LIBRARY OF THK University of California. Class A TREATISE OCTONIONS, EonDon: C. J. CLAY AND SONS, CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, AVE MARIA LANE. ffilasgob): 263, Abgyle Street. lLcip)ig: F. A. BROCKHAUS. llcto lork: THE MACMILLAN COMPANY. Bombap.: E. SEYMOUR HALE. OCTONIONS A DEVELOPMENT CLIFFOED'S BI-QUATEENIONS ALEX. M^AULAY, M.A. PROFESSOR OF MATHEMATICS AND PHYSICS IN THE UNIVERSITY OF TASMANIA. CAMBRIDGE : AT THE UNIVERSITY PRESS. 1898 All Rights resei-ved. M3 (fTambrttigc : PRINTED BY J. AND C. F. CLAY, AT THE DNIVERSITY PRESS. 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. a 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 development 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 Avords, 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) 1(>4909 to study the Ausdehningslehre (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 w^as 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 reflection 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. VH to Euclidean space; and (3) T 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 texvcui convert, convertor, axial quatet-nion (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 Vlll PREFACE. remembered that in the technical use of this treatise the term variation is always qualified by some adjective such as combina- torial or ci7xular.] 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-quaternions 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 8crev)s. Explicit references to treatises on Quaternions are almost wholly omitted, as the reader must be assumed familiar with this subject. The references to the Theoi-y of Screws and to the Ausdeknnngs- lehre are copious. Except from these subjects and treatises I have, as far as I know, received no aid from the work of others. 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. M^AULAY. University of Tasmania, 1 Autiitst, 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 {q~^) of a formal quaternion 9 4. Primary and secondary systems of formal quaternions . . 11 a 11 5. Vector 17 Quaternion 1^ Quotient of a vector by a vector 18 6. Lator .... 1!) Rotor, axis of rotor 19 Motor, axis of motor 19 Octonion, axis of octoniou 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 OF a rotor 24 8. Velocity motor 26 Force motor 26 Momentum motor . 26 Impulse motor 26 9. Axial, rotor, lator, and convertor, of an octoniou ... 27 Augmenter, tensor, additor, and pitch, of an octonion . . 28 Scalar octonion 28 Twister, versor, and translator, of an octonion .... 28 XU DEFINITIONS AND EXPLANATIONS OF TERMS. SECT. PAGE 9. Scalar octonion part, ordinary scalar, scalar converter, and convert, of an octonion 28 Motor, rotor, and lator, of an octonion 29 Converter 29 Positive and negative scalar octonions 31 Positive and negative scalar convertors 31 10. Quasi-tensor of a lator 34 Quotient of two 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 14. i, y, ^ 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, see § 30.] f (Zi, Z,); x{Z,, Z,, Z,, Z,); ^ {(, . . . . . 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. (f) cubic, (pi 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; ^,j(„ k^; Cy U; p 85 V, Vo 85 d2, d\ 86 Suffix notation for v 87 V- 87 >» 88 22. Rotor spin 90 Lator spin .91 23. »?, //, d^', d\' 92 Xo> Xr, X 93 y\r, -ir 93 24. a 96 DEFINITIONS AND EXPLANATIONS OF TKRMS. Xlll SECT. TAOE 25. Motor intensity, motor flux 97 V 97 Motor spin 99 Intensity flux, flux intensity 101 28. Combinatorial product 106 {A,A^A,A,A,A,}, jA,A,A,A,A,} 108 A„ A,, A,, A^, A„ A, 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 (n-l)th order and second degree . . . .119 IMotors 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. w_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 (p n-tic . . .157 /W 159 38. Commutative combinatorial product 166 Commutative combinatorial variation 166 Motors fully conjugate with regard to a commutative self-conjugate function IC/ Norms (with reference to a commutative self-conjugate function) of types A, B, A^, D^, C 168 Axial complex 168 [Commutative] conjugate variation 169 Ellipsoidal function 170 40. Axis of a second order complex 177 42. Tetrahedron p, p = 2Vap6 (where a and b 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 (f), 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 from the funda- 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 VaySy = aS/97 - /SSya + 7Sa/S (from which may be at once deduced in the ordinary way that pSoL^ry = aS/t^Yp -f . . . = W^ySap +...); secondly that three formal vectors i, j, k can be found such that jk = i, ki =j, ij = k, {■' =y = k- = ijk = - 1 ; § 3] FOllMAL QUATEKXION.S. and thirdly that it' ^, be defined by the eciuatioii ^^u r.)= ^(i, i) + f {./,]) + ir{k k), r.by x(r.. ?.. K: ?.) = x(^.. ^n ^ + x(?.. ?M.A.y) + x(r,- r>. z^-, /^-x and so to any niunber of pairs of f's (i/r and % being formal (juater- nion functions linear in each of their constituents) then where « is any formal vector and ^ 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

7\ = — q~^rq~^. Hence Q-^ = q-' - ^q-^-q-^ (7). [It may be noticed that from this if Q = Or, Q~^ = Hgc 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 Vf' = 0.1 §4] FORMAL QTTATERNTOXS. 13 With these definitions : — If the primari/ srjfitein (j, r, dc. is a system of formal quaternions, so also is the secondary system Q, R, cf-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 H 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+R) = {P + Q) + li, P.QK = PQ.R, (P + Q)(R + S) = PR + PS + QR + Q8.] (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. [H 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 X + riy and are therefore formal scalars. The equation (4)(«) Q=SQ + VQ is obvious. To prove the second equation (4) (a) Q = TQ\JQ, we have to prove that q+nr = (l + nSrq-') Tq{l+ nWrq-') Uq. Noticing that by definition Q, is commutative with every formal quaternion of the primary system and that f>- = 0, the expression on the right = T^U^ + n (Srq-' . TqUq + ^/rq-' . TqUq) = q + D,rq~^ •'/ = '/ + ^^'• To prove (4) (b) we have first to show that TU^= 1. Now if \JQ = q' + fir' we have by equation (o) TUQ = (l + nSr'q'-').Tq'. But by equation (G) q' = Uq, r = Wrq-' . Uq. Hence T\JQ=l + nS(\/rq \Uq. U7-') = l. 14 OCTONIONS. [§ 4 Next we have to show that T {QR)=TQTR or T(q + nr) . T ((/ + nr') =T.(q + ^r) (c/ + Or'), where q, r, q', r are any formal quaternions of the primary system. Thus by equation (5) we have to show that (1 + nSrq-^) Tq.{l + nSr'q'-') Tq = T {qq + fi [q/ + rq']) = [1 + nS . (qr' + rq') (qq')-'] T (qq'). The left of this equation is TqTq' {1 + OS (rq-' + rq''')}, and the right is T{qq')[l + nSirq-' + r'q'-% so that the equation is true. The equation (4) (c) KQ = SQ-WQ is obvious. To establish the equation KQ= {TQyQ~' it is necessary to show that Kq + nKr={(l + nSrq-')TqY'(q-' - nq-'rq-'). Remembering that \l^q .q = (Tq)" the right becomes (1 + 2nSrq-') Kq(l - ^rq-') = Kq + nKq (- rq-' + 2Srq-') = Kq + nKqK (rq-') = Kq + aK (rq-' . q) = Kg + nKr. The statements of (4) (d) except those referring to x-' and (a;2)z 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 (X~)^ = ± X when X is a formal scalar of the secondary system. Fs where Y is such a formal scalar means of course a formal scalar whose square = Y. Let Y = x + riy, Y^ = X + %'. Thus X + n?/ = (oG + n?/')- = x^ + D.1ocy'. Hence x = ± a^, y = ± ^x~-^y, \/x + ny = ± '^x (l + n ^] (8). § 4] FORMAL QUATERNIONS. 15 [This is obvious from the similarity of D to an iiifiuitcsiiDal scalar.] Hence {{w + ClyY] i = {a;- + 2D.xij)^ = + ,f (l + n |) = ±(a; + %), i.e. (ZOi = + Z. (5) is obviously true for the secondary system if true for the primary for r may be put zero. (6) consists of definitions only. CHAPTER 11. OCTONIONS AS FORMAL QUATERNIONS. 5. The meanings of certain words. The following tabular comparison of the meanings of certain 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 aocial 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 :i certain axis. A i-otor 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 07ily 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 quaternion he ought to call his rotor a vector ; i.e. his quaternion has like his rotor a definite axis fiwed 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 specificat M. o. ion 2 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 § G] OCTONTONS AS FORMAL QFATERNTONS. 19 QA where Q is an octonion and A a motor (a particnlar 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, &c. 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, &c. A lator is a quantity which requires for its specification and is completely specified by a direction and an ordinary magnitude {scalar). A rotor is a quantity tuhich requires for its specification and is completely specified by a direction, an indefinitely long straight line parcdlel to that direction and an ordinary magnitude. The line is called the acds of the 7'otor. 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 axis 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.] An octonion is a quantity ivhich requires for its specification and is completely specified by a motor and two scalars of which one is called its ordinary scalar and the other its convert. The axis 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 q' 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 cor7'esponding 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 P would obey among themselves all the laws of the § 6] OCTONIOXS AS KOIIMAI. (^UA TKKNIONS. 21 corresponding (|uaternions. As a matter of fact they do, but this must be proved.] The real meanings of the definitions however apart from theiv form are independent of the point used; i.e. if Q and R be two octonions, although is used in the definitions of QR and Q + R, these latter octonions are dependent solely on Q and R and not on 0. Since luhen we limit ourselves to axials through we are to all intents and purposes dealing with ordinary quaternions we use (/, ?■.., 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 uow 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 later parts and its two scalars are known. It is therefore completely specified by OT, 6), X, X, ij, where ct, co and A, are rotors through and x and y are ordinary scalars ; rs being the rotor perpendicular on the axis, w the rotor parallel and equal to the rotor of (the motor of) Q, \ the rotor parallel and equal to the later of (the motor of) 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 ct intersects each of these three parallels perpendicularly. Instead of is, (o and \ we may use a> and a where a is the rotor through defined by a = CT&) + X (1), which gives CT = M(T(y~', \ = (uScTft)"' (2), so that now Q is completely specified by CD, (T, X, y. 22 OCTONIONS. [§ 7 When the rotor of Q is zero, Q is still completely specified by a>, a-, X and y (i.e. by (4). It may be remarked that these transformations fail when Q and A cease to have definite axes [eq. (3) does not fail even when Q has no definite axis so long as the axial part which in this case must be a scalar SjQ is not zero], but are always legitimate (i.e. p, T and p' are not infinite) when both have definite axes. Thus QA = {l+n{p + t)} {I + np'\ q(o = \l + n(p + p + r)) qco = \l + n(p+ p')] qco + nM.rM.{l + n{p + /)} qco, the last transformation being true since qw and t are perpen- dicular intersecting rotors. Let the angle of q (the axial of Q) be called the angle of Q. By quaternion interpretation the rotor qeo is obtained from the rotor 0) by (1) rotation round the axis of Q through an angle equal to that of Q, (2) increase of the tensor in the ratio of Tq = T^Q to unity. By eq. (4) we see that multiplication hy 1 + Cl(p +p') changes this rotor into a motor whose rotor part is qo) and whose pitch is p +p'. Finally by eq. (1) § 7 we see that QA is this last motor translated through a distance equal and parallel to t. More definitely we may suppose A ={1 + ^p') w, changed succes- sively into (1) Tq(l + ap')co. (2) q (1 + np') ft) - ( 1 + np) qo). (3) {l + n{p+p')}qa>. (4) {1 -f- n (p+p')} q' = (M,A + W\,Ay = MiM + nW\,''-'AM,A (1). Hence defining e^ by the equation e'^^'u:^){pi,^^Y' (19). p" is of course assumed to be the scalar and ts' the rotor of this axial. Thus by eq. (18) p" is the pitch of A-\-B. The rotor of A -\- B is equal and parallel to «] + ;S and intersects the shortest distance between A and 5 at a distance Tct' from towards B. Eq. (19) gives /' = S (^a, + /y8) (a, + /9)-i - vsIAclS ■ («, + yS)-^ . . .(20), t^' = zjS/3 (a, + /3)- + (p -p) lAoiS . («, + /3)--^ . . .(21). In particular we may notice that if TA = TB, i.e. if Ta, = TyS, and p=p', A + B is a motor through the middle point of the shortest distance which bisects the positive directions of A and B, and A—B is a motor through the same point which bisects the 46 OCTONIONS. [§12 positive direction of A and the negative one of B. Both these statements can be deduced directly from the equations Q+KQ = 2SQ, Q-KQ = 2MQ. Hence UA ± UB are perpendicular motors passing through the middle point of the shortest distance. Similarly ATr^A ± BTr^B are perpendicular motors but they do not in general intersect. 13. Some expressions involving more than two motors. Before speaking of the products, &c. of three motors it is well to state the general theorem which is suggested by the similarity of the meanings of S^AB, M^AB, T^AB and U,AB to those of the corresponding quaternion expressions. Let us denote by the term "any formal quaternion function," any function of the octonions Qi, Q-i--- which involves beside the octonions themselves only the symbols which occur in ordinary quaternion formulae (M taking the place of V). Thus MAB, SAB are formal quaternion functions but M^AB, sAB, &c. are not. If Q be any formal quaternion function of the octonions Qi, Qo..., then q' the quaternion corj-esjionding to the axial of Q is the same function of g/, q^'... the quaternions corresponding to the axials of Qi , Qo • • • as Q is of Qi, Qo .... Put Q = q+nr, Q^ = q, + nr,, Qo = qo + iln,... where q,r, qi,7\, ... are axials through some one point 0. By § 9 q, qi, qo... are the axials of Q, Q^, Qo... translated to pass through 0. Hence q', q(, q.i ... are the quaternions corresponding to q,q^,qo.... But since H behaves (in a formal quaternion function) like an infinitesimal scalar it follows that q is the same function of qi, q^..- as q+ Hr is of q^ + firi, go+ fir.,, ... ; i.e. q' is the same function of g/, qJ... as Q is of Qj, Q., .... A particular case of the above theorem is that Wq', Sq', Tq' and Uq are the same functions of g/, q^', ... as are WiQ, SQ, TQ and UQ of Qi, Q^ .... Considering now the axials of MQ, Q-^, &c. it follows that : — Making abstraction of the positions in space (but not of the directions) of the axes, MjQ, S^Q, T^Q and UiQ have precisely the § 13] OCTONIONS AS FORMAL QUATERNIONS. 47 same geometrical relations with the axials of Qi, Q.,... tlmt V7', S^', Tq and \Jq' have luith the quaternions q(, (/./.... The first of those theorems might have been proved in § and the second in § 9. Note that a result of the second is that one intei-pretation of any octonion formula which involves only formal quaternion functions is the ordinary quaternion interpretation. Thus ^^ABG is the same function of the rotor parts of the motors A, B, C as Sa/37 is of the three vectors a, /3, y. Thus the geometrical meaning of Sjil^O, is known. Similarly the geome- trical meaning of M^ABC, except as to the position in space of the axis, may be said to be known. [It is knoAvn in the same sense as the geometrical meaning of Vo^Sy is known. To get a clear notion of this last, one way is to regard it as a linear vector function of /S.] SABC will be completely determined geometrically if in addi- tion (to the knowledge of the meaning of SiABC) we further know the meaning of tSABC. And MABC will similarly (except for the position in space of its axis) be determined when tM ABC is determined. Denote as in § 12 the angle and the distance between A and B by 6 and d respectively. Also denote the angle and the distance between MAB and C (i.e. between C and the shortest distance of A and B) by and e. The conventions as to the signs of d and e are as in § 12. By eq. (13) § 12 we have iS ABC = tS (MAB) C = mAB + XC-e tan <^. Hence by eq. (8) § 12 tS^jBC' = t^-l-t5-f-tC + rfcot6'-etan. Hence by equations (1) and (5) „ . ^, c? cot ^ — e tan (f> ,^, Since MABC = IACBA 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 ^1 = sin dn cos 2 = sin 63 cos ^3. tM {W\AB) C maybe found by a process exactly similar to that used in establishing eq. (1). Thus by eq. (8) § 12 tM (MAB) C = tl^ AB + tC + e cot , or tM {MAB) a = tA + tB + W + dcote + e cot (7). § 14] OCTONIONS AS FORMAL QUATERNIONS. 49 From the geometrical interpretation of Sjil^Cjust mentioned we see that in order that SiABC 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'ABC-S,ABCSr-ABC=Sr'ABC(l - ntSABC), we see that the conditions just mentioned are the necessary and sufficient conditions that S~^ABC should have a definite in- telligible meaning. Hence the equations ESABG = ASBCE + BSCAE + CSABE ) = MBCSAE + WJASBE + MABSCe] ' "^ ^' serve to express any motor E in terms of motors coaxial with A, B, C or with MBG, MCA, N\AB when A, B, C satisfy the conditions mentioned. By putting S-'ABC= Sr'ABC(l - ntSABC), SBCE = S,BCE(1 + ntSBCE), SAE = S,AE {1 + ntSAE), &c. it will be seen that when E is expressed in terms of (ordinary scalar) multiples of ^, B, G, CIA, [IB, CIC, the A component is AS.BCESr'ABC, and the flA component is nAS.BCESr'ABC (tSBCE - tS ABC), which may be transformed by eq. (1) to a fairly simple form. Similarly when expressed in terms of MBC, Q,MBC, &c., the MJ5C* component is MBCS,AESr'ABC, and the CIMBG component is nMBCS,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, S.,, s are all distributive, i.e. SiQ + R) = SQ + SR, S,(Q + R)=S,Q + S,B, S,{Q + R)=S,Q + S.J{, s(Q+B) = sQ + sR (1). M is distributive {and also K) but Mj, M., and m are not. M. o. 4 50 OCTONIONS. [§ l^- In § 13 we saw that T{QR) = TQTR, T,(QR) = T,QT^R, T,(QR) = T,QT,R, t(QR) = tQ + tR. The similar equation in U, viz. U {QR) = UQ\JR is true, hut those in Ui, Uo and u are not true. We have seen that fl and therefore every scalar octonion is commutative with each of the symbols K, S, M. It is not commuta- tive with Si, S,, s, Ml, Mo, m, T,Ti, To, t, U, Ui, U.,, 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=SiQ, anQ = nSiQ, SinQ = o, ns,Q = o...(2). From these we have SQ = snQ + nsQ (3), so that all the S's can he expressed hy means of s and H. A similar remark is not true of the M's; because vc\D.Q 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 . D,iA -js . D.jA—ks . DM - HisiA - fljsjA — i^kskA . . .(5). Let A be an independent variable motor given by A = xi + yj + zk + li^i + mi:ij + nnk ( 6 ), where x, I, ... are ordinary scalars. Then ^ is defined by the equation ^ = id/dl +jd/d7n + kdjdn + flid/dx + njdjdy + ^kdjdz. . .(7). This gives sdA'^ = -d (8). Hence if A^x'i' -^ ...-\- I'm' + ... where i', j, k' are any other set of mutually perpendicular inter- secting unit rotors i'djdV + . . . + ritdldx + ... = - i's . ni"^ - ... - 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 d 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, A is a rotor through the same point. (5) If spoA = where po is an arbitrary lator, ^ is a later. s . -4 2 is an expression that frequently occurs below. If ^ be a lator, A^ = and therefore sJ.- = 0. If J. be not a lator it can be put in the form (1 + fit A) M^A. Hence A'=(l + 2mA)W\M (9). In particular sA"- = 2t AMM =-2tATM (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 lator or a rotor. n motors A^, A. 2.. An are said to he independent when no relation of the form x^A-^ + . . . + x^An = ^xA = (where iCj ... a;„ are ordinary scalars not all zero) holds between them. When they are independent all motors of the form %xA are said to form a complex of order v. The complex will generally be called the complex Ai, Ao...An, and when they are not independent the complex (though then of lower order than v), 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. If Ai...As be any six independent motors, any motor in space can be expressed in the form x,A, + ... +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 later and every rotor, but no other motor is self- reciprocal. The n motors A-^...An are said to be co-reciprocal when every pair of them is a reciprocal pair. It is obvious that J-i is then reciprocal to every motor of the complex Ao...An. More generally every motor of the complex A^...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,G are motors the necessary and sufficient condition to ensure that SABG = is that either (1) two independent motors of the complex A, B,G are lators or (2) XA + YB + ZG = 0, where X, Y, Z are scalar octonions luhose ordinary scalar paints 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 -\- zG \q di lator nxA + £lyB-^nzG=0. A sufficient condition is that XA + YB + ZG=0, tuhere X, Y, Z are scalar octonions luhose ordinary scalar parts are not all zero. This is merely a part of the general enunciation. The condition is sufficient. If XA + YB + ZG=0 where S^X is not zero, A =- YX'^B - ZX-'G= Y'B + Z'G where 7', Z' are finite scalar octonions. Hence SABG=0. If ^o, Bq be two in- dependent lators of the complex A, B,G one of the three, (say G), motors A, B, G '\& independent of A a, Bq or they can all be expressed in terms of Aq and Bq in which case SA BG = 0. Expressing A and B in terms of Ao, B^ and G, SABG becomes an ordinary scalar multiple of SAoB^G, i.e. it is zero since ^o^o = 0. The condition is necessary. If SABG=0 we have [eq. (8) § 13] ASBGE + BSGAE+GSABE=0 for all motor values of E. Hence a relation of the form XA + YB + ZG=0 § 14] OCTONIONS AS FORMAL QUATERNIONS. 53 holds unless for all motor values of E S.BCE = S.GAE = S.ABE = 0. By quaternion interpretation these last give M,BC = M,CA = W\,AB = 0. Hence those motors of A, B, C which are not lators are parallel. If they are all lators either two of them are independent or they are all multiples of any one of them ; in each case the condition is satisfied. If they are not all lators let A have a definite axis. Then since B and C are either parallel to A or are lators we may put B = bA+ B', C = cA+ C, where b and c are ordinary scalars and B' and 6" are lators. [For first we can (eq. (1) § 7) by the addition of a suitable lator translate A so that its axis takes up any parallel position, say the axis of B. Then multiplying by a suitable scalar octonion, say 6 + n6', we can change it into any motor having that line for axis.] If B' and C" are independent, they are independent lators belonging to the complex A, B, C. If they are not independent or if either vanishes a relation of the form XA + YB + ZC = (i holds good. Hence the condition is necessary. In connection with this it may be remarked that if B and G have definite, not parallel, axes ; XB + YC is any motor which intersects a definite line (the axis of MBG) perpendicidarhj. For first XB + YG does intersect M5(7 perpendicularly since S{XB+YG)^BG=i). And secondly any motor which intersects lABG perpendicularly may be put in the form XB-\- YG, for if E be any motor we have, by putting A = lA'^BG in equation (8) § 13, E = M-'BGSBGE + BS . ECM-'BG - OS . EBM-'BG, of which the first term on the right is zero when E intersects MBG perpendicularly. Under the same conditions as to B and G it follows that if S^^6'= every motor of the form XA + YB+ZG 54 OCTONIONS. [§14 either intersects a definite line perpendicularly or is a lator per- pendicular to the line. 15. Linear motor functions of motors. General, com- mutative and pencil functions. will be called a linear motor function of a motor, or more frequently a general function when (j)E is such that whatever be the motor value of E, <^E is a motor, and that E+c})F={E + F) (1), whatever be the motor values of E and F. is commutative with scalar octonions and therefore in particular with SABC, Ave have for it [eq. (8) § 13] (f>E= (i>ASBGE + (fiBSGAE + CSABE) S-'ABC . . .(3). Hence when the three motors (pA, j)B, 0C are given, is com- pletely specified. Moreover whatever values be given to these motors the expression on the right of (3) is of the same form as the expression on the right of (2); [put ^SBGE = BCE + K {BGE) = BGE - EGB, &c.] ; and if any one of them be altered (])E is altered for some value of E. Hence eighteen and only eighteen scalars are required to specify a commutative function. § lo] OCTONIONS AS FORMAL QUATERNIONS. 55 Next let ^ be a general function. A, B, G being as before, we have seen (§ 13) that any motor E can be expressed in the form E = xA^-ijB + zC-{-mA ■^mVLB+nO.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(f)A + i/(fiB + z(f>C+l(nC)...(o). Hence (f> is now completely given by six motors (f>A ...0(0(7), i.e. by thirty-six scalars. And it is clear that all these are required. Let now -^{E, F) be an octonion function of ^ and i^ which is linear in the general sense in each of its constituents. Then Z^ is defined by y!r{Z„Z,) = ^lr{%,A,) (6), where the suffixes of ^ and A are to indicate the operand (A) of ^; and where ^ and A have the meanings given to them in equations (6) and (7) of § 14 above. Similarly % (E, F, G, H) being an octonion function linear in each of its constituents, Z, is defined by XiZ,, Z„ Z„ Z,) = x(>^' ^1. ^- ^U (7K and so to any number of pairs of Z's. If, as in equation (0), there is only one pair the suffix may be dropped. ^ is defined as in the case (§ 3) of formal (luateruions, 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 yjr and ^ now contemplated. Z is an invariant (§ 14). ^ 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, f 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 ' of cf) is defined by either of the equivalent equations sEcf>F=sF(}}'E (9), for all motor values of E and F or (fi'E=-ZsEcj>Z (10). In the case of the commutative function these may [eq. (3) § 14] be replaced by SE^ (12). If can he expressed in the form ^E = r,E + UE (13), where Ti is a commutative function and U is given in terms of another commutative function Tg by the equations n« = o, unco = T,(o (14), where w is an arbitrary rotor of an assigned pencil. For taking i,j, k as unit perpendicular rotors belonging to the pencil, can be expressed in the form ^E = - {AsEVli + A'sEi + BsE^j + B'sEj + CsEVlk + C'sEk) = - 2 [ASEi + {A' - Q.A) sEi} = r,E+nE, where Tj is the commutative function — 2^S( )i and Tl is the function — S(-4'— H^) s ( )i. Here we clearly have lift) = and nn&) = T.,(o where T, 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 T^ and T.^ are unique. It does not appear from the above that Tj and To are self- conjugate when ^ is. It will appear directly that in general they are not (though the two self-conjugates T, and T would involve twenty-four scalars and the self-conjugate O only twenty- one). The centre of the pencil however can be so chosen in general that they are self-conjugate. The general function can always be put in the form &) = ^ift) -I- n(f).M, nft) = <^3ft) + ni are four pencil functions with a common assigned centime and co is an arbitrary rotor through the centre. For Oft) is a linear function of co which can be expressed as the sum of two linear functions, the first being a rotor (^ift)) through the assigned centre and the second a lator (Cl(f)..(o). Similarly for nft). [If the legitimacy of this reasoning is questioned put 58 OCTONIONS. [| 15 A above = a + OX, A' = a + HV, B = ^ + flfi, &c. where a, X, &c. are rotors through the assigned centre. Then cf>,E = -laSEi, (f>,E = -l\SEi, c}),E = -^a.'SEi, (}>,E=-'^X'SEi.] 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 r, = (j>, + ncf>,, T2=(^3 + ^(<^4-<^i) (16). The general self-conjugate ^ can always he put in the form ^&) = (/)&) + Oi/ticw, ^n&) = -v|r,,w + o^'ft) (17), where -y^r^, "^o are self-conjugate pencil functions and (j> is a pencil function whose conjugate is ^' with an assigned common centre and where co is an arbitrary rotor through the centre. To prove this first suppose "*F to have the form of in equation (15) and use the equation s{(o + Ho-) ^ (ft)' + no-') = s (ft)' + no-') ^ (ft) + no-), where co, a, co', a' are rotors through the assigned centre. Putting o- and 0-' zero and leaving ft) and &)' arbitrary, we find that ^2 is self-conjugate. Similarly putting w and co' zero we find that ^3 is self-conjugate. Finally, putting co and a zero we find that (pi = 0/- ylr^ and -^jr^ involve six scalars each and cfi nine so that, since ^ involves twenty-one, the determination of yp-^, yfr^ and ^ is unique. When ^ of equation (13) is self-conjugate and equal to the present ^ we see from equation (16) that T, = (^ -f n^|ri, T., = ^2 + n (f - <^) = f, - 2nMe ( ). . .(18), where e is put for the rotation rotor ^M^i 4- n^., where ^j and ^.^ are pencil functions ivith a common assigned centre, and if T is self -conjugate ^i and 0.^ (ire 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 later it is of the form Q.^.>- Hence it reduces every later it acts on to zero and may be said only to act on rotors. Moreover it reduces equal parallel rotors to the same later. Its properties are precisely similar to those of the linear vector function of a vector corresponding to the pencil function ^.. 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 from Formal Quaternions. Confining our attention then to commutative functions we will in this section put down certain results, without proof, that flow from the fact that octonions arc formal quaternions. 60 OCTONIONS. [§16 (f>E=- BSAE - B'SA'E - B"SA"E = - ^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 (j) + cj)' = 2^ (2), 4> is self-conjugate and is called the self-conjugate part of (f). Put M^K=2^ (3). Then 'E = ^E-MHE (4). i/r (E, F) being a commutative linear motor function in both the motors E and F, t(r,K) = ^/r((/)^0 (S). [Proved at once by putting 0'^= — ^jS^^^i. Note that putting (f) = Q() Q-i it follows that ^ is an invariant when associated only with commutative functions.] When is given by equation (1) ir(^,cf>0 = HU>B) (6). If ;;^ be a perfectly arbitrary commutative function and if where ^i and 02 are commutative functions, then <^i = (f).,. And if the same equation hold when % is a perfectly arbitrary self- conjugate commutative function, then ^i = ^,. The equation proved in § 8 above for formal quaternions ESUX^S^^4tM^ = - SMU.Scf>Ecp^,cp^, (7), may be put in the form ((j>'-M"' + 3r-M)E = (8), where m=Sl;,^,^sS(f>^,4>^,cf,C, (9) 2if' =-SM^,r,M0^,(/,C.. (10), if"=-S^A, B' = (f)A', B" = (f)A" (23), [which are also given by the equation derived from equation (20), ((>E = - <})ASAE - (f)A'SA'E - (f}A"SA"E]. Hence by equations (12), (13), (14) MSAA'A" = S(f>Ac}iA'cliA" (24). M'SAA'A" = S {A4>A'c^A" + A'(^A"<^A + A"4>A(j>A') . . .