain true with the extended meaning just given to conjugate variation. Hence: — By conjugate variation a set of n independent conjugate norms remains a set of n independent conjugate norms ; and the number of positive norms remains unaltered, as also the number of negative norms and the number of zero norms. [Compare Ausd. §§155 and 391.J For the future we shall always suppose the conjugate norms to be independent. By § 30 we see that the positive norms and the negative norms must be independent. We here define the zero norms as independent. Also by § 30 the zero norms form a definite complex. In this section as in § 30 we shall confine our attention to motors belonging to a given complex of order n. Let A-, ...Ap be the positive norms, B, ...Bg the negative norms, and G, ... Cr the zero norms of a set of n conjugate norms belonging to this complex. 124 OCTONIONS. [§ 31 If any motor E of the complex is a positive or negative norm or is conjxigate to the whole complex; A^ ... 6V can he so transformed hy conjugate variation that E becomes one of the group. [Compare Ausd. § 160.] If E be conjugate to the whole complex it belongs (§ 30) to the complex Cj ... (7,.. But by the definition of conjugate varia- tion among the (7's we can transform them by such variation so as to contain any motor of their own complex. If ^ be a positive or negative norm sEwE = + 1. Let E = XiAi+ ...+XpAp + y,B^+... +ZiGi+... = lxA + lyB+'ZzC. Since sAi-07j.i = — 1, s^i^-i?! = + 1, sCictCj = 0, &c , and the n motors on the right are conjugate, sEmE=- 2«= +%== + !. If the upper sign be taken we can successively bring into the group „ _ic^Ai+_x^ „ _ x^A-i^ + a;g^2 + x^^ -~ ^{x^^+x^^)' '~ ^{xi' + x.-'+x,') • ■•■' E2 being obtained from A^ and A^, E3 from E^ and .^3, &c., and Ep from Ep^, and Ap all by circular variations. And now we can successively bring into the group , ^ txA + y A „ , ^txA+y^Pi±yA E^' being obtained from Ep and B^, E.!, from E-l and B.^, &c., and Eq from -£"g_i and Bq all by hyperbolic variations ; for not one of these denominators is zero since Sa;"— 22/- = l. Finally we can successively bring into the group E" = IxA + tyB + z,C^, EJ' = txA + lyB + zA + ^aC^, • • . , E,."=l.xA + lyB+tzC, E," being obtained from E^' and C\, E." from E," and G„ &c., and Er" or E from JS"',.-i and C',. all by linear variations. Similarly if sE-!^E== + l we can bring E into the group by beginning with the B'ii. It is to be observed that if .fi" is a zero norm (e.g. A^ + B^) which is not conjugate to the whole complex it cannot be brought § 31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 125 into the group by conjugate variation. For it cannot be obtained from Ci ... Cr since it does not belong to their complex, and it cannot be obtained by any variation that involves a single positive or negative norm since every newly introduced norm is in that case a positive or negative norm. Denote the given complex of order n by (r?) and let (m) be a complex of order m which is included in (n). Then by § 30 (m) may be expressed as a complex of m conjugate norms. When ex- pressed in such a form let «],... a„ be the positive norms and /S], ... /3i, be the negative norms. The zero norms of (vi) may have a complex in common with Ci, ... C^, the zero norms of (n). Let this complex be that of 71, ... 7^ and let the rest of the zero norms of (m) be 8], ... 8^. Thus S,, ... B^ belong to the complex A^, ...Ap, Bi, ... Bq, C], ... Gy, but the complexes Sj, ... S^ and Cj, ... G^ have no motor in common. J.1, ... Ap, 5i, ... Bq, Ci, ... C,. can by conjugate variation be transformed to Oj, ... a^, /Sj, ... /8g, 71, ... 7,-; where Oj, ... Op are positive norms, /3i, ... yS^ negative norms, and 71, ... 7^ zero norms; where a.^, ... ««, /Sj, ... iSi,, and 71, ... 7^ have the meanings jvst given to them; and where B, = X, (tta+i + A+i). ^2 = a;2 (aa+2 + ^6+2), &c (5). We first show [by a process essentially the same as that of Ausd. § 161] that A^.-.C,- can be so transformed as to bring a,...a„/3i .. /367i ... j, into the group. By the last proposition A^ ... ApB^ ... BqC^ ... C^ can be trans- formed to a^A^' ...Ap'Bi ... Bq'Gi ... Gr where the ^'s, B's, and (7s are positive, negative and zero norms respectively, a^ belongs to the complex A^... Cr and therefore to UiAJ ... G,. and it is conju- gate to «!. Hence (§ 80 above) it belongs to the complex A^' ... C,.. Hence by the last proposition this last can be transformed to a^A " ... Ap"B" . . . Bq' Gy...Gr. a^ belongs to the complex A^... G,. and therefore to a^OoAs" ... G,.. It is conjugate to a^ and eta and therefore belongs to the complex A" ... G,.. Hence this last can be transformed to as A/" ••• f'r and so on. Proceeding in this way we see that Si ... aa^i ... ^i can be brought into the group. By hypothesis 71 ... 7c belong to the complex Cj ... C^. Hence 7, . . . 7(, can be brought into the group. It remains to prove that o-a+i, A+i ••• c^" ^^ determined to form part of the group and satisfy eq. (5). 126 OCTONIONS. [§ 31 When a, ... an /3i ... /Si, 7] ... 7c have been brought into the group let the rest of the positive norms be denoted by A-l ... A'p-a, the rest of the negative norms by B^' ... B'q-i, and the rest of the zero norms by C/ ... C'r-c- B-i...Sd are all conjugate to all the motors ai.../3(,. They therefore (§ 30 above) belong to the complex J./. . A'p-a -B/- • • -B',-;, C,...Cr. Thus p-a q-b 8, = lxA'+ S 7/F + C, 1 1 where C is a motor belonging to the complex Cj ... 6^. Since Si is a zero norm we obtain tx'' - ly^ = 0. l.x' and Sy^ are therefore neither of them zero ; for if either were, all the x's and y's would be zero and S, would, contrary to hypo- thesis, belong to the C complex. Thus E, = x(A + B), where A and B are a positive and negative norm respectively given by A VS.r"- = txA', B^lf = lyB' + C. This shows that «„+! and /Sj+i can be obtained as desired. Suppose now for any value of e (< d) A^ ... B/ ... can be so chosen (consistently with the meanings just given to them) that B, = X, (A / + £/), ...B, = Xe(A/ + Be'). I proceed to show that the theorem is also true for e + 1. Let Be+, = "X ^A' + "i" vB' + C, 1 1 where G belongs to the complex Cj ... C^. Since B^+i is conjugate to S, we get ^1 = 7?i. Similarly ^2 = %. ••• ^a = Ve- Again since Bg+i is self-conjugate we get p-a q-b Here again were eiLiier 2 ^ or S 77^ zero, S^+i would, contrary e+l e+1 to hypothesis, belong to the complex B^ ... Be, C^... C,.. § 31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 127 Thus Se+i = Y ^A' + '2 T^B' + f , (^/ + ij/) + . . . + f, {a: + b:) + c = ^e+i (A + 5) + I, (^/ + 5/) + . . . + fe (^e' + -B/) + 0, where x,+, = /s^fS ^ = "s" ^^'K+i > i^ = 's' 7;S7«,+, . e + 1 e + 1 c+1 We now show how by successive conj ugate variations, ^i , f . , . . . f ^ can be got rid of from the last equation. ^1 can be got rid of by writing 5, = ^/(^ /' + £/'). S,+, = .'/•,+, {A + 5„) + ^, {A.; + BJ) + . . . + ^, {A; + £/) + c, where a.'/ = iri«e+i/\/(*e+i'-' + f 1'), ^/' = [A^ (xe+,' + |i=) + B,%' + £|;,«,+j}/,-r,+, V(^e+r + ^A £o = 5 + (^/ + £/)^,/^,+,. Here A^", B", A, £„ are a conjugate set of motors belonging to the complex A-/, £/, A, B such that s^i'V^i" = sAvtA = - s5/VB/' = - sB„^B, = -1, so that they can by a conjugate variation be obtained from A^', B,', A, B. Thus f 1 can be got rid of and similarly ^2 • • • ?? can be suc- cessively got rid of. We may therefore assume them all to be zero. We then have S, = X, (A,' + B,') ...Be = 00, (A: + Be'), Be+, = Xe+, (^'e+l + £'e+i)- where A'e+i = A, B'e+^ = B + a;«+r' C (so that B'e+j can be obtained from B, C] . . . 0,- by a multiple conjugate variation). It follows that tta+i, y9i+i... can be determined as asserted. Certain particular cases of this theorem are worth enunciating. First suppose m=n so that the complexes (n) and (m) are the same. The theorem may then be thus put : — 128 OCTONIONS. [§ 31 If E he any one of n conjugate motors of a given complex of order n, the number of motors for which sEmE is positive, the number for which it is negative and the number for which it is zero, are all definite numbers characteristic of the complex. Putting w = 1 we get : — Of n independent motors of a given complex of order n, the number with positive pitch, the number with negative pitch and the number of lators and rotors, are all definite numbers characteristic of the complex. In this case we may also add that the number of completely independent (i.e. not parallel) rotors (of the rotor and lator complex which is a definite one) is also definite. In order more easily to enunciate another result of the theorem multiply Si, S,... by (a;iV'2)~S {x^s/^)'^... and denote the new values by S,, S, Also denote by S/, 8,'... the motors defined by g, = "''+'|/^+\ g„ = °«±^+^^&c (6), Here it will be observed that Sj, S/ are obtained from aa+i, Bb+i by a negative circular variation. Hence (§29) a„+i, /S^+i are ob- tained from Si, S/ by the same negative circular variation or "'^'-~72-' ^'+^=^72"- ""+==-72"''^'' ^^^- Now change the notation as follows: — change «! ... Ua to AjA^...; /Si.../3j to BiB^...; 71 ...7c to CjOj SiSi'SA'-.. toAA'AA'.--; apap_i... (not involved in B^Dr,...) to A-^A^...; \ )----(9)- ^q^g-i . . . (not involved in A A ■■■)to B/B^' ...: 7r7r-i ■••7c+i to C/Os'... We then get the following : — If(m) he any complex of order m included in the given complex (n) of order n, a complex (n - m) of order n — m independent of (m) can be found, such that (n) and (n — m) make up {n). {m) consists of the positive norms A^Ao-.., the negative norms BiB,,. . . , and the zero norms C^C.,... D^D^. . . §31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 129 ^ (n - m) consists of the positive norms A,' A;'- ■■, the negative noi-ms B^BJ..., and the zero norms C^0„^ ... D^D.'... . AU pairs of these norms except the following pairs of zero norms (D,!*/), (D,D:). . . are conjugate. These last are such that sDiotD,' = sDoCri)/ = ... =-1 (10). From eq. (10) it further follows for these exceptional pairs that : — The pair of inotors A^', B," deduced from any such pair D,,D,' by the negative circular variation are conjugate and are respectively a positive and negative norm of (n). Therefore (n) consists of the following conjugate norms, {I) positive, A,A. . . . A^A„: ...A ,"A^". .., (2) negative, B,B. . . . 5/5.,' . . . B/'B/'. .., and (3) zero, 0,0, . . . G,'0: .... Two such complexes (ni) and (n - m) it will be observed are independent. They will be called semi- conjugate complexes. For the sake of brevity denote the complex A^A^... by {A), the complex A,A,...B,B.,... by {AB\ &c. Thus (ABCD) and (?)i) have the same meanings. Similarly (A'B'C'D') and (n-m) have the same meanings. Also order of (D) = order of (B') (12). The complex (A'B'OO'B) contains all the motors of {n) that are conjugate to {ABOD) or (m) and no others. For suppose ^ is a motor of (A'B'OO'B) and let E + A+B + B' = F be any motor of (n) where .4 is a motor of (A), B a motor of (B) and D' a motor of (B'). Expressing that F is conjugate to ^i , A.. . . we get ^ = 0. Expressing that F is conjugate to 5i, B„... we get 5 = 0. Expressing that F is, conjugate to B,, B.^... we get B' = 0. Thus if i^'is conjugate to (ABOB) it belongs to (A'B'OO'B); and it is easy to see that every motor of the latter is conjugate to the former. [It should be remembered that (00') is conjugate to the whole complex (n) and that (A) (B) (A') (B') and (BB') are all conjugate to one another; also (B) is conjugate to (ABOBA'B'O) and (B') to (ABOA'B'O'B').] M. o. 9 130 OCTONIONS. [§ 31 Thus when (n) and («i) are given, the complex (A'B'CC'D) is a determinate one. We shall call it the conjugate of (m) with reference to (n). Notice that the sum of the orders of (m) and its conjugate ■ exceeds n by the order of the complex, (G), which is common to (m) and the self-conjugate complex (CC) of (n). Note that the conjugate of the conjugate of (m) is not (m) in general but (ABCDC). We are now in a position to establish the statements in the following table. The complexes (n) and (m) and nothing more are supposed given. The first column contains a list of complexes which are then determinate. The second column contains a list of complexes which are to a certain extent arbitrary and describes the extent of the arbitrariness. Determinate Complexes Arbitrary Complexes (1) {C} (2) (CC) (C) is any complex which with (C) makes up the complex (CC) (3) {CC'D) (4) {CD) P) „ (C) „ „ (CD) (5) (ABGD) (AB) „ (CD) „ „ (ABCD) (6) (ABCC'D) (7) (A'B'CC'D) (A'B') „ (CCD) „ „ (A'B'CC'D) (8) (ABA'B'CC'D) Also (Diy) is any complex which contains (D) and is conjugate to (ABA'B') and with (ABA'B'CC) makes up the complex (n). [The second and third complex in each statement of the second column are determinate complexes. This is not true of the statement about (BD'). Hence it is not included in the table.] The statements in the first column are seen to be true by the following (now) obvious facts. (1) (ABCD) is the given complex (m) ; (2) (A 'B'CC'D) is the conjugate of (m) ; (3) {CC) and {CD) are the self-conjugate complexes of {n) and (m); (4) (6') is the complex common to {CC) and {CD) ; (5) {CC'D) is the complex containing {CC) and {CD) ; (6) {ABCC'D) is the complex containing {CC) and (m), (and indeed is the conjugate of the conjugate of (m)) ; § 31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 131 (7) (ABA'B'CC'D) is the complex containing (»i) and its conjugate. That (C) is any complex which with (C) makes up (CC) follows from the fact that what have been denoted by OjCa. . . are any independent motors composing (C); and, with this restriction, that what have been denoted by C]0« ... C/C/... are any inde- pendent motors composing (CC). (AB) has been defined as the complex of positive and negative norms of (m), when (m) is expressed as consisting of a set of conjugate norms. Now any complex which with (CD) makes up (m), can be [§ 31] expressed as consisting of conjugate norms ; these will be conjugate to (CD) since (CD) is conjugate to the whole of (m) ; and there will be among them no zero norms, since in a set of conjugate norms composing (m), (CD) contains all the zero norms. The conjugate norms of this complex may therefore be taken as our A^A^ . . . B^B^ . . .. That is, (AB) is any complex which with (CD) makes up (m). Exactly similar reasoning shows that (A'B') is any complex which with (CC'D) makes up the conjugate of (m). It remains only to prove that D^D^ ■ ■ ■ can be so chosen as to form any complex (S) which with (C) makes up (CD), and that D^Dj'D^^ . . . can be so chosen as to form any complex (S8') which contains (S), is conjugate to (ABA'B') and with (ABA'B'CC) makes up (n). DiD^... belong to the complex containing (8) and (C), and D/D^'... belong to the complex containing (SS') and (CC). The first statement follows from the fact that AA--- and (C) make up the complex (CD) as also do (8) and (C) ; the second from the fact that (88') belongs to the complex (CC'DD') since it is conjugate to (ABA'B'), and is independent of (CC) since with (ABA'B'CC) it makes up (n). Put now A = 7i + Si, A' = 7i' + S.'. A = 72 + S^.- , where 717,72. . . belong to (CC) and 8,8/8,. . . to (BB'). Thus BA- ■ • must belong to (8) [and 7,72... to (C), though we do not require this]. Since j.yi'y^--- belong to (CC) they are conjugate to every motor of (n). Hence for all values of ^ and q, sBp^Sg = sDp^Dg, sB;^B,' = sDp'mDg', s8p^8/ = sD^^D,'. Hence D^D,'D.. . . may be replaced by 818/80. . . 9—2 1 32 OCTONIONR. [§ 31 This proves all that is required, but we may as well here prove the more general theorem : — D-^D^. . . may be taken as any indepen- dent motors which form a complex which with (C) makes up the complex (CD). To prove this we have only to show in addition to what has just been proved that B^ and D^ may be replaced by any two independent motors of the complex D^, D^. If we put A, = a; (cA + sA), As = y (s'A + cA), c^-ss' =1, A] and Aj are {x and y not zero) any such independent motors. But if we further put A/ = x-^ {cD; - s'A'). a; = y-^ (- sA' + cA'), then from the facts that -Di, A. -Di', A' are all self-conjugate and all conjugate to one another except the pairs (A^/) {DJ)^) for which sDy-arB^ = sD„ztD.2 = — 1 , we deduce similar facts for Aj , As, A/, A„'. Hence A, A', A, A' may be replaced by Ai, A/, A-:, A/. As an example of the above theorems put cr = 1 ; let (n) be the complex i, fii, j, Q.j, k so that n = o; and let (m) be the complex i, (1 + ^b'D.)j, k so that m = 3. No type of motor has more than one representative here and three of them, B, A' and C", are zero. The simplest values for the typical motors are A = 6-1 (1 + J6-n)j, £ = 0, C = A;, D = i, ^' = 0, 5' = 6-1 (1 - \hm)j, c = 0, D' = m. First notice the definite complexes of the first column : — (C) and (CC) are each the complex k; (CC'B) and (CI)) are each the complex k, i; (ABCD) and (ABCC'D) are each the complex (1 + ib'n)j, k, i; (A'B'CG'B) is the complex (1 - i6^fi)j, k, i; (ABA'B'GC'B) is the complex j, %, k, i. Let us now give the more general values possible to the typical motors. The number of the motors of any type remains always the same. Hence in this case we must have A' = B = C = 0. (C) is a determinate complex, so we may put generally C = k. [We might of course put C = ck, but this does not render things clearer.] (D) is any complex which with k makes up the complex k, i. Hence we may put B = i + x-Jc. § 31] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 133 (AB) is any complex which with k, i makes up the complex (1 + \h^il)j, k, i. Hence we may put A = b-' (1 + ^b'D,)j + xjc + x^i. [The coefficient h-"^ is to ensure that s^= = - 1.] {A'B') is any complex which with k, i makes up the complex (1 - ^h^D,)j, k, i. Hence we may put B' = b-'{l-^b''a)J + x,k + xfi. (DD) is any complex which is reciprocal to A and B' and which with A, B', k makes up the given complex i, ni,j, ilj, k. To get the general value of this it is easiest to assume D' = fi + ^'D.i + r,j + 7,'ilj + i;k. Expressing the fact that sDi)' = - 1 we get f' = 1. Expressing that D' is reciprocal to A and B' we get «3 + hvh + v'b-' = 0, ^5 — ivb + 'r]'b~^ = 0, SO that B' = m - x-iA + x^B' + xek + ^ (x^- - Xe-) i, the coefficient of i being determined by the condition sO'^ = 0. It will be noticed that the sum of the orders of (m) and its reciprocal, i.e. of (1 + \b^rL)j, k, i and (1 —\h^£L)j, k, i is six, i.e. it exceeds five the order of (n) by one the order of (C). All the above theorems lose their complexity when w is an energy function as defined at the beginning of § 30. By that definition there are no negative norms in this case. This implies not only that the B's are zero but also the D's, for every D necessitates the existence of a negative norm (D — D')/>J2. It will be noticed that semvconjugacy only occurred by reason of the D's. Hence in the case of an energy function all semi-conjugate complexes reduce to conjugate complexes. Connected with this is the fact that ivhen -as is an energy function, if sEwE = for a motor E, then tsE = 0. For whatever motor F be s (xE + yF) ZT {xE + yF) = 2xysFi!7E + y'sF^F is negative or zero for all values of x and y. Hence sFvsE = or, by §14, wJS'=0. [If ts is any self-conjugate and if sEvrE—0 when E is one of a set of six conjugate motors, inE = 0, for E is then by § 30 conjugate to every motor of a complex of the sixth order, i.e. to every motor in space, so that sFuE = where F is 134 OCTONIONS. [§ 31 any motor. When isr is an energy function E need not be thus restricted.] If we. call m (< n) independent conjugate norms of a complex of order n a partial set of conjugate norms, and n independent conjugate norms a complete set, we have by the above theorems for an energy function: — Any complete set of conjugate norms can he transformed by conjugate variation so as to include any given partial set. And again hi a complex of order n, to any partial set of m conjugate norms can he added n — m independent norms which are conjugate to one another and to the given partial set. The table given above also takes a simpler form as follows: — For an energy function the complexes (C) {CC) (AG) (ACC) (A'CC) are determinate; (C) is any complex which with (C) makes up (GC), (A) any complex which with (G) makes up {AG), and {A') any complex which with (CC) makes up (A'i = 6, there cannot be any G"s in this case. § 32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 135 In this case the table above gives the following ; — The complexes (D) (ABD) (A'BD) and (ABA'B'D) are deter- minate ones; (AB) is any complex which luith (D) makes up the coviplex (ABD), {A'B') any complex which with (D) makes up (A'B'D), and (DD') is the complex reciprocal to (ABA'B'). 32. General expressions for self-coi^ugate flinctions and their reciprocals by means of sets of coi^ugate norms. Returning to the general meaning of sr, Ai...Ap, Bj ... Bq, (7i ... Gr, let n ox p + q-\-r be 6. Let Ai...G,. have the meanings with reference to A-i.-.C,- that Ai...A^ had in § 28 with reference to A^... A^. We proceed to show that VtA, = Zl, . . . TnAp = Ap, VtB, = - 5i, . . . T^rBg = - Bg, ^C,= ...=^Cr = (1). The equations otCi = . . . = ■srCr = were established in last section. They may be established also by the methods of establishing the other equations of (1). We saw in §28 that .4i ... 0^ were six independent motors (since by their definitions A^ ... Gr are six independent motors). Hgdcg CT^i = txA + l^yB + IzG. From this by § 28, s^iCT^i = — A'l, zAni^Ai — — X.,, ... sC,.-btJ.i = — z,-, i.e. «i = 1, x.2= ...= z,- = 0. Hence 'stA-^ = A^. Similarly for the rest of eq. (1). Now by eq. (21) § 28, E=- tAsEA - IBsEB - ICsEG. From eq. (1) it follows that 'uyE = -lAsEA+tBsEB (2), in which, be it observed, the total number of -4's and B's is not gi'eater than six. More particularly if th- be an energy function, say t/t, fE=-'ZAsEA (3). If yjr is a. partial energy function the number of A 'a is less than six. If it is a complete function they are six in number. 