( 2.3). W'SAA'A" = S {A'A"^A + A"A4>A' + AA'cf^A") (20). 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-4>)A{X-(I>)A' (X - 'Mcj)A'(}>A" (28). Similarly MMA'A" = 'A'ct>'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 are pencil functions with any assigned common centre. It is now important to observe that the form of i is independent of the position of this centre. By this is meant that if a, /S, ... 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 <^ = 01 + n^o = ' are equal and parallel, being the rotor of A translated to pass through and 0'. The following may also be noticed in passing. As we do not propose to use these results we give them without proof The form of (f).2 is of course altered. Let ^i, ^., be the pure parts and Mei( ), Meo( ) the rotatory parts of ,, 0.,; and let 00' = p. Then by changing the centre from to 0' the form of (p. is altered to the form of (f>, + ,( ), the form of ^o to that of ^, + 4>,Mp{ )-Mp^,( ), and e._, becomes changed to the rotor through 0' equal and parallel to eo — Mpej. On account of the similarity of 0i 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 — S^cj)^^, &c. Substituting (jj^ + 1102 for ^ in those equations we see by equating ordinary scalar and coiivertor parts that the cubic satisfied by <^i is ^' + Sjr. is real. The other two are both imaginary or both real. If any two are equal they are all real. The question now arises, — has the cj) cubic (8) § 16 always scalar octonion roots and if not always what are the conditions ? In other words, when can we put (f>' - M"(f>' + M'(j> -M=((j)- X) (-X) (<^2 - iV^'0 4- i\r) ... (6). If we put a commutative function and therefore satisfies a cubic. It is of importance to remark that this cubic is obtained from the ^ cubic by substituting ■\lr+ F for (j>. This, which is not a truism, appears from the identity (27) § 16. If two roots or three roots of the ^i cubic are equal it is not always true that there are no corresponding roots of the (f) cubic. If two roots of the (p^ cubic are equal to x and the third is not equal to x ; in order that the — x cubic must be zero. There are then three § 18] OCTONIONS AS FORMAL QUATERNIONS. 65 roots of the (f> 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 thii'd root x'' of the ^i cubic has a root x" + fly" of the (f> cubic corresponding to it. If x+ D.y is another root {<})-x- Qy) {(f> - x" - ny") is a factor of ( - x)' - M" (4> - xf + M' ((/> - x) - M. There must therefore be a third factor and this can only be — x)- — D. (y + y') (0 — x) of ((f> - x) - Cly and {j> — x) — Vly' is unaltered if y and y' are changed arbitrarily so long as their sum remains unaltered. And in the product of these and — x; i.e. the M of the (jj — x cubic is zero. Conversely if the M of the ^ — X cubic is zero it is easy to prove that there are three roots of the (f) cubic. If three roots of the ! cubic are equal to x ; in order that the ^ cubic may have a root the M of the (j) — x cubic must be zero. If ^ — X — Hy is the one corresponding factor and { - x)" -nz((})-x) + Q.z' is the other {quadratic) factor of {(f) - xf — M" {(f) — x)- + M'{(^ - x), y and z are indeterminate except as to their sum which is sM". In order that the (f) cubic may have two roots both the M and M' of the (f) — X cubic must be zero. There are then three roots of the (f> 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 , 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 aocial 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.] MO 5 66 OCTONIONS. [§ 1^ 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, /3, 7, a! &c. will stand for rotors. X, 7, Z, Y', Z' will stand for scalar octonions. x, y, a, b, 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 4> cubic corresponding to a single root of the (j>i cubic, (f> can be put in the form (E = XE. If X is a root of the (j> cubic corresponding to a repeated root of the (pi cubic there is not always an axial motor E for which 4, MjJ.', Mi5, Mi£', lA^AA', M^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 of the ^ cubic we have by the identity (27) § 16 that S(Z-)2(Z-)A + X,{X - cf>)A' + X,(X - c}>)A" = 0, where the ordinary scalars of Xj, X2 and X3 are not all zero, or else two independent motors of the complex (X-(j>)A, {X-cfi)A\ (X-(f>)A" §18] OCTONIONS AS FORMAL QUATERNIONS. 67 are lators. In the first case we may take A" for the axial motor X^A + X.,A' + X-^A", when we have A" = XA" (3). Now choose A, A', A", B, B', B" to depend on A, A', A", ((f>- X) in the same way as in § 16 A, A', A", B, F, B" depend on A, A', A", 4>', that is to say define A, A', A" by eq. (21) § 16 and in place of eq. (23) § 16 take 5 = (0-X)2, B' = {4>-X)A', B" = ((f>-X)A"...(^). Thus {-X)E = - BSAE - B'SA'E, which is the same in form as eq. (1). If now the root SjX of the tpi cubic corresponding to the root X of the (f> cubic is not repeated, one root but not two of the (<^i - S,X) cubic is zero, i.e. the S^M' of the (<^ - A^) cubic is not zero, i.e. SjM^^'M^^' is not zero by eq. (15) § 16. In the case when two independent motors of the complex (X - (f)) A, (X - (f)) A', {X - A" = XA" + n^S" (5), and now defining A, A', A", B, B', B" as before we have B' = 12/3', B" = n^" and therefore {(fi-X)E = - BSE A - n/B'SEa' - n^"SEa" (6), where a' and a" are any rotors equal and parallel to M^A' and M^A". In this case we see by eq. (13) § 16 that the S^M' of the (f> — X cubic is zero and therefore SjA" is a repeated_root oj^ the ^1 cubic. In this case we have by equation^.')) that 4>A' = XA' + a later. Hence if p is any rotor parallel to A', <^p = Xp + a later or Mi<^/3 is equal and parallel to M,Xp (unless SiA^= when (f)p is a later). Similarly for any rotor a 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 ^ <^p = Xp + a later. That there is not always in the case of eq. (6) an axial motor E for which ((p - X - D-ij) E = is seen by taking a particular case. If we put (f)E = - jSEi - nkSEj - mSEk the cubic is ^' = and it is easily proved in this case that there is no axial motor E for which ((f) — ^y) E = 0. 5—2 68 OCTONIONS. [§18 We may notice that equation (1) may be reduced to either of the forms ((f> - X) E = - BSjE - B'SkE (7), or {(f)-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 (-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 (pi cubic become respectively S^iBB' not zero and S^iAA' not zero. Since {(f)^ — M"(f)^ + M'tp - M) E =0 where E is any motor, we see in particular that when a root of the ^i cubic is repeated three times there are three completely independent axial motors B, B', B" such that for any motor E = X^B + X^B' + X^B", i4>^ — ...)E=0. Similarly when SiX is a single root of the ^i cubic we have now seen that there is one axial motor B such that if E = X^B, {(}) — X)E = 0. A particular result of the next pro- position is that when a root of the ^i cubic is repeated twice there are two completely independent motors B and B' such that if ^ = X^B + X^B', (- - N'(f) + N)E=:0 the repeated roots being the roots of x^ - xS^N' + S^N = 0. It may be remarked that X^B is any motor coaxial with B or any later parallel to it ; X^B + X^B' is any motor which intersects a definite line (the shortest distance of B and B') perpendicularly or any later per- pendicular to that line; and X^B -\- X^B' -^ X^B" is any motor whatever. Prop. II. If X is the root of the

"--N'(f>-\-N)E = (9). This is not always true if X corresponds to a repeated root of the - - N'tf) + N){(f> - X)E = for all values of E it follows that (cf>"- - iV> + iV) ( + N)((f>- X) A' = 0, i.e. (<})"'- N'(f> + N) B =0, {'- N'(f) + 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 (j>- — N'(j) + iV to 0. - (N' -ni/)4> + {N+ % (X - N% where y is arbitrary in order to take account of the arbitrari- ness of the root X + D,i/ [§17 above]) is seen by the particular case considered just now, viz. (f)E = — jSEi — D.kSEj — niSEk: The cubic is * = so that N', N and X are all zero and it will be found that (cf)'' + ni/(f)) E=-n{(k + yj) SEi +jSEk], so that in this case [<^- - (N' - Oy) -x)E=- BSEi - ni3'SEj - n^"SEk (10), tvhere B = Y(i cosd +j sin 0) (11), or {<^-x)E=- BSEi - B'SEj - n^"SEk (12), ^ohere B = Yj + Zk, B' =■ Y'j + Z'k (13), and where M^BB' is not zero. If x is repeated three times we have in the first case SjFcos ^ = and in the second SjF' = 0. [Note that eq. (13) is equivalent to the equations SiB = SiB' = (14). 70 OCTONIONS. [§18 and that the condition M^BB' not zero is equivalent to the con- dition Si{YZ' - Y'Z) not zero, and also equivalent to the condition that B and B' are completely independent axial motors.] Fig. 5 shows graphically the geometrical meaning of equations (12), (13). The meanings of A, A', A,',j' and k' will appear in the discussion. The rest of the figure is fully explained by equa- tions (12), (13). i=^ ■B'=j=-a;)E=-BSAE-B'SA'E-B"SA"E, where A, A', A" are any three completely independent axial motors, we have by eq. (12) § 16, since Siil/=0, S,BB'B" = 0. Hence (by quaternion interpretation) B, B', B" are axial motors parallel to one plane or one of them is a lator. Thus one of them (say B") can be expressed in the form yB + zB' + HyS". Substi- tuting this in the last equation {(f>-x)E = - BS (A + ijA") E- B'S {A' + zA") E - [l^"Sci"E, § 18] OCTONIONS AS FORMAL QUATERNIONS. 71 where a" is any rotor parallel and equal to M^A". Since A, A' and A" are completely independent axial motors so are A + yA", A' + zA" and a". Denoting the first two by A and A' respectively we have (-x)E = -BSAE-B'SA'E-n^"Sa"E (l.")), where S^AA'a" is not zero. Expressing now the condition that SiM" = we have by equation (13) § 16 S,MAA'MBB' = 0. Hence either M^BB' is zero or MiBB' is not zero and M^^' and MBB' are perpendicular to one another. In the first case we may put B' = z'B + 0/3' whereupon chang- ing A + z'A' to A we get ((ji-x)E = - BSAE - n^'Sa'E - 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 {j>-x)E = - BSiE - n^'SjE - nj3"SkE, (where of course the meanings of B, /8' and yQ" are in general different from their meanings in the last equation). In this form we notice that (<^ - x) {i + HMpt) = B. Hence if i' is any unit rotor parallel to i, {

-x)E=- BSA.E - B'SA.'E - LI^CScl'E, 72 OCTONIONS. [§18 where A, = A+ yna", A,' = A'+ yfla", n^/' = H (/9" -yB- y'B'). The form of eq. (15) is not thereby altered and a" is still perpen- dicular to both ^1 and ^i'. y and y' may be so chosen that not only is SjM^i^i'M^^' zero but SVAA.A^^BB' = 0, for this last equation is true if s^AA'MBB' - Si {a"MBB' .{yA' - y'A)} = 0. Here a" is parallel and MBB' is perpendicular to MAA' so that Ml {a'MBB') is parallel to the plane of A and A'. Since A and A' are not parallel it follows that y and y' can always be deter- mined, and that in an infinite number of ways, as desired. Hence we may assume eq. (15) to hold with the condition SMAA'MBB' = 0. We may therefore take i along the shortest distance of B and B' and k along the shortest distance of A and A' so that a" is parallel to k. Now express A, A', B, B' and a" in terms of i,j and k (i.e. A = — iSiA —jSjA, &c.) and collect the terms in SEi, SEj and SEk. We thus get equations of the form of (12) and (13), The condition M^BB' not zero still holds with the new meanings of B and B'. [This can be proved directly or we may notice that if it do not hold we can by the above reduce equations (12), (13) to the form of equations (10), (11).] Note that both equations (10) and (12) are of the form of eq. (15). Indeed as we see by the above proof eq. (15) with the condition SiMAA'MBB' = may be taken as giving the general form for when a; is a repeated root of the i cubic. If a; is a thrice repeated root of the <^i cubic we have further that the S^M" of the cf) — a; 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 SjF' = 0. In this case B' in fig. 5 must be drawn parallel to k. i[ UNIVERSITY J § 18] OCTONIONS a S - yOHMAt r QUATERNIONS. 73 Prop. IV. hi the case of eq. (10) if i' is any unit rotor parallel to i, {(ft - x) i' has the constant value B. If E is any axial motor perpendicular to i, Mi - x)j has the constant value B' and (cf> - x) k' = n (/3" — yB'). If E is any axial motor parallel to k, M^cpE is equal and parallel to xlA^E. We have / = (1 + ^yi)j =j + ^yk, ^' = (1 + n^O k = k - Clyj, from which the statements about / and k' follow. If £^ is a motor parallel to k, {(})-x)E is a, later, from which the statement about E follows. [We saw in making the transformation Ai = A +yQ,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 j' 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 /3'' 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), (<^ — x)i' has the constant value B, (^ — x)f has the constant value HyS', and ((j> - x) k' has the value fl {/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 /3" can always be made perpendicular to B. Prop. VII. If there is no axial motor E for which -x)E = - YjSEi - n^'SEj - n^"SEk, S,il3' = 0, SJl3"=0 (16), 74 OCTONIONS. [§18 where S^Y, S^i^" and SJc^' are not zero ; or ((/) - a-) ^ = - ( Fj + Zk) SEi - Z'kSEj - n^"SEk (17), where SjF, S^Z' and S{i^" are not zero. Conversely if these con- ditions are satisfied there is no such axial motor. Note that in this proposition <^E would be said to be coaxial with E if it were a lator parallel to E or zero. We have seen (Prop. I.) that there is always such an axial motor E except when the roots of the i cubic are all equal. Calling each of them X, (f) will be of one of the forms (10) or (12). In the case of eq. (10) we have seen that when a; is a thrice repeated root of the (^i cubic SiFcos ^ = 0. If SiF= 0, B becomes a lator 12^8 and in this case (cf) - x) E 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 Sj Y is not zero we have the first of equations (16). We have seen (Prop. VI.) that in this case we may suppose Sijf/3" = 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 + Up. [Since <^ is commutative, if E satisfies the required conditions so does the rotor M^^ or (1 - fltE) E.] Thus (-x)E=n (2//3' + z^" - YjSip). The lator on the right has to be parallel to yj + zk. It will be found by elementary algebra that when Sii/3' = Sji/S" = this condition can always be satisfied by real finite values of y, z and Sip of which the first two are not zero. [In this case indeed y and z may be given arbitrary values.] Since yj + zk is perpendicular to i we see that when S^ijS' and S^i^" are not both zero we must have y _ z _ -S,i^"~Sj^'~^' where c is not zero or infinite. Without loss of generality we may put c = 1. Thus MM/3'13" - YjSip § 18] OCTONIONS AS FORMAL QUATERNIONS. 75 is to be parallel to - jSi/S" + ^•Sl/3' ; '•^- S,t/3" S,i^' ' and this can always be satisfied by a real finite value of Sip unless Sii/3' = and neither S^i^" nor S.jjS'jS" is zero. Conversely if these conditions (S^ilS' = 0, &c.) are satisfied the equation cannot be satisfied by a finite value of p. It is easy to prove that the condition S^j/S'^" not zero may be replaced by the condition Sik^' not zero. In the case of eq. (12) we have seen that when a; is a thrice repeated root of the ^j cubic Siy' = and by taking, as we have seen we may, the point of intersection of i, j, k on B this becomes Y' = 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 + n{^i+7]j+^k). Thus (cf>-x)E=n {13" + HYJ + ^k) + vZ'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 S^i^" = 0. In the case of eq. (10) x + fly (where y is arbitrary) is a root of the (f) cubic. In the case of eq. (11) a; + fly is not always a root. For by equation (12) § 16 the J/ of - ^ is in the case of eq. (10) zero but is not generally zero in the case of eq. (12) [sec 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' = - [SkBE + nS/8" {iF -jB)], M" = - {SiB + SjB' + nSk^"}. In the case of eq. (10) these give M=0, M' =-nY {S/S' (- i sin B+j cos 6) + S^'k cos d] , M" = - {- Fcos 6' + nS ( j/3' + k^")]. Hence the cubic is ((^ - ^) {( - x) [4) - X - Q, [S/3' (i tan 6 -j) - S/3"A;]| {-x- Y cos + nt&n e Si/3'} E = 0, is ((}> - x) {(}> - X - n [Sy8' (t tan ^ - j) - Sj3"k]} i. E is however written down more simply by noticing that if {<}>-x)E=- BSE A - n^'SEa - n^"SEa", then (cf)-x)B = - BSAB - njS'Sa'B - n^"Sa"B, from which it is obvious that if we again operate by (0 — x) we shall get a scalar octonion multiple of {(f) — x) B. In fact putting n^'Sa'B + n^"Sa"B = He we find that (4)-xyB=- (SAB + nSAeS-'AB) { - x) B. Hence in our present case the desired axial motor E is {(f) — x) B, E = - BSiB - n/S'SjB - a^"SkB. The splitting of the cubic into three linear factors fails generally when x is a, thrice repeated root, i.e. when ^ = k • [i^" is a thrice repeated root if SiF = 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. M' = gives in the present case S^^'i = and then the above form of the cubic still holds and takes the simplified form ((f) - xf [{(f) - x) + nS ( j/3' + k/3")} = 0. § 19] OCTONIONS AS FORMAL QUATERNIONS. 77 For eq. (12) we have M = -n( YZ' - Y'Z) S{j3", M' = nS/3" {iZ +jZ' - kV), M" =¥'- nSk^", so that the cubic is (0 - xY +{- xy (- F + nsA,-/9") + {cj>-x)nS^"{iZ+jZ'-kY')^^{YZ'- Y'Z)Si^" = (), which may when x is not thrice repeated be put in the form [^ _ .^. _ r + nZ'Y'-"- S^"{jY' + iY)] {(4> - xf - ( - x)nY'-"- 8/3" (iYZ' +jY'Z' - kY'') - nr-HYZ' - Y'Z) s/3"i} = 0. The axial motor which is reduced to zero by the first factor is E = {^-x)B'- BnZ'Y'-'Sil3" = Y'B' + Q.Z' (/3" - BY'-'Sil^"). If X is a thrice repeated root of the <^i cubic S^M" = or SiF' = 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. S-^i^" = 0. In this case we may put ^" = h"j + c"k and the cubic is ( - x) {(cj) - xy -(Y' + nc") ((i>-x) + n (c"Y' - h"Z')] = o. In order that there may be three factors M' must be zero (§17), i.e. Si {h"Z' — c"Y') = 0. In this case the cubic is {(f> - xy { - X - (Y' + nc")} = 0. These two factorisations are true when S^i^" = and when Sii^" = and Si(b"Z' — c"Y') = respectively, whether or not SiF' = 0. If the last be true, however, we may write —Q.h"Z' instead of H {c"Y' — h"Z') in the first of the two cases. 19. The self-conjugate commutative function. We proceed to show that when ^ is self-conjugate it can always be put in the form i +j(f)j + kcf)k). In the case of eq. (16) this gives Yk + nM{j/3' + k^")=0, which is impossible when SjFis not zero. In the case of eq. (17) Yk-Zj+ Z'i + nMk^" = 0, which is impossible when SjFand Si^' are not zero. Hence by Prop. VII. there is always in the case of a self-con- jugate (f) an axial motor E for which (})E is coaxial with E. Let i be taken along the axis of this motor. Thus ^ must be given by (f3E=- XiSEi - YjSEj - ZkSEk - W ( jSkE -H kSjE). [For put <^i = X{i + Y-ij + YJc, (f>j = Y,'i + XJ + Y,k, (t)k=Y,'i+Y,'j + X,k. Making self-conjugate we get 7/ = Fj, 7/= Y^, 17= F^, and making cf)i coaxial with i we get Y^= Fg = 0.] Now Z' can always be determined so that ^ (j + Z'k) is coaxial with j + Z'k. For <^ ( j -f Z'k) = (Y + Z'W)j + {Z'Z + W) k, and this is coaxial with j + Zk if Z'-'W-\-Z' iJ-Z)- Tf=0 (2), which gives Z'=\W-' [Z- F+ \_{Z- F)^-f 4Tf^PJ. By eq. (8) § 4 above this gives a real finite value for Z' except when Tf is a converter. If If is a converter eq. (2) is satisfied by Z' = W\{Y —Z) unless Y — Z \^ also a converter. If both W and F— .2' are converters, say VLid and Vty where w and y are ordinary scalars, eq. (2) is satisfied by Z' = i?y-i {- 2/ -I- \f + ^nP^], unless w = 0. But when w = 0, TT = and both <^j and ^k are coaxial with j and k respectively. § 19] OCTONIONS AS FORMAT. QUATERNIONS. 79 j + Z'k is an axial motor which intersects i perpendicularly. We may therefore take its axis for that of j. Doing so we get eq. (1). By equations (8) and (27) § 16 wc see that the cubic of the of eq. (1) is (0_Z)((^-Z')((/,-Z") = O (3), so that when ^ is self-conjugate there are always three real roots of its cubic. In § 17 we saw that if any two, say SjX 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 + Q.y. We will now show that if for any axial motor E (l>E = YE, Y must have one of the values X, X' or X" of eq. (1). If <\iE = YE we have by eq. (1) iSiE{Y-X) +jSjE(Y- X') + kSkE(Y-X") = 0, or {Y- X)SiE = (Y - X')SjE = {Y - X")SkE = 0. Now since E is an axial motor one of the three ordinary scalars SiiE, SijE, S^kE is not zero. If this is S^iE we have F-Z = 0. Hence Y must have one of the three values X, X', X". Thus the X, X', X" of eq. (1) have determinate values when is given even when the roots of the 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 c/)., 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 ^2 + <^iMp ( ) - Mp^i ( ) are the same. Quaternion analogy suggests the examination of the self- conjugate 6 given by cf>E = MAEB (4). We have at once 4>MAB = -l^ABSAB (5), <}>{UA ±UB) = TTATB(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 ((^2 _ ^252) (<^ + S^5) = (7). It is natural now to enquire whether every self-conjugate <^ can be put in the form ^E = MAEB+YE (8). The principal roots of the cubic are Y + TATB, Y-SAB, Y-TATB, and the ordinary scalars are here in descending order. If A is parallel to B and they are both axial motors, the ordinary scalar of the intermediate root is equal to one of the others, but those of the first and last are not equal. If either A or 5 is a lator, say A, TA has no meaning. Equation (8) then becomes (f)E = YE + Si lator, so that the ordinary scalars of the roots are each equal to SjF. Suppose now the ordinary scalars of X, X' , X" are in descend- ing order. If these ordinary scalars are not all equal neither A nor B can be a lator, so that if equations (1) and (8) are the same we must have Z = F-h T^T5, X' = Y-SAB, X" = Y -TATB...{9). From this it can be shown that in one case it is impossible to express ^ in the form (8), viz. when two but not three of the roots of the ^1 cubic are equal and when the corresponding principal roots of the cubic are unequal. If SjZ = S,X' and these are not equal to S^X", X-X' = TATB + SAB = TATB(l + SUA\JB). Hence T^ATjB + S^AB is zero or A and B are parallel. Hence by eq. (10) § 12 SUA\JB= ± 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 SjZ' = SiX" and these are not equal to S^X and X' and X" are unequal. § 20] OCTCWrONS AS FORMAL QUATERNIONS. 81 In all other cases (f) can be put in the form ( = T-Q(S,Q - M,Q)] d'mQ= md'Q = S4R. mQ + sdR .W\,Q (15), d\ = sd'Q + md.'Q = sdQ + S.dR • rr\Q + sdR . M,Q. . .(IG), d' (tQ+iiQ)= d' (ry(?y-0 = qg-' {d'vQ - 7\^,q,f'd'q,^;) [since (^y, Vq, d'qq and d'r^ are all coaxial] = Tr-Q {S,Q - M,Q) [d\ - {XQ + uQ) d'q^], or d'{XQ+uQ) = J-Q{S,Q-lA,Q)[sdQ^S4R.rrxQ + sdn.M,Q-{tQ + uQ)(S,dQ + S,dR.M,Q)} (17). Hence utilising equation (10), dM,Q = ^{dRM,Q-M,QdR) + M,QS4R (18), dvnQ = i (dRmQ - mQdR) + mQS.dR + M.QsdR (19), dU,Q = UdRU,Q-U,QdR) + Tr'QM,QS,{dRSQ-dQ).U,Q(20), \ (21), duQ = ^ (dRuQ - uQdR) + Tr'QM . (S,Q - M,Q) {sdQ + mQS.dR + M,QsdR - {tQ + uQ) {S4Q + WiS.dR)} = ^(dRuQ-uQdR) + M, . Q-' {sdQ + mQSdR + MQsdR -(tQ + uQ){SdQ + MQSdR)} = i {dRqQ - q^dR) + S,dQ + f^,QS,dR (22) dqq dr. 1 (dRvQ - i\flR) + sdQ + rr\QS4R + M.QsdR .(23). Equations (18), (19), (22) and (28) may also be written dM,Q = (M + S,)dR.M,Q dmQ ={M + S,)dR.mQ + M,QsdR (25), C— 2 •(24), 84 OCTONIONS. [§ 20 dqQ = (M + S,)dRM,Q + S,dQ (26), dvQ = (M + Si) 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. 8. E. 1888-89, p. 201), d.Q- = nQ^-^dQ+l{VA-'Q-nQ^-m-^Q-){Q-dQ-dq.Q^)\ = nQ''-'dQ+{MQ>'-nQ>'-'MQ)MWi-'QMdQ ]" Also if J. be a motor d.A'' = nA^SA-'dA+MA'\MA-'dA (29), and therefore de^ = Ae^SA-'dA +Me^ MA-'dA (30). CHAPTER III. 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 i„ jo, ko. Thus i, j, k are not constant rotors but h, jo, h are. On the other hand i, j, k are independent in mean- ing of an arbitrary origin, whereas to, jo, h are not. The rotor OP will be denoted by p = ioX +joy + hz =iv+ji/ + kz (1 ). Thus 0), y, z have their ordinary Cartesian meanings. The symbolic rotor V is defined by the equation SI = idldx+jdldy + kdldz (2), and Vo will denote what may be called the value of V at the origin, i.e. iodjdx + .... 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 symbolic 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 yjr (A, B) is any octonion function of two motors A and B which is linear in each, ^o 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 d^; that is to say, if P and Q are two near points on a curve d\ = FQ and if P, Q, R are three, not collinear, near points on a surface d^ will denote the rotor |MP(J. FR 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, 6^2 should be defined as the rotor equal and parallel to ^MFQ . 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 dX will be in the direction of positive rotation round a proximate rf2, and in the second case dX will point away from the volume bounded. With these definitions we have [eq. (1) § 7 above] ^=^„+OMp^„ d\ = dp + ^Mpdp\ ^ ^' Thus dp might for some purposes be appropriately denoted by d\o. It will be observed that the properties of our present %, jo, h, p, Vo, ^0, are practically identical with those of the corresponding qua- ternion symbols i j, k, p, V, r, but it is obviously inappropriate in Octonions, on account of the arbitrary origin, to denote them by the latter symbols. With the present notation Sd\V = 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 PHYSICAI, AI'PLICATIONS. 87 that Sv^ (Mv'i;), 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^) = ()) 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 (Vii") the left V has two operands, viz. E and the right V. We cannot then treat V as a mere rotoi- 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««£',. in which Vy^. and V„b are operators both having E^ for operand and where V^ is an operator having V„^. for operand. Thus SV (V^') = SV^,^^aE, + SV^^ A--, = SV^V,,„£',. 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 inunediately succeeding it. Thus SV (MVA) = SV {VE) = SWA (6), which is nut a truism as may be illustrated by the more general equation SV(VQ)=S^VQ+V'^,,S(^„ (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 d\- d / dy 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. ^'AQa or more generally V",i Q^^ where n is a positive integer, is rather a cumbrous expression for the thing signified. It is easy to see that V%Q. = VVo-iQ[riodd]| ^^^' but this alternative method of indication is if anything still more objectionable. I shall therefore use the notation defined by V-^Qa = I>"Q (10). We have not yet found the meaning of the expression SW^E^ or SV^V((^^j,. Let yjr 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 flMpVc. Thus = ylr{^, + nMp^„ HM^oVo), or t(V,, V„)=^(r, flM^V) (11), where V on the right has the same operand as V^ on the left. As a particular result we have V'^ = (v^ + v^,) v„^ = a^M^v + >^ or V2=-2nV + >= (12). Hence SV=^' = - 2nSV^" (18). 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 jdx = 2njjd'S. (1), fIdX = (2). § 22] ADAPTATION TO PHYSICAL APPLICATIONS. 80 The truth of these arc 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 ¥0 + 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 + nMPOPQ, or P'Q + Q0 + 6P = nMOPPQ, which proves the property for the triangle. Next let pi,/3o... be the rotor perpendiculars from a fixed point on a series of rotors equal and parallel to miio, m.do... where % is a unit rotor through 0. The sum of the rotors is by eq. (1) § 7 above where p is defined by the equation p^vi = '^ tup. 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, /3, 7 be three of the rotor edges of a tetrahedron all starting from one angular point, the rotor areas of the faces are - iM/37 - V, ^^-M (/3 + 7) M/57, - \ M7a - i DM {y + a) Mjct, - iMa/3 - inM (a + /3) Ma/S, and IM (/37 + 7a + a/3) + 1 OM (a + /3 + 7) M {/By + jci + ayS), the sum of which is easily seen to be zero (since MaM/37 + W\/3Mya + WiyMoL/3 - 0). 90 OCTONIONS. [§22 Let now ^ be any linear octonion function of an octonion. Then jcf>d\=jj{2cf>{ncm) + is the spin and t is the velocity reversed, may be generalised. Let (f> be a pencil function whose centre is at P and which is a function of the position of P. I proceed to show that if we put 4>a^A = (o + Or, where co and t are rotors through P, t = M^(^^ and that the rotor to has the same geometrical relations with the coordinates of (j) that the vector ^/Vj has with the coordinates of the linear vector function (j)'. The last statement is seen at once from the equation n^^V^, = n&) 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 t = M^^^ we have merely to prove that (pa^A — nM^0^ is a rotor through P. Now by equation (4), [since, as it is easy to prove, -v/r (t^^, V^) = i/r (HM^/, ^)]. Now Si(f)d'S is an ordinary scalar. It follows that Si {(fia^.i - ^^0 = ^> and thence from statement (4) § 14 that , % by the first of each of the following three sets of equations — XoA = -SAV.p', dp'^xod^ (9), XpA = x.A + nMpxA, W = Xpd^ (10), XA = xA + nMp'x„A = xp^ + ^^VXp^' d^' = Xd^-- -(H)- The rest of the equations are easily seen to follow from the defining equations. From equations (10) and (11) it is obvious that x and Xp ^^'^ independent in meaning of the origin. In fact XpA = A + nW\vA - SAV .7] (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'P', we get dr) + dX - dX' = flM {d\ + dX') rj, whence putting dX' = xd'X = %y.fZX + nMr;;^prfA, and dr} = — SdxV .r] we get XpdX = dX + nWi^dX - SdW . 77, which proves eq. (12) for A = any rotor dX through P. Operat- ing by n and assuming that Xp is 'T- 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 Xr ^^ ^ pencil function (§ 15 above) with centre P. Hence by Tait's (Quaternions, 3rd ed., § 381, we may put XpA=qfAq-^ (13), where q is an axial (§ 6 above) through P and -\/r is a self-conjugate pencil function with centre P. By eq. (11) xA={\+^rt'n)xi'A(i+^nv)-\ Hence x^ = Qf^Q'' (1*). where Q = {l + ^nv)(j (15). 94 OCTONIONS. [§ 23 Similarly Xo^ = ^f^^~' (16), where R={1- lOp) q (17). If x', Xp ^^^ %o' ^^^ ^^^6 conjugates of %, %p, Xo we have by equations (13), (14), (16) r = XX = XpXp = x'Xo = V^S ( ) V,,Spa'p,' = ^. . .(18), say. From eq. (12) and eq. (18) ^ may be expressed in terms of 7] and its derivatives. Eq. (12) may be written xpA = - (1 + hnv) SA (V , + 1) . (va - nMvva + D (i + h^v)-' (19). Utilising eq. (12) directly ^A=A- {V^SA (va - n^VVa) + {Va " nMrjVa) SAV J] From either eq. (19) or eq. (20) we have ^A = (V,, + ^0 S^ (V^ + ^,) S iva - nMrjVa + ^^) (v,-nw\7jvb + ^2) (21). In expanding the expressions on the right of equations (19) and (21) it is understood that any term in which a ^ occurs without the corresponding ^, is put zero. It will be noticed from equations (18) and (21) that ^ is the same function of as it is of V^.. pa, ^B, pb- 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=l+(%p)' %p'=l + (%p)'' where (xp) and (%p)' are small and conjugate to one another. Thus ^ = %p%p=l+(%p) + (%p)', or from eq. (12) (^-l)A = 2{f- l)A = -V^,SAva - VaSAV^,...{22), so that in the case of small strain the expression independent of the arbitrary origin is cjuite simple. § 24] ADAPTATION TO PHYSICAL APPLICATIONS. 95 24. Strain (continued). Wo propose now to consider the straining of a volume element, the straining of a surfiice element, and differentiation with regard to the coordinates of strain. These are all now so similar to the corresponding quaternion considerati(ms that they have been marked off by being put in a separate section. The m [§ 3] or the 31 [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 X, fi, V of that definition are ordinary motors. Taking them to be three elementary rotors through the point P, — SXfxv will be the element of volume at P (contained by a parallelepiped of which X, fi, V are adjacent edges) and — S^k^tJ^x^ ^^^^ he the correspond- ing element at P'. Change X into dX and lAfxv into rf2, %X into d\' and lA^lJ^'xy into d^' so that d\, dX' have the meanings hitherto given and dS, d%' are the unstrained and strained rotor values of an element of surface. Thus mSdXdt = SdX'd^' = SdXx'd^'. Since dX is an arbitrary rotor through P it follows [statement (1) of § 14 above] that mdt = xdt' or dS' = mx'-'f^S (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)§3«] inx~^ d^ = -^{i + nv)M(va- nf^vva + ro (vu - nMr)v, + Q . (i - n^) I xScZS(V^ + W(V,, + ^,)j (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, h, c, e,f, g, where ^A=- eiSiA - fjSjA - gkSkA ) -h{a (jSkA + kSjA) + h {kSiA + iSkA) + c {iSjA +jSiA)] ] (3), are independent variables and completely specify '^. Define (I (or 4,(7" if there are more independent self-conjugate pencil func- tions than one to be dealt with) by the equation aA = - iSiAdlde - ... - {jSkA + kSjA) d/da - (4). The symbol to be operated upon by the differentiations of (I can be indicated in the usual way by suffixes. If Q be an octonion function of the independent "^ we have as in Quaternions dQ = -Q,Sdn(r.^ (5)- Should it be required we may as in Quaternions define the G corresponding to a general (i.e. not self-conjugate) pencil function with definite centre. 25. Intensities and Fluxes. The similarity of many of the above formulae even when their significance is more extended to corresponding quaternion formulae naturally leads us to ask whether an equal similarity extends to the octonion treatment of intensities and fluxes analogous to the quaternion treatment in Phil. Trans. A, 1892, pp. 689 et seq. It will be found that the similarity of the formulae, almost amounting to identity, is quite remarkable. It is probable that no great physical use could be made of the conception of a general motor intensity or a general motor flux. Nevertheless I propose to define these and examine their symbolic properties, as it is probable that it would prove convenient to regard an intensity (or a flux) sometimes as a rotor and sometimes as a lator. § 25] ADAPTATION TO PHYSICAL APPIJCATIONS. f)7 It is sufficient here to give little more than the formulae, as the proofs arc very similar to the quaternion proofs. When we say that the motor E is an intensity we mean that E is a function of the position of the point P, and E' is a function of the position of P' connected by the equivalent equations Sd\E = Sd\'E', E' = x-^E (1). Similarly i^ is a motor flux if SdtF=Sd-^'F\ F' = m-\F (2). It should be noticed that if E or i^ is a rotor through P then E' or F' is a rotor through P' (from which it follows that if E or F is a motor through P then E' or F' is a motor ^v^th the same pitch through P'). For if JS" is a rotor through P, sdXE = and therefore sdX'E' = 0. Since d\' is an arbitrary rotor through P' it follows that E' is a rotor through P' [statement (4) § 14]. Similarly when P is a rotor through P. If E is a motor through P it may be put equal to (1 + D.p) co where p is its pitch and co is its rotor part, i.e. a rotor through P. It follows that E' = {1+ nj)) x'-' ft> - (1 + ftp) co', where a is a rotor through P'. Hence E' is a motor through P' whose pitch is p. Similarly when F is a motor through P. From these statements and equations (1) and (2) it follows that x~^ ^iid iw^x reduce any rotor through P to a rotor throtigh P'. This is otherwise obvious from the equations X^=Xi'^ + ^^vxi-^> X"^ = Xr'"^ + nMnxr-'A...{^), and from remembering that Xi' is a pencil function with centre P and that 77 = 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 El, E^ are intensities ME^E. is a flux. IfEis an intensity MVE + 2nE is a flux. This statement it will be noticed is not quite identical with the corresponding quaternion statement. It might be thought M. o. * "^ 98 OCTONIONS. [§ 25 that it was inconsistent with the two preceding statements. The explanation is that E in lAVE is not in all respects analogous to E2, in lAE^E.^. To be more precise, MV^'p^'-^^^j 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 ME^E., is a flux) whereas MV'^' or MV {y^~^E) does not because of the differentia- tions of ;^'~^ 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 MV^';^,'-^ E is not. The proof of these statements is obtained by noticing that in the quaternion case X~^dp' = dp = — Sdp'V . p, and therefore x~^ - " ^1'^ ( ) Pi . whereas in the octonion case X~^d\' = d\ = dp + D^Mpdp, = - SdX'V . p + riMpx-^dx', and therefore x~' = - "^a^ i )pa+ ^X~'^ ( ) P- From this it does not follow that MV^'^„'-i^ = but by eq. (12) §21 MV_/xa-'£ = n (2V^/SEpa + M . V^'xa'-^MEpa) (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 MV^ + 2nE are fluxes. For from these we have MV (x'-'E) + 2nx-'E = m-'x (MVi; + 2nE), and M^Jx'~'^(t = m-'x^VE, from which by subtraction MV^/Xa-'E = 2n (m-^x - X'-') E (6). By a similar process or by observing the reciprocal relations between E' and E we obtain MV^,X«'^' = 2n {mx-^ -x')F' (7). 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. 00 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 %. Before going further we will deduce some more relations of the same type. From the symmetrical relations between V, p, -^ and V, p\ y^^ we can write down a similar equation to (5), viz. MV^,,;^,;^' = n (2V^,SE'pa' + M . V.,iX«'M^>„') (8). Two other relations may be written down by expressing the equa- tion Jd\ = 2njJdX in terms of the dashed letters and the equation fd\' = 2n//rfS' in terms of the undashed letters. They are Xa-'ME'V; = 2[l(m-'x£^' - X~'E') (9), XaMEV^ = 2n{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 X.MV,^;^' = (11). Similarly XoT'^^ a Xa'-' E = (12). Eq. (11) may be verified directly by expressing x i" terms of Xo by eq. (11) § 23 and putting ;^o = - S ( ) V . p' by eq. (9) § 28. Two other results may be obtained by expressing the equation JJdX = in terms of the dashed letters and Jjdl' = in terms of the undashed letters. They are ma-'Xa'^A-O (13), ^naXa-'^A = (14), each of which may be derived from the other. Of the nine equa- tions %'V' = V, (G), (7), (9), (10), (11), (12), (13), (14) only four are independent. They may be taken as %'V' = V, (7), (11), (14). If we generalise the meaning of "spin" so as to call tAVG + 2ClG the "spin" of the motor G, eq. (7) asserts that if E' is any constant motor the "spin" of x E' is the later 2Vlmx~^E'. Also the statement that lA'^E +2D.E 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 (G) et seq.]. All the corresponding (juatemion formulae are consistent with the assumption that % 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 ;j^ be 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 % be of n dimensions, m is of 3?^, and therefore mx~^ is of 2n while x is 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 ^^ be of arbitrary homogeneous dimensions, but then we must not assume that Jd\' = 2nfjdl,' where H has the same meaning as in the equation fdX = 2il JJdt. Instead we may put fd\' = 2n'jjdX', where XI' is similar to ft when it is in combination with itself and ordinary scalars but is of different dimensions from H. The relation between the dimensions of D. and O' is the same as the relation between the dimensions of the H and fl' of Phil. Trans. 1892 A, p. 691. Indeed [ibid. eq. (9)] eq. (7) above takes the con- sistent form M'^AXa^' = 2 (mx-'n' - nx') E', and the right would be identically zero if the relations between H and Q! were precisely those of that paper. This apparent anomaly in " dimensions " is ultimately due to the fact that H, 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 rj and x> c~^ will be found to occur so that if c be not of zero dimensions rj and x cannot be assumed homogeneous in dimensions. We now return to the consideration of fluxes and intensities. If E and F he an intensity and flux respectively SEFd^ = SE'F'd^', § 26] AlJAl'TAriON TO PHYSICAL APPLICATIONS. 101 where d<{ uiid dt' arc corrcs])uuding unstrained and strained elements of volume. A somewhat interesting result may be deduced t'rum this. Suppose 0-, (Ti are two arbitrary rotor intensities passing through P and T, T] are two arbitrary rotor fluxes passing through P. Define the motors G, G', H, H' by the equations G = a +nT, G' = a' +0/ (15), H=T, + n be a 102 OCTONIONS. [§ 26 pencil function Avhose centre is at the point considered and such that (f)d'^ is the force due to stress exerted on any region at the rotor element d^ of its boundary. The whole force motor exerted on the region by the rest of matter is JJ(l>dX=JjJct>aV^d,, by eq. (4) § 22, and therefore the force motor exerted on the element dl? is (^^^ ^d<;. We saw in § 22 that the motor ^S a consisted of a rotor ^^ A — nM^(f)^ through the point considered and a lator HM^^^, and that the rotor bore to the stress function <^ the same re- lation as in Quaternions the vector c^iVj bears to the stress function is such that sE'- = — 1. We are thus led to make trial of the following coreciprocal motors as primitive units — E= (i+n{)|^f2, F= (j + nj)/^/2, 0= (^' + fiA;)/V2| .^. E' = v(i-ni)l^2, F' = v(j-nj)l^f2, G' = v{k-nk)l>^2]"'^ ^' Here sE' = sE''^ sF' = sF'^- = sG' =sG'' = -l] sEE' = sEF = sEF' = sE'F = sE'F' = . . . = | ' ' •^' ^• It is now easy to see that if ili, 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 + o,G + g:G' (5), A, = e,E + eJE' +f,F +f,'F + g.fi + (/./(?' (6), where e^ &c. are all ordinary scalars. Thus [A usd. § 143] - sA,A, = e,e, I- e/e,' +fj, +fj.: + g,g, + g^gl = [A,\A.;\ = [A.AA,] (7). We are now in a position to utilise Grassmann's general 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 \_Ausd. § 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 A^, A, ...An are any n magnitudes (octonions, scalars, &c.) and if

A, ... An) is such a function of them that if any one, say A2, is expressed in the form B^ + Go we always have c{>(A„B, + G„A,...An) = 4> {A„ B„ A, ... An) + <}> (A,, G„ A, ... An), ^ (^1, Ao ... An) would by Grassmann be called a product of the magnitudes Ai...An [Ausd. § 44]. If further ^ is of such a form that when any two of the magnitudes change places (f) becomes changed to its own negative, the product is called a combinatorial product [Ausd. § 55. These are not Grassmann's definitions, but are most convenient for our definitions]. Thus if Qi, Qo, Qs are octonions and A, B, G, D, E motors, Q,Q.JQ, and MABSCDE are products, but not in general combinatorial products. Again MAB, M,AB, SGDE, sGDE are combinatorial products. Again if ^ is a general linear motor function of a motor, sG(f>AsD(f)B - sD(j)AsG(i>B is a product, but not a combinatorial product in general, oi A, B, G and D ; 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 [^i^o^3^4J.5] and [A^AoAiA^AsA,^]. 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'GG']=1 (1). (2). [A,AA.A,A.A,] ■(-•>)■ § 28] MOTORS AS MAGNITUDES OF THE b'lllST ORDER. 107 For all values of r let Ar = e,.E + e/E' +f,.F+f;F' + cj,Cr + g;G' = Xri + yrj + Zrk + Ir^i + m,.nj + llrilk Thus [eq. (2) § 27] Xr = (er + ue/)/V2, Ir = (e, - ue/)/V2, j/,- = {/>■ + 1//)/\/2, &c.. . .(3)^ Gr = (Xr + lr)/\/2, e/ = V (l,. - X,)/s/2, &LC (4) Grassmann then proves in § 63 that ' e, e.; f, f.: g, g,' es e,' f, /s' g, g,' Bi ei fi fl g^ gi ^5 65' /g /s' g^ g^ e, < /e /e' g^ g^ [It is sufficient for our purposes to take this as the definition of [^i.-.^lfi] and to notice that by the definition [-^i.-.^g] is a combinatorial product and that eq. (1) follows from this definition.] The rules for combinatorial multiplication are given in the Ausd. % 52 et 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'FFGG'] = -v [{i + ^i) (i - ni) (j + nj) {j-nj)(k + nk){k-nk)] = 8v[i.ni.j.nj.k.nk], so that [i.j.k.ni.nj.nk] = v (6). Hence [Avsd. § 03] a\ y, z, I, nil th X2 y., z., l, m., n.. x-i y-i z-i L, vh »A Xi 2/4 ^4 ^4 "^4 "4 A'-s y, z, I, m, n, x^ 2/g z. Is nis Oh 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 e(]. (7). It is unnecessary to give the proof, as the fact follows incidentally from a result we shall prove at the end of this section. [A^A.^AsA^Ar^xi^] = V .(7). 108 OCTONIONS. [§28 In Grassmann's mode of expression [^i^3/l;,^4i4g] would be the complex (Gebiet) of A^, A.,, 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.1...J.5]. For put [A,...A,] = B, where 5 is a motor. Then if ^0 ^^ J^ny motor whatever - sAoB = -sA,\B = [A,\\B] = - [A,B] (Ausd. § 93) = — [-4 0^1^ oil 3^ 4.4 gj 3 Vo '■■ » ' = s^„ 65 ... 0C5 2/5 «5 G' 0^ • • g-o vsAq m nj nk i j Xi 2/1 Zi /i Vl^ 2/5 Since ^0 is arbitrary it follows from § 14 above that [A,...A,] = BE' G' m nJ nk i J k X, 2/1 Zl k in. n. 2/5 ...(8). [For our purposes these equations may be taken as definitions. I thought it desirable to show what connection [^i...^g] and [ill ••• ^e] h'l'tl with the Ausd^^ Very often the imaginary v recurring in equations (7) and (8) is an inconvenience. We therefore define as follows [A,...A,,]^-v[A, cKl •(9), §28] MOTORS AS MAGNITUDES OK THE FIRST ORDER. 109 for the values 5 and 6 of n ; and in the definitions about to be given of {A•^... An] for values from I to 4 of ii the same e((uation may be supposed to hold. Thus {^....^1 = 1^> ^nl From these X, y, ... 71, Xo y« ... n. (10), X, 2/6 ... no m rtj...k X, If, ...ih (11). X, 7/g ...n, --sA,{A,...A, (12), .nj.nk] = 1.. (13). [j . k . m . nj . nk] = - n/ (k . i . m . nj . nk] = - nj m . nj . nk] = - nk k . nj . nk] = i k .nk.ni]=j k .m . nj] = k .(14). {J.!...^,} may be regarded as a linear motor function of A^; also as a motor function of A^, A^ linear in each; and so on. Thus = (l>,{A,,A,,A„Ar:) = ,{A„A,,A,,A„A,)] ^ '^■ With these meanings of ^i ... ^4 we may put from ?i = 1 to ?i = 4, <^„={^,...^„} (16), and <^o = {l} (17). Thus (^1...^,^- is for all values of n from to G a combinatorial product of ^, , A. 2... An. It is to be noticed that with the meanings of y^r and Z given in equation (6) § 15 above, y\r{Z,Z) = [-f{A,,[A,A,...A,]) + ylr{A,,[A,A,...A,])-...\ + >|r(A,{^^...A})]/[^^...^cF ''' where A,, Ao...A^^ are any six independent motors. For if Ave change A, into any other motor x^A^-\- x.A.-\- ... + n\,A^, (./■, not 110 OCTONIONS. [| 28 zero) independent of A„, A3...AG the expression on the right is unaltered. Similarly for the other motors involved. We may therefore change ^j... J.^ to i, j, k, Cli, ilj, Hk, whereupon the right becomes yfr (i, Cli) + -^ (Hi, i) + ... or i/r (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} = {A2As...Ae] sEA,-{A,A;...Ae} sEA„ + -{A,A....A,]sEA, (19). Define -4,, A...,A^ by the equations J _ _ { A2As...A^ ] J _ {AiA3...Ae] J _ {AiA^...A 5} ^^- {A,A,...A,]' -- {A,A,...A,}' ■" ^'- {A,A,...A^^^''^- Eq. (19) then gives E = - A.sEA, - A.sEAo - ... - A,sEA,\ = -A,sEA,-A,sEA,- ... -A.sEA,] ^ ^' Also by eq. (19) E {A, ... Z4 =A, {EA,...A,} + A, {A^EA^...!,} + ... Comparing this with the last equation we see by eq. (12) and § 14, that _ _ _ _ {AA....A ,] ^^J AA,...A 4 \A,A.:-A,}' " {A,A,...Ae}' so that the relations between the two sets -4 1^0 • • • ^b. and A^A. ...Aq are symmetrical. Again by eq. (1 2) sA.Ai = sA.A^ = sA-iAs = sA,A^ = sA,A, = sAoAq = - 1 ...(23). Again since {A^A^A3...As} = 0, s^i^.= 0. Thus sA,A. = sA.As = sA,A, = sA^I, = ...=0 (24). Ai, Ao ... A^y have been assumed independent. [Otherwise {^1 ... Ae] = 0.] Ai, A2 ... Ag 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 -4„+i ... A^ is reciprocal to the complex Ai ... An. Moreover no motor which does not belong to the former complex is reciprocal to the latter. For any motor § 2(S] MOTORS AS MAGNITUDES OF THE FIRST ORDER. Ill 6 _ can by eq. (21) bo put in the form ^5"= i!a .4, and expressing that this is reciprocal to each of the motors A^, A.... An we get by equations (2.S) and (24) Xj^ = a;.,= ... =Xn= 0, so that E belongs to the complex An+i ... -^g. Hence to every complex of order n thei^e is a reciprocal complex of order Q — n and no motor not belonging to the latter is reciprocal to the forme?-. In particular to every complex of order five there is one (and ordinary scalar multiiDles of it) and only one motor reciprocal. We have just seen that A^ ... A,^ arc independent. Hence the six motors luhich are reciprocal to each set of five out of six given independent motors are themselves independent. More generally if {ti) he a complex of order n and (6 — n) an independent complex of order (6 — w), then if {Q — n) is the complex reciprocal to (n), and (n) the complex reciprocal to (6 — n), {Tt) and (6 — n) are independent complexes. For {n) may be taken as A^...An and i^ — n) as AnJri--- -A-s- Then {n) is A^...An and (6-w) is An + l ... -4,;. If a motor is reciprocal to five oid 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 exemplifying the present methods and of showing the physical meaning of the con- nections between the two sets of motors A^...A(i and A^...Aq. [I am not sure that the second and third italicised statements are known.] If ^1, J.O ... ^6 are six independent coreciprocal motors and E=x^A^ + ... +a-«^G, we have by operating by s^i ( ), sEA^ = x^sA^-. Hence E = A.sEA JsA,' + A,sEA,lsA.^ + ... + A ,^EA,lsA,r. . .(25). Comparing this with eq. (21) we have sEAJsA," = -sEA,. Hence in this case by § 14 A,/sA{' = - A, = {A^, ... A,^J{A,A, ... A,} (20). A relation that we shall require later is the following {ABCnA'ilB'nC] = S,ABCS,A'B'fr (27), 112 OCTONIONS. [§ 28 where A,B,C, A', B', C are any six motors. The right vanishes if and only if either ^, B, C or ^', B', C are not three completely independent axial motors, li 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. If A, B, C are not completely independent axial motors one motor at least of the complex A, B, C 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, C are com- pletely independent axial motors and riA', D.B', Q.G' are indepen- dent lators. Hence xA+^jB + zC+ mA' + m^B' + nnC when x is not zero is any axial motor completely independent of B and C. If this be substituted for A both expressions are altered in the ratio of x to 1. Similarly if nA' be changed to any other lator x£lA' -I- yClB' + z^C {x not zero) independent of QB' and flC, the expressions are both altered in the ratio of x to 1. It follows that [ABGnA'nB'aC'}IS,ABGS,A'B'C' 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,Gio i,j, k and A', F, G' also to i,j, k, we see by eq. (13) that this value is unity. Since S,ABGS,A'B'G' = sA (HMSC) S.A'B'G' = s . (nA') UB'G'S.ABG, we see by equations (27) and (12) and § 14 that [BGnA'aB'nG'] = nMBGS,A'B'a (28), and [ABG£lB'nG'] = - MB'G'S.ABG + a lator. If S.ABG is not zero this lator may be put in the form n (xMBG+yMGA +ztAAB). Operating on the last equation by sA ( ) we get X = sAB'G' and similarly for y and z. Hence [ABGnB'nG'] = - MB'G'S.ABG -f n (MBGsAB'G' + MGAsBB'G + MABsGB'G')...{29), when S.ABG is not zero. This equation is also true when S^ABG is zero, for then xA + yB + zG = ii lator = ilA', § 28] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 11.'^ say, where one at least (say a-) of the ordinary scalars cc, y, z, is not zero. Operating on the last equation by nM6'( ) we get and similarly fiM^ i? = J HM^C. Thus { A BcnB'nc] = - .7-' {BcnA'nB'nc} = - .r-' nMBCS.A'B'C [eq. (28)] = - ,/•- nMBCs (a-A +yB + zG) B'C = -a [MBCsAB'C + '^MBrsBB'C' + - MBCsCB'C] x = - n [lABGsA B'C + UCAsBB'C + lAABsCB'C], which proves eq. (29) when S,ABC = 0. Eq. (29) may by eq. (8) § 18 be put in the form {ABC£IB'D.C'] = MB'C'S,ABC- MBGSAB'C -MCAS.BB'Cr -MABS.CBC (80). Putting in equations (29) and (80) B' = B, C' = C we get {ABCnBnC] = MBC(-S, + S.;)ABC (81). It will be observed that equations (28) to (31) are generalisa- tions of equations (13) and (14). Also eq. (28) is a particular case 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 E is any motor and A, B, C are three given completely independent axial motors E = ASEBCS-'ABC+ ... = A {sE(nBCS-'ABC) + QsE (BCS-'ABC)} + .... Hence if we take A, B, C, flA, VLB, QC for the A,, A.,... A, of equations (20) to (24) A = - DMBCS-'ABC, nA = - MBCS-'ABC (32), and similarly for B, ClB, G, ilG. [Notice that what was denoted in § 16 by A is here therefore denoted by ill.] The second of 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 = C. Since E = - AsEA - nAsEnA -BsEB - ..., we see that {ABGB'C} = {ABCn (AsB'nl + BsB' nB + CsB'IiC) n{AsC'IiA + BsC'nB + Cscnc)}. Equation (30) may now be utilised and we shall then have expressed {ABGB'C} in a manner independent of any arbitrary origin. Similarly for {ABCA'B'G']. The expressions are however too cumbrous to be of much use. 29. Combinatorial, linear, circular and hyperbolic variation. In § 71 of Ausd. 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 multiple, 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 A^ ... Ap ... A^ ... A^ is said to be changed by simple combinatorial variation to the group ^i ... Ap ... Afl ... An (the only two motors changed being Ap and Ag) if Ap=cAp +sAg'\ Ag=s'Ap + cAj> (1). c^ — ss' = 1 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 gTOup 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 §20] McVrOIlS AS MAGNITUDES (IF TlIP: FIRST ORDER. 115 below) we might dcfiiio it more generally. The above definition is the most convenient for our purposes. The more geneial definition is given by the equations A J,' = .v.Aj, 4- //,.■!,/, A,/ = x,A.j, + >/.,A^, x,y. - x,ij, = 1.] When c= 1 (and therefore s or .9' is zero) the vai'iation is call(>d a linear variation. When s = — s the variation is called a circular vaiiation. In this case we may put c = cos 0, s= — s' = sin 6 — (2). When .S-' = s the variation is called a hyperbolic vai-iation. In this case we may put c = cosh 0, .9 = .9' = sinh 6 (8). From eq. (1) we have A J, = cAj/ - sA^', A^ = - s'Aj' +cA^' (4). Thus ^Az:_^=_^^_ (.5), Ap, Aff Ap , — A,j which in Grassmann's notation {Ausd. § 377) expresses the f;xct that i4j/, - A^ are the same linear functions of A,,, A,, that the latter ai-e of the former. From this we see that ii A, B be changed by iiefiative com- binatorial variation to A', B' A',B' A,B ,„, -a:b-jv7i^ ^'^' that is to say that if we put A' = (f)A, B' = ^B ; 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')=\ we see from eq, (4) that Ap, A^ are obtained from Ap', Aq by a simple positive combinatorial variation. And further, if the given variation {Ap, Aq into Ap, Aq) is linear, circular or hyperbolic, the derived variation {Ap, Aq into Ap, Aq) 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 covihiiiatorial product of A^, An... An is unaltered by combinatoria I variation. In § 156 oi 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...Aq...An). Then {A, ...Ap...A^...An) = {A^ ... {cAp + sAq)...{s'Aj,-VcAq)...An) = (^1 . . . {{cAj, + sAq) - - (s'Aj, + cAg)] . . . (s'A^ + cA,j) ... A,,) = {A,...^...(s'Ap + cAg)...An) (Ar c = \A^ ... Ap ... Aq ... 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, Ao be combinatorially varied to ^/, A^, MA,A,^IAA;A.^ ; if A„ A„,A^ to A,', a:, a^,sa,a,a., = sa,'a:a^ ; if E, F to E, F' and A, Bio A', B', sE(f)AsFB-sF4>AsE(f>B=sE(f>A'sF(f)B'-sF(l>A'sEB'-sF'c(>A'sE(f)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 (f) = -sj where ct is self-conjugate. Then sA'^A'sB'ztB' - s'A'ztB' = sA^AsB^B - s^A^B...(8). A still further restricted case is obtained by putting ot = 1 when we have sA'-'sB'^ - s-A'B' = sA^sB' - s^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 2!)] MOTORS AS MAGNITUDES OF THE FIllS'l' ORDER. 117 to our U8C below of cq. (8). NcccUcsh to say the use below was suggested by his. In equations (7), (8), (9) we may put S or S, for each s. CT having the meaning just given to it wo have from c<|. (1) SA''S7A' = c"~SAv7A + 2csSA^B + s'SBtjtB] SB'-stB' = s'"-SA^A+2cs'SA^B + c"'SB^B) ^ ^' from which we deduce that s'SA'-syA' - sSB'ztB' ^s'SAt^tA -sSBthB (11). Particular cases are (circular variation) SA''G7A'+SB'^B'=SA^A+SB^B...{1-2), (hyperbolic variation) SA'x^A' -SB'-stB' = SA'!;7A -Si?OTi?...(13). Putting CT = 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 SA7;TA+-,SBz7B = 2b,, SAzyB = b, SA'x:7A' + -,SB'^B' = 2W, SA'wB' s G=c'" + ss', S = 2cs, S' = 2cs' h: .(1-5), ha h: = Gb, 4- Sb,, b.: = S'b^ + Gb,, G-' - SH' =1 (16), so that b,', h! are obtained from b„ h, by a combinatorial variation. In particular, for circular variation, = ^(SAi^A-SBi^B)coH2e + SA-:^Biim2e I ...(17), SA'^B' = - i (SA^A - SBx^B) sin 26 + SA^B cos 26) and for hyperbolic variation = 1 (S^^^ + SB^B) cosh 26 + SAztB sinh 26 [ . ..(18). SA'r^B' = J {SA-^A + SBt^B) sinh 26 + SAr^rB cosh 26] Since 8Q = S,(2 + HsQ, where Q is any octonion, we may in equations (10) to (18) write either Sj or s in place of 8. Suppose Ai,A.2... As are six independent motors and A^, A.2...Ae have the meanings defined in § 28. A^', i4/...j4/ w^ll be defined in the enunciation about to be given. Let A/,A.,'...A,^ 118 OCTONIONS. K 29 have the same relations with A^, A.,... A^ that A,, A. ... -Ag have with A^, A. ... A a. We proceed to show that : — If by a series of negative circular variations A^, ... A^^ become transformed to A,', ... A,l,Jhen Z/, ... ^Z will be exactly the same linear functions of A,, ... A^, that A,',... A^' are of A,, ... A^: or in Grassmanns notation A,', A.; , ^ /, A!, A,', A, ' ^ a;, a.;, As, a,', a j, a^ ,^^. Zi, Z,, Z,, Z4, ^5, Z« a,,a.,a.,a,,a^/a,"' It is sufficient to prove the theorem for a simple negative circular variation. Suppose then A: = A., A,' = A„A,' = A,, As =Ae .(20). A,' = cA, + sA.,, A.^ = - s'A, - cA.„ c- -ss =\] [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 A^ and A., has its sign changed, but is otherwise unaltered by this negative varia- tion. Hence - , _ - {AU-U:A.:A:] ^ [A,A,A.A-A.] ^ J ^-'~[a^a:a:a:a:a:\ - {a,a,a,amM and siibilarly for A^, Ar', A,'. Again we have -[A.: a: ... a:] _ [{ sA, + cA.:)A,... [A^A^ ...a:] -[a,a,...a:^ -J , [A^A; . . . Ai\ _ [(C^i + 6"^.,) A.,... ^e] _ J J ^^ = {^;^;7::z/] — -[a;a;7:.a^ - '^' -• Thus Z:/ = A.^ Alj= A, ,Al = A, ,A' = A, ] A,' = cA,-s'A,, A: = sA,-cA,, g— (-5)(-6'') = 1^ ■■ which expresses the connections between A/, ... A^' and Ai, ... A^ 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) Z, = cAi — s'A.,', A.2 = sAi' — cA.2 express the converse relations. In the case of circular variation s' = — s^ Hence in this case Z,', ... Z,/ are the same functions of A^, ... A^^ as Ai, ... A,! are of j,_- { A.:A:... Ae'] _ {{ sA, + cA,)A,...A,\ __ j _ ^'M)] MOTORS AS MAOXITUDES OF THE FlllSI' OllDEU. II!) A,, ... A,i. Thu theorem can be easily exLeiulcd ihnii simple lo multiple variation. 30. Conjugacy with regard to a general self-conjugate function. Let ct be a real t^oueral self-conjugate linear motor function of a motor. We are about to establish certain properties of -53-. Two particular cases will frequently be considered. (1) vs may be put equal to unity; conjugacy then reduces to reciprocity. (2) •57 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 nc^gativc and not zero for every motor E in space. It is a partial energy function when sEstE is zero for some values of E and is negative for all other values. We shall usually write yfr for zj when we restrict it to being an energy function. When sE'stE = 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 nth. 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 •ST when sEvtF=0. n motors are said to be conjugate when every pair of them is a conjugate pair. Defining a complex of order /(, as in § 14 above, we proceed to show that: — In anij complex of order ii, n real independent conjugate motors can he found. Let A^...Aa 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^, Ao so that yl / = J 1 cos ^ + A. sin 9, A.;^-A,^\nd + A,cosd. 120 OCTONIONS. [§ 30 Thus sA^'stA^' = sAisrA^fos 26— (sA^'stAj - sA.'stA.^ sin 6 cos 6, so that Ai, A.! will be conjugate if tan 26 = — -. -. j -r , and 6 can always be determined so as to satisfy this equation. When any such pair is thus made conjugate the product sA^tsA^^A.^A. ... sAn'^An 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 (s^jCT^i and sA-i-sjA,,) is by eq. (8) § 2i) diminished by the amount S^A^-sjA. which by hypothesis is not zero and since ot is real is not negative. There is an exception to this statement, viz. when one of the factors, not involving a varied motor, say 8^.3-57^3 is zero ; A.^ is then self-conjugate. The product then remains zero. But unless A-i 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 ^3 is self-conjugate and conjugate to all the other 11—1 motors it is conjugate to the whole complex. In this case A-i does not require to be varied. We now see that the following statement is true : — If any pair of ^1 ... yl„ is not conjugate we can by a circular variation make it conjugate and the product sA^thA^sA.