136 OCTONIONS. [§ 32 Thus if ■^ is an energy function it is necessarily of the form (3). Conversely if ■\/r is of the fonn (3) it is an energy function as can easily be proved by expressing any motor E in the form 'S.xA when we have sE-^E = — Ix"^. Similarly if ■sr is of the form (2) it is a self-conjugate function. If there are no G"s we have from eq. (1), ta-M, = ^i,...x3-iJi = -£i (4). It is important to remark that these values for tsr-^A^ &c. are the only ones consistent with the equation ■stzt-^E = E, where E is any motor, for if we put zr'^A^ = A^ + IxA + l,yB we get from the equation t^Ts'^E = E that ^i = ^i + IxA + lyB, so that all the x's and y's are zero. From eq. (4) and eq. (21) § 28 we deduce that ^-^E = -l,A%EA+l,BsEB (5). Similarly when ilr is a complete energy function we deduce from eq. (3) that ■<^-'E = -tA%EA (6). Thus OT~' is a general self-conjugate such that for no motor E, m~^E = 0; and y}r~'^ is a complete energy function. [That for no motor E is ot~^ E = Q may be established by putting E^'S^A+^yB.] If there are any Cs to-~' and A|r~' when operating on a general motor are unintelligible. It is convenient then to define ot_i and -v/r_i by the equations is_,E=-'2,A&EA + tBsEB (7), Vr_i-E = -2^S.E^ (8). In this case if E is any motor in the complex (AB), crcr-i E = E, and if E is any motor in the complex (AB), t^^^ts-E = E; and generally mt!T_^E = -l,AsEA-'S.BsEB (9), T!T^,^E = - tAsEA - l^BsEB (10). Thus ra- reduces any motor it acts on to the complex {ABy ^-1 » >. » ,, „ » » (■^;?) ■Brsr_i-£' is the component of E in the complex {AB) ■^-.■^E „ „ „ „ (AB)j (11). § 32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 137 It is necessary to explain here what is meant by " the com- ponent in the complex.'' This has not a definite meaning when the complex in question only is given. The term implies that a second complex independent of the first is also given. If (n) is the given complex and (6 — n) the independent complex, E can uniquely be expressed as a motor of (n) + a motor of (6 — n). The former is called the component in the complex (n). In the present case (AB) is not a determinate complex but (G) [| 30] is. Thus (AB) which is the complex reciprocal to (G) is a determinate complex, but (G) which is the complex reciprocal to (AB) is not. If however (AB) be determined in any way (say arbitrarily) the other complexes are all determinate. In this case (•BT and the complex (AB) both given) there is no ambiguity in the meaning of (11). From the above we see that when ^i ... G^ are a set of con- jugate norms for zr, then A^ ... G,. are a set of conjugate norms for •nr_i. A less general statement is that tlie six motors which are reciprocal to each set of five oat of six independent motors conjugate with regard to ■sr are themselves six independent motors conjugate with regard to w_i. Still supposing ;i=6, let m, (m), A^A., . . . B^B, . . . C^C, . . . D,D., . . . A,'A' ■■■ ^I'B.' ■■■Gi'Oi ... D.'D: ... have the meanings they had in § 31 and let A,"B,"A,"B:' ... be defined in terms of AA'A ••■ by eq. (11) § 31. First let the bar have its ordinary meaning with regard to A, ... B, ... G ... A,' ... B,' ... G' ... A,"B," .... Then by eq. (2) _ _ _ _ ^E = -XAsEA-lA'sEA' + tBsEB + XB'sEB' - 2 (A"sEA" - B"sEB"). Now remember that A, -D/ are obtained from A^", B" by a negative circular variation. Change the meaning of the bar so as to refer to the motors a,...b,...c,...a;...b;...g: ...d,d;.... By I 29 the new meanings of_2, 5, G, A', B', C" will be the same as the old, and the new A, A' will be the same functions of the old I/', A" that A, A' are of A,", A". In symbols D, = (A," + A")/V2, A' = (^/' - -B/')/V2, new A = old (I/' + A");V2, new A' = old (A," - A"),V2. 138 OCTONIONS. [§ 32 The last expression for 'o^E now gives ■^E=- tA&EA + 1.BSEB - tD'sED - XA'sEA' + I,B'sEB' - tDsEU') ^^^^' Similarly from eq. (7) ^_,E = -l,A%EA+tB&EB-lD%Eiy \ -lA'sEA' + lB'sEB'-XD'sED] ^ ^' The first line of eq. (12) gives the value of titE when E is con- fined to the given complex (m) or {ABOB), and the second line the value of -srE when E is confined to the semi-conjugate complex (6 - m) or {A'B'C'D'). Thus if E is confined to (m), tstE is confined to (ABD'). The first line in eq. (13) gives the value of ■sr_]^ when E is confined to (ABD'). If E is confined to (6 - m), utE is confined to (A'B'D). The second line in eq. (13) gives the value of vr^iE when E is con- fined to (A'B'D). Note that from equations (12), (13), we have as supplementary to equations (1), (4), (7) above ztD = D', ^D' = D ZT_,D' = D, t=-_,D = i)'j ^^*^' We may from the table of § 31 learn a good deal concerning the determinateness or arbitrai-iness of complexes involving A, B, &c. First however note the following (in which it is to be under- stood that (EJ) and [F] are independent complexes, and in which [ J is used to denote an indeterminate complex and ( ) a determinate one): — The statement that [F] is any complex which with a given complex (E) makes up another given complex (EF) is exactly equivalent to the statement that [J*] is any complex which with the reciprocal of (EF) makes up the reciprocal of (E). Let [G] be any complex which is independent of (EF) such that the sum of the orders of [G] and (EF) is 6. The reciprocals (FG) and (G) of the determinate complexes (E) and (EF) are themselves determinate. Moreover from this form of the reciprocals we see that [F] is a complex which with the reciprocal of (EF) makes up the reciprocal of (E). That by suitably choosing [F], [F] may be made any such complex is thus seen. Let § 32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 139 (1) [^] be any complex which with (0) makes up (FO) ; (2) [e] be any complex which is independent of (PQ) and is such that the sum of the orders of the two = 6 ; (3) (e), [(f)], [7] be the complexes deduced from [e], [^], (0) by the rules of § 28 above. Then (e) is the reciprocal of (FO) ; or (e) is (E). Also (e^) is the reciprocal of (0) ; or (e4>) is (EF). Thus [F] may be taken as [(f)] ; and if it is, [F] becomes [4>], which by definition is any complex which with the reciprocal of (EF) makes up the reciprocal of (E). We have proved then that if [F] is any complex which with (E) makes up (EF), [F] is any complex which with the reciprocal of (EF) makes up the reciprocal of (E). In the enunciation the converse of this is also stated, but the reader will see that this is the same proposition in other symbols ; [F], (G), (OF) replacing [F], (E), (EF) respectively. It is to be remarked that this proof depends on our complete liberty of choice of [G] as a complex of the proper order which is independent of (EF). For if [F] he taken as [(/>], then [G] must be taken as [7]. This restriction militates against the utility of the theorem for our immediate purposes. But, if [G] is not unrestricted, the above proof still shows that [F] is a complex which with the reciprocal of (EF) makes up the reciprocal of (E). The reciprocals of all the complexes of the first column of the table of § 31 are themselves determinate complexes. That is, the following are determinate. (Determinate) (ABDA'B'G'D), (ABDA'B'D')^ | (ABA'B'B'), (ABA'B'G'D'), (A'B'G'D'),[...(ro). (A'B'D'), (ABD'), (D') ) From the second column we have that (C') is a complex which with (ABDA'B'D^l \ makes up_(4fiX>^'5'C"J9') (D) „ „ „ (ABA'B'G'D')_ m_ak_es_up (ABDA'B'G'D') (IS) „ „ „ (A'B'G'D') make_s up (ABA'B'GD') (A'B') „ „ „ (ABD') makes up (ABA'B'D') , v.. .(16). 140 OCTONIONS. [§ 32 It is to be remarked that in (16) we do not say "any complex" because of the restriction in the present cases upon the {G) of the theorem just proved. All the statements of (16) are indeed obvious, and the only use of them is as a record of the connections between the possibly indeterminate complexes and the certainly determinate ones. When B7 is an energy function there are (§ 31) no B'& or D's and these lists become (Determinate) {AA'G'), {AA'), {A'G'), (I'), {A). ..{VI), (C") is a complex which with {AA') makes up {AA'C')...{1^). The other statements of (16) do not in this case require to be made, since {A) and {A') are definite complexes. If i!7 = 1 we have by equations (1) and (14) A = A, A' = A', B = -B, B' = -B', D=I>', D' = D...{19), and as we saw in § 31 G = C' = 0. In this case the list (15) only gives over again the first column of the table and (16) does not contain any information not at once obvious from the table. The following statements have a bearing on the matter of this section. Let (n) be a given complex of order n and (6 — n) the reciprocal of {n). Thus (?i) and (6 — n) are determinate complexes. Also let [?i] be any given complex of order n which is independent of (6 — n). Suppose now sEA = where A belongs to (n) and E belongs to [a] but is otherwise arbitrary. Then A must be zero. For any motor can be expressed as jE' + i^ where F belongs to (6 — n). Hence = sEA=s{E + F)A, since F and A are reciprocal. Since -E" + i^ is any motor what- ever it follows from § 14 that A=0. If {n) be the complex to which ct reduces any motor, i.e. the complex {AB) of eq. (2), {n) and (6 — n) are determinate com- plexes characteristic of -nr. In this case F being any motor of (6 — n), TsF = 0. For if E is any motor, rsE belongs to {n) and F to (6 — n). Hence = &F'r;TE=iE'srF, from which by § 14 it follows that 'stF=0. If any motor then be § 32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 141 expressed as a motor of [n] + a motor of (6 - n), ■or only affects the [n] component and reduces it to (n). Further if %E'tsF= where E is an arbitrary motor belonging to [n], tsF = 0. It may be noticed that for an energy function equations (12) and (13) reduce to ■^trE = -tlsEA -tl'sEl' (20), y}r_,E=-lAsEA-lA'sEA' (21). It is of importance to remark that when (m) is given the numbers of the motors of the different types A, B, C, B, A', B', C, D' are determinate and may be said to be characteristic of vr. For by the table of § 31 the orders of the complexes (C), (6"), (D), (and therefore {D'} which by § 31 is of the same order as (D)), (AB) and (A'B') are all determinate; and also by § 31 the number of J.'s and the number o£ B's in (ABCD) is determinate, and also the total number of A's and ^"s and the total number oi B's and B"s. Self-conjugate functions may therefore be classified according to the numbers of motors of these types. They may be classified with regard to the given complex (m) in two ways ; first accord- ing as they affect motors in (m) only, i.e. according to the numbei's of motors of the types A, B, G, D; and secondly according as they affect motors in (m) and any semi-conjugate complex of order 6 — m. It might be thought that in the first classification we ought to consider (7s and D's as of the same type since they are both conjugate to the whole of (m). But this is not so ; for by eq. (12) we see that while vtG is zero, tuB = D'. For the first classification note that the order of (A), (B) or (C) is anything from to m ; and the order of (D) is any number not greater than the less of the two numbers m, 6 — m. For the second classification we have similarly that the order of (A), (B) or (G) is anything from to m; the order of (A'), (B') or (C) is anything from to 6 — m; and the order of (DD') is any even number not greater than the less of the two numbers 2»n, 12 — 2m. We find then according to the first classification that the number of types of sr is (3 + m)l (2m-4)! (S + m)l S\ ml 3! (2m- 7)! ^^ 3! m! 142 OCTONIONS. [§32 according as m is or is not greater than 3. From this we have the following table : m 1 4 2 3 ' 5 6 Number of types of ta when classified according as it affects motors in (m) only 10 20 31 36 28 The total number of types according to this classification is 129, but it must be remembered that one of the 28 cases when m = 6 is -CT = 0. According to the second classification the number of types when m has the given value irio is the same as when in has the value 6 — wio ; for the number of types with reference to the given complex in the first case is the same as the number with reference to a semi-conjugate complex in the second case, and conversely. With this classification, rejecting the case ir=0, we shall find that the number of types of to- when m has any value from 3 to 6 is the coefficient of a^ in (1 - x)-' (1 - x')-' {l-af "-»") (1 - x'^+'f (1 - ic'-'»)^ diminished by 7, from which we have the following table: Number of types of nr when classified according as it affects motors in (m) and a semi-conjugate complex 6, 27 5, 1 208 4,2 407 477 The total number of types according to this classification is 1119. 33. The common coi^ug^ate systems of a general self- coi^ugate function and an energy flinction. If we examine the number of conditions that must be satisfied we shall find that if 13-1 and in-j are any two self- conjugate functions there are in general six motors forming a conjugate set both for ra-j and 'sr^. But by this process we are left in ignorance of the meaning of our results in certain limiting cases and also whether the six motors are real or not. Let now •sr have its usual general meaning and -^fr be an energy function. Limiting ourselves to the motors of a given complex § 33] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 143 (m) of order in let us enquire whether with reference to these two functions m such real motors exist within the complex. Let then Ai... B^ ... C^ ... be m conjugate norms with regard to ■BT, Aj... being the positive norms, 5; ... the negative norms, and Oj . . . the zero norms. We proceed to show that these can be so chosen that if the positive norms be divided into two groups ^i...ai... and the negative norms into two groups 5].../3, ... (the numbers in the a and /3 groups being the same), the following will be true. ^ ]...«!... are positive norms with regard to w, jBj.../3j... negative norms, and C\... zero norms, all conjugate with regard to zr; ^1 ... £i ... Ci ...(«! + A )/V2 • ■ . (ai - /3,)/^/2 . . . are also a conjugate set of motors with regard to yjr, and of these (a, - ^j) V2 ... are self- conjugate with regard to -v/r. Note that it is not here asserted that J 1 . . . are norms with regard to ^. Also note that the motors of types A,B,C,{'x + /S)/V2 on which alone ■•^ has any effect are a conjugate set with regard both to -bt and -v/r, being norms for isr, (a + ^)/V2 being a zero norm for m. And again the motors of types A, B, G, (a - /9)/v'2 are a conjugate set with regard both to •nr and i/r, being norms for in-, (a - /S)/\/2 being a zero norm both for w and -yjr. We arrange the proof thus : — (I) If for any motor JE, sE-\jrE = 0, then ■^E = 0. (II) For any two motors E and F '''^''^> sEfE^^fFfF ^-^ o^^E=±^F (1). (III) If any two of a set of m motors forming a conjugate set of norms with regard to ct are not already conjugate with regard to a/t they can be made so by a ^-conjugate variation unless they be a pair a /8 which are a positive and negative norm respectively with regard to -ar, and are such that if-a = i^/3. (IV) In every such transformation the product S^il/r^i ... SCryp-Cr, involving all the motors which are not self-conjugate with regard to yfr, diminishes so that this product has a minimum value which is only attained when the set of •or-conjugate norms is expressed as a series of motors A^ ... Bi ... C, .. a^ /Sj, ... as described in the enunciation. 144 OCTONIONS. [§ 33 (I) has already been proved in § 31. (II) Since yjr is an energj' function we have that s (ccE + yF) yjr (xE + yF) = w^sEfE + 2xysEylrF+ y^sF^^rF, is negative or zero for all values of x and y. Also sE->^E and sFfp-F are each negative or zero. Hence sEyfrEsFyjrF ^ s'^EfF. Now unless sEylrE = sF-\{rF {sEfE+ sFyjrFf > 4^sEy}rEsFy}rF. Hence either {sEyfrE + sFyjrFy > is'EfF or sEylrE= sF-lr(E + F) = 0, so that by (I) yjr{E + F) = 0. Hence either This proves (II). (III) If Ai and Ao are not already conjugate with regard to yjr they can be made so by the circular variation (which by § 31 is a -nr-conjugate variation) Ai = Ai cos 6 + A„ sin 6, A.! = - A^%u\0 -V A„ cos Q, tan 20= . , ■ ^ ." , . , for 6 can always be determined to satisfy this equation. Similarly if B^ and B^ are not already conjugate with regard to 1^ they can be made so by a cii'cular variation, and if 0, and G« are not already conjugate with regard to if- they can be made so by a circular variation. If A and G are not already conjugate with regard to >/r they can be made so by the linear variation (which by § 31 is a ts- conjugate variation) A' = A->rsG, G = G, s = - sAyjrClsCyfrG, for since A and G are not ilr-conjugate, yfrG is not zero, i.e. by (I) sGyjrC is not zero ; so that s can be always determined to satisfy this equation. Similarly if B and G are not already conjugate they can be made so by a linear variation. § 33] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 145 If A and B are not already conjugate with regard to yjr they can be made so except when -^frA = + yjrB by the hyperbolic variation (which by § 31 is a ■ar-conjugate variation) A' = A cosh e + 5sinh 6, F = A sinh 6 + 5cosh 6, sA'fA + sByjrB ' for by (II) except when -ijrA = + yjrB, can be determined to satisfy this equation. [If yfrA = + yjrB the equation is satisfied by ^ = + 00 but the hyperbolic variation then becomes unintelligible.] By changing the sign of B if necessary we can suppose this limiting case to occur only when yfrA =ylrB. If yfrA^ = yfrBj = yfrB^ and each is not zero, the circular varia- tion (which is a ^-conjugate variation) 5/ = (B, + B,)/^2, B: = (-B, + 5,)/V2, makes B^' self-conjugate with regard to yjr and also makes ifrA^ different from ■(/f^/. Hence this limiting case can always be disposed of except for a series of corresponding pairs of positive and negative norms — (ai/3i) (ttj/Sa) ••• — as already described. (IV) That the product sAiyfrA^ ... sCr^yCr always diminishes by such a transformation follows from eq. (8) § 29 above. The rest of (IV) now follows. Hence the theorem is true. It might be thought that it would be simpler to start with a set of norms conjugate with regard to i|r and subject these to ■<|r-conjugate variations ; for these last consist only of circular and linear variations. But it would be found that difficulties occurred with limiting cases of the linear variations not so easily surmount- able as those of the above process. The number of disposable constants in choosing m motors, apart from their tensors, is 5m. The conditions that these must satisfy in order that the m motors may belong to a given complex is shown by Sir Robert Ball (Screws, § 49) to be m (6 - m). [This is shown by noticing that each of the m motors must be reciprocal to 6 — m definite motors.] The number of conditions that must be further satisfied in order that these motors may be conjugate M. o. 10 146 OCTONIONS. [§ 33 with regard both to -ax and i/r is m (m — 1 ). Thus the total number of conditions necessarily satisfied by the 5m scalars of such a common conjugate system is m (6 — m) + m (m — 1) = 5m. Hence in general there is in a given complex of order m only one such common conjugate system. If now we take the (generally) independent and definite complex of order 6 — m which is con- jugate with regard to T|r we can in general find only 6 — m definite motors in it which are conjugate with regard both to w and -v/r. The total of six motors thus found will form a conjugate set with regard to ifr but not in general with regard to ct. Hence when the complex of order m is an arbitrary one we cannot in general find another complex which is conjugate or semi-conjugate with regard both to ot and i/r. We cannot then in the present case hope to express both ct and ■\(r in a manner similar to eq. (12) § 32, where the A's, A"s, B's and B"b have the same meanings for both ■ax and i/r. Suppose however in addition to our present m motors (forming a conjugate set of norms with regard to ■57) we take 6 — m independent motors H^, H^ ... He_m forming an indepen- dent complex of order 6 — ?)i. For the sake of definiteness we may by § 31 suppose that this complex is conjugate to (m) with regard to yfr and that all six motors are conjugate with regard to yjr. Now apply the bar introduced in § 28 above to these six motors. By the method of establishing eq. (2) § 32, we see that whm E belongs to (m) i|r£'= - taAsEA- IhBsEB- 'S,cGsEC -^■Zdid + ^)sE_(a + ^l ^E = -1{A + A') sEA + 1{B + B') sEB - ^G'sEG - 2 {(a + a') sEa - (/8 + /9') sE'^}) ■(2). where A', B', C, a', /3' all belong to the complex Hi, H^ ..., i.e. to the complex reciprocal to (m). We might generalise the expres- sion for i\rE by adding to the right of the equation — "ZhHzEH when it would be true for any value of E. Similarly the expres- sion for tsE might be generalised by adding — IKzEH to the right of the equation, K standing for any motor whatever. Again, for most uses of eq. (2) it is unnecessary to distinguish between .