^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 11 motors are conjugate. Hence n 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 J.1 ... Aiy Bi+^ ... Bn are one set of conjugate motors and At' ... Ai,/, B'„i+i---Bn are another set, none of the A's being §30] MOTORS AS MAGNITUDES OF THE FIRST ORDElt. 121 self-conjugate but all of the B's being self -conjugate, then the coinj)lex Bi+i ... Bn is the same as the complex B\n+i • ■ • B,^. [Thus in particular m = l.] For while every motor of the complex Bi^^ ...B^ is conjugate to the whole complex, no motor of the complex Ai...Bn which does not belong to the complex Bi+^...Bn is conjugate to the whole complex. Suppose that a\A, + . . . + yi+r Bi+, + ...= XxA + %i? is such a motor conjugate to the whole complex. Operating by syljTO- ( ), remembering that sA^'stA^ is not zero since A^ is not self- conjugate and that A^ is conjugate to A, ... AiBi+i ... Bn, we see that a-j = 0. Similarly a\,= ... = xi = or the motor belongs to the B complex. It follows that every motor of the complex -S'„i+i . . . Bn belongs to the complex Bi^^ ... Bn and every motor of the latter belongs to the former. Hence these complexes are identical. If we put CT = 1 we obtain the following : — In every complex of order n, n real coreciprocal motors can be found and the 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 be a conjugate set of motors not one of which is self-conjugate they must be independent, and if B be conjugate to the whole complex it cannot belong to the complex A^ ... Ai. [Com- pare Ausd. § 157.] Suppose yB^-{x,A, + ... -\-xiAi) = 0. Operating by sAizr ( ) we obtain x^ = and similarly OJn= ... = Xl = 0. If -4, ■.. Ai, Bi+^ ... Bn have the meanings they had just now: — If E be a motor of the complex conjugate to each of the motors A^, An ... Ap, it must belong to the complex Ap+i ... Ai, Bi+^ ... Bn. [Compare Ausd. § 159.] Suppose E=txA-\-'^yB. 122 OCTONIONS. [§ 30 Since E is conjugate to A^ we have by operating by sA^ts ( ), that A'l = 0. Similarly x.2= ... = ^'^ = 0, or E belongs to the complex ■^p+\ •■• Ai, Bi+^ ... Bn. By putting ■ot = 1, in both these statements we may read "reciprocal" instead of " conjugate." 31. Conjugate variation ; positive, negative and zero norms; semi-conjugate complexes. Suppose ^i and A. are two motors conjugate with regard to zj. Suppose -4/ and A/ are derived from A^, A., by the combinatorial variation A^' = cAi + sA.,, A.,' = s'A,+cA.2, c^-ss'=l (1), with the condition s'sA^TjyA, +ssA.^A., = (2). Such a variation will be called a '(^-conjugate variation or, when there is no risk of ambiguity, simply a conjugate variation. [The meaning will be very slightly extended directly.] Since sA^7jtA.2 = we have at once from equations (1) and (2) that when A^ and A.^ are conjugate and are by a conjugate varia- tion transformed to A^, A.!; A( and A.I are also conjugate, i.e. %A(mA.: = (3), and also Su4/ct^/ = s^iOT^,, ^A.lxsA-l = ^A.,^A., (4). W}ien s J.jotJ.1 = %A.^A.2 a conjugate variation is a circular variation. When sJ-i'OT^i =^ — sA-.-utA., a, conjugate variation is a hyperbolic variation. When sA.,ztA. = and sA^stA^ 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^' = AiA-sA.^, A.J = A., where s is however arbitrary ; it must not be of the type AJ = A^, A.: = A., + s'Ai. According to equations (1) and (2) if sAi-^Ai = sA..7;tA., = 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 ^ol] MOTORS AS MAONIIUDES OF I'JIK FIllST oRDKR. 12.*} arbitrary type than combinatorial variation acc(jrcling to the following definition : — If Ai and A^ are conjugate, independent and both self-cunjugate, and Ai, A! are any two independent motors of the complex A^, A.,; the variation front A^^, A. to A^, A! is said to be a conjugate variation. This may be put into symbols thus : ^/ = xA-i, + yA.,, A.! = x'A^ + y'A.,, and xy' — x'y is not zero. If a series of conjugate variations be performed on a group 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 sA-stA 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 sAa:A = — 1 ; a negative norm if sAznA = + 1 ; and a zero norm if sA-nrA = 0. Thus by § 30 any complex of order n can be expressed as a complex of 71 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 nornts 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.] 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^ ... Aj, be the positive norms, B^ ... Bq the negative norms, and C\ ... C^ the zero norms of a set of n conjugate norms belonging to this complex. varia- as 124 OCTONIONS. [§ 31 If any motor E of the complex is a positive or negative norm or is conjugate to the whole complex; A^ ... C,. can he so transformed by conjugate variation that E becomes one of the group. [Compare Ausd. I 160.] If E be conjugate to the whole complex it belongs (§ 30) to the complex Cj ... (7,.. But by the definition of conjugate tion among the C's we can transform them by such variation so to contain any motor of their own complex. If ^ be a positive or negative norm sErs^E = + 1. Let E = x,A,+ ...+Xj,Aj, + y^B, + ...+z,G, + ... = lxA+lyB + XzC. Since sA^-stAi = -1, si?it3-i?i = + 1 , sC'iotCi = 0, &c , and the n motors on the right are conjugate, sEr^E=-^x'+^f= + 1. If the upper sign be taken we can successively bring into the group _ XiAj + x^A. „ _ a^i^i + X. 2A2 + Xs As ' ~~ ^J{x,' + xi) ' ' ~ sl{x;' + xi ^-x.f) ' "" _ XiAi+ ... +XpA p Eo being obtained from A^ and A.^, E^ from E.^ and A-^, &c., and E^, from £'^j_i and Ap all by circular variations. And now we can successively bring into the group % xA+y,B, t xA + ij,B, + y,B, ^' - V(S^^ -2/r) ' ' V(S^^ - 2/f - 2/-/) ' ■ • • ' E^' being obtained from Ep and B^, E.! from E^ and B.,, &c., and Eq from £"y_i and Bg all by hyperbolic variations ; for not one of these denominators is zero since S«-— %^ = 1. Finally we can successively bring into the group E;' = IxA + ^yB + z,C„ E.!' = %xA + %yB + z,G, + z,C,, ..., E;' = IxA + lyB + tzC, E^" being obtained from E\j and C\, E" from E^' and Co, &c., and E" or E from E",.-i and C, all by linear variations. Similarly if s^ot£'= + 1 we can bring E into the group by beginning with the 5's. It is to be observed that if E is a zero norm (e.g. ^i + ^^i) which is not conjugate to the whole complex it cannot hv brought §81] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 125 into the group by conjugate variation. For it cannot be obtained from ('i... (V 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 (??) and let {in) be a complex of order m which is included in (?i). Then by § 30 {m) may be expressed as a complex of m conjugate norms. When ex- pressed in such a form let ai, ... a„ be the positive norms and /Sj, ... ySft be the negative norms. The zero norms of (??i) may have a complex in common with (\, ... C,., the zero norms of {n). Let this complex be that of 71, ... 7,, and let the rest of the zero norms of (wi) be §1, ... Srf. Thus Sj, ... ha belong to the complex A^,...Ay, Bi, ... Bq, C\, ... G,., but the complexes 8^, ... Sd and C\, ... C,. have no motor in common. Ai, ... A,^j, i?, , ... Bq, C\, ... C,. ccm by conjugate variation be tr'ans/onned to a,, ... cip, /B^, ... /S^, 71, ... 7,.; where a^, ... Up are positive norm.s-, /3i, ... /3q negative normis, and 71, ... 7,. zero norms; tohere a^, ... a„, /B^, ... /3h, and 71, ... jc have the meanings just given to them ; and ivhere §1 = x, (cta+i + ^b+i), ^2 = a-o (a«+2 + ^b+o), &c ( .5). We first show [by a process essentially the same as that of Amd. § 161] that Ai...Cr can be so transformed as to bring tti-'-OfaA .. ^bji ••■ Jc into the group. By the last proposition A^ ... ApB^ ... BqC^ ... 6V can be trans- formed to ttiA^i ...Ap'Bi ... BqG^...Gr where the ^'s, 5's, and C's are positive, negative and zero norms respectively, a.2 belongs to the complex A^... C,. and therefore to a^A.! ... G,. and it is conju- gate to «!• Hence (§ 30 above) it belongs to the complex A^' ... G,.. Hence by the last proposition this last can be transformed to 02^ .•," . . . Ap'Bi" . . . Bq" G^ ... G,.. «:! belongs to the complex A^... G,. and therefore to aitto^," ... G,.. It is conjugate to a^ and a.^ and therefore belongs to the complex A." . . . Gr. Hence this last can be transformed to a-jA^'" ... Cr and so ori. Proceeding in this way we see that ai ... aajS^ ... /3b can be brought into the group. By hypothesis 71 ... 7c belong to the complex C, ... G,.. Hence 7i ... 7c can be brought into the group. It remains to prove that a„+i, ^b+i ••• can be determined to form part of the group and satisfy eq. (5). 126 OCTONIONS. [§ 31 When «! . . . ttrt/Si . . . Ph'y\ • • • 7c have been brought into the group let the rest of the positive norms be denoted by ^Z ... A'p_a, the rest of the negative norms by B^' ... B'^^i and the rest of the zero norms by C/ ... C',—c- 8i...8d are all conjugate to all the motors a-^...l3i. They therefore (§ 30 above) belong to the complex A^' . . A'j,_„ A'- • • -S'^-'^ C,...Gr. Thus p-a q-b 8,= lxA'+ ^yB' + C, 1 1 where C is a motor belonging to the complex C, ... 6',.. Since S, is a zero norm we obtain tof-ly-^0. S.X- and ^y- are therefore neither of them zero ; for if either were, all the a;'s and y's would be zero and h-^ would, contrary to hypo- thesis, belong to the C complex. Thus h, = x{A + B\ where A and B are a positive and negative norm respectively given by A s/l,r^ = txA ', B^^lf- = lyB' + G. This shows that oto+i and ^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 8, = X, {A ; + £/), ...8,= xe {a; + b;). 1 proceed to show that the theorem is also true for e +\. Let 8,+,= X^A'+ ''l vB' + C, where C belongs to the complex C\ ... G,.. Since S,,+i is conjugate to 8i we get ^1 = ?7i . Similarly Again since S(,+i is self-conjugate we get p-a q-b Here again were either S ^- or S rf- zero, S,,+, would, contrary e + \ e + 1 to hy])()thesis, belong to the complex 8^ ... 8,., G-^ ... G.,.. § :il] MO'l'OUS AS MAGNITUDES OF TI[K FIRST ORDEK. 127 Thus Se+r = Y ^A' + '2 VB' + la (^/ + ^/) + • • • + 1. (^/ + ^/) + ^' = av+. (yi + /i) + 1, (^/ + 7^/) -f . . . + 1. (.4; + i?;) + c, p-a J) -a q-b where Xe^-^ = \/ S |'-, ^ = 2 |-47^'c+i , B = 1, rjE'/xg^i . c+l e+1 e+1 We now show how by successive conjugate variations, |i , |^o, . . . ?<, can be got rid of from the hist ecjuation. Ii can be got rid of by writing Se+, = .7V+, (A + Jl) + I, {A.; + 7?,/) +... + 1. {A; + BJ) + c, where .1/' = {^/ (^,+,^ + 1,^) + 5/lr + 5ri^v+i}/.'V+. \/(.'^.+r + f^^), 5/' = (7?;.^,+, - B^,)l^{x,^,^ + If), 5o = 5 + U/+7i/)|,/.r,+,. Here ^Z', 7?,", A, B^ are a conjugate set of motors belonging to the complex A^', 7?/, A, B such that s^/'-cT^i" = sA^A = - s/?i"ot/>V' = - sBo^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 |o . . . ^^ can be suc- cessively got rid of. We may therefore assume them all to be zero. We then have 8, = X, {A; + 7i/) ... 8, = a-,, {A: + B;\ K+. = ^Wi (^'e+1 + ^^'.+0, where J.'e+i = ^, 5'(.+, = 7? -f- a-g+i"' C (so that £'e+i can be obtained from 7?, C, ... C ,. by a multiple conjugate variation). It follows that oia+i, ^b+\--- C'l-ii be determined as asserted. Certain particular cases of this theorem are worth enunciating. First suppose m = n so that the complexes (?*) and {m) are the same. The theorem may then be thus |)ut : — 128 OCTONIONS. [§ 31 If E he any one of n conjugate motors of a given complex of 07^der n, the number of motors for which sEztE 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 OT = 1 we get : — Of n independent motors of a given complex of order n, the number ivith positive pitch, the number luith 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, So... by (iriV'2)~\ {x.2\J'2)~'^... and denote the new values by Sj, So.... Also denote by S/, So'... the motors defined by '^a+i + Hh+\ h,= ''-'^P^ , kc (6), V2 ' ' \/2 s/ = ""+^7/^+\ s;= "^+^7/^+% &c (7). Here it will be observed that Sj, S/ are obtained from ct^+i, 0b+i by a negative circular variation. Hence (§29) oia+i, A+i are ob- tained from Si , S/ by the same negative circular variation or Si + Si' _ Si-S/ _S, + S,' V2 ' ^'+^~ V2 ' "'*+'" V2 ««+i=-^, A+:=^S^, a,+,= -^7i^,&c (8). Now change the notation as follows: — change oc,...aa to ^i^o...; /3,... 13b to B,Bo...; 71...7C to C\G.,. SiS/S,S./... toDiA'AA'...; a^,a^_i , . . (not involved in D^R, . . .) to A^A.l.. .; \ J-- • -(9 ). ^q^q-i... (not involved in D^Do...) to B/B.,' .. yrjr-i •••7c+i to C/Co',.. We then get the following : — If(ni) be 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 (??). {m) consists of the positive norms A^A.,..., the negative norms B^B.,. . . , and the zero norms (\C., . . . D^D.,. . . §81] MOTORS AS MAONTTTTDES OF THE FIRST ORDER. 129 {71 - m) consists of the positive norms A^'A.!. .., the negative norms B{B.:..., and the zero norms C\'C.f ... D^'D.!... . All pairs of these norms except the folloiving pairs of zero norms (D^D^), (D.,R,')... are conjugate. These last are such that sD,v7D; = sD,vsD.: =... = - \ (10). From eq. (10) it further follows for these exceptional pairs that :— The pair ofrnotors A(' , B(' deduced from any such pair D,, D,' hy the negative circidar variation ^' =-^2"' ^' = V2' ^^^^ are conjugate and are respectively a positive and negative norm of {n). Therefoi^e {n) consists of the folloiving conjugate norms, (1) positive, A,A,... A;a:... a;' a;',.., (2) negaUve, B,B.,... B,'B.: ... B/'B./'..., and {S) zero, C,a--.G/a:.... Two such complexes (m) 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 {m) have the same meanings. Similarly (A'B'C'D') and {n — m) have the same meanings. Also order of (D) = order of (D') (12). The complex (A'B'CC'D) contains all the motors of (n) that are conjugate to (ABCD) or {m) and no others. For suppose E is a motor of (A'B'CC'D) and let E + A+B + D' = F be any motor of (n) where ^ is a motor of (A), B a motor of (B) and D' a motor of (D'). Expressing that F is conjugate to -4i, ^o... we get A=0. Expressing that F is conjugate to B^, B^... we get B = 0. Expressing that F is conjugate to D^, D.,... we get D' = 0. Thus if F is conjugate to (ABCD) it belongs to (A'B'CC'D) ; and it is easy to sec that every motor of the latter is conjugate to the former. [It should be remembered that (CC) is conjugate to the whole complex (/;) and that (A)(B)(A')(B') and (DD') are all conjugate to one another; also (D) is conjugate to (ABCDA'B'C) and (D') to (ABCA'B'C'D').-] M. o. 9 180 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) ivith reference to (n). Notice that the sum of the orders of (m) and its conjugate exceeds n by the order of the complex, (0), 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 (?i) 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 {€') makes up the complex (CC) (3) [CC'D) (4) {CD) (i>) .. (C) „ „ (CD) (5) (ABCD) {AB) „ {CD) „ „ {ABCD) (6) (ABCC'D) (7) {A'B'CC'D) {A'B') „ {CC'D) „ „ {A'B'CC'D) (8) {ABA'B'CC'D) Also (DD') is any complex which contains (D) and is conjugate to (ABA'B') and luith (ABA'B'CC) makes up the complex (??,). [The second and third complex in each statement of the second column are determinate complexes. This is not true of the statement about (DD'). 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) (C) 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 MAONITUDES OF THE FIRST ORDER. 131 (7) (ABA'B'CC'D) is the complex containing (in) and its conjngate. That (C") is any complex which with {C) makes up (CC) follows from the fact that what have been denoted by G-^Co... are any independent motors composing (0) ; and, with this restriction, that what have been denoted by C\C., . . . G(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 (in), (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 (CG'D) makes up the conjugate of (in). It remains only to prove that D^D., . . . can be so chosen as to form any complex (8) which with (C) makes up (CD), and that DiDi'D. . . . can be so chosen as to form any complex (SB') which contains (B), is conjugate to (ABA'B') and with (ABA'B'CC) makes up (w). D^D,... belong to the complex containing (h) and (G), and D^D.^... belong to the complex containing (hh') and (CC). The first statement follows from the fact that D^D.,... and (C) make up the complex (CD) as also do (8) and (G) ; 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 D, = ^, + 8,, A' = 7/ + V. D, = ^.,-^8,..., where 717/7-.. belong to (CC) and SiS/^.,--- to (88'). Thus 8^8.,... must belong to (8) [and 7170... to (C), though we do not require this]. Since ji^yij,... belong to (CC) they are conjugate to every motor of (n). Hence for all values of ji) and q, s8p^8,j = sDp^D^j, s8p'^8^' = sDp'^D^, s8p^8^' = sD^v^D^. Hence D^D^'D.... may be replaced by 8,8/80... 9—2 132 OCTONIONS. [§ 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 tvhich with (C) makes up the complex (CD). To prove this we have only to show in addition to what has just been proved that D^ and D. may be replaced by any two independent motors of the complex A, D.. If we put A,=a; (cDi + 5 A), ^2 = y (s'A + cA), c--ss' = 1, Ai and A2 are (x and y not zero) any such independent motors. But if we further put A/ = x-^ {cD; - s'D,'), a; = y-' (- sA' + c A'), then from the facts that A, A, A'. A' are all self-conjugate and all conjugate to one another except the pairs (AA') (AA) for which sDi-srDi = sD.^D! = — 1 , we deduce similar facts for Aj , Ao, A/, A,'. Hence A, A', A. A' may be replaced by A,, A/, A„ A/. As an example of the above theorems put ot = 1 ; let (n) be the complex i, Qi, j, Oj, k so that n = o; and let (m) be the complex i, (1 + i^'H) 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 = h-' (1 + i^^H) j, B = 0, G=k, D = i, A' =0, B'= h-' (1 - W^)j, C =0, D'= m. First notice the definite complexes of the first column : — (C) and (CC) are each the complex k; (CC'D) and (CD) are each the complex k, i\ (ABCD) and (ABCC'D) are each the complex (1 + ^ft-H) j, k, i; (A'B'CC'D) is the complex (1 - ^b'n)j, k, i; {ABA'B'GG'D) 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—G'=0. (0) is a determinate complex, so we may put generally G = k. [We might of course put G = 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 D = i + xjc. §31] MOTORS AS MAGNITUDES Ol>' THE FIRST OIJDER. 133 (AB) is any complex which with k, 1 makes up the complex (1 + i-6-n)J, k, i. Hence we may put A=h-'{\-\- Wa)j + x.,k 4- x.,i. [The coefficient h~' is to ensure that sJ.'- = — 1.] {A'B') is any complex which with k, i makes up the complex (1 — hh-Vl)], k, i. Hence we may put B'=b-^{1 - ^b-D,)j + xjc + x^i. (DD) is any complex which is reciprocal to A and B' and which with A, B', k makes up the given complex i, Vli^j, D,j, k. To get the general value of this it is easiest to assume D' = ^i + f m + 7)] + vnj + ^k. Expressing the fact that sDD' = — 1 we get ^' = 1. Expressing that D' is reciprocal to A and B' we get x-i + ^vh + v'h-^ = 0, ■''3 ~ hv^ + r)'h~^ = 0, so that D' = ni — x-jA + X5B' + xjt + ^ {X3- — xf) i, the coefficient of i being determined by the condition sD'-= 0. It will be noticed that the sum of the orders of (m) and its reciprocal, i.e. of (1 + ^b'-^)j, k, % and (1 —^h'-D.)j, k, i is six, i.e. it exceeds five the order of (n) by one the order of ((7). All the above theorems lose their complexity when ct 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')/V2. It will be noticed that sey/ti-conjugacy only occurred by reason of the D's. Hence in the case of an energi/ function all semi-conjugate complexes reduce to conjugate complexes. Connected with this is the fact that wJien zr is an energy function, if sEwE = for a motor E, then otA' = 0. For whatever motor F be S {xE + }jF) CT {xE + ijF) = 2xi/sF-stE + ij-sFr::F is negative or zero for all values of x and i/. Hence sF-stE = or, by §14, ■stE={). [If CT is any self-conjugate and if sEzxE = when E is one of a set of six conjugate motors, rsE = 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 sFx^E — O where F is 134 OCTONIONS. [§ 31 any motor. When ct is an energy function E need not be thus restricted.] If we call m{, {^B)[ OTCT_iii' is the component of E in the complex (-45) | ^-r^E „ „ „ „ {AB)) (11). § :)2] MOTORS AS MAGNITUDES OF THE FIRST ORDER. I:i7 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 (vi). In the present case (AB) is nut a determinate complex but (C) [§ .SO] is. Thus {AB) which is the complex reciprocal to (G) is a determinate complex, but (0) 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 (-OT and the complex (AB) both given) there is no ambiguity in the meaning of (11). From the above we see that when Ai ... C\. are a set of con- jugate norms for -sr, then A^... 0,. are a set of conjugate norms for CT_i. A less general statement is that tJie six motors which are reciprocal to each set of five out of six independent motors conjugate with regard to tn- are themselves six independoit motors conjugate with regard to ct_i. Still supposing /t=6, let in, (m), A^A.,.,.BiB,...C\C.,...D^D..... A I A.! . . . ByB! . . . C/C/ . . . D/B.,' . . . have the meanings they had in § 31 and let A,"B,"A.:'B.," ... be defined in terms of D,D,'D., ... by eq. (11) § 31. First let the bar have its ordinary meaning with regard to A^ ... B, ... C^ ... Ai ... B/ ... (',' • • . A/'B," .... Then by eq.(2) ^E = - ^A^EA - 'S.A'sEA' + tBsEB + 'S.B'sEB' - S {A"sEA" - B'sEB"). Now remember that D^, D/ are obtained from J./', B^" by a negative cii'cular variation. Change the meaning of the bar so as to refer to the motors A, ...B,...(\...A,' ...B; ...V: ...D,D^ .... By § 29 the new meanings of A, B, G, A', B', G' will be the same as the old, and the new D^ , D/ will be the same functions of the old Ai". B" that A, A' are of J./', B". In symbols D, = {A:' + 5/')/V2, A' = (^i" - B;')i^|±, new A = old (J," + 5,") \/2, new D,' = old (1," - ^/'),V2. 138 OCTONIONS. [§ 32 The last expression for -utE now gives ^E=- SZs^l + IBsEB - tUsED _\ -XA'sEA' + %B'sEB' -SDsEd] ^ ^^' Similarly from eq. (7) 7^_,E = -^AsEA+%BsEB-lDsED' ] - :^A'sEA' + IB'sEB' - %D'sED\ ^ ' ^' The first line of eq. (12) gives the value of t^E when E is con- fined to the given complex (m) or (ABCD), and the second line the value of 'sxE when E is confined to the semi-conjugate complex (6 - m) or {A'B'C'D'). Thus if E is confined to (m), stE is confined to {ABD'). The first line in eq. (13) gives the value of -ur^-^E when E is confined _to (ABD'). If E is confined to (6 - m), tsE is confined to {A'B'D). The second line in eq. (13) gives the value of ct_i^ 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 rsD = D', tjtD' = D yD' = D, ^_,D=D'] ^^^^' We may from the table of § 31 learn a good deal concerning the determinateness or arbitrariness of complexes involving A, B, &c. First however note the following (in which it . is to be under- stood that (E) and [F] are independent complexes, and in which [ ] is used to denote an indeterminate complex and ( ) a determinate one) : — The statement that [F] is any complex luhicJi tvith a given complex (E) makes up another given complex {EF) is exactly equivalent to the statement that [F] 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 tlie sum of the orders of [G] and (EF) is 6. The reciprocals {PG) 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], [P] may be made any such complex is thus seen. Let §32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 139 (1) [0] be any complex which with (G) makes up {PG)\ (2) [e] be any complex which is independent of (PO) and is such that the sum of the orders of the two = 6 ; (3) (e), [) is (EF). Thus [F] may be taken as [] ; and if it is, [P] becomes [^], 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), [P] 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 ; [P], {G), (GP) 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] be 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 [P] 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) {A BD A' BCD'), {ABDA'B'D')^ \ (ABA'B'D), (ABA'B'G'D'), {A'B'G'D')}- ...{lo). {A'B'D), {ABD), {D') ) From the second column w-e have that (0') is a complex which with {ABDA'B'D^l ' makes xxi)_{ABDA'B'C'D') {D) „ „ „ {ABA' BCD') makj.'s^up ( A BDA BCD' ) I {AB) „ „ „ (A'B'G'D') >...(10). niakes up {ABABU'D' (A'B') „ „ „ (ABD') makes xxyt (ABA'B'D) 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 ((?) 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 ST is an energy function there are (§ 31) no Fs or D's and these lists become (Determinate) (AA'C), {AA'), {A'G'), {A'), (Z)...(17), {G') is a complex which with {AA') makes up {AA'G')...{\^). The other statements_of (16) do not in this case require to be made, since {A) and {A') are definite complexes. If CT = 1 we have by equations (1) and (14) A = A, A' = A', B = -B, B' = -B', B=D', 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 {u). Thus {n) and (6 — n) are determinate complexes. Also let [n] 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 [n] but is otherwise arbitrary. Then A must be zero. For any motor can be expressed as E + F where F belongs to (6 — n). Hence = sEA=s{E + F)A, since F and A are reciprocal. Since E + F is, any motor what- ever it follows from § 14 that A =0. If (n) be the complex to which ^ reduces any motor, i.e. the complex (AB) of eq. (2). (n) and (6 - n) are determinate com- plexes characteristic of ot. In this case F being any motor of (6 - n), -stF^ 0. For if E is amj motor, -stE belongs to (/«) and F to (6 — n). Hence Q = sFt;tE=sEv7F, from which by § 14 it follows that -S7i'^=0. If any motor then be §32] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 141 expressed as a motor of [ii] + a motor of (fi - n), w only aft sE^e7sW f''-' °'- +''= *^'' ^'>- (III) If any two of a set of m motors forming a conjugate set of norms with regard to -sr are not already conjugate with regard to yfr they can be made so by a •nr-conjugate variation unless they be a pair a /9 which are a positive and negative norm respectively with regard to -nr, and are such that yfra = -v^r/S. (IV) In every such transformation the product sA,fA,...sC,y(rCr, involving all the motors which are not self-conjugate -mth regard to yfr, diminishes so that this product has a minimum value which is only attained when the set of t3--conjugate norms is expressed as a series of motors A^ ... Bi ... Gj .. Oj /8i, ... as described in the enunciation. 144 OCTONIONS. [§ 8r> (I) has already been proved in § 31. (II) Since yjr is an energy function we have that s (xE + yF) yjr (ccE + yF) = cc^sEfE + ^xysE^^F+ y'sF^|rF, is negative or zero for all values of cc and y. Also ^E-y^E and sFyfrF are each negative or zero. Hence sEfEsFfF ^ S'EfF. Now unless sEyJrE=sFylrF (sEylrE+ sFfFy- > 4>sEylrEsFyfrF. Hence either {sE^|rE + sFylrFT- > ^s'E^F or sEyjrE^ sFfF= ± sEfF. If the latter be the case we have s (^ + F)ylr (E + F)=0, so that by (I) 's{r{E + F) = 0. Hence either 1> r^^fK.^ >-^ or .lrE= + fF This proves (II). (III) If Ai and A.^ are not already conjugate with regard to -x/r they can be made so by the circular variation (which by § 31 is a -57- conjugate variation) A^' = A^ cos + A., sin 0, A.' = - A^ sin 6 -{- A., cos 6, tan 2^= - , , . ^ / , J, , sAiyjrA^-sA.sjrAz for can always be determined to satisfy this equation. Similarly if B^ and B. are not already conjugate with regard to -v/r they can be made so by a circular variation, and if C\ and Co are not already conjugate with regard to ^|r they can be made so by a circular variation. If A and C are not already conjugate with regard to yjr they can be made so by the linear variation (which by § 31 is a -sr- conjugate variation) A' = A + sG, C = G, s = - sA^|rG/sC^|rC, for since A and G are not -vlr-conjugate, yjrG is not zero, i.e. by (I) sGyjrG is not zero ; so that .9 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 ORDKH. 14;") If A and B arc not already conjugate with legard to ^ they can be made so except when ^A = ±<^B by the hyperbolic variation (which by § 31 is a ^-conjugate variation) A' = A cosh e^B sinh B, B' = A sinh B^B cosh 6*, tanh2^-- ^^^^~ %A/rC,. 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 -v/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 (Screivs, § 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 -nx and -v/r is m{m—l). Thus the total number of conditions necessarily satisfied by the 5??i 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 Q — m which is con- jugate with regard to i|r we can in general find only 6 — m definite motors in it which are conjugate with regard both to -or and -v/r. The total of six motors thus found will form a conjugate set with regard to -v|r but not in general with regard to -sr. 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 t/t. We cannot then in the present case hope to express both ct and i/r in a manner similar to eq. (12) § 32, where the A's, A"s, 5's and B"s have the same meanings for both -cr and yfr. Suppose however in addition to our present m motors A,...B,... C\ ...tti/Si ... (forming a conjugate set of norms with regard to ct) we take 6 -TO independent motors H^, U^...H^^m forming an indepen- dent complex of order 6 — m. For the sake of definiteness we may by I 31 suppose that this complex is conjugate to {in) with regard to ■^^ and that all six motors are conjugate with regard to -t/r. Now apply the bar introduced in § 28 above to these six motors. By the method of establishing eq. (2) § 32, we see that when E belongs to (m) ^E=- talsEA- thBsEB- ^cGsEG ] -itd_(d + ^)sE_{a + ^)_ ^E = -%{A+A')sEA + %iB + B')sEB - XC'sEG - S {(a + a') sEd - (fi -\- yS') sE^}j where A', B', C, a!, /3' all belong to the complex H^, H^..., i.e. to the complex reciprocal to {in). We might generalise the expres- sion for ^\rE by adding to the right of the equation — XhHsEH when it would be true for any value of E. Similarly the expres- sion for -uyE might be generalised by adding — IKsEH 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 ■(2), .(3), V .(4). ^'or •(o). §33] MOTORS AS MAGNITUDES OF THE FIHST ORDER. 147 the three types A, B, C of motors. Writing G tor any (not zero) scalar multiple of any one of them we get zyE=-l{fG+ G')sEG - 2 {(a_+ a') s Aa j -(^ + ^')sE^]-lKsEH } where G,, Go...ai,^i,(/.,... form the given complex (7?i)^ Hi, H..... form an independent complex (6 — ?)i) G-y', ... a/. /3/ ... belong to the reciprocal of (?7i) (ie.H^H,...) Ki, K. ... are any motors whatever gi, g»... di, d., ... are positive or zero scalars /,,/2 ... are any scalars whatever For the important case when )n = 6 there is no H, G', a' K, and we get the simplified form ^^E = - IgGsEG - ^^d (a + y8)_s^_(a + ^) 'stE=- 'S./GsEG - S (dsEa - ^sE/S) 6?!, Ga, ...«!, /9i, 0^2, ■•• six independent motors: gi, g2 ■■■ di, do ... positive or zero scalars ; f^,/., ... any scalars For application to the dynamics of a rigid body there is one very simple case. Suppose ts- and ^|r are both energy functions either both complete or both partial in the following way.— (1) If E be any motor- in a certain complex neither sE-yJrE nor sEvrE is zero ; (2) both ct and yjr reduce every motor they act on to a second complex of the same order as the first. The case of ct and -^ 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 ^ for ^ (a - /S) = 0, so that putting E = a- ^, sEyjrE is zero. No g can be zero, for if g^ = 0, sG^yfrG, = 0. G,,G,... are (§ 28) independent, and therefore form a complex of order m. This must be the second complex mentioned above, since any motor Sa(? of this complex is a motor obtained by operating by yfr on some value of E, viz., E = Xxg-^ G. Hence every h must be zero. Turning now to ■BT it follows that every/ must be positive and not zero and every G' and K must be zero. We have then ^|rE=-'S.gGsEG, ^E = -XfGsEG (6), 10—2 148 OCTONIONS. [§ 33 from which with the meanings of yjr^i and t3-_i of equations (7) and (8) of § 32 ylr^,E = -lg-'GsEG, ^^,E = -l/-'GsEG (7). From this again putting ^p-^iOT = (f) (f>E = yfr_,i;yE = - Ig-'fGsEG, 4>G, = gr\fiG, ,... 4>G,n = gm'^fmGm (8)- Also if F{(\>) be any algebraic function of <\> (such as <^^)- F{(l>) G, = F{gr'A) • ^1' &c., so that F{cj>)E = -^F(g-'f).GsEG (9). Another important case is when m = 6 and ot = 1. The motors conjugate with regard to zr are then reciprocal motors. It is in this case convenient to retain the -4, 5 notation. We saw in § 31 that the number of motors of any type A, B... was definite. Now when m = 6 and w = 1 there can be no G's, for otC = 0. Also all the motors in space belong to the complex of six reciprocal motors (1 ±pn)i, (1 ±p^)j, (1 ±P^) ^ of which three have positive and three negative pitch. It follows that the Aa group must be three in number and B^ group also three in number. Also as in § 32 I=A,B = -B. Hence t^= - S {aAsEA + bBsEB)-lld (a- /3) sE {a- ^)...(10), where A, B, a, ^ represent reciprocal motors such that sA"' = sar = - sB' = - s/32 = - 1. From eq. (10) ^|rA=aA, ylrB=-hB, i|r(a + /3) = (^(a-/3), ^(a-/3) = 0...(11). 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 ther^e 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), ylrE = - anisEm - bnjsEnj - cnksEnk (i3). 34. Scalar ftinctions 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 ORDER. 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) § lo] f(E) = -/(Z) sEZ = - sEA, where A =Zf{Z) (2). Let f{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) = -sE(t>F (3), where (^ is a general linear motor function of a motor. For f{E, F) = -f(Z, F) sEZ = - sE(f>F, where F = ZfiZ,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 in a 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) = -sE^E (5), where ct is a general self-conjugate, w being in fact the self-con- jugate part ^(4> + 4>') of ^. The consideration then of such homogeneous quadratic func- tions may be made to depend on that of cr. More particularly such a scalar function which is always positive or zero has properties depending on what we have (§ 30) called an energy function, partial or complete according as the scalar function is zero for some motors or for none. For instance (see § 30 above) in § 146 of Screios, Sir Robert Ball defines a motor complex of the (n — l)th order and second degree as consisting of all those motors of a complex of the nth. order which satisfy the ecjuation sE^E=0 (6). Thus they are those motors of the complex which are self-con- jugate with regard to ct. What Sir Robert Ball calls the polar of any motor E of the complex is simply what wo denote by r^E. 150 OCTONIONS. [§ 34- In § 92 of Screws he uses the word "central" to denote any motor of the complex which is conjugate to the whole complex. What in the same section he terras conjugate screws of the complex are simply motors which are conjugate with regard to CT. With the exception of the statement about the discriminant which will only appear in our work when we have considered the sextic satisfied by -or, all the theorems enunciated in § 92 of Screws are obvious from the above work. Most of them are not true in general for real motors, though they are for imaginary motors. A curious fact however is that just when the complex of the second degree itself becomes imaginary by reason of or becom- ing a complete energy function, the theorems he gives concerning sets of screws are true of real screws. An example of such a complex is examined in § 93 of Screws. We treat it here by the present methods as it serves as an illustration of some of the remarks above. The necessary and sufficient condition that a motor E should have its pitch equal to /t is by § 14 above that s^- = - 2hTi'E = 2hsEnE. Hence it belongs to the motor complex of the fifth order and second degree for which ct = 1 - 2/ifl. The polar nrE of E is therefore E — 2hQ.E, i.e. it is the coaxial motor with the same tensor as E and with pitch equal to that of E diminished by 2h. If E belongs to the complex so that sE^E = 0, the polar is thus the reciprocal coaxial motor of equal tensor. When E is any motor, sE^E is negative or positive according as the pitch of E is greater or less than h, so that ur is not an energy function. On the other hand since it is commutative with n, it is a commutative function. Two coaxial motors are conjugate with regard to -sy if the arithmetic mean of their pitches is h. If then we take three such pairs with axes on three mutually perpendicular intersecting lines they will form a conjugate system with regard to ot. In this case then we may put A, = a-'[l + il {h + ^a')] i, B, = a" { 1 4- H (/t - ha')] i, A, = h-'[l + n(h + ^¥)] j, B, = 6-1 [1 +n{h- ^¥)] j, A, = C-' [1 + n {h + hy)} k, B, = c- [1 + a (// - }jc')] k. B,= -(/-> ;i-n(A + ^a-)|'i, B,= -b-^{\-n(h + ^¥)}j, B.^ -C-' {i-n(h + hy-)]k. §34] MOTORS AS MAGNITUDES OF THE FIHST ORDER. 151 Ai is reciprocal to A.,, A.^, 5,, B.,, i?.,, and is such that S^,Z, =- 1. Hence A, = a-'{l-n{li-^a'}\i, A, = b-'{l-n{h-hb')]j, Equations (2) and (5) § 32 give -orE = — AiSEAi — A^sEA. - AsSEA^ + B.sEB, + B,sEB, + B,sEB„ vr-'E= - A.sEA, - A.^EA, - A,sEA, + B.sEB, + B,sEB, + B.sEB,. Put a = h = c. Suppose E = 2a-^ + 2y5. Then is the necessary and sufficient condition that E should belong to the complex sEnrE = 0. If we examine the geometrical meaning of this we shall find that : — Any motor in space ivhose pitch is h is the resultant of two motors through a given point luhose tensors are equal, and the arithmetic mean of whose pitches is h; and con- versely any motor which is the resultant of two such motors has h for pitch. The following considerations are of importance in connection with complexes of the second degree. Suppose that for every motor E of the complex {m) or {ABCJD) of eq. (12) §32, sEtjjE = 0. Every A and B must be zero (since - sA-ctA = sB-stB=1). But the G's and B's are conjugate to the whole complex. Hence if E and F are any two motors of the complex sE'!:7F=0. [This may be proved directly from eq. 12 §32. For we may put E = S.t6' + XyD, F = S*-'6' + ^y'D, so that 7zF='%y'D'. Now every D' is by § 28 reciprocal to every D. Hence sJ£'t!ri^=0.] Putting now for ct in this statement ct — tir' where tn- and ct' are two self-conjugates we get the following theorem : — //' sETffE=sE^'E for every motor E belonging to a given complex, then sE-otF = sEsr'F when E and F are any two motors of the complex. In particular: — 152 OCTONIONS. [§ 3-i If sE'ujE = sE-ar'E for every motor E belonging to a given complex, then any set of motors in the complex ivhich are con- jugate with regard to tjt are also conjugate with regard to is-'. Let (m) and (6 — rn) be two given independent complexes of orders m and 6 — m respectively, and let {6—m) and (m) be the reciprocals of (m) and (6 — m) respectively. If -UT he a given self-conjugate, a self-conjugate ot' can he uniquely determined so that sE^zr'E^ = sEi-syEi , zr'E. = ; where El is any motor belonging to (m) and Eo, any motor belonging to (6 — m). Also -st'E belongs to (m) tvhatever motor value E have. Let Gi...Gm be m independent -^-conjugate motors of (m) and Hi ... ^c-m, 6 — m independent -sr-conjugate motors of (6 — m). Using the bar introduced in § 28 with reference to these six motors, (m) will be the complex of Gi...Gm, (6-m) that of IIi...II,_,n, (m) that of ^1 ... G,n and {6-m) that of H^ ...H,_,,,. By the method of establishing equations (1) and (2) § 32, we have ■^Gi ^giGi + Gi', . . . TsG.,, = gmG,a + G,n', ^H, = hjli + H^, . . . where gw- gm, fh, •■• /'e-m are ordinary scalars, and where (r/. . . GJ belong to (6 - m) and H^ ... II\_m to {m). Thus isE = -^(gG-\- G') sEG - S {hH +H') sEH. Define -ot' by the equation is'E = -lgGsEG, so that vr'Gi = gA, i!T'G, = gM,,---, i^'Hi = ^'H, = ... ^0, -57' will then satisfy the conditions of the enunciation. For with this definition it is evident that -ot' is self-conjugate, that 'st'E2 = and that nr'E is confined to (?»). Also since G/ belongs to (6 — m), sEiZTGi=giBE^Gi-=sEi'UT'Gi, and similarly for Gz-.-Gm, fi'oni which it follows that si57,-sr£'i = s^j-ar'^i. Lastly no other self-conjugate ot' + OTo satisfies the conditions of ot' in the enunciation. For if it did we should have sE^tjt^^Ei = 0, -5ro£'o = 0, from which s(Ei-\-E,)i!Jo(Ei-i-E,) = 0, or sEtjToE = for all motor values of E. From the general form of a self-conjugate given in equations (1) and (2) §32, it follows that OT|, — 0. § So] MOTORS AS MAGNITUDES OF TUK FUIST ORDER. 15:3 Putting ct' = CT — ct" we have that ct" can be determined uniquely to satisfy the conditions sEiZt"Ei = 0, 'st"E., = -srE-i. Also from the forms given above for -bt and ■sr' we see that -ct"^j is confined to (6 — vi). Changing here m into 6 — m, and therefore interchanging E^ and E.,, and writing ct' for the present ct", we get the theorem: — If-uT be a given self -conjugate, a self -conjugate ur' can be uniquely determined, so that ■st'E-^^'stE^, sE.2'st'E.^ = 0, where E^ is any motor belonging to (m) and E.^ any motor belonging to (6 — m). Also -ut'Eo, is confined to (/»). As a matter of fact when -cr is given as above we shall have -ut'E = -1{gG+ Q') sEG - XH'sEH. [When VT as given above is self-conjugate, ot' as given by this equation is also self-conjugate, since (-57 — ^')E= — ^hHsEH.] From these two theorems it is easy to see that -cr' can be determined in an infinite number of ways to satisfy the two conditions s£'iot'£'i = s^^ot^i, sE.,tjt'Eo = 0. 35. The general function when not necessarily self- conjugate. Let ^ be a general linear motor function of a motor. In the discussion of the properties of (f> we start by practically reproducing for our own case § 390 of Ausd. There are two or three reasons for here reproducing that section. It has been seen that many of Grassmann's theorems are not applicable to our case, and it is perhaps just as easy to reproduce the proof as to show by general reasoning why this particular proof is valid. Moreover, Grassmann's theorem if immediately applied would not give quite so general a result as we seek, as it would only apply to the case where n below = 6. Moreover, the theorem is of such fundamental importance that this alone is sufficient reason for giving a proof here. Let (n) be a complex of the nth order and (G — n) some inde- pendent complex of order 6 — n. Let at first J-i ... An be any n independent motors of (n) and let ffn+i . . ■ Hg he 6 — n independent motors of (6 — n). Suppose (ft operating on any motor of (n) reduces it to a motor of (n). The (f) of equation (8) § 33 above is an example of such a function. 154 OCTONIONS. [§ 35 Consider the equation of the nth. order in x, [{j>-x)A,...{<^-x)AnHn+....H,\==0 (1). In the first place, if ^^ be changed to any other motor XiA^ + ... +XnAn of (?i) which is independent of An,.. An (so that x-^ is not zero), the equation is unaltered. In fact the two expressions [{-x)A,...{-x)AnHn+....H,], and [A^.. AnIIn+i...Hs] are both thereby merely altered in the ratio of x^ to 1. We may thus change A^.-.A^ to any n independent motors B^...Bn of (n) and we shall have {{-x)A,...((f>-x)AnHn+i'-H, \ \A,...Ar,Hn^,.,.H,] _ [{4> -x)B,...{4>-x)BnHn^,...H,] ^2). {^1 . . . BnHn+1 ' • • He} [We may clearly on the right further change Hn+i ...Ha to Gn+i ... ^6 any other 6 — n independent motors of (G — n), but this is unnecessary for our purpose.] Let the n roots of (1) be a repeated ?' times, b repeated s times, c repeated t times, &c. Thus {(x-cf>)A^...(x-) AnHn+i ...Hs] (x — ay (x — by where {A,...AnH^^,...H,\ 1' «"-!+ . . . + (-y-^m'x + (-)"m ..AnHn+,...H,\ "- [A. ..A^Hn+,...H,} .. 2{^. ...ApAp^,...AnHn+^...H,] \A,...AnHn^,...Ha] ,.(3). •(4), where the summation sign implies that while Ay ... An are always written in the same order the ^'s are to be applied to every possible set of 7i—p out of the n motors A^... An- From the identity (2) it follows that the expressions on the right of equations (4) are the same whatever independent motors Ai...An be of the complex (n). This can also be shown inde- pendently. § 85] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 155 The theorem in §390 of Ausd. may be thus enunciated. — In the complex {n), r + s + t ... (= «) independent motors A,...ArB,...BsC,...Ct... can be found such that (f)A^ = aAi, Ar = aA,. + A," cf>B, = bB„ B, = hB, + B:, . . . (f>B, = hB, + B: ■ ..(5), luhere Ap stands for some motor of the complex A^, A.^ . • • Ap^i and similarly for Bp, d-c. To prove this, § 66 of the Ausd. will be required. In our case it asserts that if for six motors D^ ... i),; {A...A} = o, Di ... Dg are not independent. [If they are independent the primitive units E^...E^ can be expressed in terms of them. Expressing these units in this form we get [Ei ... E^ = [D^ ... Dgj X a finite scalar = 0. But by § 28 {E^ ... E^] is not zero.] Since a is a root of eq. (1) we have {(<^ -a)A,...(-a) AnH,,+, . . . H,] = 0. Hence some relation of the form %{-a)A+ 2 zH=0 must hold, where all the ^'s and zs are not zero. By hypothesis ^y{ — a) A belongs to {n) and IzH to (6 — n). Hence these two motors must separately vanish. Since Hn+i...He are in- dependent, all the ^'s are zero and therefore Si/{ — x)Ai = (a — x) Ai, {(f) —x) A.i = {a -x) Ai + A2, ... ( - x) Ap+, ...{-x) AnHn+, ...H,}=0. But the root a occurs more than p times. Hence {A,...Ap{(ji-a) Ap+, ...((t>- a) AnHn+i ...H,]=0. Hence some relation of tiie form iyA -\-'iz{4>-a)A-^l,wH = 1 V+\ M+l holds, where the ya, zs> and w's are not all zero. As before the terms in H separately vanish, so that %jA^- S^(<^-a)^=0, where the ?/'s and ^'s are not all zero. Further the zs, are not all zero, for otherwise A-^... Ap would not be independent. Suppose ^p+i is not zero. Then instead of Ap^i we may take % zA. We then have V { - a) Ap+, = - Xi/A = Ap+„ where A'p+i belongs to the complex A^ ... Ap. From this it follows that r independent motors A^ ... A,- can be found in (n) to satisfy eq. (5). Similarly s independent motors Bi... Bg and t independent motors C^.-.C), &c., can be found to satisfy eq. (5). It remains to prove that these n motors are all independent of one another; i.e. for instance that not only are Gi ... Ct, t independent motors, but these t motors are also inde- pendent of the r + s motors A^ ... Ay B^ ... B^. We shall not prove this last in the most direct way, but shall adopt a process that gives us incidentally two other useful pro- positions. ((l> — ay acting on any motor of the complex A■^ ... Ap reduces it to zero, p being less or equal to r. For (^ — a)-4i=0. A.! belongs to the complex A^. Hence (^-a).4/ = 0. But {(p ~a)A.= A.:. Hence (-cty operating on any motor of the complex A^, A.. reduces it to zero. Hence (^ - ay ^Z = 0. But (<^ - a) A^ = A^'. Hence (<^ - af As = 0. The proposition is now obvious. If p be any positive integer and e any scalar different from a, (^ — ey A is not zero and belongs to the complex A^... Ar whenever A is any motor belonging to the complex A^... A,. Note first that {(^-e) A must belong to the complex A-^... A,, by eq. (5). Also {^-e) A is not zero. For suppose A =x^A^-k- ... +X1A1, where xi is the x with highest suffix which is not zero. Then from eq. (5) {j)-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^...Ay reduces it to a non- evanescent motor of the same complex. It follows that (<^ - ey A is a motor of ^1 ... A,, not zero. In particular (^ - by /? = if jB 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 5i . . . Bg are independent. Similarly all the complexes (^1 ... Ar) (A ••• Bs){C\ ... Ct) ... are independent of one another, i.e. the ?• + s + ^ + . . . motors A^ ... B^ ... C\ ... are independent. Let now E be any motor of the complex A^ ... ArBi ... 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 (^(j) _ ey E is a non-evanescent motor belonging to the complex A^... ArBi...Bg: for ((f>- e^A is not zero and belongs to Ai...Ar and ( — aY(A+ E) = 0. Hence E must be zero or the motor A + E must belong to the definite complex hitherto denoted by ^i ... A^. If ((f> — e)P E = for every motor E of a complex included in (n) of order q, e must be a root of (I) repeated at least q times and the complex must be included in the complex corresponding to this root. For we have seen that unless e is a root (<^ — e)^ E is not zero for any motor E of (n). We may then assume that e = a. In this case ((f>— ey E is zero only if E belongs to the complex Ai...Ar. Hence the complex must be included in Aj^...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 (n) can be expressed as A + B + C + ... where A belongs to the complex corresponding to the root a, B to that corresponding to b, and so on. Now we have seen that ( - aY A = 0, (cf) -by B = 0, &c. It follows that (-by{(f)-cy ...E = o. We may here as usual leave the operand E out, understanding that it is to be restricted to the given complex (n). Thus by the identity (3) we find that + (-)" m=0 (6). When w = 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 w-tic satisfied by (f> we can always find the complexes corresponding to these roots by the following simple property, (cf) — by {

) be any algebraic function of (f) we see by equation (5) that /(0) A, =f{a) A, \ f{^) A2 =f(a) A2 + a motor of the complex A^, f ,_, /(0) As =f(a) ^3 + a motor of the complex A^, AJ'" But putting f{(f>) = ((/)- by ( - c)« . . . , /(a) is not zero. Thus in this case f(E = -(f>A ,sEA, - (f>A „sEA, - ... Hence (f)E = -a [A^sEA^ + (a^.A, + A.^ sEA., + ... + {ar,A, + ... + a, .r-^A,_, + Ar)sE%] - b \ ]- (2). Hence if 'Ai = a (A-i + a^iA^ + . . . + anAr)\ L 'A, = air [§36 .(3). The symmetry of the coefficients in (1) and (3) may be noticed. The expression for (f)'E may be written (f)'E =-a l(2i + aaiZo + ...+ anAr) ^EA^ + . . . + (l,.-i + ar.r-iA,) SEAr-i + ArSEA,} - (4). These show (a) That there is a complex (n) of order n and a complex (6 — n) of order 6 — n standing towards ' as the complexes (n) and (6 — 7i) stand towards (f). (n) is the reciprocal of (6 — n) and (6 — n) is the reciprocal of (?i). (b) That the <^' ?i-tic is the same as the ^ ?i-tic, since they have the same roots repeated the same number of times. (c) That if the complexes corresponding to the different roots are reciprocal to one another and to (6 — n). [For since ^' = ^ the complex A^ ... A,, corre- sponding to the root a of the cf) n-tic is the same as the complex ^1 ... Ay corresponding to the root a of the ' n-tic] (d) That if cf) be self-conjugate every motor in the complex corresponding to one root is conjugate with regard to <}> 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 ?i-tic is zero. Then 0"~i E where E is any motor of (n) has a definite meaning, namely, that one motor belonging to (n) which when operated on by -'A, = a-^A^, i-4,). ^37] MOTORS AS MACiNITFDES ()V THE FIRST OPDEI}. Ifil Proceeding in this way we obtain -'A„ = a"{-a,,A, + A.^ -'A,= cr'A, (f>-^A. = a-" (- A.^ + aAo) ^-1^ ., = 0-- {choA: - ^3' + a^z) [- (0). ^-^^4 = a-" [- (043^32 - f'42) ^2 + «43^3' - -4/ + aAi] From these we see (a) That ^~' is a linear motor function of a motor. (6) That the roots of the ^-^ /?-tic are the reciprocals of the roots of the n-tic. (c) That the complex corresponding to any root of the ^~' ?i-tic is the same as the complex corresponding to the correspond- ing root of the n-tic. (d) That <^'~i is the conjugate of ^-'. 37. Some properties of self-conjugate ftinctions 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 wdiich it always ( UNIVERSITY 162 OCTONIONS. [§ 37 satisfies are reciprocal to one another. It is not true however that for a real self-conjugate ot they are always real. For instance, put zyE = x (nisEm - isEi) + y (D-jsEnj -jsEj) + z{nksEm'-HEk) (1), where cc, y, z are real ordinary scalars. Here Tsi = — xfli, tjtCH = xi. Hence if v be put for the imaginary V(— 1); we have -nr {i ± vfli) = ± vx (i ± vHi). Hence vx and - vx are roots of the ot sextic, and the complexes corresponding to them are those of i + vfli and i — vni. The sextic is {^'' + x')(7;T-' + y')(^' + z"-) = (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, fli A similar statement may be made of the general real (f>. 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, a real complex of the fourth order for each motor E of which [{<\>-aJ^a"'']-E = 0, and so on ; and that for every motor £" of the complex of the 2rth order, {( - a')- + a"'}'' E = 0. Returning now to eq. (2) we see that there is no motor E in this case for which ^E = 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 {Ausd., § 391) in that the sextic may have imaginary roots. In the present properties it is the energy ^ :37] MOTORS AS MACJXITUDES OF THE I'lHST OHDEK. M)'^ function yjr which is the complete anah)gne 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 § 3o the root corresponding to A is a, to B, — b, and to a + /3 and a — j8 zero twice repeated. This last is seen by comparing the equations with eq. (o) §35. We can clearly identify the present « — /3, a + ^, zero, with the A^, 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 A^ and A., for which ct^i = aAi, 'ctAo = aA.. This is illustrated by the self-conjugate commutative function. The general form of this is given by eq. (1) § 19 above. Changing the (f) of that equation to ot we have ^E = - XiSiE - X'jSjE - X"kSkE (3), where X, X', X" are scalar octonions. Put X = a; + nrj, X' = x + %', X" = x" + D.ij", where x, y, &c. are ordinary scalars. Then Tjri = (,r -}- D.y) i, -sTQi = xD.i. We may therefore identify the present Hi, i, x Avith the A^, Ao, a of eq. (5) § 35. Hence by § 35 x is a root of the ct sextic twice repeated. The sextic is (ot - .rV- (ct - x'Y (w - x")- = (4). Also the complex corresponding to the root x is that of fit, i. There is one motor E, viz. Cli (and scalar multiples of it) for which (f)E = xE, but there is no other (except Avhen y = 0) as we can see by trying E = ai -f bm. 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. [§ 37 It is easy to express the U7 of eq. (8) in the form -lAsEA + lBsEB [eq. (2) § 32]. When a- is not zero we have - XiSEi = - A.sEA, + B.sEB, (5), where A^ = cr^x-^ {x + ^n (-y + a-)} i B, = a-'x-'{x+m{-y-a')]i\ ^^^' Ij = a-' {x + ^n(y + a')} i | ^ B, = -a-'{x+^n(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 Ai and ^1 are the most general values for two conjugate norms in the complex {, m corresponding to the repeated root x of the sextic. If a; = 0, we have -XiSEi=-ynisEni (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 C\ are respectively a negative and zero norm, where A = V2/.ni c, = i j ^ ^' Bj = i/s/(-y), C\ = nil B,=^s/{-y)Mi, G, = i \ We get the general expressions for the norms in the complex i, Hi in the ease of e(i. (9) by the linear variation A,' = A,+ pQ^Jy = (1 + £2^9) ijs/y, C,' = C, -. ^ .^^^^ A,' = A, = sjym, a; = - pAJs/y + Oj = (1 -pQ.) i) 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 -57 of eq. (3) has been expressed in the form — l.AsEA + 2BsEB. 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. §87] 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 later function. [For unless x = oc' = x" = there are B'h.] 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 -crAi = .vAj where x is an ordinary scalar. But we have just seen that except when y = y' = y" = there are not six independent motors for which this equation is true with the present ct. It may be noticed that the ot considered at the end of § 84 above is a function of the kind now under consideration for which {V = x' = x" = 1, u = y' = y" = — 2/'. The sextic is therefore In the present case (ot — 1)'- reduces every motor it acts on to zero. The statement made about the roots of the or sextic occurring in pairs is true of any commutative function. In fact, the sextic of a commutative <^ is what in § 17 was called the ^i cubic squared and equated to zero. For let A, B, C he any three completely independent axial motors (§ 18). If the 0i cubic is ^i" — «"]■- + »'<^i — n = 0, we have by in\. (!) § 17 and v([. (27) § IG, a-i - m'V -1- n'x -n = S, {.r - ) B (./; - (/>) (^Sr'ABC. By e(|uations (1) and (6) § 35, the sextic is (^^ — iii'c^-' + ///"^■* — in"'^^ f m"(^- — in'^ + in — 0, where a^ — VI' x^ + ... + m ^ {{x-)A(x-(f>)B{ .r - c}i)C(x-cl>)riA{x-cl>)nB{x-ci>)nC} [ABCilAilBilC\ 166 OCTONIONS. [§ o7 Hence by eq. (27) § 28 of — tii^x^ + . . . + 111 = («" — '}("x- + n'x — }if, or ^^ — m'(^^ + in"<^^ — m'"^" + m"^- — ni'(j) + m = {(!>■■•' -)i '(f)' + )i(j)-nr (12), which proves the proposition. It may be noticed that in this case the Ai,Bi ... of eq. (5) § 85 may invariably be taken as lators, for if a is a root of the sextic we have by what has just been shown Si ( - a) B ( — a) A, (-«) B, (^ — a) C are parallel to one plane or one is a lator. In either case n (<^ - a) {yA +zB + luC) - 0, for some ordinar}^ scalar values of y, 2, w not all zero. Putting then E = n {yA + zB + wC), (f)E = ciE, or E may be taken as A•^. 38. Another method of dealing with self-conjugate commutative functions. We give a sketch here of a method of dealing with commutative self-conjugates suggested by the methods above used for the general self-conjugate. A commutative combinatorial product of two motors A and B is one which is commutative with fl. Thus yfr {A, B) is a com- mutative combinatorial product of A and B if it is a commutative linear function of both A and B such that -^ (A, B)=- ^jr {B, A). A simple commutative combinatorial variation of any group of motors J-i ...An is one in which two of them Ap, A,^ are replaced by Ap, Aq where Ap = cAp + sA,^, Arj = a'Ap + cAq (1), where c, s, s' are any scalar octonions which satisfy the equation C--SS' = 1 (2). Similarly for a negative and a multiple comnmtative combinatorial variation. If c = 1 and therefore either s or 6-' is zero or both are con- vertors the variation is called linear; if s' = — 6- it is called circular; and if s' = s it is called hyperbolic. For a circular variation we may put c = cos e - rtr sin 6, s = - s' = sin $ + Or cos 6? (3), §38] MOTORS AS MAGNLl'UDES OF THE FIRST ORlJFR. 1G7 where r is any ordinary scalai-. Thus c- + s- = 1 j c^-s- = co»2d-n2rsin20r (4), 2cs = sin 26 + il2r cos 26) so that c' — s- and 2cs are obtained from c and s by changing /•, 6 into 2r, 26. For a hyperbolic variation we may put c = cosh 6 + fir sinh 6] s' = s = sinh 6 + fir cosh 6\ .(5), from which c- — s- = 1 ] c- + s- = cosh2^ + n2rsinh2^[ (6). 2gs = sinh 26 + n2r cosh 26) Thus cr + s- and 2c6- are obtained from c and s by changing ?•, ^ into 2r, 2^. Circular variation may also be defined by the equations A,'^{- YA, + XA,)/^{X-^+Y"-) (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; = {xa,+ ya,)/,/{X'^-y% A,; = (YA, + XA,)!^{X'^-Y^) (8), where X and Y are any two scalar octonions for which the ordinary scalar of X is numerically greater than that of Y. 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 ot be a commutative self- conjugate function and \{ A, B be varied to A', B' QA''syA'SB'-:^B' - S^A'-utB' = SA-^^ASB-utB - S'A^B. . .Q)), from which by taking the ordinary scalar part S.^VA'S.fi'T^B - SM'^^' = S.AtjtAS.BvtB - SM^Ji- ■ -(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 fidly conjugate with regard ti) -sr when S^ot5 = (11). 168 OCTONIONS. [§ 88 Thus they are conjugate in the ordinary sense and are also such that Si^ot£ = 0. If A and B are not already fidUj conjugate tuith regard to ot they can invariably be made so by a commutative circular va7'iation, and in this case the product SiA-srASiB-uyB diminishes except ivhen S^A'utB = tuhen the product remains unaltered. Three completely independent axial motors can always be found which arefidly conjugate with regard to ot. For by the commuta- tive circular variation just mentioned we can go on diminishing the product S^A'stAS^BsrBS^CtsC unless S,B^C= Sfi-ujA = S.A^B = 0. In this case if we vary ^, J5 to A', B' by a circular variation so as to make SA'-stB' = we shall find that A' = A + flrB, B' = — O^rA + B, and therefore that SA'vTC = SAinC, SB'^C = SButC. Hence in this case we can make each pair conjugate without affecting the conjugacy or otherwise of the others. If SA^A = - 1, SB-syB = 1, S^o^-^o = - H, SBo^B, = n, SC^C = (12), where A, B, A^, B^, G are axial motors, the motors will be called norms of types A, B, Ao, Bo, C respectively. A set of three real norms fully conjugate with regard to ct can always be found. The norms of types QA^, H^o, C, ClC form a definite complex. The norms of types QA, flB, A„ nA^, B^, nJ5o, G, HC, also form a definite complex. -otAq and htB^ are lators and -srC* = 0. By the axial complex of any number of motors A,B... is meant the complex of ^, Hxl, B, CIB.... 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 5„ in a set of conjugate norms it belongs to the complex of H J.o or D,Bo and the axial com- plex of the other norms. [Since every motor is fully conjugate to a G we have no corresponding property for a motor which is fully conjugate to a C] §88] McrroKs as magnitude.s of the fiust urdeu. Kil) A motor of the form )LXA +^YB+ SXc4o + SFoi^o + ^^C where X...Z are scalar octonions cannot form one of a set of conjugate norms unless (1) SZ"^- SF"^ + n(SZ„^- SFoO= ± 1 ; or (2) all the S.X's and S.Y's are zero and S, (SXo-- SF,r) = ± 1; or (3) all the X's, Y's, SiXo's and SiF„'s are zero. What is here meant by a ^-conjugate variation may be thus explained. A pair of zero norms C\ and C, may be varied to any two completely independent motors C\', CV belonging to the axial complex of C, and G., i.e. they can be varied (§ 14) to any two not parallel axial motors intersecting the shortest distance of Cj and C, perpendicularly. Any other pair of norms E and F are varied according to the equations E' = cE + sF, F' = s'E ^cF, \ c2 -ss =\, s'SE-^E + sSF^F =0] ' ^' A multiple conjugate variation consists of a series of simple variations. Any motor luhich satisfies one of the three conditions mentioned in the last enunciation can be 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 cliaracteristic of the function -or. 