(3), § 33] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 147 the three types A, B, C of motors. Writing G for any (not zero) scalar multiple of any one of them we get - 1 2d (« + /3) s^(a + ^) - IhHsEHl ^E = - S{fG + G')sEG - 2 {(a_+ a') sEci - (/3 + ^') sE0} - tKsEH where (?!, ^2 ... Oi, /3i, a, ... form the given complex (m)\ Hj, H^... form an independent complex (6 — m) Gi, ... «,', ,Si' ... be]ong to the reciprocal of (m) (i.e.H„H,...) V (4). K^, K2 ... are any motors whatever Qi, g2--- di, d^, ... are positive or zero scalars fi,fi--- are any scalars whatever For the important case when vi = 6 there is no H, G', a', y3' or K, and we get the simplified form fE = - IgGsEQ - \td (a + B)_sE(ci + ^) ^E=- IfGsEG - 2 (asEa - /9s£/3) (?!, G^, ... tti, /S], 02, ■■• six independent motors; gi, g^ ... di, d^ ... positive or zero scalars; /u/a, . any scalars For application to the dynamics of a rigid body there is one very simple case. Suppose •et and i|r are both energy functions either both complete or both partial in the following way. — (1) If E be any motor in a certain compleos neither sEyp'E nor sEzrE is zero ; (2) hoih ■sr and, y]r reduce every motor they act on to a second complex of the same order as the first. The case of ■or and ■\jr both complete energy functions is clearly a particular one of the other. Take the first complex as the complex (m) of eq. (3). We see that there can be no a or /S for ■yjr('i — ^) = 0, so that putting E=a- /3, sEyjrE is zero. No g can be zero, for if ^r, = 0, sG^fG^ = 0. Gi,Gr,... are (§ 28) independent, and therefore form a complex of order m. This must be the second complex mentioned above, since any motor l.a:0 of this complex is a motor obtained by operating by t/t on some value of E, viz., E = Xa;g~^ G. Hence every h must be zero. Turning now to w it follows that every/ must be positive and not zero and every G' and K must be zero. We have then ■^E = -l^gQsEG, ^E = -'ZfGsEG (6), 10—2 ■(•5). 148 OCTONtONS. [§ 33 from which with the meanings of i/r_i and ■Br_i of equations (7) and (8) of § 32 y}r_,E = -lg-'OsEG, '!!T.,E = --Zf-'GsEO (7). From this again putting T^_ior = (f> j>E = y\r_,^E = - Ig-'fGsEO, Gm = gm~^fmGm (8). Also if F((f>) be any algebraic function of (such as i) F ((/)) (?, = F{gr'A) ■ Gi, &c., so that F(4>)E = -^F(g-^f).GsEG (9). Another important case is when m = 6 and ct = 1. The motors conjugate with regard to ct are then reciprocal motors. It is in this case convenient to retain the A, B notation. We saw in § 31 that the number of motors of any type A, B... was definite. Now when m = 6 and •nr = 1 there can be no C's, for -arC = 0. Also all the motors in space belong to the complex of six reciprocal motors (1 +jofl)i, (1 ±pQ.)i, (1 ±pD,)k of which three have positive and three negative pitch. It follows that the Act group must be three in number and B^ group also three iu number. Also as in | 32 A = A,B = -B. Hence ylfE=-t (aAsEA + bBsEB) - ^Id {a - ^) sE (a- /3)...{10), where A, B, a, represent reciprocal motors such that sA' = Sa' = -s& = -s^'=-l. From eq. (10) ^yA=aA, y}rB=-bB, -f (a + ^) = d(a -/3), f{a-/3) = 0...in). Unless the a, /3 terms are absent there are not six real co-reci- procal motors which are conjugate with regard to i/r, nor are there six co-reciprocal motors for any one (say E) of which y]rE is coaxial with E. If these terms are absent there are six such motors. As examples the two cases may be considered yjrE = - aisEi - bjsEj - cksEk (12), ■^E = - anisEm - bnjsEnj - cnksEnk (is). 34. Scalar functions of motors. Complexes of the second degree. A linear scalar function f{E) of any motor E is defined as a scalar function such that for any two motors E, F f{E + F)=f{E)+f{F). § 34] MOTORS AS MAGNITUDES OF THE FIRST ORDEK. 149 From this it is quite easy to prove that / is commutative with ordinary scalars. The most general form of such a function is f{E) = -sEA (1), where A is some constant motor. For [eq. (8) § 15] f{E) = -f{Z)sEZ = -%EA, where A =Zf{Z) (2). lietf(E, F) be a scalar function of the two motors E, F, linear in each. The most general form of such a function is f{E,F) = -sEcl>F (3), where (^ is a general linear motor function of a motor. For f(E, F) = -f(Z, F) sEZ = - sEcpF, where ^F = Zf{Z,F) (4). A homogeneous quadratic scalar function f{E) of a motor E is defined as the function which is obtained by putting F=E ins, scalar function of E and F which is linear in each of them. From eq. (3) we see that the most general form of such a function is f(E) = -sEwE (5), where ct is a general self-conjugate, w being in fact the self-con- jugate part ^((f) + (j}') of -x)A,...(-x)AnH„^,...H,\ = (1). In the first place, if ^i be changed to any other motor XiAi+ ... +XnAn of (n) which is independent of Ai...An (so that x^ is not zero), the equation is unaltered. In fact the two expressions {(-x) AnHn+i ...He}, and {Ai.. AnHn+i ... Hg] are both thereby merely altered in the ratio of a;i to 1. We may thus change Ai...An to any n independent motors Bi...Bn of («,) and we shall have [{j>-x)A^...{4>-x) AnHn+i ...H,\ \Ai . . . AnHn+i . • • He\ _ {( 4>-x)Bi ...(- x)BnH„+i... H,} ^2^ [By... BnHn+i ■ . . He} [We may clearly on the right further change Hn+i ... H^ to Gn+i ■■. Gs any other 6 — ?i independent motors of (G — n), but this is unnecessary for our purpose.] Let the n roots of (1) be a repeated r times, b repeated s times, c repeated t times, &c. Thus [^l ... Jl.n-"-n+i ■•• -"6t ...(3), = «"- m<»-ii a;"-'+ . . . + (-y-^m'x + (-)"ml where {Ai. .4>AnHn+i...He\ [A,. ■ AnHn+i . . . He} 2{^.. ..Ap(j)Ap+,...A, = aAu 4>A^ = aA, + A,', ... ^Ar = aAr + Ar') B, =hB„ B,=bB,-\-B,', ...tfyB, =bB, +B,' ■■■(o), where Ap stands for some motor of the complex A^^, A^ ... Ap^y, and similarly for Bp', die. To prove this, § 66 of the Ausd. will be required. In our case it asserts that if for six motors D^ ... D^ {A...A} = o, Di...De are not independent. [If they are independent the primitive units Ei...Es can be expressed in terms of them. Expressing these units in this form we get {£'1 ... E^] = {A ••• A) X a finite scalar = 0. But by § 28 {E^ ... Eg] is not zero.] Since a is a root of eq. (1) we have {(-a)A,...(- a)AnHn+,... H,} = 0. Hence some relation of the form 11 6 ly{^-a)A+ S zH=0 1 «+i must hold, where all the y's and ^■'s are not zero. By hypothesis Sy ((f> — a)A belongs to (n) and -zH to (6 — n). Hence these two motors must separately vanish. Since Hn+i-Sg are in- dependent, all the z's are zero and therefore l.y{^ — a)A =0 where all the y's are not zero. Suppose that y^ is not zero. Then instead of A^ we may take ^yA, and we shall have (0 -a)Ay = 0. Hence a motor Ai can be found to satisfy eq. (5). Now suppose p independent motors A^... Ap can be found to satisfy eq. (5), p being less than r. We proceed to show that p+1 such independent motors Ai ... ApAp+i can be found. Take the Aj... Ap of eq. (1) to be our present A^ ... Ap. Thus { — x)Ai = (a — x) Ai, ((j) — x)A,. = {a -x)A2 + Aa', ... ( - a) Ap+i ...{-a)A„=A:. Hence (<^ - a)M, = (^ - a) J./ = 0. From this § 35] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 157 we see that (6 — ay operating on any motor of the complex A^, A^ reduces it to zero. Hence (^ - of A^ = 0. But ((^ -a)As = A^. Hence (<^ — of A^ = 0. The proposition is now obvious. If p be any positive integer and e any scalar different from a, ( — e)A = (a — e) xiAi + a motor of the complex A^ ... Ai^^. The two motors on the right are independent, and the first is not zero since neither a—e nor xi is zero. Hence {<^—e) A is not zero. Thus — e operating on any motor of the complex A^...Af reduces it to a non- evanescent motor of the same complex. It follows that (^ — eY J. is a motor o{ A^... A,- not zero. In particular ( — 6)* 5 = if B is any motor belonging to the complex B^ ... Bg. But b is different from a. Hence B cannot belong to the complex A^^ ... A,.. Hence the complexes A,,... A,. and B^ ... Bg are independent. Similarly all the complexes (Ai ... Ar) (Bi ... Bs){Ci ... Gt) ... are independent of one another, i.e. the r + s + t+... motors A^ ... B^ ... C^... are independent. Let now E be any motor of the complex Ai... ArB^ ... Bg. It can be put in the form E = A + B where A and B have the mean- ings just given to them. If then e is equal neither to a nor b (0 _ e)P .£■ is a non-evanescent motor belonging to the complex Aj ... ArBi... Bg-, for ( — eyA is not zero and belongs to A^. . .Ar and (^ — ey B is not zero and belongs to B^ ... Bg. Proceeding in this way and calling the complexes (Aj^...Ar), {B^... Bg) ... the complexes corresponding to the roots a,b,... respectively, we get the following generalisation of the last theorem : — If e be a scalar different from each of any assigned group of roots, (<^ — ey operating on any motor belonging to the complex con- sisting of the complexes corresponding to those roots is a motor not zero belonging to the same complex. From this we have the following important theorem ; — 1 58 OCTONIONS. [§ 35 The complex corresponding to any root is a perfectly definite complex. [The ^'s themselves may be chosen in various ways. Thus instead of .^2 we may invariably write Ar^ + yA^ where y is any scalar. The theorem asserts however that the whole complex A^... Aris & definite complex characteristic of (^.] Suppose A is any motor belonging to the complex A^ ... A^. Then any motor of (n) may be expressed s& A-\-E where E belongs to the complex Bi...BgCi... Ct ..- If this belongs to a complex corresponding in the sense just used to the root a we have (^ — aY(A+E) = 0. Hence E must be zero or the motor A + E must belong to the definite complex hitherto denoted by Ai ... Ar- If (<}) — e)P E = for every motor E of a complex included in (n) of order q, e must be a root of (1) repeated at least q times and the complex must he included in the complex corresponding to this root. For we have seen that unless e is a root (^ — e)P E is not zero for any motor E of (n). We may then assume that e = a. In this case (cf) — e)P E is zero only if E belongs to the complex Ai...Ar. Hence the complex must be included in A^.-.Ar. q is therefore not greater than r, and therefore the root e since it is repeated r times is repeated at least q times. Any motor E belonging to the complex («) can be expressed as A + B + C + ... where A belongs to the complex corresponding to the root a, B to that corresponding to h, and so on. Now we have seen that (^ - a)'' A = 0,{<^-hyB = 0, &c. It follows that (^ - a)' ((^ - 6)» (" - ml"-') + (-)" m = (6). When TO = 6 the given complex includes every motor in space. Hence invariably satisfies a sextic whatever motor the operand may be. When we have found the roots of the n-tic satisfied by <^ we can always find the complexes corresponding to these roots by the following simple property. ((/> - 6)* (^-c)* ... E where E is any motor belonging to (n), and where all the roots except a are involved, is a motor belonging to the complex corresponding to the root a, for operating on it by (<^ - ay it is reduced to zero by 36] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 159 eq. (6). Moreover, by giving E different values any motor in this complex can be expressed in the above form. To show this, it is only necessary to show that { — b)' . . . Ai, () be any algebraic function of we see by equation (5) that f{4>)A,=f{a)A, f{) A2 =f(a) A^ + a, motor of the complex A^, f{4>) A3 =f(a) Ai + a motor of the complex ^1, A,^ ■in But putting /(- by {-cy ..., f{a) is not zero. Thus in this case f((f)) Ai,f{^)Ai, ... are r independent motors of the complex Ai ... Ar- 36. G-eneral forms of , its conjugate and its reci- procal. Express Ap in terms of .4, ... Ap-i as follows: ^Ai = aAi (pA^ = a (a^Ai + A^) B, = bB, B, = b {h,B, + B,) .(1). [If a = these expressions are illegitimate. Except however when A^sEA^ - ... Hence E = -a [A^sEA^ + {a,,A, + A^) sEA^ + . .^ + (ar,A, + ...+a,.r-.Ar-, + Ar)sEAr]-b\ ]- (2). Hence if ' is the conjugate of <(> 'A^ = a (^2 + a^s + . . . + a^Ar) (f>'Ar = aA^ The symmetry of the coefficients in (1) and (3) may be noticed. The expression for 'E may be written ' as the complexes (n) and (6 — n) stand towards . (n) is the reciprocal of (6 — n) and (6 — n) is the reciprocal of (w). (6) That the <^' n-tic is the same as the n-tic, since they have the same roots repeated the same number of times. (c) That if (^ be self-conjugate so that <()' = the complex A^...Ar corre- sponding to the root a of the (p n-tic is the same as the complex A^...Ar corresponding to the root a of the <^' n-tic] (d) That if to every motor in the complex corresponding to a different root. [For if A be a motor in the complex corresponding to the root a, (f)A belongs to the same complex and is therefore reciprocal to B any motor belonging to the complex corresponding to another root b ; i.e. A and B are conjugate.] Suppose now that none of the roots of the n-tic is zero. Then -^ E where E is any motor of (n) has a definite meaning, namely, that one motor belonging to (n) which when operated on by di gives E. -^ may be obtained from equations (1) by treating each motor in succession thus, <}>-'A, = a-'A„ -'(a^Ai + A^) = a-'A^, whence <^~M2 = a~^ {A2 — OaAi). §37] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 161 Proceeding in this way we obtain 4>~^A„ = a-i (- ao,A-i + A„) 4>~^A3 = a-i {(asM.^ - asi) J., - a^A„_ + ^3} 4>~^A^ = rt-i {- (a„a.3^a„, - a^jaai - a^^a.^ + a„) A^ + (atsasi - 049) A^ - a^^As + A^}! .(5). -'B, = b-'B, It is unnecessary to go further as the coefficients can always be at once written down in the form of determinants. It is easy apart from determinants however to see the law of the coefficients. For instance the coefficient of ^] in -^A,= a-' (a^A^' - Ai + aA^ \ (6). <^-' J.4 = a-^ {- (a43a32 - ^42) M + ^43-4 3' - Al + aA^ From these we see (a) That -' is a linear motor function of a motor. (6) That the roots of the (/)-^ n-tic are the reciprocals of the roots of the ^ ??,-tic. (c) That the complex corresponding to any root of the -^ w-tic is the same as the complex corresponding to the correspond- ing root of the <^ ?i-tic. (d) That <^'-» is the conjugate of <^-^ 37. Some properties of self-conjugate flinctions and commutative functions. In this and the following sections we propose to notice certain miscellaneous properties of different kinds of linear motor functions of motors. In § 36 we saw that when is self-conjugate the complexes corresponding to the different roots of the sextic which it always M.O. 11 162 OCTONIONS. [§ 37 satisfies are reciprocal to one another. It is not true however that for a real self-conjugate ■bt they are always real. For instance, put ■57 J? = X {D.isEili - isEi) + y {D,jsED.j -jsEj) + z {nksEnk - ksEk) (1), where x, y, z are real ordinary scalars. Here m = — x€Li, zTfli = xi. Hence if v be put for the imaginary \/(— 1). we have ■ST (i ± vHi) = ± vx (i ± vfli). Hence vx and — vx are roots of the •et sextic, and the complexes corresponding to them are those of i + vfli and i — i/Ht. The sextic is (^= + «^)(^=' + 2/0(^' + ^') = (2). The complexes corresponding to the imaginary roots + vx are both imaginary, but the complex consisting of these two imaginary complexes is itself real, being in fact that of i, D,i A similar statement may be made of the general real (ji. For the coefficients of its sextic are real. Hence, if an imaginary root a'+va" occurs r times where a' and a" are real, another imaginary root a' — va" also occurs r times. It is quite easy to prove that the complex of the 2rth order consisting of the two imaginary complexes each of the rth order corresponding to these roots is itself real ; that included in it is a real complex of the second order for each motor E of which {(^ — a')- + a"^} E = 0, & real complex of the fourth order for each motor E of which and so on ; and that for every motor E of the complex of the 2rth order, [{^ - aj + a"^Y E = 0. Returning now to eq. (2) we see that there is no motor E in this case for which t:tE = aE where a is real, for if there were a would by § 35 be a root of the sextic, whereas in this case all the roots are imaginary. Thus in Octonions a self-conjugate ot differs from the quaternion self-conjugate function and also differs from Grassmann's self- conjugate function {Amd., § 391) in that the sextic may have imaginary roots. In the present properties it is the energy § 37] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 16S function -f which is the complete analogue of the other self- conjugate functions. For the roots of an energy function sextic are always real. This is shown by eq. (11) §33 above, in which by § 3.5 the root corresponding to A is a, to B, - h, and to a + /3 and a - /S zero twice repeated. This last is seen by comparing the equations t (a - /3) = (a - /S), A/r (a + y3) = £^ (a - /9) + (a + ;8) with eq. (5) §35. We can clearly identify the present a - /3, a + ^, zero, with the ^i, A^, a of that equation. There is an allied property in which the present self-conjugate differs from the quaternion and Grassmann's self-conjugates. If a repeated root a is real or imaginary it does not follow that there are two independent motors ^i and Ao. for which -sr^i = aA^, ■tsAn = aAo. This is illustrated by the self-conjugate commutative function. The general form of this is given by eq. (1) § 19 above. Changing the <^ of that equation to -ur we have vrE = - XiSiE - X'jSjE - X"kSkE (3), where X, X', X" are scalar octonions. Put Z=«4-%, X' = x' + ily', X" = a;" + ny", where x, y, &c. are ordinary scalars. Then m = (.x + VLy) i, ra-ni = xD,i. We may therefore identify the present Hi, i, x with the ^i, A„_, a of eq. (5) § 35. Hence by § 35 « is a root of the rs sextic twice repeated. The sextic is (VT - Xy {^ - Xj {-ST - X'J = (4). Also the complex corresponding to the root x is that of Hi, i. There is one motor E, viz. Sli (and scalar multiples of it) for which E = xE, but there is no other (except when i/ = 0) as we can see by trying E =ai + bD,i. The analogy in this case breaks down even for an energy function with a twice repeated zero root such as we considered just now, though it does not break down for any other root of an energy function sextic. The reason that the present self-conjugate differs from Grass- mann's is again that Grassmann assumes his extensive magnitudes to be such that if their "numerical values'' are zero they are themselves zero. 11—2 164 OCTONIONS. K 37 It is easy to express the -bt of eq. (3) in the form [eq. (2) § 32]. When x is not zero we have - XiSEi = - JiS£'A + B^'&EB^ (5), where B^ = a-'x-^[x + lD.{-y-a?)]i\ ^ '' 1,= a-'{x + \n{y + a')]i\ .^. B, = -a-'[x+in{y-a')]i] '' where a is any scalar. Moreover it is easy to show by hyperbolic variation (§ 31) that with a arbitrary the above values for A-^ and B^ are the most general values for two conjugate norms in the complex i, VLi corresponding to the repeated root x of the sextic. If a; = 0, we have - XiSEi = -ynisED,i (8), so that if y is positive, A-^, Cj are here a conjugate positive and zero norm respectively in the complex i, fli, and if y is negative Bi and Cj are respectively a negative and zero norm, where A, = il^/y, C^ = ni\ Ai = ijy . D,i, Gi = i ^ Bj = il^(-y), C, = ni} B, = ^{-y)Mi, G, = i i We get the general expressions for the norms in the complex i, m in the case of eq. (9) by the linear variation A,' = A + pcyvy = (1 + np) ii^y, c; = c, = riii a; = a, = >jyni, c; = - pAj^y + e, = (1 - pfi) i J ■ ■ "^ '' the second of these lines being written down by the transformation at the end of § 29 above. A similar treatment may be accorded to eq. (10). We may now treat — X'jSEj and — X"kSEk in the same way as we have treated — XiSEi. Thus in all cases the in- of eq. (3) has been expressed in the form — ^AsEA + IBsEB. The following deductions may be made from these results : — For a commutative self-conjugate function (a) The roots of the sextic are all real and consist of three pairs of equal roots. § 37] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 165 (b) The three complexes corresponding to the three roots are three sets of coaxial motors. The axes of these sets are generally determinate and always form a set of three perpen- dicular intersecting lines. (c) The function can never be a complete energy function and can only be a partial energy function when it degenerates into a lator function. [For imless .r = «' = a," = there are B's.] The roots of the sextic are in this case all zero. (d) Except when sX = sX' = sX" = there are not six co- reciprocal motors forming a conjugate set also. For suppose in eq. (1) § 32 the A's and B's are co-reciprocal. By eq. (26) § 28 A^ is an ordinary scalar multiple of A^. Hence ■stAi = xAi where x is an ordinary scalar. But we have just seen that except when y = y' = y" = Q there are not six independent motors for which this equation is true with the present in-. It may be noticed that the ot considered at the end of § 34 above is a function of the kind now under consideration for which X = x' = x" = \, y = y' = y" = — 2h. The sextic is therefore (t^ - ly = 0. In the present case {■sr — ly reduces every motor it acts on to zero. The statement made about the roots of the ct sextic occurring in pairs is true of any commutative function. In fact, the sextic of a commutative <^ is what in § 17 was called the (/>j cubic squared and equated to zero. For let A, B, C he any three completely independent axial motors (§ 18). If the cpi cubic is 01^ — ?i"^r + n'(j)i — n = 0, we have by eq. (1) § 17 and eq. (27) § 16, ^ _ n"x^ + n'x -n = S,{x-)A{x-)B(x- (/>) CSr'ABC. By equations (1) and (6) § 35, the sextic is ^fi _ ni'cp' + m^cf)* - in"'' + m"^'' - m'4> + m = 0, where 01? — m'oi? 4- . . . -f- m _ \{x - )A (:c - 4>)B(.T - (^) C(a' - ) nA {x - (f>) D.B{x - (/>) fiC) = ■"" ~ [ABUD.AilBilV\ ■ 166 OCTONIONS. [§ 37 Hence by eq. (27) § 28 afi — mV ■\- ...+m = {a? — n"oi? + n'x — n)-, or <^« - m'^" + m"'' - m"'^ + m"<^'' - m'(f> + m = {^ - n"(f>^ + n'(f> - ny (12), which proves the proposition. It may be noticed that in this case the Ai,Bi ... of eq. (5) § 35 may invariably be taken as lators, for if a is a root of the sextic we have by what has just been shown S,{-a)A{(j>-a)Bi — a) A, {- a) B, ( — a) C are parallel to one plane or one is a lator. In either case n(J(X-' + Y% A,' = (- YA, + XA,)/^(X-^ + FO (7), where X and Y are any two scalar octonions for which not both the ordinary scalars are zero. Hyperbolic variation may be defined as a; = (XAp + F^,)/V(Z^ - F=), a; = (ya^ + xa,)!>J{X-^-y^) (8), where X and F are any two scalar octonions for which the ordinary scalar of X is numerically greater than that of F. A commutative combinatorial product of any number of motors is unaltered by a multiple commutative combinatorial variation. From this it may be deduced that if to- be a commutative self- conjugate function and if J., £ be varied to A\B' SA'^u^A'SF'TsB' - S^A'vtB' = SAvrASBmB - S^AvtB. . .