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 w and an ellipsoidal function.] Then we have: — If A, B, C be any three completely independent axial motors luhich are fully con- jugate loith regard to -or, i.e. are such that SBxffC = SC^A = SAi;tB = : the numbers of the scalar octonions SAxsA, SBt^B and SCwC which are (1) positive scalar octonions, (2) negative scalar octonions, (3) positive convertors, (4) negative convertors, (5) zero, are the same as the numbers of the principal roots of the ct cubic of the same types. [This connection with the principal roots is obvious from lYO OCTONIONS. [| 38 the fact that i, j, k of eq. (1) of § 19 are fully conjugate with regard to the (f) 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 ot imply geometrically, take the corresponding quaternion case. VT being a self-conjugate linear vector function of a vector we have, rejecting the case •57 = 0, nine types of •sr 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 OT whose sextic was considered in | 37. If i/r is a commutative self- conjugate function such that S^E^^E is negative and not zero for every axial motor E in space, we will call t/t an ellipsoidal function. [We only require this not very appropriate name temporarily.] Thus as a particular case -v/r may be put equal to unity. A set of tJiree real completebj independent axial motors can invariahly he found wJdch are fully conjugate with regard both to tjT any commutative self -conjugate and yfr an ellipsoidal function. The axes of these motors are in general unique. Putting -v/r = 1, we get : — A set of three mutually perpendicular intersecting rotors can always he found which are fully conjugate with regard to ct. These are of course the i,j, k of eq. (1) § 19. If i)i> A', Ai are three conjugate norms with regard to -cr, and if the bar have the meaning with regard to these three that it §39] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 171 had with reference to A, A', A" in § IG (not the meaning it had with reference to six motors in §2H), the term in -syE corresponding to A k_-ASEA; to B, BQEB : to A„ -ilA.SEA,] to 5,,, nSoSEBo ; to C, zero. Thus ^E=%{±DSED) + nX{±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 ylrE = - A.SEA, - A.JSEA, - A,SEA, (15). For the common fully-conjugate system of ot and -v/r we shall have equation (15) and also ^E = - a,A,S^A, - ajL.JSEA, - a.A.JSEA, (16), where tfj, a.,, O3 are any scalar octonions. From eq. (15) we have always ^|r-'E=-A,SEA,-A.JSEA,-^3SEA, (17), and from ec[. (16) when the ordinary scalars of «i, a.,, a-, are not any of them zero ^-^E = - ar'A.SEA, - a.r'A3EA, - a,-'A,SEA,..\\d,). Expressing equations (15) and (16) in full by means of § 16 in terms of the three conjugate norms (w^ith regard to a/t) A^, A.,, A-. we have ■s\rES'A,A.A, = - lAAAJSEA.A, - \AA,A,QEA,A, - MA,A.SEA,A,. . .(19), ^ES'A.A.A., =-a,^AA3EA.,A,- aMA,A,SEA,A,- a}AA,A.BEA,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 j>E be a general linear motor function of a mntur E and -\^{E,F) be any octonion function of the motors E, F, linear in the general sense in each, y^{Z, (f)Z) = f{(P'Z, Z) (1), 172 OCTONIONS. [§ 39 where Z has the meaning detiued in eq. (6) § 15. For ylr((l)'Z,Z) = -f(Z,sZ(f)Z,,Z) [eq. (10) § 15] = -f{Z„ZsZZ,) [eq.(8)§15]. Particular cases of eq. (1) are MZ(I>Z = - MZcfi'Z, SZ(f>Z = SZ(f>'Z (2). When (f)E = - XBsEA, we have (f)E=-^BsEA, s^riZ, (f>Z) = 'S.f(A, B) (3), for r/r {Z, Z) = -^\r (Z, ^BsAZ) = - S^ {ZsAZ, B) = Xy}r {A, B). By eq. (21) §28 (}>E = - (f>A,sEA, - ... - <})AsSEA,. (4), where A^ ... Aq are any six independent motors ; or E=-B,sEA,-...-B,sEAe (5), where _ _ _ B, = cf>A„ B, = (f>A,, ...Be = (f>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 Ai ... A^ to be indejDendent so that [A^ . . . Ag} is not zero. The equation {cf>A,...c},A,} = m{A,...A,} (7), gives a value for m independent of the particular values of J.i . . . A^: for if we change A^ to any motor x^A^ + ... + x^A^ (x^ not zero) independent of A^.-.A^, both {(f)A^ ... (pA^} and {A■^...AQ] 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^ ... As are zero.] In particular, {Z,...Z^[cl>Z,...Z,\ = m{Z,...Zs\{Z,...Ze\ (8), {Z,...Z,} {cl,E4>Z,... Z,] = m [Z, ... Z,] [EZ, ... Z,] ...(9). The Z's may be eliminated from the right of these equations. To do this, first note that by eq. (3), {Z, ... Z,]{cj>Z, ... cfiZs] = Ql {A, ... A,}{B, ... Bs] (10), \Z,...Z,][Z,\ = 51 S 1^, ... A,]\(f>EB,... B,l..(l\), §39] MOTORS AS MAGNITUDES OF THE I'llJST oHDEU. 173 there being six terms under the siuntnation sign, obtained by omitting A,, A.,... A,-, successively. A particular case of the last two equations is obtained by putting i, j, k, m, nj, flk for A^...A^ and Hi, Hj, flk, i,j, k for J5i...5«. Thus Z,...(^Ze} = -6!m (14), \Z,...Z,]{^E<\>Z,...<^Z,\= o\E (15), or, eliminating m, E[Z,... Z,][Z,... Z,] = - 6 {Z, ...Z,}{cf,Ecf>Z, ... cf>Z,]...(lQ), which gives E explicitly in terms of ^-E", i.e. gives, except when m is zero, the reciprocal of ~^ of . Again, from equations (10) and (14) m = -{A,...A,]{B,...B,} (17), mE=%{A,...Ae}{cf>EB,...B,} (18), from which again E{A,...Ae]{B,...Bs] = -t {A,. ..Ae}{EB,...Bl.. (19). When m is not zero {B^.-.B^} is not zero. In this case we have by equations (12) and (20) § 28, E = - A,S(f>EB, - ... - A,S(f)EB, (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 di of (f)' is the same as the m of 0, where 'A,...'A,] = m\A,...A,] (21), or by eq. (12) § 28, sAs'A,...(ji'A,] = 77isAa{A, ...Ai\. 174 OCTONIONS. [§39 Hence by § 14, (f> {'A,} = m {A, ... A,] (22), or -'{A,...A,},=m-'{(f>'A,...ct>'A,:, (23), which shows that to obtain (fi~^E we have only to determine A^...Ar^ so that E = {A-^ ... A^], and this can always be done explicitly in an infinite number of ways. By a similar process we have (f>'-'{A,...A,] = w-i{^i...(^^5} (24). The -E. [Z, ... Zy [Z, ...Z,cf>Zn^,... 4>Z,] I E,: = sZ.rE . [Z, . . . Z,] [Z,... Z,4Z,,^, . . . 4>Z4 j • • -^ ^• We proceed to show that E, = nE,' + {Q-n)E'n^, (26). By eq. (19) § 28 E,, = {Z,... Z,Z,\ [( [Z,... ^4 ^Z,4>-E -... ±[Z,...Zn-,Zn+,...Z,]sZ,,4>^^E) + (+ {Z, . . . ZnZ,^, ...Z,] sZ,^,-E)]. By interchange of suffixes we see that each of the first w terms is equal to the first, and each of the last 6 — 71 to the last. Hence L\ = nE,: + (6 - n) sZ,(f>>'E .{Z,... Z,} {4>Z,Z, . . . Z.^Z,^, . . . 4>Z,}, where (jiZ^ has been shifted five places and therefore the sign has been changed. Now by eq. (1) change Z^, (f>Ze into ^'Z^, Z^, write s.^6^""^^^ instead of S(/)'Zs0"£', 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^ = QE,', E^ = 6E,'. Also En' = E,,/n-E'n+,{6-n)/n. Putting n=l, 2 ... 5 successively r-aF' ^F '^-^F'-^F 5.6 4.5.6 E,-6E,=-E,-^ ^^-i^^-r:2^^+"r.2 ^^=' = . . . = e^", - 15^„ + 20Es - 15^4 + 6^5 - E„ or E, - GE, + 1dE,- 20E, + loE, -QE, + E,= (27). Substituting for the ^'s from eq. (25) we get the sextic (f)'' — m"^' + 7/^"Ze} l]5lm' = -{Z,...Z,]{Z,Z,...cf>Z,] .(29). p ! (Q-p)': m^P^ = -[Z,... Z,] [Z, . . . Z,,Z,^, . . . <^Z.,] j We have from the above 5! m' = - sZ, [Z,... Z,] . sZ, |^, ... Z,] = s.[Z,... Z,.] \<\>Z,...<\>Z:, or o\m' = s.[Z,...Z,]\Z,...4>Z,] (30). Also by eq. (13) .5 ! 111- = - sZ, \Z, . . . Z,\ . S(}>Z, {Z,... Z,\ = - 5 ! sZ,^Z„ or m'=-sZ4,Z (31). I do not see how in the case of ?u<^' generally to get rid as in these two cases of Z, ... Z^. We may deduce from the sextic now obtained the n-tic satisfied by

.... Remembering (§ 31) when a conjugate varia- tion is circular and when hyperbolic we see by equations (12) and (13) § 29 that SSA^A - tSB^B has the same value for any set of conjugate norms in the complex. But for a positive norm sA'stA = -1, and for a negative norm sB-stB= 1. Hence denoting any norm of the conjugate set by H we have that *" sH^H 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 (S^HtzHjsH'urH) is constant, or finally St-^SF-ar^ = const (1), for any set of conjugate motors in the complex. By eq. (9) § 14, or eq. (8) § 13, iH- = 2tH. Hence putting -BT = 1 we get that 'E,t~^H = constant (2), for any set of co-reciprocal motors of a given complex. Eq. (2) is proved in Sc7-ews, § 136. Let A', B', C be any motors of a complex of the third order given by A, B, C. Expressing A', B', C in terms of J., B, C we get SA'B'C' = xSABC (3), where x is an ordinary scalar. Hence tSA'B'C =tSABC (4), or iS ABC 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 hj A, B MA'B' = xMAB (5), where oo is an ordinary scalar ; and in particular tMA'B' = mAB (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 of 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 -utA, 'utB, -utG, tjtA' , stB', •stC' in place of A, B, C, A', B', C". But if ^, B, C belong to a complex of the third order it is obvious that ^A, ^B, otC 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, (f> we have tA +tB + tC +dcotd-etsin(f) = const (7), §40] MOTORS AS MAGNITUDES OF THE FIRST ORDER. 177 for any three motors of a complex of the third order, and tA + t5 + rf cot ^ = 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 v ^ ttoOoCo OoCo + Cotto + floOo \ ( ^ ^ )• ft 4- 6 + c + c? cot ^ — e tan ^ = Oo + 6o + Co ' Similarly for a complex of the second order • 2 /3 _ i^ _ ^ + ^ ] ~ ao&o ~ ao + ^0 [ (13), rt + 64-c?cot = ao + &o' but eq. (13) is most easily deduced from eq. (12). Eq. (12) is fully expressed by saying that a, h, c are the roots of the equation (so - ao) (oc - ho) (a; - Co) + oc"" {(cosec- 6 sec= (f) - l)(x - Uo -h-Co) + cosec- 6 sec- ^{dcotd —e tan 0)} = (14), and eq. (13) by saying that a, h are the roots of {x-ao){x-ho) + x''cot''0 = (15), and that d = i («„ + 6o) sin 20 (16). It may be noticed that eq. (10) gives for three perpendicular motors of a complex of the third order a + h +c = const (17), and similarly for two perpendicular motors of a complex of the second order a + h = 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 ill 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 MAGNITUDKS OF THE FIRST ORDER. 170 Thus the complex of the first order is that of one motor. If this is ii lator the complex is that of fli, 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 + aD.)i. Thus the complex may always be expressed as that of (« + u'Cl)i, where a and a' are ordinary scalars, not both zero. The complex of the fifth order reciprocal to this is that of (a — rt'n)t and of (6 + b'D.)j and (c + c'ri) 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 h, h', 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'D.)i, (b ± h'il)j, (c + c'fi) k, where i,j, k are any set of mutually perpendicular intersecting unit rotors and a, a', h, h', 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 1 G O g' 1 'o « "o i 1 O 0) 8 -T ■| f +1 g S i +1 a +1 i'- + pi + 2Sp^po = p-^ + p/ + '2Spspi. Equating the converter parts, Spip. = spspi. This is quoted in the same place by Sir Robert Ball as a theorem due to Chasles. Similarly we have Spip-i = Spo_Pi, S/)iP4 = Spop-i. Hence Spop^sp^PiSpsp^ = Sp^psSp.piSpsPi = ... = .,., each group being obtained by writing down all the products from which one suffix is absent. Dividing by pi'pip-ipi we get 23 . 24 . 34 _ 13.14.34 _ 12.14.24 _ 12.13.23 J} ~ P-? pi ~ pi ' another result quoted as due to Mobius. A result somewhat similar to the above Sp^p. = Sp^.p^ is obtained by squaring both sides of the equation - pi = p2 + ps + Pi and equating the converter parts. It is spop.j + sp..pi + spipi = 0. This is not independent of the two previous results (%Sap = and spip. = ^Pspi), 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 a. For then the small displacement MJ. ( ) will be compounded of the two small rotations Mp ( ) 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] KXAMPLES OF THE API'LICATION OF OCTONIONS. 185 To prove this wc have by § 11 above to show that any octoniou Q for which TQ = 1 can be expressed in the form qr, where q and r arc (§ 6) axials for which Tq = Tr = 1. [Note that an octonion Q in general cannot be expressed in the form qr where q and r arc axials, since TQ is not in general an ordinary scalar, whereas T ((/r) = TqTr is.] Looking upon Q as an operator on motors wc sec that it can always (when TQ = 1 ) be expressed as pa--^, where p and a are two rotors whose tensors are equal. Now let t be a?ii/ rotor whose tensor is the same as that of p and that of o- and which intersects both p and 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 always he effected in an infinite number of luays by two semi-revolutions. The following suffices to obtain the axes in all cases. The axis 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 giving to the latter half the given twist. If the given twist degenerates into a translation this be- comes : — // the twist 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 serai-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 (7(7 = 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 (f-^ -t- Q,pMq~^ for Spq~^ = 0. Hence the whole displacement is (ry-i + D.pMq-^) q or I + flpWiq-' . q. Hence (§11) the displacement is a translation equal and parallel to 2pMq-' . q. Since q-' = Kg, we have 2pM(/-' .q = -2K {pMq-' .q} = - 2KqMq-' . p = q-' (q - q"') p = (I - q-') P- Hence the translation is compounded of two translations, the one ..§43] EXAMPLES OF THE APPLICATION OF OCTONIONS. 1S7 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 ivhen it has been first rotated with the second rotation and then 'revised. 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 twists take two lines 1 and 2 such tJiat Jialf the first twist brings 1 into coincidence luith the shortest distance between the axes of the twists and half tlie second twist brings the shortest distance into coincidence witlt 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 axis which ivill bring 1 into coincidence luitli 2. 43. Geometrical properties of the second order com- plex. Passing now to the text of Screius 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 (/8) 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 fii'st. First consider the singular cases of (a). If a = 6 = 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 latur parallel to the plane and perpendicular to the rotors belongs to the complex. All the motors are self-reciprocal and any two arc reciprocal. Similarly for the case b' = a = 0. If b only =0 the complex consists of a plane of parallel motors all of the same pitch. Also the later parallel to the plane and perpendicular to the motors belongs to the complex. This later 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. [§ 48. If a' = h' = 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 =0 the complex is that of (1 + n^)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 + Up) i cos 6 +j sin 6} = r[l + n{p cos- 6 - kp sin 9 cos 6)] (i cos 6 +j sin 6), where r is an ordinary scalar. Thus all the motors intersect k perpendicularly. If 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 6. They all lie on the ruled surface whose ordinary Cartesian equation is 2 {x- + %f) + pocy = 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 j ; 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 0P'\ 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 (/8) if d = \iT 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 n(icos^+jsin^) [which corresponds to a' = go ]. §43] EXAMPLES OF THE APPLlf'ATIOX OF OCTONIOXS. IS!) The complex that remains for consideration is that of (1 + rrll)/, (1 + bCl)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 quatcnnon 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) '\^ by the equation '^E = -aiSiE-h}SjE (1). Thus any motor belonging to the complex can be expressed as o) + fl-^Q), where co is any rotor through perpendicular to k. The equation Spylrp = -1 (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 Spfp = l (3) also represents a conic in the same way. When both conies are real they are conjugate hyperbolas. In this case Spy}rp = (4) is the equation of their asymptotes. Since (§ .9) Ti (w + n-/rft)2. 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 Scretus, § 136. The cause of the error will be easily enough seen by the reader.] Suppose ft) + D,(T, where ft) 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 m + Cla: [The virtual coefficient of two motors A and J5 is — sAB/T^AT^B.] The virtual coefficient of two screws ro + D,a and &)' + fla' is - S (&)o-' + aoo'). Hence the virtual coefficient of the screw co + Her and the screw of our complex (1 -f- aft) i cos ^ -I- (1 + bn)j sin 6 is — S [ft) (ai cos + hj sin 6) + a (i cos 6 +j sin 6)} = — cos ^S?" (riM + a) — sin ^S; (bm + a). § 4o] EXAMPLES OF THE APPLICATION OF OCTONIONS. 101 The two values of for which this is zero and a maxinium or minimum differ by ^tt, i.e. the two corresponding screws of the complex are perpendicular {Screws, § 87). In§ 25 oi Screws it is proved that there is a cone of the second degree with vertex at any point on which lie all the motors through the point which are reciprocal to a given complex of the second order. And in § 97 it is proved that these motors have pitches ranging from + oc to — x and that the later (motor of infinite pitch) is parallel to the nodal line of the cylindroid. To prove these statements let the point be taken for origin and let two of the motors of the complex to which the motors are to be reciprocal be tWi + Ho-i and (Oo-\- Vla-n, where Wj, to.,, o-j, o-, are rotors through the point. The nodal line is the axis (§ 40 above) of M {co^ + Oo-j) (&),_, + Ho-o), i.e. it is (§ 12 above) parallel to M&)i&)o. Let {\-\-Q.p)o) be one of the motors whose pitch is p, where w is a rotor through the point. Expressing that it is reciprocal to both co^ + Vla^ and w., + Her..,, we have ;)Sa)ft).+ Sa)o-, = 0] ^ ^' These give as a necessary and sufficient condition m = xlA (p<«i + o-i)(/)&)2 + o-a) (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 Oft) when we get to = x^Aay^a).,. This shows (1) that one motor (and ordinary scalar multiples of it) and only one can be found for each value of p from + oo to — oc , 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 w satisfies the homogeneous quadratic equation SwWiS(ii(7., — SwCTiSoXWo = (8), i.e. it lies on a cone of the second degree. The first of equations (6) proves other facts. Foi- instance in § 80 of Scretus it is stated that every line in space serves as the residence of a motor reciprocal to a given motor. Given ft) and «! + Ho-,, the first of equations (6) serves to determine p so that {\-\-Q.p)(o 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 co, so that co 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', h, h', c, c is zero. With the present as with the second order complex we shall change the notation so that the complex is that of (1 + afl) i, (1 + hVl)j and (1 + cH) k. Define the self-conjugate pencil function i/r by the equation f ^ = - aiSiE - bjSjE - ckSkE (1). Since the reciprocal complex is that of (1 - an) i (1 - hn)j, (1 - en) k, the corresponding pencil function for it is — yfr. Any motor of the present complex is co + n-^co, where co is any rotor through 0. The two quadrics Spy^rp = abc (2), Spyjrp = — ahc (3) will be called the pitch quadric and the conjugate pitch quadric respectively. Their common asymptotic cone is Spyfrp = (4). These quadrics have a fundamental property exactly similar to that of the pitch conies in the second order complex. For the pitch of the motor co + nyp-co of the present complex is Soy-^yfrco, whence we have that if the p of (2) be parallel to the motor the pitch is ahcp~^, and if the p of (3) be parallel to the motor the pitch is — ahcp~". 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 + O-v/ro) is Sco'^^^co. Hence if this is p, co is confined to a cone of the second degree, as is evident from the equation Sft) (i/r -jL») ft) = 0. § 44] EXAMPLES OF THE APPLICATION OF OCTONIONS. 193 The rotor p from to any point on this motor is by § G above jO = M (yfro) + x) oi~^, where x is any scalar. Thus the extremity of p is confined to a ruled surftice. We will show that this surface is a quadric. Putting i/r — jt) = OT, we have SeOCTft) = 0, p = M (-SJOJ + x) (0~' = (vTOi + x) (t)~^. Hence Mp&) = ctw. Hence ft) = '33-~^Mp&) = ???^~^MOTpCTa), where nip has the usual meaning with regard to the pencil function -ot, i.e. nip = — SctiotJotZ; = (a - p){h — p){c - p) (5). Substituting in the last equation M/ow for -sro), we have ft) = mp~^lA-uTplAp(o = ft)??i^~^Sp'33-p, since Swurp = Sprs-co = SpMpoi = 0. Thus finally Sp if -p)p = Sp^p = vip = (a -2)){b- p){c -J))... (6). Putting in this p = 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 -v/r into — t/t, 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 oue 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 Sco'^co = 0, we have for the motors of the given complex p = (ZTCO + X) ft)~l, and for those of the reciprocal complex P = — (otW + X) ft)~^ 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^p = — 1, where (fy is any self-conjugate pencil function, are given by p = ± [{— m)~^(f)a + x] a~'^, where a is any rotor gene- M. o. 13 194 OCTONIONS. [§ 44 rating line of the asymptotic cone Server = 0. That (— m)~^ may be real all three roots or one and only one of the (f) 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 jj 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 Mpw = -ara) which gives o> (as to direction but of course not as to tensor) when p and therefore -cr is known. A general method of obtaining m from this equation is to choose that one of the two values of (o given by the equations Scozjo) = 0, Sp^co = which makes Mpw = -srw. Another general method is to write down the c\ibic of (f), where (f>(0 = ■SJfO — Mptt), 03 _ Jf "<^2 ^ Jf 'j^ - if = 0, and notice that M must be zero, since there is a rotor co for which a/3 = aSf^M-^ ct^Q - /SSfaM-^ a^. 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 From the last two we have a = .rMe-v/re, and x is now obtained from the first of the three equations. Putting in that equation /S = a~'e, thus making ^ a rotor in the plane perpendicular to a, we have Sa~^e^e~^ = 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~^'\/re. Similarly the perpendicular on the axis of the cylindroid similarly related to the reciprocal complex is — Wie'^yjre, 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-v/^w of the given complex is — M&)~^-v/r&). Hence putting co = 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 w perpendicular to e, we see that the motor of the complex parallel to co is in the plane through perpen- dicular to e if Se&jA/r&) = 0. But this is the necessary and sufficient condition that co 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,... A 6 are six independent motors. Then we may express any motor E by the equation E = oojA^+ ...+ocQAQ='S + E'% 2Q-i = -^' + V(4 + A"^) (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. (o). 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 § 4G] 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 = QGoQ-\ 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 ^o. ^o, -^o 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 E^ and E^, fur 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 hodg of ivhich tJie 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 + ^cr 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 = G is a, linear function of « + fla = F hy the definition in § 15 above of a linear function. Putting ma = i/r (co + Ho-) we have S (o)' + no-') ylr{Q) + no-) = mSaa' = S (o) + Cla) yjr (o)' + fla'), so that -v/r 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 (ft) + Xlo") (ft)' + Ho-') = S (&)o-' + ft)V).] Hence with the meanings of F^ and G^ given just now G^o = to^o (10), where y^rf^ is a constant self-conjugate energy-function. Also if T is the kinetic energy 2T=-sFoGo=-sFoy{r,Fo (11). When the velocity motor of a rigid body is a lator {velocity of translation) the momentum motor is a parallel and similarly directed rotor equal to M x the lator, ivhere M is the mass of the body. This only asserts that the sum of a series of parallel similarly directed rotors is a similarly directed rotor whose tensor is the sum of the tensors of the components. This was incidentally proved by octonion methods in § 22 above. Without assuming any of the well-known geometrical properties of the position of the centre of mass it is easy to prove by octonion methods from this proposition alone, that this momentum rotor must pass through a point fixed in the rigid body. For if Mp, Ma be the two momentum rotors due to the two velocity lators Up, Xlcr; the momentum rotor due to the velocity lator xfLp + yrio; must be M{xp + y(T) for all ordinary scalar values of x and y. Hence xp + y(T is a rotor for all values of x and y. Hence Mp and Ma and similarly all other momentum rotors due to velocity lators must intersect, i.e. they must pass through a point fixed in the rigid body. This point is of course what is known as the centre of mass, and will here be denoted by 0. Now let x^~^^- Since both -v/r^ and % are self-conjugate energy-functions there must, by § 83, be six independent motors forming a conjugate system with regard both to ->|^„ and x- Two axial motors which are conjugate with regard to ;y must be perpendicular. Since not more than three motors can be mutually perpendicular and since not more than three lators can be independent, it follows that the six motors must be three mutually perpendicular axial motors and three independent lators. Since every lator belongs to the complex of three in- dependent lators, it follows that there are three mutually per- pendicular axial motors A, B, G which together with any three §46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 201 independent lators which form a conjugate system with regard to i/to, form a system of six conjugate motors with regard to i|r„. Denote any later by D.p where p is any rotor through 0. From the last proposition it follows that yfr^np = Mp. Since A and Op are conjugate with regard to -»/r^ = sA-fp.p = MsAp. Hence by statement (4) § 14, J. and similarly B and C must be rotors through 0. Take i, j, k as the unit coaxial rotors. It is easy to see that £li, Hj, nk are conjugate to one another with regard to t/tq. Hence i, j, k, Hi, flj, ^^■ are a conjugate system of motors with regard to -v/r^. Since sO/o-vlrdt = 0, i/^o* is a later; and since sjyjr^i = skyp-^i = this later is parallel to i. Thus since -yJTo is an energy-function yjr^i = J/a^ni, fo^i = ^^i "^» j = ^1/^'^y", &c (12). Putting the velocity-motor equal to Hi, i, &c. successively we get all the ordinary theorems concerning moments of inertia, principal axes, and their connection with the kinetic energy of the rigid body. Interpreting the meaning of momentum motor we also get all their connections with moment of mo- mentum. i,j, k are of course the principal axes and il/a'-, Mb', Mo- the moments of inertia about them. From equation (8) G = Qfo{Q-'FQ).Q-^ = irF (13), which defines the self-conjugate energy-function yjr in terms of yfr^ and Q. Thus whereas -v/r^ is of constant form i/r is not. We may also note that 2T = -sFG = -sFylrF (14). Eq. (10) when expanded in full by aid of equations (12) gives Cr, = f^F, = - M{lsiF, + a'nisniF, +jsjF„ + &c.). ..(15). It may be noticed that SlFo^|^o^o = - -^1/ (sniF.siFo + siljF.sjF, + siU-F„skFo) = ^MsF,\ From this, since SFG=SF,rro, F-' = F,^ 202 OCTONIONS. [§ 46 we have Si^o^^o = ^MsF,' - 2nT\ = SFG=^MsF'-2nT] ^ ^" i, j, k, m, nj, flk are a conjugate set of motors of -^o) ^^ we have seen, but they are not a co-reciprocal set. The set which are both conjugate and co-reciprocal we may call the absolute principal motors of t/tq. These are what in Screws (§ 105) are called " the absolute principal screws of inertia." [A set of "conjugate screws of inertia" is a set of motors conjugate with regard to i|r(,. If a complex of order n be given, those n screws of it which are co-reciprocal and form a conjugate set with regard to t/tq are " the principal screws of inertia " of that complex. Thus the absolute principal screws of inertia are the principal screws of inertia for the case n = 6.] The absolute principal motors of i/^o are (i±aa)i (i±60)j, (l±c^)^• (i7), for these are co-reciprocal and are such that y\r,{l±a^)i=±Ma{l±a^)i,^c (18), from which it easily follows (from the fact that they are co-reci- procal) that they are also a conjugate set. Equation (18) shows that the roots of the ■^^, sextic are ±Ma, ±Mb, ±Mc. The sextic is (^Ir-^ - APa') (f „-• - M'b') (f o' - M-'c"~) = (19). From these we see that the absolute principal motors of '\lr are (1 ± aO) QlQ-\ (1 ± 60) QjQ-\ (1 + cH) QkQ-' (20), and that the roots of the t/^ sextic are the same as the roots of the -v/to 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 = XH'. The " internal " forces of the system contribute zero to the sum XH', 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 § 4G] EXAMPLES OF THE AI'PLICATION OF OCTOXIOXS. 