(9), from which by taking the ordinary scalar part S^A'v7A'S,B''stB' - S,^A'^B' = Q^A'^^^AS.BwB - SM^B. . .(10). [This of course may be generalised to statements similar to those in § 29 above, but equations (9) and (10) are sufficient for our purposes.] Two motors A and B are said to be fully conjugate with regard to •or when SAi^B=0 (11). 168 OCTONIONS. [§ 38 Thus they are conjugate in the ordinary sense and are also such that S,A'stB = 0. If A atid B are not already fully conjugate with regard to vr they can invariably he made so by a commutative circular variation, and in this case the product SiAwASiBtitB diminishes except when SiA-btB = when the product remains unaltered. Three completely independent axial motors can always be found which are fully conjugate with regard to -bt. For by the commuta- tive circular variation just mentioned we can go on diminishing the product SiAvsAS^BTsBSyC-urC unless Q,BztC= SfiizA = Si4w£= 0. In this case if we vary A, B to A\ B' hy a, circular variation so as to make SA'^rsB' = we shall find that A' = A-\-D,rB, B' = - D,rA + B, and therefore that SA'-5rC = S4wC, SB'^G = SB'btG. Hence in this case we can make each pair conjugate without affecting the conjugacy or otherwise of the others. If SA'stA = - 1, SB7!tB= 1, SAo^A, = -n, S5„w5„ = n, SCv!G = Q (12), where A, B, A^, Bo, C are axial motors, the motors will be called norms of types A, B, A^, B^, G respectively. A set of three real norms fully conjugate luith regard to sr can always be found. The norms of types Q^o. ^-60, G, ^G form a definite complex. The norms of types HA, flB, A,,, H^o. -Bo, HB^, G, flG, also form a definite complex. •stAo and ztBo are lators and ■btC = 0. By the axial complex of any number of motors A, B... is meant the complex oi A, ClA, B, D,B If a motor be fully conjugate to an A or a B in a set of con- jugate norms it belongs to the axial complex of the other norms. If a motor be fully conjugate to an A^ or B„ in a set of conjugate norms it belongs to the complex of HA^ or D,Bo and the axial com- plex of the other norms. [Since every motor is fully conjugate to a C we have no corresponding property for a motor which is fully conjugate to a C] § 38] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 169 A motor of the form IX A + SFjS + 2Z„^„ + 2FA + ^ZG where X...Z are scalar octonions cannot form one of a set of conjugate norms unless (1) IX^'-I.Y^ + n{lXo'- lYo'')=± 1 ; or (2) all the S.X's and S.Y's are zero and S, (SX„=- 2F„=) = ± 1; or (3) all the X's, Y's, S,X,'s and S,Y,'s are zero. What is here meant by a in--conj ugate variation may be thus explained. A pair of zero norms Cj and C^ may be varied to any two completely independent motors (7/, C/ belonging to the axial complex of d and G^, i.e. they can be varied (§ 14) to any two not parallel axial motors intersecting the shortest distance of Oi and G^ perpendicularly. Any other pair of norms E and F are varied according to the equations E' = cE + sF, F' = s'E+cF, ] c=-ss' = l, s'SEwE + sSF'mF=o] ^ ^' A multiple conjugate variation consists of a series of simple variations. Any motor which satisfies one of the three conditions mentioned in the last enunciation can he brought into a group of fully conju- gate norms by a multiple conjugate variation. Any group of conjugate norms can be obtained from any other by a multiple conjugate variation. The number of norms of any type in a set of conjugate norms is a definite number clmracteristic of the function ct. This important proposition may be enunciated otherwise. Refer back to § 9 for the meaning of positive and negative scalar octonions and positive and negative scalar convertors. Also refer to § 19 for the mean- ing of the principal roots of the cubic. [By the method of the present section we have not yet established the existence of these principal roots. That existence is however immediately deducible from the proposition below about the common conjugate systems of in- and an ellipsoidal function.] Then we have : — If A, B, G be any three completely independent axial motors which are fully con- jugate with regard to w, i.e. are such that SB'uyG = SCu^A = SAzyB = ; tlie numbers of tlie scalar octonions SA'usA, SB-stB and SGztG which are (l) positive scalar octonions, (2) negative scalar octonions, (3) positive coyivertors, (4) negative convertors, (5) zero, are tlie same as the numbm^s of the pnncipal roots of the ot cubic of the same types. [This connection with the principal roots is obvious from 170 OCTONIONS. [§ 38 the fact that i, j, k of eq. (1) of § 19 are fully conjugate with regard to the <^ of that equation.] If the self-conjugate commutative functions themselves be classified according to the numbers of conjugate norms of different types appertaining to them, rejecting the case ■sr = 0, it will be found that there are thirty-four kinds. To see by analogy to a certain extent what these different kinds of isr imply geometrically, take the corresponding quaternion case. ■ET being a self-conjugate linear vector function of a vector we have, rejecting the case •sr = 0, nine types of in- according to the numbers of roots of the ot cubic that are positive, negative, or zero. Interpreting this geometrically we have nine kinds of central conicoids not passing through their own centres repre- sented by the quaternion equation Sp-srp = — 1. These are (1) the ellipsoid, (2) the hyperboloid of one sheet, (3) the hyperboloid of two sheets, (4) the imaginary quadric which is not cylindrical (or conical), (.5) the elliptic cylinder, (6) the hyperbolic cylinder, (7) the imaginary cylinder which is not a pair of planes, (8) a pair of real parallel planes, (9) a pair of imaginary parallel planes. It might be thought that by analogy in the case of the general self-conjugate, the number of conjugate norms of any type (positive, negative or zero) would prove to be the same as the number of roots of the sextic of the same type, but curiously enough, this is not the case. This can easily be seen in the case of the com- mutative ZT whose sextic was considered in 1 37. If i/r is a commutative self- conjugate function such that SiE^jrE is negative and not zero for every axial motor E in space, we will call tjr an ellipsoidal function. [We only require this not very appropriate name temporarily.] Thus as a particular case i|r may be put equal to unity. A set of three real completely independent axial motors can invariably he found which are fully conjugate with regard both to vx any commutative self-conjugate and ■v|r an ellipsoidal function. The axes of these motors are in general unique. Putting ilr = 1, we get ; — A set of three mutually perpendicular intersecting rotors can always he found which are fully conjugate with regard to sr. These ai-e of course the i,j, k of eq. (1) § 19. If Di, Da, D-j are three conjugate norms with regard to w, and if the bar have the meaning with regard to these three that it I 39] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 171 had with reference to A, A', A" in 1 16 (not the meaning it had Avith reference to six motors in 1 28), the term in vtE corresponding to_A is_-ASEA; to B, B8EB , to A„ -nloSEA,; to £„, nBoSEB, ; to G, zero. Thus '7TE=t{±DSEB) + ni,(±D'SED') (14), where the total number of terms under the two summation signs does not exceed three. For an ellipsoidal function we have always ^ffE=- A.SEA, - AJSEA, - A,SEA, (15). For the common fully-conjugate system of tir and ffr we shall have equation (15) and also V7E=- a,A,SEA, - a^JS^A. - a^s^EA, (16), where Oj, a.^, a^ are any scalar octonions. From eq. (15) we have always yfr-'E = - A.SJEA, - ASEA-2 - A,SEA, (17), and from eq. (16) when the ordinary scalars of a,, a^, a^ are not any of them zero -sT-^E = - ac^A^EA^ - af-'A^JEA^ - ar'ASE A,... (18). Expressing equations (15) and (16) in full by means of § 16 in terms of the three conjugate norms (with regard to \jr) Ai, A„, A, we have y}rE8'A,A^A, = - fAA,A,SEAJ., - M^3^i8.E'^3^i - MA,A,SEA,A^. . .(19), ■stES'A.A^As =-a,MA^.,SEA,A,- ajMA^A^SEAsA,- a,MA,A,SEA,A,{20). 39. Another method of dealing with the general linear motor function of a motor. Just as the first method of dealing with the general self-conjugate function suggests a method of dealing with the commutative self-conjugate, so the first method of dealing with the commutative function suggests a method of dealing with the general function. If E be a general linear motor function of a motor E and yjr(E,F) be any octonion function of the motors E, F, linear in the general sense in each, y}r{Z,(j>Z)^^lr{(f>'Z,Z) (1), 172 OCTONIONS. [§ 39 where Z has the meaning defined in eq. (6) § 15. For ./r(f Z, Z) = -^lr{Z,sZZ„ Z) [eq. (10) § 15] = -yjr(Z„ZsZZ,) [since yjr is commutative with ordinary scalars] = ^fr(Z„Z,) [eq.(8)§15]. Particular cases of eq. (1) are MZ^Z = - MZ4>'Z, SZ|r (ZsAZ, B) = 2>|r {A, B). By eq. (21) § 28 _ <^E=-^A^%EA,- ...-A,sEA, (4), where A^... As are any six independent motors ; or 4>E = -B,sEA,- ...-B^sEA, (5), where _ _ _ B, = A„ B, = cf>A„...Bs = cj>As (6), so that the expression for in equations (3) is a perfectly general form of even when the number of terms is limited to six. We shall in what follows, as just now, always suppose A^ ... A^ to be independent so that {A^ ... A^} is not zero. The equation {A,...A,} = m{A,...As] (7), gives a value for m independent of the particular values oi Ai...As; for if we change Ai to any motor a\Ai + ... + x^A^ (^i not zero) independent of A„...Ae, both {(pA^ . . . cpAgl and {.^,...^.5} are altered in the ratio of x^ to unity. [The equation remains true for the same value of m when A^... A^ are not independent, for then both the expressions of eq. (7) involving A^... A^ are zero.] In particular, {Z,...Z,]{Zs} = m{Z,...Zs}{Z,...Zs} (8), {Z, ...Z,] {E4,Z, . . . Z,] = m \Z,... Z,} [EZ, ...Z,].. .(9). The Z's inay be eliminated from the right of these equations. To do this, first note that by eq. (3), [Z,...Zs}{Z,...Z,]^%\{A,...A,}[B,...B,} (10), {Z,...Z,]{.^EEB,...B,\...(U\ § 39] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 173 there being six terms under the summation sign, obtained bj- omitting ^4,, ^,> ... A^ successively. A particular case of the last two equations is obtained by putting i j, k, m, nj, flk for A,...Af, and m, ilj, D,k, i,j, k for B,...B,. Thus Z,...Z,}=^-6lm (14), {Z, ...Z,]{EZ, ... Z,] = 5\E (15), or, eliminating m, E{Z,... Z,} [Z, ...,f>Z,} = -6{Z,... Z,} {cf>EZ, ...Z,].. .(16), which gives E explicitly in terms of . Again, from equations (10) and (14) m = -{A,...A,}{B,...B,] (17), mE = -ZlA,...A,}{EB,...B,} (18), from which again E {A,... A,} {B,...B4 = --Z {A,... A,] {EB,...B,]... (19). When m is not zero {Bi... Bg] is not zero. In this case we have by equations (12) and (20) § 28, E = -A,S(f>EB^-... -A,scj>EBe (20), which can easily be verified from equations (6) of the present section and (21) of § 28. This method of course is the simpler one of establishing eq. (20), but it does not lead to eq. (19) in the exceptional case when m is zero. From equations (1) and (14) we see that the m of ^' is the same as the m of 'A,} = m{A,...A,} (21), or by eq. (12) § 28, sA,(f) {(/)'^i . . . (f>'A,} = msAe [Ai... A,}. 174 OCTONIONS. [| 39 Hence by §14, 4,{'A,...cf>'A,]=m{A,...A,} (22), or -'{A,...A,}::=m-'{(j>'A,...'A,} (23), which shows that to obtain '-'{A,...A,}=m-'{A,...4>A,} (24). The (p sextic can be obtained in the following way. Put £•„ = r^ • {^1 • • • -^el 1^. • • • ZnZn+, ■ ■ ■ Z.} ) .go) E„' = sZ,rE .{Z,... Z,} {Z,... ZnZ,,+, ...i>Z,}}--^ We proceed to show that E„ = nE„' + {6-n)E'r,+, (26). By eq. (19) § 28 E^ ={Z,... Zn4>Zn+, . . Z,] [{{Z, ...Z,] sZ,rE -... ±{Z,...Zn_,Zn+,...Z,}sZ„-E) + {+{Z,... ZnZn+, ...Z,} sZ^+^rE +...-{Z,...Z,} sZ,«E)]. By interchange of suffixes we see that each of the first n terms is equal to the first, and each of the last 6 — m to the last. Hence E^ = «^„' + (6 - n) sZ,4>-E .{Z,... Z,] {Z,Z, . . . Z^Z„+, . . . Zg has been shifted five places and therefore the sign has been changed. Now by eq. (1) change Z^, (fiZg into ^'Z^, Z^, write sZe^^+'jE" instead of s<})'Ze4>^E, and finally change the suffixes from 6, 1, 2, ... 5 to 1, 2 ... 6. Eq. (26) follows. Putting in that eq. n = and 6, E, = 6E,', E, = 6E,'. Also Er:=-En/n-E'„+,{6-n)/n. Putting «= 1, 2 ... 5 successively E, = eE,-J-E.-^'E.'=lE,-'^^E. + '^E: = ... = 6E,-15E,+ 20E,-loE,-\-6E,-E„ or E, - 6E, + 15E, - 20E, + loE, - 6E, + E, = (27). Substituting for the E's from eq. (25) we get the sextic 4>' - nf4>' + iv}'4>' - m"'4>^ + 7n"(f>' - m'cj) + m = (28), § 40] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 175 where remembering that {Z^... Z^] [Z-^ ... Zs\ = -Q\ 6\m = -{Z,...Z,}{,i>Z,...4,Z,} \\h\m' = -{Z,...Z,]{Z,<\,Z,...4>Z,] pl{Q-p)lm'^^=- [Z, ... Z,} {Z, ...ZpZp+, ... ^Z, ^ (29). We have from the above 5lm' = -sZ, {Z,... Z,] . sZ, {Z, ... .^^.j = s . {Z,... Z,] {Z,] (30). Also by eq. (13) 5!m' = - %Z, \Z, ...Z,}. s,}>Z, \Z,... Z,] = -5lsZ,4>Z„ or 7n' = -sZ^Z (31). I do not see how in the case of m'^' generally to get rid as in these two cases of Z^ ... Zp. We may deduce from the sextic now obtained the m-tic satisfied by ^ when acting on any motor of a complex of order n, when is related to the complex in the manner explained in § 35. But we have given enough to show the essential features of this method of dealing with ^. 40. Some f\irther deductions from combinatorial varia- tion. A few miscellaneous properties of motors readily deducible from combinatorial variation will now be established. In a set of conjugate norms of ■ar, a general self-conjugate function, in a given complex let there be no zero norms (§ 31), and let the positive norms be denoted by 4j.42... and the negative norms by BiB^ Remembering (§ 31) when a conjugate varia- tion is circular and when hyperbolic we see by equations (12) and (13) § 29 that %SA^A - tSB^B has the same value for any set of conjugate norms in the complex. But for a positive norm sAvtA = — 1, and for a negative norm sB'1!tB= 1. Hence denoting any norm of the conjugate set by H we have that sHvtH has the same value for any set of conjugate norms. Since the value of this expression is unaltered when H is multiplied by any scalar, the expression is constant for any set of conjugate motors. 176 OCTONIONS. [§ 40 This again is fully expressed by saying that % (S^H-stH/sH'stH) is constant, or finally St-'S^'srir= const (1), for any set of conjugate motors in the complex. By eq. (9) § 14, or eq. (3) § 13, tR- = 2tH. Hence putting OT = 1 we get that 2t-'i? = constant (2), for any set of co-reciprocal motors of a given complex. Eq. (2) is proved in Screws, § 136. Let A', B\ 6" be any motors of a complex of the third order given by A, B, C. Expressing A', B', G' in terms of A, B, C we get SA'B'C' = xSABC (3), where *■ is an ordinary scalar. Hence tSA'B'C =tSABC (4), or tSABC has the same value for any three motors of a complex of the third order. Similarly if A', B' are any two motors of the complex of the second order given hy A, B MA'B' = a:MAB (5), where x is an ordinary scalar ; and in particular tMA'B'^tMAB (6). Equations (3) and (4) are exactly equivalent, but eq. (5) expresses in addition to what is given by eq. (6) that the shortest distance oi A' and B' is the shortest distance of A and B ; i.e. all the motors of a complex of the second order intersect a definite line perpendicularly. [These equations might be apparently general- ised by writing to-^, ib-jB, stG, t^A' , stB', ctC" in place of A, B, C, A', B', G'. But a A, B, G belong to a complex of the third order it is obvious that ■utA, vtB, ■nrC also in general belong to a complex of the third order, so that we gain no additional information by the change.] The geometrical interpretation of these results is given by eq. (1) § 13 and eq. (8) § 12. Adopting the notations of those sections for d, 6, e, we have XA +XB + tC-\- dcoid -e\i&-a. = const (7), § 40] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 177 for any three motors of a complex of the third order, and tA+tB + dcotd = const (8), for any two motors of a complex of the second order. Eq. (8) can clearly be- deduced from eq. (7) if it is assumed as just proved that the shortest distance of any two motors of a complex of the second order is a definite line. Another result may be deduced by combining equations (7) and (8). Call the definite line just mentioned the axis of the complex. Thus e is the distance and ^ the angle between C and the axis of the complex A, B. Suppose now we are given a com- plex of the second order A, B and a complex of the third order including the first, viz. A,B,G. Then not on ly is t^ + 1 jB + d cot constant for any two motors of the complex of the second order, but tC —e tan (j) is constant for any motor of the complex of the third order. Passing now to combinatorial variation, if A', B', C are derived from A, B, C by this process the x of eq. (3) is unity, and similarly for A', B', A, B,a; in eq. (5). Let A, B, C (or A, B as the case may be) be conjugate norms of CT and let the variation be a conjugate one. Then sA'i^A, sBiffB and sG-btC remain individually constant. It follows that S'ABC sAvtAsB'stBsC'utC is constant for any three conjugate motors (not necessarily norms) of a complex of the third order. Assuming the truth of eq. (4) this only in addition gives us the information that SMBC/sA^AsB^BsCz7C is constant. Putting a, b, c for t^, tB, t(7 we have when isr = 1 sin^^cos^c^ ocabc (9). Eq. (7) similarly expressed gives a + b + c + dcot6-e tan ^ = const (10) and eq. (2) a-i + 6-1 + c-^ = const (11). We shall see directly that in general there are three mutually perpendicular intersecting motors belonging to the complex. M.O. 12 178 OCTONIONS. [§ 40 For these d = e= = and = ^tt. Calling their pitches ao,bo, Co the above equations give abc be -\- ca + ab \ ''°'^°''' *'' = ^A^„-6„c„ + c„a„ + a„6o (12). a + b + c + dcot 6 -eta,n4> = ao + h + Col Similarly for a complex of the second order . „ ab a + b sm^ a = -^ = .(13), aobo do + bo a + b + dcot6 — ao + bo but eq. (13) is most easily deduced from eq. (12). Eq. (12) is fully expressed by saying that a, b, c are the roots of the equation (x - a„) (x - bo) {x - Co) + a^ {(cosec- 6 sec^ )) = (14), and eq. (13) by sa3ring that a, b are the roots of {x - ao) {x - bo) + a^ coV" 6 = (15), and that d = ^ (a„ + &„) sin 2^ (16). It may be noticed that eq. (10) gives for three perpendicular motors of a complex of the third order a+b+c = const (17), and similarly for two perpendicular motors of a complex of the second order a + b = const (18). 41. Analysis of complexes of all orders into their simplest elements. We proceed to express every complex as a complex of reciprocal motors; the motors being whenever it is possible both perpendicular and intersecting. Every motor (including rotors and lators) has a coaxial recip- rocal motor, the latter having a pitch equal in magnitude and opposite in sign to that of the former. From this, by writing down in the most obvious way the complexes which are reciprocal to the complexes of orders zero, one, two, we shall obtain general forms of the complexes of orders six, five, four. § 41] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 179 Thus the complex of the first order is that of ODe motor. If this is a lator the complex is that of Hi, where i is any unit rotor parallel to the lator. If it is an axial motor, the unit rotor i may be taken on the axis and the complex is that of (1 + a'fl) i. Thus the complex may always be expressed as that of (a + ail) i, where a and a' are ordinary scalars, not both zero. The complex of the fifth order reciprocal to this is that of (a — aTl) i and of (6 + b'il) j and (c + c'D.) k ; where the five motors are reciprocal to one another and with axes on three mutually perpendicular in- tersecting lines ; provided that j and k are any perpendicular intersecting unit rotors, intersecting i perpendicularly, and that not one of the ordinary scalars b, b', c, c' is zero (though they are otherwise arbitrary). Similarly the complex of the sixth order is that of the six co-reciprocal motors (a + a'fi)t, (b ± b'D,)j, (c + c'fi) k, where i,j, k are any set of mutually perpendicular intersecting unit rotors and a, a', b, b', c, c' are any ordinary scalars of which not one is zero. We proceed now to show that every complex can be expressed as consisting of one of the lists of motors contained in the follow- ing table. It is to be understood that i, j, k are three mutually perpendicular intersecting unit rotors, and when in the table it is said that two, or three, of these are arbitrary, it is meant that they are arbitrary within the limits imposed by this condition. 12—2 s •J5 ^ ■*S J (V i +1 -^ i ^ 2 ■ — ^ a €" 60 e^-H E» 1 c O a) c o 1 o 1 8 CO 15 £ ■* .S ^ 1 +1 -« ^ J3 ;:-, 2 CO lO ■* M 1" 5 • =c ^ -si •^ ■E 2. + a 3. 'e "S "w ,— *— ^ -1^ 1 3 + « S^-S -4- « ■0 -^^ ni § .s ^ ^ -§ i si 1 1 «- ^.^^^-iZ ill's + |. -§ 2 l^.sui a ^ - „ ^ .-S 3 .S i^ 8 1 o 03 V i 1 . e i e +1 bo u 8 ^— ' * cfl © + 5£ 1 1 o p-H i/)3 = ^PiPi, S/J1P4 = ipipa. Hence ^PiPi^Pipi^PiPi = ^Pips&PiPtSpsPi =... = ..., each group being obtained by writing down all the products from which one suffix is absent. Dividing by pip^PsPi we get 23 ■ 24 . 34 _ 13 . 14 ■ 34 _ 12 ■ 14 . 24 _ 12 . 13 ■ 23 Pi' ~ p.' ~ Pz' ~ Pi' another result quoted as due to Mobius. A result somewhat similar to the above Sp-^p^ = $^5^4 is obtained by squaring both sides of the equation — Pi = Pi + Pi-^ Pi and equating the convertor parts. It is S/32/33 + S/32/34 + S/33/J4 = 0. This is not independent of the two previous results (Ssap = and ^PiPi = 8/33/34), as can be seen by putting a = p.^. On p. 179 is quoted from Mobius that: "Any given displace- ment of a rigid body can be effected by two rotations." From the title of Mobius' paper this apparently refers only to small displacements and rotations. In this form (§11 above) it is only necessary to prove that any motor A can be expressed as the sum of two rotors p and cr. For then the small displacement lAA ( ) will be compounded of the two small rotations M/a ( ) and Mo- ( ). As a matter of fact, A can be expressed in an infinite number of ways as p + cr. We will however prove the more general theorem for a displacement of any magnitude and rotations of any magnitude. We will show that such a displacement can always be effected by one such rotation followed by another, and this in an infinite number of ways. § 42] EXAMPLES OF THE APPLICATION OF OCTONIONS. 185 To prove this we have by § 11 above to show that any octonion Q for which TQ = 1 can be expressed in the form qr, where q and r are (§ 6) axials for which Tq = T»- = 1. [Note that an octonion Q in general cannot be expressed in the form qr where q and r are axials, since TQ is not in general an ordinary scalar, whereas T (qr) = TqTr is.] Looking upon Q as an operator on motors we see that it can always (when TQ = 1) be expressed as pa-~'^, where p and o- are two rotors whose tensors are equal. Now let t be any rotor whose tensor is the same as that of p and that of o- and which intersects both p and a. Then Q = pa~^ = pr~^Tcr~^ = qr, where q = pT~^, r=Ta~^. Here since p and r are intersecting rotors with equal tensors q is an axial for which Tg' = 1, and similarly r is an axial for which Tr = l. This proves the theorem. Note that t is at right angles to the axes of both q and r, i.e. it is along the shortest distance of these axes. Starting with q and r it is easy to show that any two rotations which suffice to produce the given displacement can be obtained by the above construction. The construction can be given in language not explicitly involving octonions. To do this, first define the double of a given twist as the twist obtained by doubling both the trans- lation and rotation of the given twist (when that is reduced to the standard or canonical form of a translation along and a rotation around one and the same axis). Similarly for a half twist, a double rotation and a half rotation. Then we have the following : — A twist can always be effected in an infinite number of ways by two rotations. The following construction suffices to find any two such rotations. Take any line 1 intersecting the axis of the twist perpeiidicularly. Let 1 become 2 when it is subjected to half the given twist. Take any transversal S of I and 2. Then double the rotation which converts 1 into 3 followed by double the rotation which converts 3 into 2 will effect the given twist. A particular case of this is interesting. Let p and cr above be unit rotors and put a-~^ = p'. Thus Q = pp'> 186 OCTONIONS. [§ 42 so that a particular case of the above is obtained by putting q = p, r= p. In this case the axials, viewed as operators, are quadrantal versors, and therefore the two rotations are rotations each through two right angles, or as we may call them semi-revolutions. A semi-revolution is completely specified by its axis. Thus we have : — A twist can ahuays he effected in an infinite number of ways by two semi-revolutions. The following suffices to obtain the axes in all cases. The aocis of the first semi-revolution is any line inter- secting the axis of the twist perpendicularly. The second axis is obtained from the first by giviyig to the latter half the given twist. If the given twist degenerates into a translation this be- comes : — If the tvjist be a translation the first axis is any line at right angles to the direction of translation, and the second axis is obtained by giving to the first half the given translation. It is interesting to note that of the four semi-revolutions thus possible to effect the twist obtained by superposing one given twist on another, the two intermediate ones can be made to cancel by taking the shortest distance between the axes of the twists as the axis of the second semi-revolution of the first twist and also as the axis of the first semi-revolution of the second twist. In the same part of the Appendix we find: — "Two equal parallel and opposite rotations combine into a translation." This also is apparently meant only to apply to small displacements, but we will prove it for finite ones. It might be treated as a particular case of the above, but we will treat it separately. Let q be the axial {Tq = 1) which expresses the first rotation and let p be a rotor perpendicular from any point of the first axis on the second. Thus by eq. (1) § 7 the second rotation is q~^ + D,pMq~^ for Spq"'^ = 0. Hence the whole displacement is (^-' + Q,pMq-^) g or 1 -I- flpMq-'^ . q. Hence (§11) the displacement is a translation equal and parallel to 'iplAq"^ . q. Since q~^ = Kg, we have 2pMq-' .q = -2K {pMq-' . q) = - 2K.qMq-' . p = q-'{q-q-')p = {l-q-')p. Hence the translation is compounded of two translations, the one § 43] EXAMPLES OF THE APPLICATION OF OCTONIONS. 187 being equal and parallel to the perpendicular from the first axis on the second, and the other equal and parallel to the same per- pendicular when it has been first rotated with the second rotatim and then reversed. In connection with this subject we may put into a form not explicitly involving octonions the statement that the displacement QR is the displacement obtained by superposing the displacement Q on the displacement R. It may be deduced from the remark above about the corresponding four semi-revolutions. To combine two tunsts take two lines 1 and 2 such that half the first twist brings 1 into coincidence with the shortest distance between the axes of the twists and half the second twist brings the shortest distance into coincidence with 2. Then the axis of the resultant twist is the shortest distance between 1 and 2, and the twist itself is double the twist about this aacis which will bring 1 into coincidence with 2. 43. Geometrical properties of the second order com- plex. Passing now to the text of Screws let us first establish the chief geometrical propositions there enunciated with regard to the complexes of different orders. Beginning with the complex of the second order, case (/S) of the table of § 41 and any case of (a) where one or more of the scalars a, a', b, b' is zero may be called singular cases. These are all simple. Let us dispose of them first. First consider the singular cases of (a). If a = b = the complex consists of all lators parallel to a certain plane. Every motor of the complex is self-reciprocal and any two motors are reciprocal. If a =b = the complex consists of a plane of parallel rotors. Also the lator parallel to the plane and perpendicular to the rotors belongs to the complex. All the motors are self-reciprocal and any two are reciprocal. Similarly for the case b' = a = 0. If b only = the complex consists of a plane of parallel motors all of the same pitch. Also the lator parallel to the plane and perpendicular to the motors belongs to the complex. This lator is reciprocal to every motor of the complex and is the only motor of the complex which is reciprocal to any given motor of the plane of motors. Similarly for the case when a only = 0. 188 OCTONIONS. [§ 43 If a =b' = the complex consists of a plane pencil of rotors passing through a fixed point of the plane. All the rotors are self-reciprocal and any two are reciprocal. If b' only = the complex is that of (1 + D,p) i, j, where p is not zero. The rotor j is self-reciprocal and is the reciprocal of any motor of the complex. Any motor of the complex can be put in the form r {(1 -1- Up) i cos 6 +j sin 6} = r{l +n{p cos^^ — kpsinO cos 0)} {i cos +j sin 9), where r is an ordinary scalar. Thus all the motors intersect k perpendicularly. If 6 is the inclination of one to i, its pitch is p cos^^ and the perpendicular on it from the point of intersection of i, j, k is —kp sin 6 cos d. They all lie on the ruled surface whose ordinary Cartesian equation is z («' -I- y^) + pxy = 0, which surface lies between the limits z=±\p [the maximum and minimum values of — ^sin^cos^]. If a rectangular hyper- bola with i and j for asymptotes be drawn and also a straight line through any point of i except parallel to _;'; and if any line OPQ be drawn through in the plane of i, j cutting the hyper- bola in P and the straight line in Q ; the pitch of a motor of the complex parallel to OPQ is inversely proportional to OQ^ and the distance of the motor from is inversely proportional to OP^. This case can easily be looked upon as a particular form of the non-singular case. Similarly for the case when a' only = 0. This disposes of all the singular cases of (a) of the table. In case (yS) if = ^tt we get the case just considered when a' = 6 = 0. When ^ = we get a set of coaxial motors. The reciprocal of any motor is the motor with equal and opposite pitch. There are two self-reciprocal motors, the coaxial rotor and the parallel lator. When 6 is neither zero nor ^tt the chief geometrical properties of the complex are contained in the table and represented in fig. 7. The two motors represented are reciprocal motors, and by varying a' we can get any motor of the complex except the lator £l(i cos 6 +jsvii6) [which corresponds to a' = x J. § 43] EXAMPLES OF THE APPLICATION OF OCTONIONS. 189 The complex that remains for consideration is that of (1 + aD,){, (1 + bn)j, where neither a nor b is zero. In discussing the properties of this and also those of the complex of the third order in the next section, octonion methods for the most part are to all intents and purposes quaternion methods. The examples now to be considered, if they serve no other purpose, will show how quaternion methods are practically a particular form of octonion methods. The point of intersection of i, j, k will be denoted by and called the origin. The rotor from to any point P will be denoted by p and may be called the coordinate rotor of P. It will be used in a manner practically identical with that of using the quaternion p when that represents a coordinate vector. Define the self- conjugate pencil function (§ 15) yjr by the equation ■^E = -aiSiE-hjSjE (1). Thus any motor belonging to the complex can be expressed as (o + fl-v/ro), where to is any rotor through perpendicular to k. The equation Sp^^p = -l (2) represents a cylinder which is completely specified by its trace — a conic — on the plane of i, j. We shall refer to this conic as the conic (2). In fact (2) is the equation of the conic when p is confined to being perpendicular to k (i.e. Spk = 0). Similarly the equation Sp^|rp = l (3) also represents a conic in the same way. When both conies are real they are conjugate hyperbolas. In this case Spylrp = (4) is the equation of their asymptotes. Since (§ 9) Ti (w + ft-f co) = Tw, the pitch of w + il^oo is by eq. (10) § 14, ^s (a + n^jray/w" = Scoylrw-^ Hence if the p of eq. (2) is parallel to the motor a + fli/reu of the complex, the pitch of that motor is - p-^ ; and if the p of eq. (3) is parallel to the motor the pitch is p-^. (2) may be called the pitch conic, and (3) the conjugate pitch conic. [In Screws the pitch conic is defined as the conic Spyjrp = -H, where H is any ordinary constant scalar.] 190 OCTONIONS. [§43 The pitch then of any motor of the complex is the inverse square of the parallel semi-diameter of the pitch conic, and is with sign reversed the inverse square of the parallel semi-diameter of the conjugate pitch conic. Also the two motors of the complex parallel to the asymptotes (4) have zero pitch, i.e. are rotors. (Screws, § 20.) The equation of the cylindroid (the ruled surface to which motors of the complex are confined) and the distribution of pitch on it may be obtained in the usual form thus, 0) -I- n-v^w = (1 -f- fli^ft) . w/ffl^) a> = {l + D,(p + zk)] CO, where p and z are scalars given by p = Sio-tlra/co", zk = — Mftji/rctf/to^. Putting (o — osi + yj, x, y, z will have their usual Cartesian mean- ings [eq. (1) § 7 above] and -p will be the pitch [eq. (8) § 9]. Thus ac^^Vhf ■ {h-a)xy , ^ a^-(-?/2 ' oi? + y^ ^ '' [Otherwise — by eq. (2) § 6 the pitch =S'^ww~^ and the perpen- dicular rotor from on the motor = Mi/rtoto^^] If two motors of the complex Wi + Oi/rwi and co^ + Hy^to^ are reciprocal, = S . (tOi -f fli/ra)j) (ft)2 -1- Xii^Wj) = 2Swii/rt02. Hence they are parallel to a pair of conjugate diameters of the pitch conic. [Screws, § 42. There is an error at the end of this section where it is stated that the sum or difference of the reciprocals of the pitches of two reciprocal motors of a complex of the second order is constant. It is the sum in every case that is constant, as we saw in § 40 above and as Sir Robert Ball himself proves in Screws, § 136. The cause of the error will be easily enough seen by the reader.] Suppose a -I- fla, where w is a unit rotor and a any rotor through 0, is a given " screw," i.e. a motor with unit tensor. Required the " screw " of our complex that has a given " virtual coefficient" with w + D^a. [The virtual coefficient of two motors A and £ is — iABjT^AT^BJ] The virtual coefficient of two screws !, Wj, o-j, a^ are rotors through the point. The nodal line is the axis (§ 40 above) of M (wj + flo-i) (&)2 + Ho-a), i.e. it is (§ 12 above) parallel to MwiWa. Let (l+OjD)ft) be one of the motors whose pitch is p, where a> is a rotor through the point. Expressing that it is reciprocal to both a^ + fici and m^ + Ho-o, we have jsSwwj + Swo-o = 0| ^^-*" These give as a necessary and sufficient condition w = a;M (paji + o-j) (pwj + 0-3) (7), where x is an ordinary scalar. This can be satisfied for all values of p not infinite, but for this value we of course take the motor to be flo) when we get w =a;M(B]Q)2. This shows (1) that one motor (and ordinary scalar multiples of it) and only one can be found for each value of p from +00 to — 00 , and (2) that the motor of infinite pitch is parallel to the nodal line of the cylin- droid. Also eliminating p from equations (6) we find that a satisfies the homogeneous quadratic equation StUCUiSoXTg — SwffiSww.i =0 (8), i.e. it lies on a cone of the second degree. The first of equations (6) proves other facts. For instance in § 80 of Screws it is stated that every line in space serves as the residence of a motor reciprocal to a given motor. Given a and Wi + Ho-], the first of equations (6) serves to determine p so that (l+Op)w shall be thus reciprocal. Again in § 80 it is stated that all the motors of given pitch which pass through a given point and are reciprocal to a given motor lie in a plane. 192 OCTONIONS. [§ 43 This is obvious from the same equation above since now we must suppose p given when the equation becomes a homogeneous linear equation in w, so that to lies in a plane through the origin. 44. Geometrical properties of the third order complex. The complex of the third order can be treated in a way very like that of the second order. We shall consider the case of no singularities only, i.e. case (7) of the table of § 41 where not one of the six scalars a, a', b, b', c, d is zero. With the present as with the second order complex we shall change the notation so that the complex is that of (1 + a£l) i, (1 + b^j and (1 + cfi) h. Define the self-conjugate pencil function i|r by the equation ^E-=- aiSiE - bjSjE - ckSkE (1). Since the reciprocal complex is that of (1 - oD,) i, (1 - 60) j, (1 - cXl) k, the corresponding pencil function for it is — i/r. Any motor of the present complex is + ilyfrco of the present complex is Sw^i^o), whence we have that if the p of (2) be parallel to the motor the pitch is abcp~^, and if the p of (3) be parallel to . the motor the pitch is — abcp~^. Also the motors of zero pitch are those parallel to the generating lines of the asymptotic cone. But in the present case not only is this last true but the motors of zero pitch, i.e. the rotors of the system, actually lie on the pitch quadric. To prove this we will find the locus of the motors of given pitch p. The pitch of w + Hi/ro) is Sw'i/rtB. Hence if this is p, m is confined to a cone of the second degree, as is evident from the equation 8ftj(l/r -p) ft) = 0. § 44] EXAMPLES OF THE APPLICATION OF OCTONIONS. 193 The rotor p from to any point on this motor is by § 6 above p = M (y^ro) + x) O)-', where x is any scalar. Thus the extremity of p is confined to a ruled surface. We will show that this surface is a quadric. Putting i|f — p = •57, we have Scora-co = 0, jO = M (cto) + x) a)-^ = (wca + x) a)~^ Hence lApw = isai. Hence as = ta-~'Mpa) = m^~^Mw/3TO-o), where rUp has the usual meaning with regard to the pencil function in-, i.e. nf(p = - S'tsiTajwk = (a - p)(b -p){c-p) (5). Substituting in the last equation Mpw for ra-oj, we have since Swsrp = Sp'srw = SpMjOw = 0. Thus finally Sp{yjr —p)p = Sp'!!Tp = mp = (a -p){b - p){c-p)...(6). Putting in this ^ = we get eq. (2), which shows, as stated above, that the pitch quadric is the locus of the rotors of the complex. Since for the reciprocal complex we have merely to change yjr into — yjr, eq. (6) is also the locus of those motors of the reciprocal complex whose pitch is —p. It is obvious from the fact that two motors, the sum of whose pitches is zero, are reciprocal only if they intersect that the motors of the given complex lie on one set of generators and the motors of the reciprocal complex on the other set. This also follows from the fact that though for both sets of motors Scc-nrm = 0, we have for the motors of the given complex p = (•cro) + x) o)"', and for those of the reciprocal complex p = — (■sj'w + os) (o~^. Thus a plane through the centre containing a motor of the given complex also contains at an equal distance on the opposite side of the centre a parallel motor of the reciprocal complex. [It is deducible from the above that the generating lines of Sp(f>p = — 1, where (/> is any self-conjugate pencil function, are given by /3= + {(-m)-i^o- + a;} o-S where a is any rotor gene- M. o. 13 194 OCTONIONS. [§ 44 rating line of the asjonptotic cone Saifxj- = 0. That (— m)~i may be real all three roots or one and only one of the ^ cubic must be negative. That the quadric and the asymptotic cone may them- selves be real all three cannot be negative. Hence one and only one is negative or the quadric is an hyperboloid of one sheet.] Eq. (6) may be looked at in a different way. Regarding p as given it represents, as we have seen, a quadric on which the motor must lie. But regarding p as given it is a cubic for p, the pitch of a motor of the complex passing through a given point. Thus there are three such pitches. And there are three such motors. These are given by the equation Mpco = ■sro) which gives 01 (as to direction but of course not as to tensor) when p and therefore ot is known. A general method of obtaining a> from this equation is to choose that one of the two values of a given by the equations Scovtq} = 0, Spts-co = which makes Mpw = cro). Another general method is to write down the cubic of (p, where (fia = -srra — Mpio, ^-M"' — M"i>p-o)2 = 0, i.e. the three motors are parallel to a triad of conjugate diameters of the pitch quadric. That a motor a + [lyfrco of the complex may be parallel to a fixed plane, we must have where a and /3 are given rotors through 0, and x and y are arbitrary scalars. The motor is therefore x(a + n^}ra) + 2/ (/3 + D.ylr^), i.e. it belongs to a complex of the second order. The axis and centre of the cylindroid of this complex can easily be found in terms of e, the rotor through perpendicular to the plane. For by § 40 above, the axis of the cylindroid is that of the motor M (a + fif a) (^ + nf /3) = Ma/3 + DM (ax|f/3 - pyjra). § 44] EXAMPLES OF THE APPLICATION OF OCTONIONS. 195 Thus it is (of course) perpendicular to the plane of a, /3, and the rotor perpendicular from on it is (§ 6 above) M . M (ai/r/3 - ySi/ftt) M-'a/3 = aS^/r/3M-lay8 - /SSi/raM-'a/S. Take this rotor to be the same as a. Thus the coefficient of a in this expression must be unity, and the coefficient of /3 zero, or Si|r/3M-iajS = l, Se>/fa = 0, S€a = 0. From the last two we have a = xfA€^fr€, and x is now obtained from the first of the three equations. Putting in that equation /3 = a~^e, thus making ^ a rotor in the plane perpendicular to a, we have Sa~^€-\{r6~^ = 1, which gives Thus the rotor perpendicular from the origin on the axis of the cylindroid of those motors of the given complex which are per- pendicular to 6 is Me~'i|re. Similarly the perpendicular on the axis of the cylindroid similarly related to the reciprocal complex is — Me^^i/re, so that these axes are situated symmetrically on opposite sides of the origin. There is another meaning in connection with the complexes to be attached to these rotors + Me^'i/re, where e is any rotor through the origin. For the rotor perpendicular from the origin on any motor co + n-yp-co of the given complex is — M&)~^-v/f&). Hence putting <» = e, we see that the axis of the cylindroid of all motors of the complex perpendicular to a given direction is the residence of a motor of the reciprocal complex. This is other- wise obvious from the facts already proved: (1) that any motor whose axis is the nodal line is reciprocal to every motor of the cylindroid, and (2) that by giving this motor a suitable pitch it can be made reciprocal to a given third motor of the given complex. Next supposing co perpendicular to e, we see that the motor of the complex parallel to o) is in the plane through perpen- dicular to e if Sea-^m = 0. But this is the necessary and sufficient condition that a should be parallel to one of the principal axes of the section of the pitch quadric which is perpendicular to e. Hence there are two such motors in this plane parallel to the two principal axes; these motors are therefore perpendicular; hence they are the principal motors of the cylindroid. Hence 13—2 196 OCTONIONS. [§ 44 the plane through perpendicular to the axis of the cylindroid (i.e. perpendicular to e) cuts the axis at the centre of the cylindroid. These examples suffice to show how such purely geometrical results can be obtained by octonion methods which are essentially quaternion methods. In the examples now to be given the methods will be more characteristically octonion. 