203 time) values of G 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 SQQ-^^i) (22). Hence by eq. (8) § 11 F=2QQ-' (23). Hence by eq. (8) of the present section Fo = 2Q-'Q (24). Hence H = d {QGoQ-')ldt = MFG + QG,Q-' [§11 above], or H, = Q-'HQ = UF,G, + G,. Thus Ho = ylrJ\ + MF,f,F, (25). This can be expressed in terras of Q instead of F^, for by eq. (24) F, = 2Q-^Q-2Q-^QQ-^Q = 2Q-^Q-^F^ Taking the scalar-octonion and motor parts we have ^SQ-^Q=Fi = F-^ (26), F,= 2MQ-Hi (27). Hence our equation of motion to determine Q when H^ is given is H, = 2fMrQ + 4>^-Q-'Qfo{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 ^|ro 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. i|ro 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 -v^o is not a commutative function, -\|r(,- is. For by equation (18) ^|r,^ (1 ± an) i = M'a' (1 ± aO) i, from which yjr^i = M-'aH, y}r,^m = M ^^a^ni, and similarly for j and k. Thus i/r^^ is a self-conjugate pencil function whose centre is 0. Define -stq by the equation ^oE = il/-^ fo'^ = - {aHSiE + b'jSjE + c'kSkE) (29). The relation between t/t'-^ and -v/tq^ is the same as the relation between ^|r and -v/tq, so we naturally define ot by the equation ^E = M-"-f'E = Q^o {Q-'EQ) -Q-' (30). Now let Fo = co, + nao, F=co + n(r (31), where co^ and a^ are rotors through and co = Q(OoQ~^, cr = Q(TqQ~'^, so that ft) and a are rotors through the point to which has been transported at any time. The relations between F,^, w^, a^, -yJTQ and -570 will clearly be precisely similar to those between F, co, a, yjr and -ST. We restrict ourselves therefore to the former set. Next notice the peculiar properties of the two (not self- conjugate and not commutative) linear functions -v/^uO and Hi/ro. By equation (15) ilf-ifl-f oi^o = Ho-o, M-'f,nFo = o), (32), that is to sa,y if any motor he expressed as the sum of a rotor through and a lator, J/^^Hi/tq operating 07i the motor picks out the lator and If-i-v/rgQ picks out the rotor. We shall understand eq. (32) to have this general meaning and not to have special reference to F^, the velocity motor. As a result we may notice that M-'{a^\r, + ^\r,n) = l (33), M-^n^^,VL = n (34), so that both the linear functions on the left are both self-conjugate and commutative. Since il/'^-v/roH reduces any motor to its rotor part translated so as to pass through a given point, we see that if E be any formal quaternion motor function [§ 13 above] of the motors § 4G] EXAMPLES OF THE APPLICATION OF (KTONlONS. 'lO'i Ai, Ao..., then M~^ylr^O.E is the same function of M'^^^^^VlAi, M-^y^^VlA., .... For instance if A and B be any two motors M-^Uylr,nA^\r,Q.B = M-'^\r,ilMAB (3.5), and indeed each of these is the rotor through parallel and equal to the rotor part M^AB of lAAB. ■^^Fq may be expressed in terms of w^, a^ and ■CTu, for by equation (15) M-'f,o), = nCT„fy„, M-'f^na, = 0-0 (36) (the second of which is a particular case of eq. (32)). This again is true of any rotor and later and will be quoted as meaning that. From this equation we have G, = ^|r,F, = M((To+n^,(o,) (37), which expresses Gq in the form of a rotor through + a later. The physical meanings of the four rotors and lators a>o, fio-Q, J/cTy, il/HOTotOo into which Fq and (t„ have been decomposed are obvious from the meaning of a velocity motor and a momentum motor (§ 8 above). The instantaneous motion of the body has been decomposed into a rotation (rotor) Wq round an axis through and a translation (lator) Oo-o. The momentum in the ordinary sense is the rotor ifo-o through 0, and the moment of momentum about is the lator Mfl'^oCOo. [We should naturally call the rotor momentum not Mo-q but the equal and parallel, but not coincident, rotor MiG^o = -^^^i (o'o + fiOTotWo).] As it is important that the geometrical relations between the rotors Wo, cto and CToWq and the motors Fo and Go should be clearly grasped, I have endeavoured in fig. 8 to render them evident to the eye. First note that Mi^oG^o = MM (wo + fio-o) (^0 + flt^o^u) = MwoG^o (38), so that the shortest distances of Fo and G^o and of , &c. The point of the figure is now not a fixed point but the position of the centre of mass at any time. Regarding the figure in this way, since G and a are both constant, the lines of the figure G and a- are absolutely fixed in space. Thus the centre of mass moves along a fixed straight line with uniform velocity. Also the later flOTft) is constant. Hence the rotor marked otw in the figure though not fixed in space is fixed in direction and tensor. The sphere twists about always inside the fixed cylinder, a portion of which is indicated in the figure. The mode of its twisting depends on the manner in which (o and F vary. Poinsot's construction, which (Tait's Quater- nions, 3rd ed., § 407) is immediately deducible from the fact that OTft) is constant in magnitude and direction, indicates how co varies and the relations shown in the figure are then sufficient to determine the variation in F. We shall investigate directly to a certain extent how the foot of the perpendicular from on F moves, but for the present return to the equation (39). Since G is G^o displaced and -nrw is -CToWq displaced and a is ctq displaced, it is easy to see that four independent integrations of this equation can be effected. Three of these are obtained with ease. Operating by S(ro( ) we at once obtain G."^ = const. § 46] EXAMPLES OF THE APPLTCATIOX OF OCTONIONS. 200 This is equivalent to two ordinary scalar equations, so it involves two integrals. Operating by sFo ( ) we get = 2sF4Go = 2sF„yjr4F, = dsF,^|roF„ , so that S^j-v/tqFo = const, is another integral. The fourth integral may be obtained thus : — By eq. (32) By equation (38) MF.Go = MojoG^o = il/Mo), (a, + fi^.o),,). Hence by eq. (39) sdGoyjron^jr.Go = - sF,G,yfronf,Go dt = — M^ScOo {(To + n-OToWo) tn-oWo = 0. Hence Const. = sG,^|r,nfoGo = sH (foGof = S, {^froG,)-. By expressing Go, &c., in terms of 0-^, &c., we can see what well-known result each of these integrals corresponds to. The integral sFo^foFo = sFoGo = const, gives o-fl" + Styo-CToWo = const., i.e., it expresses that the kinetic energy remains constant ; Si (iroGoY = const, gives i^ofOof = const., i.e., the ordinarily called moment of momentum is constant. Go^ = const, gives the two equations (To" = const., ScTo-nToWo = const. From the first of these the translatory energy is constant ; and since the whole is constant, the rotatory energy is also constant. From the last three integrals it follows also that the angle between o-Q and -sToCOo is constant. Since SGodG.^^O we have sGodGo = and snGodGo = 0; i.e. dGo is reciprocal to Go and flGo- Similarly the other two integrals express that dGo is reciprocal to Fo and ylr^i^HFo, i.e. to Fq and CToO),,. But equation (39) gives both the axis and pitch of dGo, i-e. expresses that dGo must be reciprocal to five motors, so that one motor independent of the four just mentioned can be found to which dGo is reciprocal. As a matter of fact since from that equation SGodGo = 0, SFodGo = 0, sdGoM-'FoGo = 0, M. O. 14 210 OCTONIONS. [§ 46 we see that dG^, is reciprocal to the five motors (tq, n6ro> ^^o, ^^o and M-^i^oG^o- Either of the two last may be taken as the fifth motor independent of the four given by the four integrals. Again, the statement that dGo is reciprocal to each of the motors i^o. Gq, flGo and ylr^ClyjroGo may be expressed by saying that Fq and dFo are conjugate with regard to each of the self-conjugate functions t/tq, -^q^, ^^o^i^o and ^frQ'n^fra^ respectively; or that Go and dGo are conjugate with regard to each of the functions 'v/^o'~S 1> ^^: i/Tufi-v/ro respectively. Put 'STCO=fl, S(0'ST(0 = — Ci, fl" = Soi-UT'OO = — C.2 (41). Thus /A is a rotor of constant magnitude and direction, and Ci and Cg are constant scalars ; /m being the ordinarily called moment of momentum, Ci twice the rotatory energy and c, the square of the moment of momentum. We may notice that if m, m, m" are the coefficients of the tn- cubic for all positive integral values of n from zero. This equation enables us to express Cn — — S&)'n7"^&) in terms of c.2, Ci and Cu= - 6)^ i.e. in terms of constants and the square of the angular velocity. For it gives Cn = — ScoST^^Q), Cn+3 — m"cn+2 + m'cn+i — mCn = (42). From this in particular we deduce a result required later, viz. — Ci% + 2C1C2C3 — Ca^ = {c^^ {m'm" — m) — Ci^c, {m"- + m) + CiC.?2m" — c./j - . . .(48), + mci (m"ci — 2C2) (o^ and generally _ _ oi^ h-c" + Ci (b" + c^) - C2 2n _ ta^c'c^^ + Ci (c^ + a") - c^ , ^^ ^"~ {a''-b'){a'-c"~) ^ (6— c-)(6— a^) ft)-tt^62 + Ci (a^ + 6') - Ca „,. .(44), (c^ - a-) (c^ - If) since a"-, ¥, c- are the roots of the -cr 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 § 4(J] EXAMI'LES OF 'I'llK AIM'MCATION OK OCTONIONS. 211 and p = QpoQ~^ the corresponding rotor through the point to which has been transported at any time. Then the two equations Sp'urp = — Ci, Spo-^opu = — 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 — 0) of this ellipsoid is S/j/x = -c, (46). Since p, is a rotor through the centre of mass of constant tensor and direction this plane is simply moving with the velocity cr 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 co, 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 (4.5) 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 co (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 S/3CT-p = — c... 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) Sp (Ca — Cjot) '!jTp — 0. This cone will be called the jJolhode 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 T/ji~% from the centre; i.e. which touch the sphere p" = Ci^/j,~- = — Ci-/c2. [The rotor perpendicular from the centre on the plane (46) is — Cj/a"'.] It will be observed that &> and tWo are not related in the same way as w and coq; i.e. we have not &> = Q6)i,Q~\ But Qw^Q''^ i« a rotor through the centre of mass at any time which we have occasion to consider, so we define ,Q-' (47). Thus eo' is the value of (o^ 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) G>0 = — '53-o~^M&)o'5ro«0, which gives the following expressions for w &)'= — '57~^Mw-33-&) = — 7n~^M-57ft)-5r-&) (48), or &)' = — trr-iM&)yu, = — ?/l"^M/iOTyU, (49). The first of each of these expressions shows that w is along that diameter of the Poinsot ellipsoid whose conjugate plane contains the angular velocity and the moment of momentum. Call the centre of mass at any instant 0' , the point of contact of the ellipsoid and contact-plane P, and the foot of the perpendicular from 0' on the contact plane T. Then the conjugate plane of the diameter along co' 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 fjb and -nr/i. That it is perpendicular to jx 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 2M(u'&) can be expressed as a self-conjugate pencil function y^oi of a where -v/r is fixed in the rigid body ; i.e. where -y^ = Q-»/^o {Qr^ [ ] Q) • Q~^ "^0 being an absolutely constant self- conjugate pencil function. For 2Mw'ft) = 2wi~^ McoMOTfu-sr-ft) = 2??i~^ (— OTwSwcT-ct) + OT-eoSwOTft)) = y^w\ " where -(l^ = 2m~^(c2 — Ci'ot)^ (SI)- [In the rest of this section and in the following section (§ 47) we shall have no occasion to refer to the i/r and -v/rg of equations (10) to (39) above. The former ^^r was not like the present •\/r, a pencil function, though it was self-conjugate.] The polhode cone is Spy\rp = (52). As P traces out the polhode the extremity of p = y^w, a rotor drawn from 0' traces out another curve. The point on this curve corresponding to P we will call Q. Thus OT = ft), CrQ = fay (53). § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 213 Since [eq. (52)] Scoyfrco = 0, O'P and O'Q are perpendicular. And further, the normal to the polhode cone at P is parallel to -v|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 SfO-V^^O) = 0, ScOZTQ) = — Ci, or Sylroi\lr~^ ylrco — 0, S-»^co-v^~-OT\^fi) = — d, the ecpiation of the normal cone is Spf-ip = (.54), and the equations of the normal cone curve are (54) and Spyjr-'^-uyp = — Ci (55). From the equation Sco^-co = — Ca we also have for any point on the normal cone curve Spylr--,;,y = -Co (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 + ^nat)-' R{1 + ^nat) ^R- tnWiaMR. Let us then define Ro, Rj, R' and R" by the equations R^ = Q-^RQ, R,^R - tnMaMR ] R' = QR,Q-\ R" = R, + mMaMR,] ^ ^■ Thus i?o 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 pusitiun of the rigid b"di/. [If the 214 OCTONIONS. [§ 46 body is subject to external forces and Ht is its integral translation at any time we ought to write Or in place of flat in the last equation. In this case flf takes the place of Her in eq. (58).] By eq. (6) § 11 we have R = R' + MFMR = R' + nW\aMR (58), i.e., the actual rate of change R which is noted by the outsider consists of the rate R' observed by the resident combined with the rate MFMR due to the resident's motion ; and also consists of the rate R" observed by the plane-resident combined with the rate D,M(xMR due to the plane-resident's motion. It will be noticed that since MFw = OMcreo, the meaning [eq. (47)] above given to w' is consistent with the present meaning of R' and is also such that (o' = co" (59). If we define the axials q and r by the equations a(o~^ = q=f+ 13, r = 'UTa)a~^ = ixa~^ = 9 + 7 (60), /3 and 7 being rotors and / and g scalars ; then Sg or / is the pitch of F, and Sr or g is the pitch of G ; and Mq or /3 is the rotor perpendicular from 0' on F, and Mr or 7 the rotor perpen- dicular from 0' on G. It will be noticed that rg = /XQ)~^ = -STWCO"^ (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 (7 is a variable. For instance by eq. (49) co' = o)-'ST~^W[/j,u)~^ = co"z7~^W\rq, which gives (o' = oy'^-' (My/3 + g/3 +fy) (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) irw'^ = - 2Mft)OT-^ MyLto)-! = - 2nr^zTMfj,Mfxco-\ § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 215 Putting ill this equation Mfico-' = Mrq = M7/3 + gl3 +fy, wcget y}r(o-' = -2iir'^Mfj,{Myl3 + gl3+fy) (63), which gives an expression for i/rct)"' in terms of the relative posi- tions of 0', F, G, the constant rotor fx and the ellipsoid. Agaia putting in the last fj, = ra = (y + g) a, we get -f «-' = - '2m~' OT |- 0-^7 (gl3 +fy) + ^ W - 7^) /S + {fg + S/37) 7]] (C4), which gives an expression in terms of the same things except that the constant rotor fx has been replaced by the constant rotor ' + flMo-co for &>, since &>' = eo". Now a(o~^Maco . a)~i = — o-Mo-&)~\ Hence by equation (58) q" = — <7(o~^ ft)' ft)~i = — q(o'o)~^ (C5). Taking the scalar part we get / the rate of change of the pitch of F. Thus _/= — Saco~^a>'co~^ — — Sqa>'co~^ (66). Each of these expressions for / admits of simple interpretation. o)co'q)~^ is co' turned conically round co through two right angles. Call such a conical rotation 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 cd, multiplied by — o)~^. Putting in the second expression q =f+ /S we have /= -fSto'co-' - S/3Mft)'ft)-i (67). Here Sft)'ft)"^ is the rate of increase of the logarithm of Tco, and M(o'(o~^ is the rotor angular velocity of the point of contact about 0'. This gives a second interpretation. Remembering that Mco'(o~^ = ^\lro)~^ 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) /= — So-ft)-57~^M/ift)~l . ft)-l [eq. (61)] where ot^ is what ot would become if the rigid body were revolved half round co, i.e. '^oj^ = ftJOT ((o''^Eco) .(o~^ (68). Thus substituting for Miq in terms of ^, &c. f=Sa^^-\My^+g^+fy) (69). Again in the expression — Scr(o~^^~^W\rq . co if we substitute q for o-ft)~^ and q~^a for co we get /= — Saq-sx'^Mrq . q~^, or /= - ScTtTT.-i Mqr = - Sazr,-' (M/Sy + g/3 +/y) (70), where -sr,^ is what -ot would become if the rigid body were turned § 46] EXAMPLES OF THE APPLICATION OF OCTONIONS. 217 in the plane of a, w through twice the angle between these two ; ^,^E = qv7(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 jS" . Taking the motor part of equation (65) we get /S" = — (o~-McrQ)(o'(o~^ (72). Thus this velocity is the rotor product of the velocity a of the rigid body, and the velocity co' of the point of contact revolved half round w, multiplied by — (o~-. Another interesting expression for /3" may be obtained thus. By eq. (48) 'MOT(y-v|r&>, or by eq. (51) «' = |cr^M/i,\|ra) (73). Now 'i/rft) is [eq. (50)] perpendicular to w. Hence (oylrco . oi~^ = — ylru). Hence w"^ w'eo"^ = — ^Cj"^ M . (o/j.a)~^ ylrco~^, so that ^" = ^cr^MaMco/j.o)~^yJrw-^ (74). Here cofico''^ is /x revolved half round to, and A|r&}-^ may be substi- tuted for from equation (63) or (64). Again taking the second expression of eq. (65) and putting it in the form (since co = q~^ a) q" — — U)~- qw q~^ -^ = 2M (o)' -f- HMo-ft)) w-i + wFco-\ by equations (58) and (59). Now ViMaoy = MFco since F=q} + fla: Hence 2HMo-&) . co-' + ojFo)-^ = 2MFo) . co'' + (oFo)-' = F. Hence F^ = F + 2Moj'(o-' ^ F + ^|r(o-' (79). Fg may be obtained independently or from this result. For the displacement q( )q-^ = (Tco-^( )Q)a-\ Hence the displace- ment qQ( ) Q~^q~'^ may be obtained by first making the displace- ment Q)~^ Q ( ) Q"^ (o [or coQ{ ) Q~^a)~'^] and then the displacement o- ( ) cr-\ Using eq. (78) for this case Fg = 2M(7<7-1 + (tF^<7-\ or since & = 0, Fq = aF^a-' (80). The decomposition here of the rotation q{ )q~^ into the two semi-revolutions ~^ and o- ( ) a~^ is a particular case of the more general decompositions of § 42 above. If the velocity motor F^ be decomposed into a rotation about 0' and a translation, the rotation is w + 'y^af^ and the translation Ilo- that of the rigid body. Similarly decomposing Fq the rotation is cr (o) + -vlro)"^) a~^ and the translation riar. 47. The polhode and the normal cone curve. The frequent appearance above of the function -^ [eq. (51) § 46] leads us to enquire into the geometrical properties of what was called the normal cone curve and its relations to the polhode. § 47] EXAMPLES OF THE APPLICATION OV OCTONIONS. 219 In this section we shall drop all consideration of co as repre- senting the rotation of the rigid body and also all consideration of the motion of the body. We shall regard co merely as the coordinate rotor of any point on the polhode and co' (though as a matter of fact it is equal and parallel to the velocity of the point of contact of the polhode with the contact plane) as a rotor parallel to the tangent of the polhode at the point co. For the sake of clearness we here briefly recapitulate the geometrical properties and equations established in § 40. ■57 is a given self-conjugate pencil function whose centre is the origin 0. Its cubic is •53-^ — ??i"-S3-'- + Vl'ST — m — (1). The roots of this cubic which are all positive will be taken as a, |S, 7. [They were a", 6^ c", in § 46.] Thus a + /8 + 7 = m", /3y + yx + al3 = m\ a^y = m (2). We shall have occasion to require the half-sum of the roots, so define s by the equation 2s = x + /3 + ry = >ii" (3). -v/r is a pencil function defined by the e(]uation f = 2m-' (a, - 0,7:7) 7^7 (4), where c, and c. are given positive constants. The polhode is defined as the curve given by the two equations SwOTO) = — Ci , Sco-st'-(o = (cto))- = — c, (o). [Thus the perpendicular from on the tangent plane of the ellipsoid Scomco = — Cj at the point co has when this point is on the polhode the constant value Ci/Vc2-] Thus the polhode lies on the quadric cone S(ofco = (6). The normal cone curve is defined as the curve any point of which has p= yjrco for coordinate rotor. Thus F being a point ou the polhode and Q the corresponding point on the normal curve Ol'^co, UQ = p = fco (7). Byeq. (6) Spco = (8), so that p is perpendicular to co. It is also perpendicular to the 220 OCTONIONS. [§ 47 tangent plane at P of the polhode cone (6). It therefore lies on the cone normal to the polhode cone. Substituting for to in terms of p equations (5) and (6) give Sp-^jr^'^p = — Ci, S/Oi/r-'-CT-p = — Co, SyD-»/^~^/? = (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 Sp-^"^ p = 0. When 0) is given p^ can be written down from the following: — p^ — Scoyjr'^co = 4??i~-S(i) (C2 — Ci^)" ot-co = 4fm~- (— Ca^ — 2ciC2Sct)S7*&) + Ci-ScoOT^o)). Now by eq. (1) Swtn-^tu and Sco-ar^o) can (§ 46) be expressed in terms of w^, Ci and Cg. The relation that we thus get is eq. (43) § 46 which for present purposes may be written p2 = 4^ + const (10). If w is a rotor through parallel to the tangent of the polhode at P we have by equations (5) Sw'otci) = 0, StuV-o) = 0. Thus &)' = .^•MOT(yOT2(w where x is any scalar. If x is put equal to — m-^ [e(i[. (48) | 46], w 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 0)' = — Wi^^MtrrwOT-ft) = — •n7~^Mft)t«7ft) (H)- By substituting here for ct- in terms of -»/r and ot by means of eq. (4) we get ft)' = |^Ci~^MOTft)'v/r&) = — \c{~^\ApiT(i> (12). From the last expression we have by equations (8) and (5) 2Mft)'ft> = p (13). These results except (10) have already been proved in § 46. As P moves along the polhode with velocity w , Q moves along the normal cone curve with some definite velocity p'. Thus p' = ^frw. [This velocity is of course the velocity as it appears to the resident (of § 46). The velocity as it appears to the plane- resident = p" = p + M&)/9.] Thus by the first of equations (12) p = — ^cr^m^Mftj-v/r-'wco = — ^cr'/y^^Mftj-v/^^'-CT/? (14), § 47] EXAMPLES OF THK APPLICATION OK OCTONIONS. 221 where m^ is the product of the roots of the yfr cubic, so that by equations (1) and (4) m^ = Siir" (c? - c.,%m" + cx.'m - cM) (1 -'O- From the first of equations (14) we have by eq. (8). Thus by eq. (7) and eq. (5) 2W\p'p = Dl^CO (16). Thus from eq. (18) ^=-.u, (17), i.e. T/3-' X the angular velocity of Q about the origin bears a con- stant ratio to Tft)^ x the angular velocity of P about the origin ; or Tp X the areal velocity of Q about the origin bears a constant ratio to Too x the areal velocity of P about the origin. From eq. (10) J^=477i-ic,(m"ci-2c.,) (18). Combining the last two equations, or better combining (13), (16) and (18) g."," I cT^ = 4!Vi-hn^-^c^ (m"ci - 2c.>) p/(o (19), Oft) CO I op p or the tangent of the angle between the radius vector (from 0) and the polhode at P bears a ratio to the tangent of the angle between the radius vector and the normal cone curve at Q which is a constant multiple of the ratio of Ttw"^ to Tp-\ This, along with the fact that Q lies on the cone normal to the polhode cone, suffices to geometrically construct the normal cone curve when one point on it is given and the polhode is given. These relations it will be observed are symmetrical with respect to the two curves. This symmetry is further borne out by a comparison between the two equations of each pair of the following which have been established above or which may be at once deduced from the established results. 222 ocTONioNS. [§ 47 S&)-v/rft) = 0, Sp-v/r-'OTp = 0, &)' = — 7?i~^ M-OTCOOT-Q), p' = — ^Ci~hn^'M'\jr~^p-ylr~--UTp, 6)' = — OT^^Mw-OTCo, /a' = — ICi^^-^/rMpilr-iOT^, w' = ?»tn-~- MwOT"^ o), p' = ^Ci~^ mm^~^ yjr-'UT~^ MpyJrvT~^ p, 2M&}'ft) = p, 2Mp'p = m^o). This symmetry suggests that perhaps the normal cone curve is a polhode on some quadric and that the original polhode is its normal cone curve. We will now prove that something very like this is the case. If four scalars y, z, y', z' can be determined such that {y + z^)- = -v/r^OT-i (?/' + /-D3-) (20), we shall have (y\lr~-'ST + Z'\\r~- -st")' = y'y\r~"'aT + zy^r'-t^- (^21). Thus if we put OTi = y\^~-'aj 4- zyjr'-'ST" (22), the equations of the normal cone curve can by equations (9) be put in the form SpsT^p = — yci — ZC.2 = — Ci (•23) (tn-jp)'- = Spzjj^p — — y'ci — z'c.2 = — Cn] so that the normal cone curve is a polhode on the quadric S/3OTip = — Ci' whose contact plane is at the distance cZ/Vco' from the centre. Assuming that y, z, y and / can be so determined, Wj the product of the roots of the tn-j cubic is by equation (22) given by Wi = mm^- {y^ + in"y-z + m'yz^ + mz^) (24), and we have 2???.i~^ (Co' — Ci'oti) -OTi = 27»i~' [Co (y + Zth) — c/ {\J + z'-si)] yjr-~zT = 2mr^ (yz' — y'z) {c^ — c^vt) -v/t-^ot = mi~^ m{yz — y'z) ■\^~^ = x'>^~^ = i/ri , say, by eq. (4). Thus the normal cone curve of this new polhode is given by the equation Pi = "^iP = 00(0 (25), where p^ is the coordinate vector of Qi the point on the new normal cone curve corresponding to Q on the new polhode. Thus § 47] EXAMPLES OF THE APPLTCATIOX OF OCTOXIONS. 223 OQiP is a straight line ; and OQJOF has the constant value sc. 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 y'^: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 c, is of two dimensions and Ci of one and m of three in ct, -^ is of no dimensions in ct. Similarly i^i is of no dimensions in CTj. -^ \s, however of two dimensions in w and -\/rj of two dimensions in p. This is illustrated in § 4G where •\|r6)~' 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 CTi are real each of the quadrics Sp-sr^p = — c/ and Spr^^-p = — Cg' must be real, for each equation is satisfied by an infinite number of real values of p, namely the coordinate rotqrs of the (real) normal cone curve. Under these circumstances the quadric Spzj^-p = — c.,' must be an ellipsoid and c.,' must be positive since the roots of the -sri" cubic are all positive. [It is necessary to note this because ^ may be real and yet the quadric Spcpp = — c imaginary. This last is the case if all the roots of the (ji 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) Sp-sr^p = — c^ is a sphere and therefore (2) S/a-sn-j-p = — Co 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) yjr = 2m~^ (c, — Cj-ct) -st it becomes (y + z-otY = -inr- (cj — Cjot)'- ot (y + z'zs). 224 OCTONIONS. [§ 47 This can be satisfied by constant scalar values of y, z, y', z if and only if (y + Z'ury — 4: i.e. if a; = Si, a;' = ai/3i7i' 1 /'29^ so that these vahies of w, x and y satisfy the required conditions and we have -S7o = OTi (Si — CT]) (30). From eq. (30) the roots a,, /S,, 7. of the ot, cubic are given by oi,= a,{s,-a,\ ^, = /3As,-^i), y, = yi(s,-%) (31), and these are proportional to the roots ai, /Si, 71 of the OTj cubic if and only if «i = A = 7i> i. e. if and only if the quadric Spt^^p = — c/ is a sphere. This proves that the normal cone curve is a polhode of both the quadrics SpCTj/o = — c/ and Spwop = — 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 w, = a.,/3,7,) i/To = 27>i.2~^ (Ca" — C/V.,) -572 is a constant scalar multiplied by yp-i [eq. (25)]. As a matter of fact it is quite easy to prove from the above results that The direct determination of the ratios y'- '. z- : y' : z' is tedious. The following process is therefore preferable. Changing the above ct to d-cr the original polhode has for equations ScO'STO) = — 1, S&)t3"-&) = — 6 (33), where 6 is put for c^\c^. This is of course equivalent to taking in the above Ci = l, c. = h. According to eq. (4) <^ would now be 27/1-1 -57 (6 - -57), but we will take the slightly more general form ■f = e//t-'OT(6-'5r) (34), and we shall assume that the normal cone curve is given by p = ylro) with this new value of yjr. Thus the roots of the ct 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. O. 1 ■'> 226 ocTONiONS. [| 47 hyperboloid ; and b is the inverse square of the perpendicular on the contact plane. Define ^o by the equation OT (g - ct ) .„-. ^«--^3y ^^^> By equation (33) Sco-cj-oW = — 1. Also (s - by W = -ST (ot=» - 25-57^ + s-ot), or by eq. (1) tn-fl- = OT [m — {ni — s") ot}/(s — 6)- (86). Hence Sco'stoCo = — 1, Swotq-o) = — 6o ('^^)' where bo= {-b{m' - s") + m}l{s-by (38). Hence the given polhode is also a polhode on the quadric Sa)OTo&) = - 1 and the inverse square of the perpendicular on the new contact plane is b^. If the -sTo cubic is stq^ — 2so'U7(f + Wo'^o — wio = (39), we have 2so ={0L(s-a) + /3{s-^) + y(s- y)}/{s -b) = 2 (m - s^l{s - b), 7n: = {l3y{s-/3){s~y)+...]/(s-by = [s-m' — s (2ms — Sm) + (m'^ - 4^ms)]l(s - b)-, mo = a/37 (s - a) (s - ^) (s - 7)/(s - by = m (s* — 2s^ + m's — m)l{s — by, m - s^ , — s^m —ms + m ' ) I (40). or So = r- , mo — - , , s „ 5-6 {s — by 111 (— s^ + m's — m) From the first of these and equation (38) , s{m'—s^) — m ,.^. From this we may notice in passing that mo m .-b.r^i <''^>- § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 227 Also So — -u^o — ("^' — •^■" — '^'^ + ■^")/('^' ~ ^)> whence = OT {(s — 7at) (in' — s") — CT (s— ot)-}/!^ (lit — S-) ■in or by eq. (1) ^o(5o-^o) ^43^_ So — ^0 Comparing this with equation (35) we see that the relations between CTq, Iq and the coefficients of the OTo cubic on the one hand and OT, b and the coefficients of the zr 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 -570 = ^ and the two quadrics are the same. When s = b eq. (35) gives CTo = 00 . In this case there are not two different quadrics on which the given polhode is a polhode. Putting however -sj' = CT (5 — -ot) = e~^ »i-v/r (44), [eq. (34)] we have Swtn-'ft) = 0, Soo-ut'-co = s (m' — s") — m ^is-a){s-^){s-y) (45). From this it follows that T-\/ra) 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 Sp-r^^p = const, is a sphere. From equations (35) and (43) we have CTo-57~^ = {s — '57)/(s - b), ^-sy-^ = {So - •ro-o)/(So — b^) (4G). Hence (5 - ^) (So - tiTo) = (s - b) (.90 - 60) = [s (m - s"-) - m]/(s - b). . .(47), by eq. (41). This very simple relation between s — -cr and 5,, — CTo makes it more convenient for many purposes to regard (j) = s — ■ur and o = So — -BTo as the two fundamental pencil functions, -ot is a determinate function of (f), for s is the sum of the roots of the cubic. We will not here however work with (f), <^o aiid their cubics. 15—2 228 OCTONIONS. [§ 47 By eq. (36) OTo ih - ^o) {s - by = ZT [bo {s - by (s - -57) - (5 - b) [m - ot (w' - s"-)]} = -or {[m - 6 (m - s~)] {s-^)-{s-b) [m - -cr {m' - s"")]] = — OT (6 — ot) {s (m' — s^) — m}, or by eq. (40) 7?^o~' ^0 {bo -'^o) = - 'ni~^ -srib-'ST) (48). Hence if we define 'fo by the equation -^0 = e^iih-^ ^0 (bo - ^o) (49), -v/to and yjr will have the same value if eo = -e (50), and therefore p = >|^o• In connection with this note that h,-b = (m - 6wi' + 2sb' - ¥)/{s - bf = (a-h){^-b){r^-b)l{s-br, or (6o -h){s- by = (a - 6) (/3 - 6) (7 - 6) = - m'm^e-'. . .(57). To determine y first notice that Sw-sTCToO) . (s — 6) = SwCT- (s — tir) ft) = Sco (sCT^ — 2sCT- + ??i'OT — ?/i) ft) [eq. (1)] = sb — ni — may- [eq. (33)]. Thus by equation (55) (ctjp)" = Sw [x{-!s + CTo) + y}- ft) = Sg) [x' (ot^ + -BTo^) + 'ixy (■uT + OTo) + 2x^'aT'STo + y'] ft> = - a^' (6 + 60) - 4a;2/ + 2^;^ (s6 - m')l(s - h) + [y- — 2x^m/(s — b)] cd'. Putting then y = x^/{2ml(s-b)] (58), we shall have SpCTj-p = — bi, SpCT.^-p = — 62 ('59), where 6^ = x^ {{b + &«) + 2 (s + s^) + 4 V[2m/(s - 6)]]) 62 = a;'^ ((6 + 60) + 2 (s + 5o) - 4 V[2W(s " &)]}) " ' ' Since [eq. (50)] eo = — e, e (b^ — b) may be regarded as expressed symmetrically in terms of ct and -stq. It will be seen that x, bi and h, are also thus symmetrically expressed when m/{s — b) is so expressed [see eq. (42) above]. By equations (40) and (41) we have m/{s-b) = sSo-(s-b){s,-h) (61). From these we have ^, = x{^ + ■:^,+ J[2nl!(s-b)]]^{r-' (62), 230 ocTONiONS. [§ 47 and ^2 is obtained from uti by altering the sign of the radical. Putting in this expression the value of x, substituting for -»/r in terms of ot — -cto, and utilising equations (57) and (61), we have e '57 + OTo+\/{2sso-2(s-6)(So-6o)} .