45. Miscellaneous simple results. Many of the results (besides those of the last three sections) of Screws have already in the present treatise been explicitly enunciated. Several others are almost obvious consequences of what has already been said. Suppose A^... As are six independent motors. Then we may express any motor E by the equation £ = XiAi+ ... +XeAe = l,xA. Thus E^ = ^a^A' + 2ta;,x^8 A^A^. Taking the ordinary scalar part and the converter part, we get Ti=.E'= Sa^TiM - 2Xa;,xS,A,A^, which is § 37 of Screws, and tET.'E = laftATM - l,x^x^%A^A^, which is § 32 of Screws. li Ai... Af are co-reciprocal, tET:,^E=%x'tATM, which is § 33 of Screws. li F be another motor which is equal to ^yA and the A's are co-reciprocal, sEF = txysA^ = - 2l,xytATM, which is § 35 of Screws. 1 41 of Screws asserts that {Ai ... A^} (§ 28 above) is reciprocal to each of the motors A^, A^... A^ and also that the condition that Ai...Aa belong to a complex of lower order than the sixth is that {Ai ....44 = 0. § 45 asserts that if sEAr=... = sEA„ = 0, then S.E' (x,Ai + ...+ a;„^„) = 0. If for some ordinary scalar values of x and y, xA + yB is § 46] EXAMPLES OF THE APPLICATION OP OCTONlONS. 197 reciprocal to both A' and B', it follows that for some ordinary scalar values of x and y', x'A' + y'B' is reciprocal to both A and B, for the necessary and sufficient condition for either of these is that ^AA'sBB' = sAB'sBA'. This is § 88 of Screws. § 89 asserts that if A and B be two motors and x and y two ordinary scalars, the four motors xA, yB, xA ± yB are parallel to the four rays of an harmonic pencil. This is obvious (§13 above) from the fact that a similar statement is true of vectors. § 99 asserts that the anharmonic ratio of a pencil with rays parallel to the four motors A, xA + yB, x'A + y'B, B is the same as the anharmonic ratio of the pencil with rays parallel to the four motors ■ST A, ts {xA + yB), OT (a; .4 + y'B), inB, where ct is a linear motor function of a motor (self-conjugate in the case considered in Screws). This is obvious from the fact that the anharmonic ratio of the pencil with rays parallel to the four vectors a, xoL + y^, ic'a + y'A /3 is x'yjxy'. § 137 asserts that there is one line such that a motor of any pitch having that line for axis belongs to a given complex of the fourth order. This is of course k of cases (a) and (/3) of the table of § 41 above. Similarly in a complex of the fifth order any line intersecting perpendicularly the motor reciprocal to the complex is the axis of a motor of arbitrary pitch belonging to the complex. 46. Equation of motion of a rigid body. Free body- subject to no forces. We proceed now to the dynamics of a single rigid body. And first we shall not make the assumption that the motion is small or that the body is subject only to instantaneous impulses, though later one or other of these re- strictions will be imposed and several propositions of Screws applying to these cases will be proved. 198 OCTONIONS. [§ 46 Take a certain position of the rigid body which may be con- sidered the initial position as a standard of reference. Let Q ( ) Q~^ be the operator (§11 above) which brings the body at one operation from its initial position to its actual position at any time. We assume, as by § 11 we may, that TQ=1, and that Q = e^/^ (1), where E is a, motor. Thus the axis about which the body must be twisted is that of E (and also that of Q), the angle of the twist is TiE (and also twice the angle of Q), and the translation is M^E (and also is twice the translation of Q). E is therefore called the displacement motor. Another motor E' will be used for some purposes defined by the equation E'=2MQ = 2B (2). It will be observed that when the displacement is a small one Q = l + ^^=1+|^ (3), so that E and E' are in this case the same small (i.e. with small tensor) motor. It may be noticed that since TQ = 1 KQ = Q-^ (4), 2B = E' = 2MQ = Q - Q-i = 2 sinh {^E), 2SQ = Q + Q-' = 2 cosh i^E) (5). From these since cosh= (i-fi") = 1 + sinh^ (^E) we have SQ = V(1+-B^), Q=5 + V(l+-B^), Q-' = -5+V(l+^)...(6), or 2SQ = >^{i + E'% 2.Q = E' + >^{i + E'% 2Q-' = -E' + ^{i + E'') (7). Notwithstanding that these equations are simpler when B is used instead of E', E' is the more convenient motor for our purposes mainly on account of the consequences of eq. (3). E' instead of E may be taken to specify the displacement since Q which specifies it is given by means of equations (7) in terms of E. Let F, G, H be the velocity motor, the momentum motor and § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 199 the force motor respectively of the rigid body. These terms have all been explained in § 8 above. Also let F=QFoQ-\ G = QG,Qr\ H = QH,Q-' (8), so that Fo, Go, Ho may be spoken of as the corresponding motors relative to the standard position or relative to the rigid body itself In fact, in using in our equations F^, Go, Ho instead of F, G, H, we are doing what is analogous in the Cartesian treat- ment of a rigid body with one point fixed to the well-known reference to axes fixed in the body. We do not require two analogous symbols Eo and Eo, for since Q, E and E' are coaxial E = QEQ-\ E'^QEQ-' (9). We might assume from elementary Rigid Dynamics the existence of moments of inertia and principal axes of a rigid body. But we make a digression here to prove this, as a rather instructive series of octonion examples is involved. Since impulses combine like forces, being of essentially the same nature (except as to time dimensions), we see that the momentum motor of a system of moving particles is the sum of the individual momentum rotors of the particles. If G' is the momentum rotor of a single particle of a moving rigid body of which the velocity motor is F, twice the kinetic energy of the particle is — sFG', and if the position in space of the particle is given G' is a self-conjugate linear {energy-) function of F. If m is the mass of the particle, and to + Her is the velocity motor F, where w and a are rotors through the particle, the momentum rotor G' is ma: Hence - sFG' = - ma^ = twice the kinetic energy. Also if the position of the particle is given ma = (?' is a linear function of w -I- ^a = F hy the definition in § 15 above of a linear function. Putting ma = yjr{(o + Ha) we have S (ft)' -I- no-') -\|r (ft) + na) = mSaa' = S (oj -H Oo") i^ (w + ila), so that yjr is self-conjugate. From the last two propositions it follows that : — If the velocity motor and momentum motor of a rigid body be F and G respectively, twice the kinetic energy is - sFG, and if the position of the rigid body is given G is a self -conjugate linear 200 OCTONIONS. [§ 46 {energy-) function of F. [This might be proved if we assume some very elementary facts of Rigid Dynamics from the equation S (<» + S1|r are (1 ± an) QiQr\ (1 ± 611) QjQ-\ (1 ± cfl) QA;Q-' (20), and that the roots of the -^ sextic are the same as the roots of the a/tj sextic. The equation of motion of a single particle is G' = H\ where G' is its momentum rotor and H' the force rotor acting on it. Hence if G be the momentum motor of a system of particles G = Xif '. The " internal " forces of the system contribute zero to the sum 1,H', so that an equation of motion of any system of moving matter is G = H (21), where G is its momentum motor and H the force motor of the system of " external " forces. For our rigid body eq. (21) is the equation of motion, since it suffices to determine consecutive (in § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 203 time) values of when H is given at any instant, and therefore also suffices to determine consecutive values of Q (as will appear more obviously directly). Since TQ = 1 we have by eq. (5) § 20 SOQ-^ = (22). Hence by eq. (8>§ 11 F=2QQ-^ (23). Hence by eq. (8) of the present section F,= 2Q-'Q (24). Hence H=d (QOoQ-')ldt = MFG + QGoQ-' [§11 above], or H„ = Q-'HQ = MF,Go + G,. Thus i?o = W + Mi^ot„i^„ (25). This can be expressed in terms of Q instead of F^, for by eq. (24) F, = 2Q-'Q- 2Q-1 Q Q-^ = 2Q-' Q - \F,\ Taking the scalar-octonion and motor parts we have 4SQ->Q = i^„^ = J^^ (26). /, = 2MQ-'Q (27). Hence our equation of motion to determine Q when H„ is given is H, = 2i|roMQ-' Q + 4M . Q-Qa^o (Q- Q) (28). It will be noticed that these equations are very analogous indeed to the quaternion equations which express the motion of a rigid body one point of which is fixed. It is worthy of remark however that the analogy here is not quite that of the general analogy so largely used above between Octonions and Quaternions. This is because -^i, is not a commutative function (§15 above). The quaternion equations can be deduced from these. Without actually making this deduction it will be evident from what immediately follows how it may be made. ■x/to is by no means of the most general type of energy-functions, so we may expect it to have some special properties. These will now be examined. 204 OCTONIONS. [§ 46 In the first place though yjro is not a commutative function, i/tj^ is. For by equation (18) f,' (1 ± ail) i = MW (1 ± an) i, from which y{r,H = M^aH, f,mi = M wm, and similarly for j and k. Thus a^o'' is a self-conjugate pencil function whose centre is 0. Define oto by the equation ^,E = AI-^ ^Ir.'E = - (a^iSt^ + ¥jSjE + c^kSkE) (29). The relation between yjr' and -v/r/ is the same as the relation between i|r and i^o; so we naturally define •sr by the equation -hjE = M-fE = Qt^o (Q-'EQ) . Q-i (30). Now let F„ = w, + a„, the point on the axis at the point of contact of Go and the sphere slides along the sphere. [If we give to Go the twist — Fodt we must add to this rotation a small translation D,aodt. This of course slides the old point of contact off the sphere but leaves Go still touching the sphere since the translation mentioned is a mere sliding of Go along itself.] In the present method as in the ordinary dynamical methods we gain much insight in the case of no external forces by con- sidering the equations involving the actual motors F, G, &c., instead of the same referred to the rigid body. Thus in eq. (21) H is zero, so that G is an absolutely constant motor. But, more than this, a is an absolutely constant rotor ; and therefore since G = M (/roO'v|r(, and yjro'D.yjfo' respectively ; or that G^ and dGo are conjugate with regard to each of the functions '^//'o~^ 1. ^. yJTail'yJro respectively. Put v7eo = fi, Sa3'S7a) = — Ci, fi^ = Saym'^eo = — C^ (41)- Thus /x is a rotor of constant magnitude and direction, and Ci and Cj are constant scalars ; fi being the ordinarily called moment of momentum, Ci twice the rotatory energy and Ca the square of the moment of momentum. We may notice that if m, m, in" are the coeflScients of the in- cubic ^n+s _ ,yi'V"+2 + m'iir"+' - mi3-" = 0, for all positive integral values of n from zero. This equation enables us to express c„ = — Sa)-!3-"£B in terms of Ca, Ci and Co= - w^ i.e. in terms of constants and the square of the angular velocity. For it gives Cn = — SwTs^co, Cn+3 - m"c„+2 + m'c„+i — mCn = (42). From this in particular we deduce a result required later, viz. Cj C4 "T ^OiC^C^ -~ Cg = {ci^ (m'm" - m) - c^^c^ (m"" + m) + CiCi2m" - d] ■ . . .(43), + mCi (m"ci — 2C2) a)" and generally since a^ b^, d' are the roots of the w cubic. We may remind the reader here that the quaternion process (Tait's Quaternions, 3rd ed. § 407) of establishing Poinsot's con- struction is as follows. Let po be a coordinate rotor through § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 211 and p = QpoQ~^ the corresponding rotor through the point to which has been transported at any time. Then the two equations Sp'Tsp = - Ci , S/aoOTopo = - Ci (45), are exactly the same and therefore each is the equation of an ellipsoid fixed in the rigid body. The tangent plane at the point p = a) oi this ellipsoid is S/3/i = -Ci (46). Since /u, is a rotor through the centre of mass of constant tensor and direction this plane is simply moving with the velocity a of the centre of mass. Since relative to the centre of mass the rigid body, and therefore the ellipsoid which it carries, is moving with the angular velocity o), the point of the ellipsoid in contact with the plane (46) is at rest relative to the plane. Hence the ellipsoid rolls on the plane. The ellipsoid (45) will be referred to as the Poinsot ellipsoid, or when there is no ambiguity the ellipsoid, and the plane (46) will be referred to as the contact-plane. The curve traced out on the surface of this ellipsoid by the extremity of w (the rotor drawn from the centre of mass represent- ing the angular velocity) is called the polhode. It is by equations (41) the intersection of the Poinsot ellipsoid with the ellipsoid Sp'BT^p = — C2. For our purposes it is more convenient to regard it as the intersection of the Poinsot ellipsoid with the quadric cone (vertex at centre of mass) S/3 (Ca — Ci'or) 'STp = 0. This cone will be called the polhode cone. The usual definition of the polhode is that it is the locus of points on the Poinsot ellipsoid whose tangent planes are at the constant distance Tp,~% from the centre; i.e. which touch the sphere p'' = C)>~'' = - CiVca- [The rotor perpendicular from the centre on the plane (46) is - CiIj,~\] It will be observed that w and (»„ are not related in the same way as w and &)(,; i.e. we have not w = Q' = QwoQ-' (47). Thus w' is the value of Wo when the standard fixed position is taken as that of the position at the instant under consideration. It is parallel and equal to the velocity relative to the contact 14—2 212 OCTONIONS. [§ 46 plane of the point of contact of the Poinsot ellipsoid with the contact plane. By equation (40) Wo = - ■oro~^Ma)(,'srot<)o, which gives the following expressions for w O)' = — ■nr~^M(»'Brfi) = — WT^H^vswrn^a (48), or 6)' = — ■ar~'Ma)/4 = — m~'Myai!r/i (49). The first of each of these expressions shows that ' is O'PT; in other words the normal at the point where w cuts the ellipsoid is perpendicular to the plane O'PT. The second expression shows that w' is perpendicular both to (i and tff/i. That it is perpendicular to yu, of course follows from the fact that it is by its definition parallel to the contact plane. For applications to be made immediately it is necessary to notice that 2Ma)'ei) can be expressed as a self-conjugate pencil function i/roj of w where ■v/r is fixed in the rigid body ; i.e. where '>/^ = Q^o(Q~^[ ]Q)-Q~S ^0 being an absolutely constant self- conjugate pencil function. For = 2m~' (— ■orwSeBCT^ft) -f ■nr^wSwcTfi)) = i^w) " where i/r = 2m~^ (ca — Ciw) ct (51). [In the rest of this section and in the following section (§ 47) we shall have no occasion to refer to the i/r and ^^ of equations (10) to (39) above. The former i/r was not like the present ■y^, a pencil function, though it was self-conjugate.J The polhode cone is Spy}rp = (52). As P traces out the polhode the extremity of p = i/rw, a rotor drawn from 0' traces out another curve. The point on this curve corresponding to P we will call Q. Thus OT = a), WQ = f(o (53). § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 213 Since [eq. (52)] Stoi/rw = 0, O'P and O'Q are perpendicular. And further, the normal to the polhode cone at P is parallel to -\|r(», i.e. to O'Q. Hence Q lies on the cone (vertex 0") of lines normal to the polhode cone. This cone will be referred to as the normal cone and the curve traced out by Q will be called the normal cone curve. Since 8wv|f&) = 0, SwSTO) = — Ci , or Silrco-\lr~^-\jra) = 0, S-\lfw\fr~''-s7-\jrM = — Ci, the equation of the normal cone is Spi/r-ip = • (54), and the equations of the normal cone curve are (54) and Spyfr-'-urp = - Cj (55). From the equation Swar^w = — C2 we also have for any point on the normal cone curve Spf-'^'p = -c, (56), and its equations may be taken as any two independent combina- tions of the three last equations. We will examine the geometri- cal properties of the normal cone curve later. Meanwhile we return to the dynamics involved. Suppose now R is any octonion and suppose its time-changes are observed by three observers — the first, whom we will call the outsider, being fixed in space ; the second, whom we will call the plane-resident, moving with the velocity of the body's centre of mass ; and the third, whom we will call the resident, residing on the rigid body. We may suppose the plane-resident to be un- conscious of his own motion and also the resident to be unconscious of the motion of the rigid body on which he resides. The evolu- tions of R will appear to the resident to be those of the actual octonion Q~^RQ. To the plane-resident they will appear to be those of (1 +^D.(Tt)-' R{1 + ^Ha-t) = R- ifiMo-MiJ. Let us then define R^, R^,, R' and R" by the equations Ro = Q-'RQ, R, = R -taMaMR) . R' = QR,Q-\ R" = R, + tnMaMR,] ^ ^' Thus Ro and R^ are the aspects of R to the resident and plane- resident respectively, reduced to their respective initial positions ; and R' and R" are the apparent rates of change of R to these observers reduced to the actual position of the rigid body. [If the 214 OCTONIONS. [§ 46 body is subject to external forces and Xir is its integral translation at any time we ought to write fir in place of D,crt in the last equation. In this case fif takes the place of D,a in eq. (58).] By eq. (6) § 11 we have R = R' + MFMR = R" + nM~^ = ■sj-eoo)"^ (61). These equations enable us to express various rotors, &c., in terms of the relative positions of 0', F and G. Note that r is a constant except as to the position of its axis which always passes through 0' ; but 2' is a variable. For instance by eq. (49) ft)' = Q)^'m~^Mfia)~^ = 0)^1)7"' Mrg', which gives ft,' = 0,^-1(1^7^-^5^/3 4-/7) (62). The expression in the brackets admits of quite simple interpreta- tion by means of figure (8) ; thus we have an expression for the velocity of the point of contact of the ellipsoid with the contact- plane in terms of the relative positions of 0', F, G and the ellipsoid. Again by eq. (50) iro)-^ = - 2M&)w-iMyiift,-i = - 2;?i-''CTM/iM/i&,-\ § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 215 Putting in this equation Mfio)-^ = Mrq = M^/S + g^ +fy, we get yfrco-^ = - imr^^Mfi (M7/3 + g^+fy) (63), which gives an expression for ■\fra)~^ in terms of the relative posi- tions of 0', F, G, the constant rotor fi and the ellipsoid. Again putting in the last fj, = ra- = (ry + g) ~\ Hence by equation (58) q" = — aar^(o' (o~^ = — q(o'Q)~^ (65). Taking the scalar part we get / the rate of change of the pitch of F. Thus /= — Saeo~^a)'a)~^ = — Sqa>'a>~'^ (66). Each of these expressions for f admits of simple interpretation, wto'o)"^ is tu' turned conically round a through two right angles. Call such a conical rotatioa a semi-revolution (§ 41 above). By the first expression of eq. (66) then, / is the scalar product of the velocity of the rigid body a, and of the velocity of the point of contact revolved half round co, multiplied by — co~^. Putting in the second expression q =/+ /S we have f=-fSco'w-'-S0Mo}'(o-' (67). Here S(o'co~^ is the rate of increase of the logarithm of Tcd, and Mctf'ft)"^ is the rotor angular velocity of the point of contact about 0'. This gives a second interpretation. Remembering that Mtu'tB"^ = ^y}rci)~^ we get other simple expressions from equations (63) and (64) which can be written down by the reader. And again we may substitute from equa- tions (48) and (49) for co'. Thus taking the first expression of eq. (49) f= — S(7C0Z7~^MfJ,(0~'^ . C0~^ [eq. (61)] where •nr^ is what ot would become if the rigid body were revolved half round co, i.e. ZT„E = COZT (cO-^Eco) . &)-! (68). Thus substituting for Mrq in terms of y8, &c. f=Sa^^-^(My0+gl3+fy) (69). Again in the expression - Scrco-^'m-^Mrq.co if we substitute q for (TO)"' and q~'^cr for co we get f= — Saq7!T~^Mrq . q~^, or /= - So-tn-g-i Mg-r = - So-ot,-i (M^Sy + g^ +fy) (70), where OTj is what vr would become if the rigid body were turned § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 217 in the plane of o-, a> through twice the angle between these two ; i.e. vTjE = qzT{q-^Eq).q-' (71). The velocity of the foot of the perpendicular from 0' on F as observed by the plane-resident is equal and parallel to /9". Taking the motor part of equation (65) we get ^" = _o)-^Mo-6oa)'a)-' (72). Thus this velocity is the rotor product of the velocity a of the rigid body, and the velocity w' of the point of contact revolved half round a>, multiplied by — w~^. Another interesting expression for /3" may be obtained thus. By eq. (48) m = — m~'Mi3-Q)TO-^ = MFco since F= w + D,a: Hence 2D,M(tco . to-^ 4- aFm-"^ = 2lAFw . q)-i + coFo)-^ = F. Hence F^ = F +'m(c'a-'- = F +■^(0-'^ (79). Fg may be obtained independently or from this result. For the displacement q{ )q~'^ = crw~^ ( ) aa"^. Hence the displace- ment qQ( ) Q~^q~^ may be obtained by first making the displace- ment a>~' Q( ) Q~^ (o [or Q)Q ( ) Q~^co~^] and then the displacement = (6). The normal cone curve is defined as the curve any point of which has p=-\jra> for coordinate rotor. Thus P being a point on the polhode and Q the corresponding point on the normal curve 0P = (o, OQ = p = ylr(o (7). Byeq. (6) Spa = (8), so that p is perpendicular to co. It is also perpendicular to the 220 OCTONIONS. [§ 47 tangent plane at F of the polhode cone (6). It therefore lies on the cone normal to the polhode cone. Substituting for a in terms of p equations (5) and (6) give Sp-^-'-srp = - d , Spf-'^iff'^p = - C2, Spyjr-^p = (9), and any two independent combinations of these three equations may be taken as the equations of the normal cone curve. Thus the equation of the normal cone is Spifr~^p = 0. When CO is given p^ can be written down from the following: — p^ = Sw\fr'co = ismr^Sa (Cj — C-^vsy ■sr^co = 4m~^ (— Cj^ — 2ciC2Sa)cj-'a) + Ci^Scbct^w). Now by eq. (1) Sco-sr^o) and Swut^co can (§ 46) be expressed in terms of (o% Ci and Ca- The relation that we thus get is eq. (43) § 46 which for present purposes may be written p2 = 4m-% (m"ci - 2C2) &)2 + const (10). If a>' is a rotor through parallel to the tangent of the polhode at P we have by equations (5) Sco'vTco = 0, StoW^w = 0. Thus o)' = xWlTSTcctn-'^co where x is any scalar. If x is put equal to — m-^ [eq. (48) § 46], a is the velocity of P along the polhode if P is always the point of contact of the ellipsoid and contact-plane. We therefore put &)'= — m~' MCTCtfCT^O) = — •5r~^Mw5r&) (11)- By substituting here for ■w^ in terms of i/r and ct by means of eq. (4) we get to' = ^Ci~^Mww\lreo = — ^Ci~^ Mpww (12). From the last expression we have by equations (8) and (5) 2Ma)'lr''^-^{y' + z'i!^) (20), we shall have (yf-^-ST + z-^lr-'wy = y'y^-'-ur + z'^f-^'-ur'' (21). Thus if we put ■nri = yyjr-'sr + zyjr-'-ur'' (22), the equations of the normal cone curve can by equations (9) be put in the form SpCTi/) = — yci — ZC2 = — Ci" (^1/3 )' = SpWp = - y'ci - 2^C2 = - Ca'l so that the normal cone curve is a polhode on the quadric Sp^ip = — d' whose contact plane is at the distance Cilfjc^ from the centre. Assuming that y, z, y' and a! can be so determined, mi the product of the roots of the CTj cubic is by equation (22) given by wii = mmf-'^ (y^ + m'y'z + m'yz" + mz^) (24), and we have 2?)ir' (ca' - CiVi) in-i = 2mi-' {ca' (y + zis) — c/ {y + z'zr)] i^-^^ = 2mr^ {y2/ — y'z) (Cj — C^vr) yfr-^'ur = mi~' m{yz' — y'z) '^~^ = xyfr~^ = -\^j , say, by eq. (4). Thus the normal cone curve of this new polhode is given by the equation Pi = fip = i)sa) (25), where p^ is the coordinate vector of Q^ the point on the new normal cone curve corresponding to Q on the new polhode. Thus § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 223 OQiP is a straight line ; and OQ^jOP has the constant value x. Hence the new normal cone curve is similar and similarly situated to the original polhode, the centre of similarity being the origin. It may be noticed that though only the three ratios f-:z'':y' : z' are determined by eq. (20) the quadrics of eq. (23) are not altered by altering the absolute magnitudes of these four quantities. Thus we cannot by such an alteration reduce x of eq. (25) to unity. Also since Cg is of two dimensions and Cj of one and m of three in to-, a/t is of no dimensions in in-. Similarly -v/ti is of no dimensions in CTi. i|r is however of two dimensions in w and i/tj of two dimensions in p. This is illustrated in § 46 where -^ur^ appears as an angular velocity which is therefore of the same dimensions as w. It is to be observed that if y and z and therefore -nri are real each of the quadrics Spt^ip = — c-[ and ^pta^p = — c^ must be real, for each equation is satisfied by an infinite number of real values of p, namely the coordinate rotors of the (real) normal cone curve. Under these circumstances the quadric SpT^i-p = — c^' must be an ellipsoid and Cg' must be positive since the roots of the CTi^ cubic are all positive. [It is necessary to note this because (f> may be real and yet the quadric Sp(pp = — c imaginary. This last is the case if all the roots of the (j) cubic have the same sign as — c] We shall find that the determination of the ratios y' : z" : y' : z' depends on the solution of a quadratic equation. This equation has in general two roots which give rise to two sets of values of the ratios. There are thus two and only two values which satisfy the required conditions except when the roots of this quadratic equation are equal, when there is only one. We will now show that in the limiting case when the roots are equal (1) Spiuip = — c/ is a sphere and therefore (2) Sp'BTi'p = - ci is also a sphere as that the two equations (23) [since they are both satisfied by at least one and the same value of p] become identical ; and (3) we will find the second quadric which always exists when Sp'sr^p = — c/ is not a sphere, of which the normal cone curve is a polhode. Putting in eq. (20) ■^fr = 2m-' (c, - Ojin-) is- it becomes (y + zts-y = 4«i-' (C2 - Cits-y Tz {y + ^V). 224 OCTONIONS. [§ 47 This can be satisfied by constant scalar values of y, z, •>/, z if and only if (y + zvif - 4m-2 (cj - c^isf vs {y' + sf'or) = (y" + z'ts) (ot= - m'W + m V - m), where y" and z" are two other constant scalars. Equating coefficients of the powers of -or it is easy to see as stated above that we are eventually led to a quadratic equation for the ratio of y : z. This shows that there are not more than two values of vs-^ of the form (y + zts) yjr~^'S7 satisfying the required conditions. We now show that there are two functions of this form except when Sp-sT^p = — Ci' is a sphere. We shall postpone till later the proof that when the two values of the ratio y : z are equal the quadric in question is a sphere. Suppose one set of values of y, z, y , z', c/, c/, -ts-^ has been determined to satisfy the required conditions. Then the normal cone curve is given by the two equations S/OOTi/) = - Ci', Spra-i^p = - C2', for these are always independent except when the roots of the CTi cubic are equal, i.e., except when S/j'sri/a = — Cj' is a sphere. [It will appear later that this is always a real sphere.] Let tti, ySi, 7i be the roots of the CTi cubic and Sj their half sum so that {-m^ - a,) (^1 - (8,) K - 7i) = 0, s, = i (a. + /3i + 7i) (26). If the constant scalars x, x and y' can be determined to satisfy {xw-, - ts^y = (icVi - y'lsyi"), the normal cone curve is also given by the two equations Sp'^iP = - Ci" = - (xCi - Ci), Sp^ip = - ci' = - (x'c' - y'ci)\ (^^)' where cts = a;wi - ■bti" (28), and expressing zj^ and 1:71= in terms of ot and i/r, it is clear that to-j and ct/ will be expressed in terms of -nr and i/r by equations of the same form as those for tB-i and Wjl Now Wi {x - ■uT^f - {x' - y'-sr,) = (13-1 - Hi) (^1 - /3,) (■S7i - 71), if x = s^, a? + y' = ^^y^ + y,a^ + a,^^, x' = a^fy,, § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 225 i.e. if a;=Si, x' = a^^^r^-^, 2/' = - i {(«! - aa)= + {s, - A)= + (Sa - 70" - Sil .(29), so that these values of x, x and y' satisfy the required conditions and we have CT„ = CTi (S, - tB-i) (30). From eq. (30) the roots a^, /3o, 7^ of the ct„ cubic are given by "2 = «! (Si - Ki), ^2 = A (si - /3i), 72 = 7i («i - 7i) (31), and these are proportional to the roots aj, ySj, 7, of the -m^ cubic if and only if Ml = ^a = 7i. i. e. if and only if the quadric Sp'us-^p = — c/ is a sphere. This proves that the normal cone curve is a polhode of both the quadrics Sp'STip = - c/ and S/j^jb-j/j = - c/', and that these quadrics though coaxial are not similar except when the first is a sphere. Also when one quadric is real the other is also. It is clear from the above that (if we put m^ = oufi^'y^ is a constant scalar multiplied by t/ti [eq. (25)]. As a matter of fact it is quite easy to prove from the above results that ^2 = -^i (32). The direct determination of the ratios if'.z^-.y': z' is tedious. The following process is therefore preferable. Changing the above ts to CjOT the original polhode has for equations Seo'STO) = — 1, SwB7°ci) — — h (33), where h is put for c^lc^. This is of course equivalent to taking in the above Ci=l, c^^h. According to eq. (4) i/r would now be 2m~^ ST (b — ij), but we will take the slightly more general form ■\^ = em~^'S!(h — ot) (34), and we shall assume that the normal cone curve is given by p = yjra with this new value of yjr. Thus the roots of the ■as- cubic [equations (1) and (2)] are now the inverse squares of the axes of the original quadric, if the ordinary conventions be adopted as to the imaginary axes of an M. 0. 15 226 OCTONIONS. [§ 47 hyperboloid ; and b is the inverse square of the perpendicular on the contact plane. Define ■bto by the equation •57 (s — w) .(35). «'0 — I. S—0 By equation (33) Swoto® = — 1. Also (s - by W = w (otS - 25^^ + s^ct), or by eq. (1) w„=' = '=r{m-(m'-s=)w}/(s-6)^ (36). Hence SwsrjtB = — 1, Swtn-o^w = — 60 (37), where bo= {-b{m' -^) + m}l(s-by (38), Hence the given polhode is also a polhode on the quadric StuCToW = — 1 and the inverse square of the perpendicular on the new contact plane is bo. If the Wo cubic is W-2s„W + mo'-^o-mo = (39), we have 2s„ = {a (5 - a) + /3 (s - ^) + 7 (s - y)}/(s - 6) = 2 (m' - s')/{s - b), < = {y87(s-/3)(s- 7) +...}/(« -6)= = {s'm' - s {2m's - 3m) + (m'" - 4ms)}/(s - 6)=, Too = ayS7 (s - o) (s - 0) {s - y)l(s - by = m(s'-2s' + m's- m)l(s - by, m — s^ , — ^m — ms + m'^ \ or ^0= J-, Too = s_6 ' '-»- (5_i,)= _ TO (— S^ + to's — to) e-o- .(40). From the first of these and equation (38) , _ s{m'-^)-m '"-"' (s-by From this we may notice in passing that Too _ TO ■% — b„ s-b .(41). .(42). § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 227 Also So - Wo = (m' -s'-t!rs+ w2)/(s - b), whence OT„(So-OTo)/(So-6o) = -sr {(s - ■57) (m' - s^) - TO- (s - «-)2j/{s (m' - s'') - m], or by eq. (1) '-^t^ m. Comparing this with equation (35) we see that the relations between i3-„, 60 and the coefficients of the ^^ cubic on the one hand and ■ST, b and the coefficients of the ot cubic on the other are symmetrical. When the given quadric is a sphere b must be equal to ct in order that the polhode may be real. In this case TOo = ct and the two quadrics are the same. "When s = 6 eq. (35) gives CTo = cx> . In this case there are not two different quadrics on which the given polhode is a polhode. Putting however w' = TO- (s — TO-) = e~^ my}r (44), [eq. (34)] we have SwTO-'o) = 0, Swto'^o) = s {m —s^) — m = (s-a)(s-^)(s~ry) (45). From this it follo-ws that Tyjro} is constant or the normal cone curve lies on a sphere. As we shall see directly this is the limiting case mentioned just now, when S/stoi/) = const, is a sphere. From equations (35) and (43) we have to-oTO-1 = (s- to-)/(s - b), to-to„-i = {So - w-„)/(so - bo) (46). Hence (s - to) (so - -EJ-o) = (s-b) (so - bo) = {s (m' - s^) - m]/(s - b). . .(47), by eq. (41). This very simple relation between s — to- and So — ■sj-,, makes it more convenient for many purposes to regard = s — to and ^0 = So — '='■0 ^-s the two fundamental pencil functions, to is a determinate function of <^, for s is the sum of the roots of the ^ cubic. We will not here however work with ^, (^0 and their cubics. 15—2 228 OCTONIONS. [§ 47 Byeq.(36) -sTo (bo - ^o) {s - hy = CT {bo (s - by (s - -sr) - (s - &) [m - -=3- (m' - s^)]} = -nr {[m - 6 (m' - s^)] (s - w) - (s - 6) [m - ot (to' - s")]} = — -57 (6 — ot) {s (jyi' — s^) — to}, or by eq. (40) m,,""^ i!7o (60 — ■CTo) = — m~^ in- (6 — -bt) (48). Hence if we define ilr, by the equation •\/f„ = eoTOo"-' t3-(, (6„ - OTo) (49), i/tj and 1^ will have the same value if eo = -e (50), and therefore p = i/^ow- Thus the normul cone curve will have the same meaning for both the quadrics of which the given polhode is a polhode. [If we put 60 = 6 the same statement is true, but the point Q on the normal cone curve corresponding to the point P on the polhode when that curve is defined with reference to the quadric Sm-srw = — 1, is diametrically opposite to Q„ the point on the curve corresponding to P when the curve is defined with reference to the quadric SwctoW = — 1,] By equation (35) we have 13- + i!7„ = CT (2s - & - •=r)/(s - 6) ; (51), OT — ■CTo = •CT (or — b)j(s — b) = — ■^mje (s — b) (52). Now define the pencil functions Wj and ts-j by the equations ^, + ^,= 2x{^ + ^,)f-^ = - f'^^ J + -^o (53^ e (s — 6) OT — -ST,, ^ ^' ^, - ^, = 22/V^-^ = - -J^ -i e (s — 6) •Ej- — in-i .(54), where x and y are ordinary scalars to be determined directly by certain conventions. Since p = i/rta, OTi/3 =[x(v7 + tD-o) + y} CO, -GT^p = [x (m + CT„) - y] w. . .(55). Also S/3 (isr, - •572) p = 2ySu]yjrco = 0. § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 229 X is determined by making each of the equal quantities Spv,,p and S/sOTa/j equal to — 1. Thus - 1 = ISp (oti + ffl-j) p = a;So) (•sr + Wo) i/rto = -xe{s-h) mr^ Sco (w' - Wo^ o [eq. (o2)J = xe(s- b) m-' (6 - 6„) [equations (33) and (37)]. Hence * x = — /Ka\ e(s-b)ib,-b) (56). In connection with this note that bo-b = (m-bm.' + 2s6= - 6')/(s - bf = (oc-b)(^-b)(y-b)/(s-by, or (b, -b){s- by = (a-b)(^-b)(y-b) = - w?m^e-\ . .(57). To determine y first notice that Sta-BTOToft) ,{s—b) = SWTO-" (S - w) ft) = Sft) (sw'' — 2sCT^ + m'-nr — m) w [eq. (1)] = sb-m' - map [eq. (33)]. Thus by equation (55) {■BTipY = StD {a; (cj- + ra-j) + y}^ o) = Sft) {ic^ (w'' + ■BTa') + 2xy (■ST + uTo) + 2*Vwi, + y^} w = -af(b + b„)- 4*2/ + 2a;'' (sb - m')/(s - b) + {y^-2aPm/(s-b)}o>\ Putting then y = x 'J{2ml(s -b)} (58), we shall have Sp'BTi'p = — 6i, S/)Ws=p = -62 (59), where ^ = 0^ {(b + 6„) + 2 (s + s„) + 4 V[2m/(s - b)]}\ b, = a? {{b + 6„) + 2 (s + So) - 4 V[2W(s - 6)]] j " " " ^ ^■ Since [eq. (50)] esi = — e,e (b^ — b) may be regarded as expressed symmetrically in terms of ts and CTq. It will be seen that x, b^ and bi are also thus symmetrically expressed when ml(s — b) is so expressed [see eq. (42) above]. By equations (40) and (41) we have m/(s — b) = sSo - {s-b)(so-bo) (61). From these we have Wi = a;{«r+^„+ V[2m/(s-6)]}t-' (62), 230 OCTONIONS. [§ 47 and Wj is obtained from •sti by altering the sign of the radical. Putting in this expression the value of x, substituting for -^ in terms of or — ■ctj, and utilising equations (57) and (61), we have e CT + 'nro+ V{2sSo-2(s-t)(So-6o)} /co\ ■STi = VO''/- m^ •B7 — ■ETo This is a symmetrical expression in ts and CTo for e = — eo and • /S > 7. We assume that none of these are zero. That the quadric Sco-sTco = — 1 may be real a must be positive. Also (by geome- trical interpretation) that the original polhode may be real b must be positive ; it must lie between a and 7 when Swoto) = — 1 is an ellipsoid, be greater than /8 when the quadric is an hyperboloid of one sheet, and be greater than a when it is an hyperboloid of two sheets. y, and therefore both the quadrics on which the normal cone curve is a polhode, are imaginary if to (s - b) is negative ; i.e. if s-b is negative when Swotw = - 1 is an ellipsoid or hyperboloid of two sheets, and if s-6 is positive when this equation represents an hyperboloid of one sheet. From this it is quite easy to see in any particular case between what limits b must lie in order that the two normal cone curve quadrics may be real. Thus if the 232 OCTONIONS. [§ 47 original quadric is an hyperboloid of two sheets b> a and there- fore s — b is negative. In this case, then, the two normal cone quadrics are always imaginary. When s=h, X and y both vanish, but as we have already seen 13-0 becomes infinite. Thus equations (53) and (54) become un- intelligible and this case requires separate consideration. We have for it [equations (44) and (45)] p^ = ^m-^{s-a){s-^){s-r^) = -p{-'- (71), say. Thus as already remarked the normal cone curve is a polhode on a sphere ^PPiP = - 1- That p^ is constant can also be seen from equation (10) since the coefficient of ay^ in that equation vanishes. The normal cone curve is in the present case generally not a polhode on any second quadric. For if it lie on the quadric Spvr^p = — 1 the normal cone must have for equation Sp{-ar2-pi)p = 0, so that this equation is the same as the equation Spylr~^p = 0. Hence 1:^2—^1 is a simple multiple of \^~'. We may therefore put Hence ■s^^p = (pi'^ + z) co, so that (®-2 p)' = Pip' + -S'^ta'- Here p' is constant, but co' is constant only in the extreme case when the original quadric is a surface of revolution. For if = or Sw (sp — ■BT^) (0=0. Hence is —pis a. simple multiple of sp—m^. Hence a relation of the form xw^ -)- j/tD- + 2 = 0, where x, y, z are constant scalars, holds good. Now ■m satisfies such a quadratic equation if and only if two of the roots of its cubic are equal, i.e. if and only if Scuto-w = - 1 is a surface of revolution. When it is a surface of revolution (wjp)- above is constant for any constant value of z. Hence in this case there is § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 233 a whole family of quadrics on each of which the normal cone curve is a polhode. It is not hard to see by similar reasoning that the last case is a particular one of a more general case. If Seozro) = - 1 is a surface of revolution w= is constant whatever constant value b have and therefore by equation (10) p^ is also constant. Both the polhode cone and the normal cone are coaxial right circular cones; and the polhode and normal cone curve are generating circles of these cones. Hence each of them is a polhode on each of a whole family of quadrics of revolution having the given centre for common centre. Returning to the singular case when b = s and Smra-o) = — 1 is not a surface of revolution we may notice that s cannot be equal to a or 7 but it may be equal to /3. We then have 7 = ^ - a, so that 7 must be negative and a positive. Moreover since b is positive and = /3, /S is positive. The surface is therefore an hyperboloid of One sheet and co is along the greatest axis. In this case we see by geometrical interpretation that the polhode reduces to a point. We also have p = yjrco = 0, so that the normal cone curve reduces to a point at the centre. This again is a particular case of a more general one now to be considered. The only other singular case when the above general solution breaks down is when x and y both become infinite by reason of b being equal to one of the roots of the w cubic. When the given quadric Sw^srco = — 1 is an ellipsoid it is easy to see geometrically that when 6 = a or 7 the polhode reduces to a point. Similarly when the quadric is an hyperboloid of one sheet and 6 = /3, or when it is an hyperboloid of two sheets and b=a, the polhode reduces to a point. In each of these cases, (b — zj) CD = and therefore the normal cone curve vanishes at the centre of the quadric. The only other possible cases of b being equal to a root of the w cubic are when & = yS for the ellipsoid and when 6 = a for the hyperboloid of one sheet. In both these cases one root of the yfr cubic is zero and the other two are of opposite signs so that the polhode cone Swi/ra) =0 becomes a pair of planes, i.e. the polhode consists of two plane sections of the given quadric whose intersection is the axis of the 234 OCTONIONS. [§47 quadric corresponding to the root of the vr cubic which is equal to b. These planes are equally inclined to a second axis of the quadric. The normal cone curve therefore reduces to the two straight lines through the centre which are normal to these two planes. It is quite easy to prove that the distance of Q a point on the normal cone curve from the first of these axes varies as the distance of P the .corresponding point on the polhode from the same axis. We now return to the cases when the general solution is applicable. There is an important reciprocal relation between the pair of quadrics CTi, •btj and the pair to-, i!7„ which shows (as we might otherwise anticipate) that the original polhode and its two quadrics are related to the normal cone curve and its two quadrics in exactly the same way as the latter are to the former. If we substitute from equations (.56) and (58) in eq. (67) for y we get rrh-'e, (s, -h,) = -e (h, - b) {(s - b)/2m]K But by eq. (60) 6i - 62 = 2e-' (60 - b)-' {2ml(s - b)}K ei (Si - h) (h - bi) e(s- b) (6„ - b) Hence If then we put e(s-b)(bo-b) 2m 2m = 1 = 9 .(72). .(73), and similarly for go, g^, g^, we shall have 9=9o, gi = g2, ggi = 'i- (74), ^ = 45'-^ (75), ■5r(6 — ■BJ-) ^ (76), ■\}r = yjro = 2g ^1 = ^2 = ■f~'^ = 2gi (s-b)(bo-b)' •CTl (&1 - TOl) {s.-bOib,-b,)) ■ra- — Wo .(77). ^ = '^-b-K If we put g = 1 and therefore ^Tj = 1 we may call the resulting normal cone curve the principal normal cone curve. Thus the given polhode is the principal normal cone curve of its own principal normal cone curve when the last curve is regarded as a polhode. Or we may call in this case the given polhode and its normal cone curve reciprocal or conjugate polhodes. 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 23c To give a notion of the possible magnitudes of the quantities appearing above, consider the four quadrics corresponding to each of two polhodes on one ellipsoid for which a = 8, /3 = 4, 7 = 2, 6 = 5 and |. On this ellipsoid we have real polhodes for all values of b from 8 to 2, but since s = 7 their normal cone curves and in particular their conjugate polhodes are not polhodes on real quadrics for values of b between 8 and 7. Some of the eight quadrics in the above two cases are ellipsoids and some are hyperboloids of one sheet. They cannot possibly be hyperboloids of two sheets, as we have already seen. The following tabulated results are easily calculated from the equations above. Specification in two cases of the two polhode quadrics AND the two conjugate POLHODE QUADRICS. Quadric (Semi-axes -^ (Perp.)