„^, i^x = V^o). m^ -nr — ■STo This is a symmetrical expression in ot and -OTq for e = — e^ and t/t = -v^Q. Again -cti may be expressed explicitly in terms of ot, e, h and the coefficients of the -sr cubic. Modifying in this way every- thing but the scalar m^, we get by equations (51) and (52) '53'i = 7 TT \y^)- m^ 'uriyj — b) Since Spvr^p =—\, Sp^-^p = — hx (65), and similarly for OTo, it follows that the normal cone curve is a polhode of the quadric Sptn-jp = — 1 and also of the quadric Sp'STop = — 1. When the normal cone curve is thus regarded as a polhode, its own normal cone curve is the original polhode magnified in a definite ratio, as we see from the above. It is well, notwithstand- ing, to show that this follows from the results just established. From the equations Spijj^p = — 1, Sp'S7i^p = — bi, the equation of the normal cone must be Sp (6i — -sTj) OTip = 0; and from the equations Sp-sTip = — 1, SpOTop = — 1, it must be Sp (otj — ot^) p = 0. Hence the pencil function zr^ — OTo must be a simple multiple of the function (6i — -CTi) OTj. It follows that OTo is of the form 2/OTi + Z'os^. From this again it follows by the above that OTs can only have the unique value OTj (.Sj — 'S7i)/(si — h^, where s^ is the half-sum of the roots of the tn-j cubic. If then we put •\|^i = ei7^ii~^CTi (6i — -OTi) (66), where rrn^ is the product of the roots of the -OTj cubic, the relations between ■^^, -OTi, CTo, e^, tn-^, b^, Sj must be exactly the same as the relations between yjr, •ur, ■ur^, e, m, b, s. Hence by eq. (52) -v/^l = — 7»i~^ei (Si — by) (-37] — OT.), § 47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 231 or ■x/tj = — 2^-v|r-' '//<,-' e?! (6*1 — &i). Hence if" we put e, = - in,l2y {s, -h,) = - (2y)- [s,s, - {s, - h,) (s, - 6,)} . . .(67), [eq. (Gl)] we shall have yjr, = ylr-^ (68), and the normal cone curve of the original normal cone curve becomes identical with the original polhode. Let us now examine the cases where the above results become unintelligible by reason of certain of the constants becoming infinite, zero or imaginary. For this purpose it is convenient to write the values of x and y [equations (56) and (58)] in the form [eq. (57)] w = m{s-b)/{e{a-b){^-h)(y-b)} (69), y = m V{2m {s - b)}/{e (a - 6) (/3 - b) (7 -b)] . . .(70). We shall assume that the original polhode and quadnc are real and that e is not zero, infinite or imaginary. Thus the normal cone curve is always real. It is never infinite (if we exclude the possibility of the original quadric being a cylinder) but may be evanescent. Of the roots a, /3, 7 of the ct cubic let a > /S > 7. We assume that none of these are zero. That the quadric Styro-ft) = — 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 Swsroi = — 1 is an ellipsoid, be greater than /S 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 noniial cone curve is a polhode, are imaginary if m (s — b) is negative ; i.e. if s — 6 is negative when Swvrw = — 1 is an ellipsoid or hyperboloid of two sheets, and ii s — b 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 6 > 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 OTo 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'^ = e''m-^{s-a.){s-l3){s-ry) = -p{-^ (71), say. Thus as already remarked the normal cone curve is a polhode on a sphere That /)2 is constant can also be seen from equation (10) since the coefficient of w^ 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 8/3t3-2/J = — 1 the normal cone must have for equation Sp{zj2—pi) p = 0, so that this equation is the same as the equation Spylr~^p = 0. Hence ta-a— ^i is a simple multiple of -\|r-\ We may therefore put Hence vy^p = (pi^jr + z) co, so that (^2/o)' = PiY + zW'. Here p^ is constant, but co'^ is constant only in the extreme case when the original quadric is a surface of revolution. For if G)2 = _ p-i, a constant, the normal cone has for equation either Sft) (ot —j)) ft) = or Sco (sp — CT-) ft) = 0. Hence ot — ^ is a simple multiple of sp — m". Hence a relation of the form iCOT^ + ym + = 0, where x, y, z are constant scalars, holds good. Now ts satisfies such a quadratic equation if and only if two of the roots of its cubic are equal, i.e. if and only if Swusw^ — X is a surftice of revolution. When it is a surface of revolution (tn-.p)- above is constant for any constant value of z. Hence in this case there is >i:s i)F ■['[IE Al'l'l.lCATloN OF uctonkjns. 238 a whole family of quadiies on each of which the normal cone curve is a polhode. It is not hard to sec by similar reasoning that the last case is a particular one of a more general case. If Sw-cjoy = — 1 is a surface of revolution co^ 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 Swww = — 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 = /3 - a, so that 7 must be negative and a positive. Moreover since b is positive and = yS, /8 is positive. The surface is therefore an hyperboloid of one sheet and to is along the greatest axis. In this case we see by geometrical interpretation that the polhode reduces to a point. We also have p = i/ro) = 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 zj cubic. When the given quadric Sw-crw = — 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 — vf)(o = 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 m cubic are when 6 = /3 for the ellipsoid and when 6 = a for the h}q3erboloid of one sheet. In both these cases one root of the ^jr cubic is zero and the other two are of opposite signs so that the polhode cone Sooyjrco =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 ct 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 OTj, •sTg and the pair ct, •utq 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 mr'e, (s, -h,) = -e (h - h) {(s - h)l2m]l But by eq. (60) b, - b, = 2e-' (h - h)-' {2mJ(s - 6)}l Hence ^.^-^(6.-6.) .(.-6)(6.-6 )^^ 2mi 2m If then we jDut e(s-b) (bo - b) ^ ,^ . and similarly for g^, gi, g'>, we shall have g=go, gi = g2, ggi = ^ • (^4), ^=h-' (75), CT (6 — ct) I . . 1 « '^l(bl— -53"!) .(76), t = 2^^" (")■ If we put g = 1 and therefore 7, = 1 wc 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 ciu've reciprocal or conjugate polhodcs. §47] EXAMPLES OF THE APPLICATION OF OCTONIONS. 2:j5 To give a notion of the possible magnitudes of the (luantities 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 f . 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 hjperboloids 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^ -2 (Perp.)-2 Semi-axes Perp. EUip. OT 8 4 2 5 •35355 -50000 -70711 -44721 Hyp. OTo -4 6 5 29 4 [ -50000] -40825 -44721 -37139 Hyp. tJTi 9 16 81 16 45 16 261 16 [1-33333] -44444 -59629 -24759 Hyp. ^.^ 3 16 9 16 3 16 5 16 2-30940 1-33333 [2-30940] 1-78886 2nd CASE. EUip ■ux 8 4 2 5 2 -35355 -50000 -70711 -63246 Hyp. ^0 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 EUip. ■uj.. 1 216 11 216 55 216 31 216 14-69694 4-43130 198173 2-63965 286 OCTONIONS. [§47 Under the heading " serai-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 OCTOXIONS. 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 polhodos intersect the parts of the planes included in the octant arc indicated by thick marks. The conjugate polhode it will be remembered is obtained by taking (/=l. The curve first denoted above, "the normal cone curve," was the one for which e = 2 and therefore (f = m~^ (s — b) (bo — 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 •stj and OTo in the ratio of G9"8i 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 CTj and sr.^ being all smaller than the smallest of the axes of -cr and zjq. We conclude this section by establishing certain results which are symmetrical with regard to the four quadrics. Remembering [eq. (75)] that o:: = ^g~^ we get from eq. (60) 2g^ (b, + 6.,) = b + b, + 2(s + 5„). Similarly since ggi = 1 2(b + bo)=g'{b, + b, + 2{s, + s,)}. By cross addition and subtraction we obtain from these f {3 ih + h)+2 (s, + s,)] = S{b + bo) + 2(s + s,).. .(78), g"- [2 (s, + s.,) - (6, + b,)} + {2(s + s^) - (b + b^)] = . . .(79). Again from equation (60) 2_ 1 2m Similarly ^"^" = 2^^^. ^^^^- Multiplying and dividing (^■-*')(*-'")=VfeA} <«'>' 6,_6, = i /- fV s 238 OCTONIONS. [§ 47 The symmetry of the last two results is rendered more evident by substituting from equation (61) ??ii/(si - bi) = S1S2 - (5i - bi) (S2 - b^), m/{s -b) = sso -(s-b) (sq - h). Again by eq. (77) p = 2g (b — &o)~^ ('^ — ^ti) <»• Hence p2 (6 - boYl^g- = — b-bo— 2S&)OTCTotu = -{b + bo) + 2 [mco- + (m' - s") + s (s - b)]/{s - b), by the value found above for Sco'sj-ctqCo. Thus by equations (40) and (80) ^y ~zh, = """'s^b + ^ ^' + '^^-^^ + ^°^ •• ^^^^' which by eq. (79) may be written <"%4?^ = »' f%7^ + 2 («. + ^J -(«-.+«} (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.] A symmetrical form of these equations is ^^ f%7?6/2 (.,+..) -(6, + ^ ,2 i!!L , 2 (s + So) - {b + bo)[ ...(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) = (^ + y-b){y + a-b)(a + ^-b)/(s - by...{86). Since the left is symmetrical in w and CTq, and by eq. (79), we have § 4-7] EXAMPLES OF THE APPLICATION OF (K'TONIONS. il3+y-b)(y + a-b){a + /3-h) {s-by 2:VJ .o(A + 7i (So-bof b,)(a,+^,-b,) > («7). .../A±h: {s.-b,y b.^ {s,-b.;)"- Thus if any one of these vanishes they all vanish ; i.e. if b is equal to the sum of two roots of the ct cubic, b^ is equal to the sum of two roots of the ot,, cubic &c. In this rather curious case we see by eq. (88) that Tp/Tco is constant. From eq. (79) it follows that at least one of the four quadrics tsr, cTo, -BTi, CT, when all are real is an hyperboloid, i.e. as we have already seen, an hyperboloid of one sheet. For if -or is an ellipsoid, s is positive, and if OTi and zTa are real, s — 6 is also positive. It follows that if all four were ellipsoids 2{s + So)-(b + bo) and 2 (Si + So) — (^1 + ^2) 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 /S are positive and 7 negative. Assuming this only we have for the present case flo and 7o positive, /So negative, «« > 70 > A. b-s, a-s, s - /3, and s — 7 all positive, and s„ — 7o all positive. S„, So -/So Of the four quadrics «r, ztq, ts^, nr^ we may take the first pair such that 2 (s + So) -(b + h) is positive. Assuming this b is between a + 7 and a + /S. [If we assumed that 2 (s + So) -{b + h) was negative we should have b between s and a + 7 or between a + yS and + x .] 6 is not equal to bo, for if it were we should have the singular case where b is equal to a root of the zj cubic. We may then 240 OCTONIONS. [§ 47 choose the quadric -sr such that h>ho. Assuming this we shall have 2a + 7 > a + /3- > 6 > a > fto > ^'o > ao + /3o > To, «! and /3i positive, 71 negative, «! >/3i > 71, a^ and 72 positive, ySg negative, o.<,>^2> ^-i- Thus the real smaller semi-axes of all four hyperboloids are coincident (a, a^, a^, a.^. The larger real semi-axis of tir is co- incident with the imaginary semi-axis of OTo and conversely. A similar statement also holds with regard to the semi-axes of zTi and CTj. These statements may be illustrated by taking a = 50, y8=6, 7 = -16, 6 = ^, when it will be found that 18 ,«^ ^ 18 ^ 18 ,^ «o= 49-125, /3o = -^g.7, 70 = 49-48, ^18 23.37 ^18 851 " 49 ■ 7 ~ 49 • 7 and also 206.277 _ 206.24 , 50.23 ^"^ = -21749-' 9~^^= -2174-9' ^"^^=-21:49' 206.193 „^ 206.16 , 50.19 ^"^=^1749"' ^'^^ = -109 ' ^"^'^^ 21749' 61 and 62 can also be found from eq. (60). They are greater than ai and a^ respectively. From equation (78) we can get another symmetrical result similar to eq. (87). It is -6)-^{(/3 + 7-6)(7-|-a-6)(a-|-/3-6)-i-4(a-6)(/3-6)(7-6)}4-86 = (^o-&o)-M(A + 7o-6o)()() + 4(«„-6o)()()} + 86o = 5'H*i-&i)-M(^i + 7i-M()() + 4(«,-60()()}+^^86, (88). 48. Motion of rigid body resumed. Potential. Free- dom and constraint. Reactions. Impulses. Return now §48] EXAMPLES OF THE APPLICATION OF OCTONIONS. 241 to the discussion of the mechanics of a rigid body as considered in equations (1) to (28) § 46. ^0, 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+2m^^.ME-'E.e-'^ (1), Fo = ESE-'E+2e-'^^'Mei^.ME-'E (2). This last may also be written F,, = E + (l-E-e-^)ME-^E = E+(-^- + ^^_-~+ ..}jlAEE (3), from which i\ = E + {\-E-e-E)lAE-^E-[E + {2-2E-2e-^-Ee-E) X SE-' E + Me- ^^ M^-' 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 2SQQ''' [eq. (8) § 11]. The work done in this displacement is - 2sHBQQ-\ Hence by eq. (8) § 14 2sBQQ-' II = Bv = - sSE"^v = - 2s8Q^v, where now the independent variable motor implied by ^ is E'. Since 8QQ~^ is an arbitrary infinitesimal motor we have by § 14 H = -MQ^v (5). Hence by eq. (8) § 46 H, = -M^vQ (6). These may be substituted for the force motor in the variou.s equations of § 46. It may be noticed that v = - 2sQ^i; = - sFQ'^v = sFH. M. O. 16 242 OCTONIONS. [§ 48 Also by eq. (11), § 46 T = - sF,^lrJ\ = - sF,ff, [eq. (25), § 46] = - sFH [eq. (8), § 46]. Thus v + t=0, i.e. the sum of the potential and kinetic energies remains constant. When the displacement is small we ma}^ in all these equations putQ=l. Thus H=H, = -^v (7). We have seen (§ 46) that in this case E' = E, so that we ma}^ 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 fhe reactions. Of these P is due to the field. Hence in place of equations (5) and (6) we have P = -MQ^v, P, = -M^vQ..: (9). Also if R= QRf,Q~^ is the force motor due to the reaction of the constraints, H = P+R, H, = Po + Ro (10). On account of the constraint F (and therefore also ^o, and (7o) 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 Fq, Go 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 J?o complex, &c. Eq. (25) § 46 and the last equation give P, + Ro = ylr,K + MF,yfr„F, (11). Here Po and -v/tq are given, P„ is confined to a definite complex and Ra to the reciprocal complex. Thus Fo and Ro together involve six unknown scalars ; and equation (11) is equivalent to six scalar equations ; it is therefore sufticient to determine Ro and Fo and therefore the motion. If P' = QP^'Q-^ is a given external impulse motor and R' = QRoQ~^ the impulsive reaction motor, R' is still confined to the reaction complex. If Fo and Fo + AFo= Q-'iF + AF)Q, Go and Go + ^Go = Q~^ (G + AG) Q are the values of Fo, G^, just before and just after the impulse, AP and AG are confined to the velocity and momentum complexes respectively. Eq. (11) gives in this case P«' + Po' = ^«APo = AG„ (12). This admits of the following simple interpretation : — AG the increment in the 'momentum motor = the component of P' the given impulse in the momentum complex when P' is expressed as such a component + a component in the reaction complex. In other words AG — P' is reciprocal to the velocity complex. This last statement is sufficient to determine AG since it involves the same number of scalar equations as there are disposable scalars in AG (the number of degrees of freedom). By the definition of the momentum complex every motor of that complex can be expressed as i/rP where P is a motor of the velocity complex. By the definition of the reaction complex si2P= where R is any motor of that complex. Now - ^sFyjrF being the kinetic energy for a possible motion is not zero. Hence no values of P and R can be found for which R = -^F; i.e. there is no motor common to the momentum and reaction complexes, or: — The momentum and reaction complexes are independent. This is otherwise evident since P' above is any motor and is resolved into the two finite components AG and — R in the momentum 16—2 244 OCTONIONS. [§ 48 and reaction complexes ; and by their definitions the sum of the orders of these complexes is six. In order to apply the general methods of dynamics to the motion of a constrained rigid body it is convenient to re-define ^. Let the number of degrees of freedom be n and let us denote the velocity complex when Q = 1 by (n). Let ^.j ... ^„ be ?? indepen- dent motors of (??), and B^... B^^n 6-n independent motors of any given independent complex (6 - n) of order 6 - n. Let the bar introduced in § 28 be used with reference to these six motors. Let C, any independent variable motor of {n), be given by the equation G = x,A, + ...+XnAn = ^xA (13). Then define ^ by the equation ^=Sla/a^ (14). By equations (23), (24) § 28 we have sdG^ = -d (15), and therefore by eq. (8) and statement (2) of § 14, when n = 6 the present and former meanings of '^ are the same. It is important to notice that except when n = 6 this meaning of ^ does not depend only on the complex (?i) but also upon the independent complex (6 — n). (n) is fixed by the constraints. (6 — n) may be any complex independent of (n) which is convenient for the matter in hand. (6 — n) will denote the complex reciprocal to (n), i.e. the reaction complex which is perfectly definite, (n) will_denote the complex reciprocal to (6 — n). Since Zi . . . u4„, 5i . . . BQ_n are (I 28) always independent we may either take (n) as any complex convenient for our purpose which is independent of (6 — n) or as above we may take (6 — n) as any complex independent of (n). Thus, as the momentum complex is independent of (6 - n), we may take (n) as the momentum complex when Q=l. This we shall frequently do below. If y is any scalar function of G, "^y is confined to (n). When (n) contains no motors reciprocal to the whole of (n) it is often convenient to take (6 — n) as the reciprocal of (n) since this reciprocal is then independent of (n). (n) and (n) are then the same complex, as also are (6-w) and (6—n). Even when this is not the case it is often convenient to take (Q—n) as a § 48] EXAMPLES OF THE APPLICATION OF OCTONIGNS. 245 complex "semi-reciprocal " (§ 31) to (n). We may remind the reader that («) and (6 — ?i) are semi-reciprocal when (n) is the complex of the motors J.i-42 . . . A A • • • and (G — n) that of B^B^ . . . A' A' • • • ; where all pairs of these motors are reciprocal except the pairs (AA') (AA)--- which are such that sAA' = sAA' = •.. = -!; and where A, A'. A--- are all self-reciprocal. The complex AA ••• is a definite complex ; and when (6 — n) is given so also is the complex AA'---- With this notation and that of § 31 for (AD), &c., we here have: — the complex (n) is (AD), (6-7i) is {BD'), (n) is (AD'), and (G^Ti) is (BD). Also by eq. (26) § 28 and eq. (14) § 32, if G=XxA+tyD, '^ = -t(As-'AKd/dx) + t(D'd/dy) (16). In § 32 TA is chosen so that sJ.'- = — 1. Q is a function of n independent variables. It is not in general such that E or E' is confined to the definite complex (n). But Q may be supposed a single-valued function of G and the functional form may be so chosen that when Q-1 is small, E = E' = G (17). Thus when, as in next section, the displacement is small through- out all time, the new independent variable G is not required. E the displacement motor and E the velocity motor in this case both belong to the given complex (n). Let L be the Lagrangian function. L is thus a single- valued function of G and C only, and so far as it depends on G it only contains a positive quadratic term T. This term may by § 34 (p. 1.52) be put in the form T=-^sC^fr,C (18), where -^^ is a partial energy function such that (1) ■\/r,C' = 0, where G' is any motor belonging to (6 - n), (2) sG-^iG is negative and not zero for any motor G belonging to (n), and (3) V^iC" belongs to (n) for all values of C. The connection between ■\/r, and yp-,, of § 46 may bo given. Since [eq. (15)] Q = - sC^ . Q, 246 ocTONiONS. [§ 48 we have by equations (11) and (24) § 46 T= - 2s (Q-'sC^ . Q) to (Q-^s(7^ . Q), so that y|r, = 4>%s()%sQ-'Q,^|ro(Q-'Q,) (19), for yjr^ as defined by this equation is (1) self-conjugate, (2) such that T=- ^sCf.C, (3) such that ylr^C" is confined to (n) [eq. (14)]. Using now the general dynamical principle h{Ldt = 0, where the initial and final positions are not varied, we get = [ s (S(7ti(7 + hC^L) dt = ls8G{'^L-dylr,C/dt)dt. Here ^L—dylr^C/dt is a motor belonging to (n) and 8G is an arbi- trary motor belonging to (n). Hence by § 32 (p. 140) T=^^ (^«)- This equation is equivalent to w scalar equations and is there- fore sufficient to determine C and therefore the motion as a function of the time. C may appropriately be called the generalised velocity motor and ylr^C the generalised momentum motor. 49. Small motions. Let now the rigid body never depart more than a small wa}' from an absolutely stable position. The potential energy v is then a positive quadratic function of the displacement. We may therefore put v = -^sL\'ST,E, (1), where Eq is the displacement motor and zj^ is a given complete energy function. The force motor due to this displacement is by eq. (7), §48, -t^o^;. It is convenient here to recapitulate and in some slight degree alter our notation. [The alterations refer only to Eq and yjr. 'y^ as defined directly is for the case of small motions-, our previous tIti and not what has hitherto been denoted by -y^. For small § 49] EXAMPLES OF THE APPLICATION OF OCTONIONS. 247 motions our previous -v/tq and ^|r have the same meaning aixl we continue to use ^fr„ with that meaning.] When the body is ft^ee the displacement motor, the velocity motor, the momentum motor and the force motor are denoted by Eo, Eq, yJroEo, — 'OToEfy respectively, where i/r„ and OTo are complete energy functions ; the kinetic and potential energies are respec- tively — ^sE^^jrJiJ^) and — ^sE^)Zj,,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— w) is any complex independent of («) ; (n) and (Q — n) are the reciprocals of (6 — n) and (n) respectively and are independent of one another. En and Es_n will be used to denote a^^hitrary motors of (n) and (6 — n), respectively. The complex to which '>^(^En is confined is called the momentum complex, that to which tSoEn 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). (n) 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, -v/r^E" and —ss^E. The reaction motor is R. The partial energy functions -v/r and ot are uniquely (§ 34) defined by the equations sE„ylrEn = sE,f,En, ^/.^o-n = (2), sE,,^E,, = sE„^,E,„ ^E,_„ = (3). By § 34 it follows that yfrF and z:rF are both confined to (n) what- ever motor value F have. -yjrE and — zjE are called the general- ised momentum motor and the generalised force motor respectively. They are for small motions the -yfr^C and "^L of eq. (20) § 48. For constrained motion the kinetic energy = — ^SjE'-v/t^E' = — IsE-^jrE, and the potential energy = — \^E-a^E = — ^sE-sjE. From these definitions we see that to-^ and i/r„ may be regarded as particular forms of zj and yjr, viz. when n = 6. The equation of motion is by cq. (20) § 48 fE=--!zE (4), or E = — ■^_^TsE = — (pE (5). 248 OCTONIONS. [§ 49 See equations (6) to (9) | 33. The general real solution of this equation is E = cos (t ^cf)) . E, + sin (t ^/ . E^ (6), where E^ 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, -^^ or CTo, -v/t or zj, E^ and -fi'e-ji may be identified with the m, ts, ts' , E^ and ^2 of the last proposition but one of § 34. It follows from the forms there given for •nr and ot' that, since E and E are confined to (n), •\/r£' and tsE are the components of i/roE" and OTq^' respectively in (h) when the last motors are each expressed as an (n) component + a (6-?i) component. Thus the generalised momentum motor is the (») component of the actual momentum motor, and similarly for the force motor. The important result follows that when (») 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 ■^. By § 34 they also form a conjugate set with regard to i/tq. They are what Sir Robert Ball calls the n principal motors of inertia. Similarly the n real coreciprocal motors of (?i) which form a conjugate set with regard to OT (and therefore also with regard to -us^ are what he calls the n principal motors of the potential. Again by § 33 there are n real motors {Q^O^... of equations (6) to (9) § 83) which form a conjugate set both with regard to ot and ■^. These are what he calls the n harmonic motors. The important property which gives them their name is at once proved from eq. (9) § 33 and eq. (6) of the present section. § 49] EXAMPLES OF THE APPLICATION OF OCTONIOXS. 249 Most of the theorems of Screius not ah-cady 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 y^(tE only. If it is constrained the answer is -^^ + R where R is any motor of the reaction complex (6 — n). If A^, A. ... An are n conjugate motors of inertia and if the velocity motor E is given by E = x-^A^ + ... = ^xA, the kinetic energy is - 1 sE^^rE = - ^ ^x^sA^A. 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 G-^ and Go be the former com- ponents. The kinetic energy acquired is = -h ^G^ytr^G, - sG.frG, - h sG,ylr,-^G,. It is therefore the sum of the kinetic energies due to P/ and P/ separately if and only if G^ and (r, are conjugate with regard to •\|ro~S i.e. if and only if the velocity motors ^Jto'^Gi and yfro'^G^ acquired are conjugate motors of inertia (i.e. conjugate with regard to a/to). 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 A. By the method of § 48 (that AG-F' is reciprocal to the velocity complex) we see that if xA is the velocity motor acquired xy{r,A = P' +R, where R is reciprocal to A. Thus xsAyJToA = sAP', or X varies directly as sAP' and inversely as sA^ItqA, 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 inchided in (n) 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' — yjroF is reciprocal to every motor of (n) and P'-yfr^F' is reciprocal to every motor of (m). They are both therefore reciprocal to F' which belongs to (m) and therefore to (n). Thus sF'foF = sF'P' = sF'yJToF', or s{F-F')ylr,F'=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 Screivs 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 | 65 that " one screw can always be found upon a screw complex of the wth order reciprocal to n — 1 screws of the same complex" is unsound. For the Q —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 {n — 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 (w) and (7 — n) must contain at least one motor in common, for otherwise we should have 7 independent § 49] KXAMPLKS OF TIIK APPLICATION OF OCTONIONS. 251 motors. Thus there is at least one motor of (n) which is re- ciprocal to (n — 1). This does not prove that this ?vth 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, fli which is reciprocal to Cli is fli itself (and ordinary scalar multiples of it). The main proposition of § QG 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 (Screius, § G6) 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 (C — n) above of (n) is independent of (n), i.e. w^hen (n) contains no motor which is reciprocal to (n) (i.e. no rotor or later 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 (?i) ; and then (h) becomes identical with (n) and (6 — ?i) with (6 — n). Further, in this case our generalised foi'ce 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 (Ti) 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 momentum complex when P is expressed as such a component + a component in the reaction complex. Similarly the virtual impulse motor G 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 docs not hold for the virtual force motor. 252 OCTONIONS. [§ 49 When 71 = 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 % = 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 n = 3, and suppose as in § 44 above that the case is not what is there called a singular one. Thus the complex (n) contains no motors which are reciprocal to the whole complex, so that in this case (n) and (ri) may be taken as identical. Putting 'X^ for the i/r of § 44, so as to enable us to retain our present meaning of yjr, we see that every motor of (n) can be expressed as (1 + II;^;) ft), where co is an arbitrary rotor through a definite point and % is a given self-conjugate pencil function with this point for centre. Thus we may put E,, = il+nx)co (7). If En be any other motor (1 + Cl-^) co' of (n) we have sEnfE,,' = Sft) (1 + nx) ^{1 + ^x) "' = ^^' (1 + ^x) i^0- + ^x) ^> i.e. sEn'^En is a symmetrical function of co and co' linear in each. Hence sEnylrEn' = S(OTco' (8), where r is a definite self-conjugate pencil function with the point just mentioned for centre. The ellipsoid SpTp = -l (9) is what is called the ellipsoid of inertia. [That it is an ellipsoid follows from the fact that sEn'^E^ is always negative and not zero.] Similarly SEn'^En = ScOVCo' (10), where v is a function of the same nature as t. The ellipsoid Spvp = -\ (11) is what is called the ellipsoid of the potential. When sEn^frEn = 0, ScoT(o'=0; i.e. two conjugate motors of inertia are any two motors of (n) which are parallel to a pair of conjugate diameters of the ellipsoid of inertia. Similarly two conjugate motors of the potential are any two motors of («) which § 40] EXAMPLES OF THE APPLICATION OF OCTOXIONS. 253 arc parallel to a pair of conjugate dianiotors of tho ellipsoid of the potential. Twice the kinetic energy is — sE-\lrE', or if £^ = (1 + ft^) co, — SoTO). It is therefore the square of the tensor of the velocity motor multiplied by the inverse square of the parallel semi- diameter of the ellipsoid of inertia. Similarly twice the potential energy is the square of the tensor of the displacement motor multiplied by the inverse square of the parallel semi-diameter of the ellipsoid of the potential. En and E,/ are conjugate with regard to yjr when S&)T(u'= 0, and are reciprocal when S&);i^&)' = [§ 44 above]. Hence the principal motors of inertia are those three of (n) which are parallel to the system of common conjugate diameters of the ellipsoid of inertia (t) and the pitch quadric (x)- Similarly the principal motors of the potential are those three of (n) which are parallel to the system of common conjugate diameters of the ellipsoid of the potential (v) and the pitch quadric (x)- En and i/V are conjugate with regard to i/r when S(OTQi'=0, and are conjugate with regard to ct when S(iyvco'=0. Hence the harmonic motors are those three of (n) which are parallel to the system of common conjugate diameters of the ellipsoid of inertia and the ellipsoid of the potential. CAMBBIDQE: PBINTED by J. and C. F. clay, at the DNIVEKSIXY FKE8S. RETURN CIRCULATION DEPARTMENT TOh^ 202 Main Library 642-3403 LOAN PERIOD 1 HOME USE ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 1 -month loans nnay be renewed by calling 642-3405 6-month loans may be recharged by bringing books to Circulation Desk Renewals and recharges may be made 4 days prior to due date DUE AS STAMPED BELOW SANTA BARBAE fNTERLlBRARY LOAN •^ SEP 3 1976 AUiO.UlSC. njL « 1988 C3i _- a' p,pon< ATION 'AUTQ^drSC. NOV 2 0b'nH 1^<:-rldtdtU-];(^- Jjuu,^-9-9i FORM NO. DD 6, 40m, 6'76 UNIVERSITY OF CALIFORNIA, BERKEl BERKELEY, CA 94720 U.C. BERKELEY LIBRARIES CODblflMSBS A M3 I (• too a