-2 Semi-axes Perp. Ellip. or 8 4 2 5 •35355 -50000 -70711 -44721 Hyp. OTq -4 6 5 29 4 [ -50000] -40825 -44721 -37139 Hyp. OTi 9 16 81 16 45 16 261 16 [1-33333] -44444 -59629 -24759 Hyp. cTa 3 16 9 16 3 16 5 16 2-30940 1-33333 [2-30940] 1-78886 2nd CASE. Ellip. or 8 4 2 5 2 -35355 -50000 -70711 -63246 Hyp. tzTo 16 9 8 3 20 9 62 27 [ -75000] -61237 -67082 •65991 Hyp. cTi 13 216 11 24 473 216 2335 216 4-07620 1-47710 [ -67577] -30415 Ellip. I3-2 1 216 11 216 55 216 31 216 14-69694 4-43130 1-98173 2-63965 •236 OCTONIONS. [§47 Under the heading "semi-axes" the square brackets indicate the imaginary axes of the hyperboloids. In figure 9 is given an indication of the relative positions of the four quadrics of the second case. It represents, roughly to scale, three of the semi-axes of each quadric, and the traces of each Fig. 9. quadric in one of the corresponding "octants." To distinguish between the polhode quadrics on the one hand and the conjugate polhode quadrics on the other, the traces of the latter are put in dotted lines. Since in the present case, of each of these two pairs § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 237 of quadrics one quadric is an ellipsoid and one an hyperboloid of one sheet, the four quadrics are easily distinguishable. The four points where the two polhodes intersect the parts of the planes included in the octant are indicated by thick marks. The conjugate polhode it will be remembered is obtained by taking g=l. The curve first denoted above, "the normal cone curve," was the one for which e= 2 and therefore g = m-' (s - b) (6„ - b). It will be found that to obtain this curve and its corresponding quadrics in the second case just considered we must diminish each axis of the two quadrics Wj and CTj in the ratio of 6981 to unity. The figure that we thus get shows about the same dis- proportion between the dimensions of the two pairs of quadrics as does Fig. 9, but the disproportion is in the opposite direction, the axes of Wj and zr^ being all smaller than the smallest of the axes of IS and ■sf„. We conclude this section by establishing certain results which are symmetrical with regard to the four quadrics. Remembering [eq. (75)] that a; = ^cr^ we get from eq. (60) 2g' (b, + b,) = b + bo + 2(s + s,). Similarly since ggi = 1 2 (6 + bo) = g' {b, + b, + 2(s, + s,)]. By cross addition and subtraction we obtain from these g" {3 {b, + h)+2 (s, + s,)} = 3 (6 + 6„) + 2 (s + s„) . . .(78), . g' {2 (s, + s,) - {b, + h)} + {2(s + So) - (6 + 6„)} = . . .(79). Again from equation (60) Similarly b-h = 2g'^^^^ (80). Multiplying and dividing (h-mb-b^)=s^{^,^] (8i)> 238 OCTONIONS. [§ 47 The sjTnmetry of the last two results is rendered more evident by substituting from equation (61) wii/(si - bi) = S1S2 - (si - &i) (S2 - ^2). ™/(« -i) = sso -(s-b) (so - bo). Again by eq. (77) p^^gQ} — 6o)~' («^ - 'sf 0) «• Hence p" (b - hyi^g'' = -b-bo- 2Sco^'Boa) = - (& + 60) + 2 {may' + {m'-^) + s{s- b)]/(s - b). by the value found above for Smwzjow. Thus by equations (40) and (80) 9y .frV, = <"= ^j, + 2 (s + 5„) - (6 + to) (83), which by eq. (79) may be written ^' ^^ = f[p'^^ + 2{s. + s,)-ib, + h)^ (84). [Both equations (79) and (80) may be deduced from eq. (83) and the one it is deduced from by reason of the symmetry of the relations between the two pairs of quadrics.J A symmetrical form of these equations is f{p'^^ + ^(s^+S.)-{br + h)^ = |«= ^j^ + 2(s + So) - (6 + 6o)l . . .(85). The last three equations are of course only eq. (43) § 46 in a simplified form. They may in fact be deduced from that equation. It is not hard to prove from equations (40) and (57) that 2(s + So)-(b + bo) = (y8 + 7 - 6) (ry + a - 6) (a + /3 - b}/{s - by. . .(86). Since the left is symmetrical in w and Wo, and by eq. (79), we have § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. (^ + y-b)(y+a-b){a + ^-b) . 239 (s - by ^ (A + 7o - ^o) (70 + «o - ^0) («o + /3o - bp) {So-bof ^_ , (^1 + 7i - ^1) (71 + «i - &i) («i + A - h) x7 \ (87). ^,(^i±7^ {s,-b,r ■ h) (72 + Oa - 62) (a2 + A - b^) {s,-b,y Thus if any one of these vanishes they all vanish; i.e. if b is equal to the sum of two roots of the is cubic, bo is equal to the sum of two roots of the OTo cubic &c. In this rather curious case we see by eq. (83) that Tp/Tco is constant. From eq. (79) it follows that at least one of the four quadrics TST, Wj, iffi, CTj when all are real is an hyperboloid, i.e. as we have already seen, an hyperboloid of one sheet. For if •nr is an ellipsoid, s is positive, and if CTj and vr^ are real, s — 6 is also positive. It follows that if all four were ellipsoids 2 (s + s„) — (b + b„) and 2 (si + Sa) — (61 + 62) would both be positive and not zero, and this is impossible by eq. (79). They may however all be hyperboloids though the case is sufficiently rare to require careful artificial construction. The following properties of this case are stated for brevity without proof. As usual above, a > /3 > 7 so that a and /3 are positive and 7 negative. Assuming this only we have for the present case flio and 7o positive, /So negative, Oj > 70 >/8o, b — s, a — s, s — /S, and s — 7 all positive, bo — So, Ko - So, So-^o, and «„ - 70 all positive. Of the four quadrics w, cto, crj, is^ we may take the first pair such that 2 (s + Sj) - (6 + 60) is positive. Assuming this b is between a + 7 and a + /9. [If we assumed that 2 (s + So) - (6 + ^o) was negative we should have b between s and a + 7 or between a + ^ and + 00 .J 6 is not equal to 60, for if it were we should have the singular case where b is equal to a root of the w cubic. "We may then 240 ocTONiONs. [§ 47 choose the quadric is such that h>h^. Assiiming this we shall have 2a + 7>a + y3>6>a>ao>6o>ao + /3o>7o. ffi and y8i positive, 71 negative, «! >/3i > 71, a^ and 72 positive, /S2 negative, Oa > 72 > /Sa. Thus the real smaller semi-axes of all four hyperboloids are coincident (a, ao, a^, Oa). The larger real semi-axis of ts is co- incident with the imaginary semi-axis of m^ and conversely. A similar statement also holds with regard to the semi-axes of ■BTj and 13-2. These statements may be illustrated by taking a=50, /3=6, 7 = -16, h = ^ , when it will be found that 18 „ ^ 18 ^ 18 ,„ «„=|g.l25, ^o = -j9.7, 70 = 49.48, h =— 23 ■ 37 _ 18 851 " - 49 • 7 ~ 49 ■ "T and also 206.277 _ 206.24 ^ 50.23 ^"' = ^1749-' ^^'= 109' ^'^' = -21749' 206 ■ 193 ,- 206.16 , 50.19 ^"^=^09"' ^'^^ = -109 ■ ^"y^^ 21749' 61 and 62 can also be found from eq. (60). They are greater than «! and 02 respectively. From equation (78) we can get another symmetrical result similar to eq. (87). It is (s-6)-={(^-f7-&)(7-fa-6)(a-|-/3-6) + 4(a-6)(/3-6)(7-6)}-|-86 = (So-6a)-M(A + 7«-6o)()() + 4(a„-6„)()()} + 86„ = ^n^i - 6i)-M(A + 7i - 61) ( ) + 4 (a, - 6,) ( ) ()} + ff'8&, =g' fe - h)-' {(A + 72 - 62) ( ) ( ) + 4 (02 - &.) ()} +^^862 (88). 48. Motion of rigid body resumed. Potential. Free- dom and constraint. Reactions. Impulses. Return now § 48] EXAMPLES OF THE APPLICATION OF OOTONIONS. 241 to the discussion of the mechanics of a rigid body as considered in equations (1) to (28) § 46. F^, Go, F, and their rates of variation may be expressed in terms of E the displacement motor and its time derivatives. The formulae are too complicated to be of much use in a general discussion, so I content myself with writing them down without proof [See eq. (30) § 20.] F = ESE-'E^2Me^^ME-'E.e-i^ (1), F, = ESE-^E+'2.e-^^lAe^^.lAE-'E (2). F, = E + {\- E -e-^)ME-' E = E +[-^^ + -^^--^ + ...]\AEE This last may also be written ^ ( \ E E'' \ ^+l-2!+3-!-4!+-) (3), from which F, = E + {\-E-e-^)H\E-^E-{E + {2-2E-2e-^-Ee-^) X SE-' E + Me-^ME-' E] ME'' E (4). This might perhaps prove useful in writing down to any required degree of approximation equations of motion when E is small. Suppose that the external forces have a potential v. v may be regarded as a function of E but it is simpler to regard it as a function of 2MQ or E' [eq. (2) § 46]. If Q be varied by the infinitesimal SQ the small displacement that the rigid body receives is 2hQQr' [eq. (8) § 11]. The work done in this displacement is - 2sHBQQ-y Hence by eq. (8) § 14 2sBQQ-' H=Bv=^- sBE"^v = - 2sSQ^, where now the independent variable motor implied by ^ is E'. Since SQQ-^ is an arbitrary infinitesimal motor we have by § 14 H = -MQ^v (5). Hence by eq. (8) § 46 Ho = -M^vQ (6). These may be substituted for the force motor in the various equations of § 46. It may be noticed that v = - 2s0^t; = - sFQ^v = sFH. M. o. 16 242 OCTONIONS. [§ 48 Also by eq. (11), § 46 > = - si^oto^o = - sF,H„ [eq. (25), § 46] = -si^^[eq.(8),§46]. Thus v + f=0, i.e. the sum of the potential and kinetic energies remains constant. When the displacement is small we may in all these equations put Q = 1. Thus H = H, = -'^v (7). We have seen (§ 46) that in this case E' = E, so that we may suppose the independent variable motor implied by ^ to be the displacement motor. H and H^ were originally defined as the external force motor. If the body is constrained it is necessary to state whether or not the reaction of the constraints is to be included under the term " external." Let us suppose that it is so included. Then equa- tions (21) and (25), § 46, remain true for a constrained body but equations (5), (6) and (7) of the present section do not. Let P be the external force motor exclusive of the reactions and let P=QPoQ-' (8). In this case the system of forces is due to two causes, the " field," as it may be called, with potential v, and the reactions. Of these P is due to the field. Hence in place of equations (5) and (6) we have P = -MQV P, = -M'irvQ (9). Also if i?= QRoQ~^ is the force motor due to the reaction of the constraints, H = P + R, H, = Po + Ro (10). On account of the constraint ^(and therefore also Fo, Q and Oo) is for a given value of Q confined to a definite complex whose order is the number of degrees of freedom. Let us call the complex to which F is confined the velocity complex, and that to which G is confined the momentum complex. The constraint is assumed to be smooth. Hence the reaction is such that it would do no work on the body whatever were its instantaneous motion consistent with the given constraints. Hence -R is confined to the complex reciprocal to the velocity complex. Call this re- ciprocal complex the reaction complex. § 48] EXAMPLES OF THE APPLICATION OF OCTONIONS. 243 The complexes to which F^, Oa and R^ are confined may similarly be called the velocity, momentum and reaction complexes referred to the standard position respectively. More shortly we may call the complexes, the F complex, the R^ complex, &c. Eq. (25) § 46 and the last equation give P„ + i2o = 'fo-P'o + Mi^„l/r„i^„ (11). Here P,, and a^o are given, F^, is confined to a definite complex and P„ to the reciprocal complex. Thus F^ and R^ together involve six unknown scalars ; and equation (11) is equivalent to six scalar equations ; it is therefore sufficient to determine R^ and F„ and therefore the motion. If P' = QPo'Q~^ is a given external impulse motor and R' = QRo'Q~^ the impulsive reaction motor, R' is still confined to the reaction complex., If F, and Fo + AF^ = Q'^ (F + AF) Q, Qo and /r. T|r as defined directly is for the case of small motions, our previous y]ri and not what has hitherto been denoted by i/r. For small § 49] EXAMPLES OF THE APPLICATION OF OCTONIONS. 247 motions our previous i/r,, and yjr have the same meaning and we continue to use yfr^ with that meaning.] When the body is free the displacement motor, the velocity motor, the momentum motor and the force motor are denoted by Eo, i'o, "^oEa, — •ni'o-^o respectively, where i|ro and xb-q are complete energy functions ; the kinetic and potential energies are respec- tively - |s£'„A|r^'„ and - ^sE,is,E,. When the body is constrained the displacement motor (and therefore also the velocity motor) is confined to the given complex (n) called either the displacement complex or the velocity complex. (6— n) is any complex independent of (n); (n) and (6 — n) are the reciprocals of (6 — n) and (?i) respectively and are independent of one another. En and E^-n will be used to denote arbitrary motors of (n) and (6 — n), respectively. The complex to which ■\}r„En is confined is called the momentum complex, that to which ^o^n is confined is called the force complex, that to which the reaction motor due to the constraints is confined is called the reaction complex. The reaction complex is (6 — w). (h) may but will not always be taken as the momentum complex. The displacement motor, the velocity motor, the momentum motor and the force motor are E, E, yjrj: and -iSoE. The reaction motor is R. The partial energy functions ■\jr and w are uniquely (§ 34) defined by the equations sEn^rEn = sEnfoEn, fE^ = (2), sEn'nrEn = sK^o£'„, ^-^e-n = (3). By § 34 it follows that yjrF and 'tsF are both confined to (n) what- ever motor value F have. -ylrE and — 'stE are called the general- ised momentum motor and the generalised force motor respectively. They are for small motions the f^C and ^i of eq. (20) § 48. For constrained motion the kinetic energy =-^sEyp-J: = -^sEy}rE, and the potential energy = - ^sEisoE = - ^sE-ssE. From these definitions we see that ra-„ and i/^o may be regarded as particular forms of w and -«|r, viz. when n = 6. The equation of motion is by eq. (20) § 48 ■^E^-isE (4), or E = -f-,T^E=-4>E (5). 248 OCTONIONS. [§ 49 See equations (6) to (9) | 33. The general real solution of this equation is E = cos (t ^(t>) . E, + sm(t >J)/^/(}) . E^ (6), where i\ and E.^ are the initial values of E and E respectively. A physical definition of the generalised momentum and force motors may be given. The present n, ^jro or nr,,, -^ or •or, En and Ee_n may be identified with the to, w, w' , E^ and E^ of the last proposition but one of § 34. It follows from the forms there given for ■m and ■m' that, since E and E are confined to (n), ■y^E and 'stE are the components oi -^frf^ and 'stoE respectively in (7?) when the last motors are each expressed as an (n) component + a (6 — n) component. Thus the generalised momentum motor is the (n) component of the actual momentum motor, and similarly for the force motor. The important result follows that when (n) is taken as the momentum complex the generalised momentum motor is the actual momentum motor. The force complex is not necessarily independent of the reaction complex (6 — n), so that (n) cannot always be identified with the force complex. If it can however, by so identifying it, the generalised force motor becomes the actual force motor. It will be observed that only in very special circumstances, viz. when the momentum complex and force complex are identical, can the generalised momentum motor and the generalised force motor be simultaneously regarded as the actual momentum motor and the actual force motor re- spectively. By § 33 there are always n real coreciprocal motors of (n) forming a conjugate set with regard to yjr. By | 34 they also form a conjugate set wilh regard to yfr^. They are what Sir Robert Ball calls the n principal motors of inertia. Similarly the n real coreciprocal motors of (n) which form a conjugate set with regard to or (and therefore also with regard to z:^) are what he calls the n principal motors of the potential. Again by § 33 there are n real motors (OiO^... of equations (6) to (9) § 33) which form a conjugate set both with regard to •sr and i/r. These are what he calls the n harmonic motors. The important property which gives them their name is at once proved fi-om eq. (9) § 33 and eq. (6) of the present section. § 49] EXAMPLES OF THE APPLICATION OF OCTONIONS. 249 Most of the theorems of Screws not already explicitly con- sidered above now follow almost obviously. I content myself with considering a few of the less obvious. In § 55 of Screws the question is asked — When the velocity motor E is given, what motors will serve as impulses each of which will generate E from rest ? If the body is free the answer is yfrJE only. If it is constrained the answer is •v/r(^ + R where R is any motor of the reaction complex (6 — n). If Aj, Ai ... An are n conjugate motors of inertia and if the velocity motor E is given by E = «iJ.i + ... = IxA, the kinetic energy is - i sEfE = - i lx"-sAfA. This is part of § 58 and § 59. In the rest of § 58 is considered the effect on the kinetic energy of the superposition of two impulse motors Pj' and P^'. Resolving these each into two com- ponents (§ 48 above), the one in the momentum complex and the other in the reaction complex, let ffj and (?„ be the former com- ponents. The kinetic energy acquired is = - i s G,fr G,-s G.fr (?2 - i sG.ir,-^ G,. It is therefore the sum of the kinetic energies due to P/ and P^' separately if and only if Gi and G^ are conjugate with regard to i|r(,~', i.e. if and only if the velocity motors ■yjriT'^Gi and ■^^^-'^G^ acquired are conjugate motors of inertia (i.e. conjugate with regard to •\|r„). In § 60 is considered how the twist velocity acquired due to a given impulse P' when the body is constrained to twist about A depends on ^. By the method of § 48 (that A(r — P' is reciprocal to the velocity complex) we see that if xA is the velocity motor acquired xyir,A = P' +R, where R is reciprocal to A. Thus xsAy^tiA = sAP', or X varies directly as sAP' and inversely as sAy^r^A, which is the theorem enunciated for this case. 250 OCTONIONS. [§ 49 In § 64 a theorem due to Euler is proved, viz. that if (m) be a complex included in («) the kinetic energy acquired by a given impulse P' is greater when E is allowed the freedom of (n) than when it is further restricted to (m). Let F' be the velocity motor acquired when the freedom is (m) and F when the freedom is (n). Thus P' — •v/fo ^ is reciprocal to every motor of (n) and P' — i^„F' is reciprocal to every motor of (m). They are both therefore reciprocal to F' which belongs to {m) and therefore to (n). Thus si?"i|r„P = sFP' = sF'^jr.F', or s (F - F') y^ToF' = 0. Hence F — F' and F' are conjugate motors of inertia. The kinetic energy due to F is therefore (see above) the sum of those due to F' and F — F', i. e. (except when F= F' and therefore the restriction to (m) is really no restriction) the kinetic energy due to F is greater than that due to F'. In §§ 65, 66 of Screws there are certain errors due to the assumption which is explicitly stated in § 66 but which is not true, viz. that " a given wrench can always be resolved into two wrenches — one on a screw of any given complex and the other on a screw of the reciprocal complex." If the given complex contains any screw which is reciprocal to the whole complex this is not true, though it is otherwise. For instance if the given complex is that of the third order consisting of all the rotors through a given point, the reciprocal complex is identical with the given complex and therefore the statement is obviously untrue. Thus the attempted proof in § G5 that " one screw can always be found upon a screw complex of the nth order reciprocal to 71 — 1 screws of the same complex" is unsound. For the 6 — n screws reciprocal to the given complex that are taken are not necessarily independent of the n — 1 screws of the given complex. The statement itself is true, but sometimes more than one screw can be found since there may be [§ 30 above] a complex (of not higher order than the third) in the given complex which is reciprocal to the whole complex. The proposition may be proved thus: — Let (n) be the given complex of order n and let (m — 1) be the complex included in (n) consisting of the given n—1 motors. Let (7 — n) of order 7 — ?i be the complex reciprocal to (n — 1). Then the two complexes (n) and (7 — »i) must contain at least one motor in common, for otherwise we should have 7 independent § 49] EXAMPLES OF THE APPLICATION OF OCTONIONS. 251 motors. Thus there is at least one motor of (n) which is re- ciprocal to (?i - 1). This does not prove that this wth motor is independent of the given n-1 motors. And as a matter of fact it is not necessarily thus independent. For instance the only motor of the complex i, D.i which is reciprocal to £Li is D,i itself (and ordinary scalar multiples of it). The main proposition of § 66 is erroneous, viz. that "a wrench which acts upon a constrained rigid body may always be replaced by a wrench on a screw belonging to the screw complex which defines the freedom of the body." For instance if the complex is that of i the freedom enjoyed is that of rotating about i. But the only wrench on the screw i is a force along i, and this cannot replace any given system of forces, such as a couple whose plane is perpendicular to i. Thus (Screws, § 66) Sir Robert Ball's "reduced wrench" is not always intelligible. When it is intelligible it is a useful conception. It is intelligible when the reciprocal (6 - n) above of (n) is independent of (n), i.e. when (n) contains no motor which is reciprocal to (n) (i.e. no rotor or -lator intersecting every other motor of the complex perpendicularly). In this case (6 — n) may as we saw in § 48 be defined as the reciprocal of (n) ; and then (n) becomes identical with (n) and (6 — n) with (6 - »). Further, in this case our generalised force motor becomes identical with Sir Robert Ball's reduced wrench. The main object apparently of the introduction of the reduced wrench is to obtain a definite motor function of the given force motor which has the same mechanical effect as the force motor itself. This may be done in a way that is always intelligible, viz. by identifying (n) above with the momentum complex. The generalised force motor that we then get may be called the virtual force motor. It may be defined as the component of P the force motor in the mom-entum complex when P is expressed as such a component + a component in the reaction complex. Similarly the virtual impulse motor (? of a given impulse motor P' may be defined as the component of P' in the momentum complex when P' is expressed as such a component + a compo- nent in the reaction complex. The virtual impulse motor is then the given impulse combined with the impulsive reaction, but a similar statement does not hold for the virtual force motor. 252 OCTONIONS. [§ 49 When w= 2 Sir Robert Ball establishes in §§ 102 and 103 the existence of what he calls the ellipse of inertia and the ellipse of the potential. Similarly when m = 3 he establishes the existence of the ellipsoid of inertia and the ellipsoid of the potential. Since by present methods (see §§ 43, 44 above) the treatment of these two cases — n = 2 and n = 3 — are very similar, we content ourselves with the consideration of the ellipsoids only. Suppose then re = 3, and suppose as in § 44 above that the case is not what is there called a singular one. Thus the complex (w) contains no motors which are reciprocal to the whole complex, so that in this case (Ti) and (n) may be taken as identical. Putting p^ for the >|f of § 44, so as to enable us to retain our present meaning of i/r, we see that every motor of (n) can be expressed as (1 + ri'x) w, where