XXVI. . . ψν = Γλνμβλμ – Γλμβλνμ - g (λSμν + μλν) + gr ; and therefore XXVII. . . mFv = Sv↓v = Sλ vμ v SXμ + (Sλvμ)2 — 29S\v Sμ v + g3v2 = (g2 − X2 μ2) v‚2 + X2 (Sμv)2 + μ2 (S\v)2 – 2gSXvSμv; which last, when compared with 360, VI., is seen to be what we have called a bifocal form its focal lines a, a' (360, (1.)) having here the directions of A, μ, that is of what may be called the cyclic lines of the form XXIII. The cyclic and bifocal transformations are therefore reciprocals of each other. (7.) As another example of this reciprocal relation between cyclic and focal lines, in the passage from fp to Fv, or conversely from the latter to the former, let us now begin with the focal form, XXVIII. . . fp = Spøp = (Vap)2 + (Sẞp)2, in which a and ẞ are supposed to be given and real vectors. (11.), XXIX... and therefore, 359, XXX., We have now, by 359, [v=pp = - aVap+BSẞp, m = a2($aß)2, XXX... mFy = a2 (Saẞ3)2 Fv = Sr&v = SavßvSaß + (a2 − ß2) (Sav)2 =-v2 (Saẞ)2 + Sav ((a2 – ẞ2) Sav + 2SaßSẞv) an expression which is of cyclic form; one cyclic line of F being the given focal line a of fp; and the other cyclic line of Fy having the direction of ± (a3 + Buß), and consequently (by 359, LXV.) of a', where a' is the second real and focal line of fo. (8.) And to verify the equation XVIII., or to show by an example that the two functions fo and Fv are equal in value, although they are (generally) different in form, it is sufficient to substitute in XXX. the value XXIX. of v; which, after a few reductions, will exhibit the asserted equality. 362. It is often convenient to introduce a certain scalar and symmetric function of two independent vectors, p and p', which is linear with respect to each of them, and is deduced from the linear and self-conjugate vector function Op, of a single vector p, as follows: I. . . ƒ (p, p') =ƒ (p', p) = $p'¢p = SpÞp'. With this notation, we have They are in fact (compare the Note to page 468) the cyclic normals, or the normals to the cyclic planes, of that surface of the second order, which has for its equation fp = const.; while they are, as above, the focal lines of that other or reciprocal surface, of which v is the variable vector, and the equation is Fv = const. II...f(p+P) =ƒp + 2ƒ (p, p') +ƒp'; and as a verification, VII. . . ƒ (xp) = x2fp, 485 a result which might have been obtained, without introducing this new function I. (1.) It appears to be unnecessary, at this stage, to write down proofs of the foregoing consequences, II. to VI., of the definition I.; but it may be worth remarking, that we here depart a little, in the formula V., from a notation (325) which was used in some early Articles of the present Chapter, although avowedly only as a temporary one, and adopted merely for convenience of exposition of the principles of Quaternion Differentials. (2.) In that provisional notation (comp. 325, IX.) we should have had, for the differentiation of the recent function fp (361, II.), the formulæ, dfp-f(p, dp), f(p, p')=28p'pp; the numerical coefficient being thus transferred from one of them to the other, as compared with the recent equations, I. and V. But there is a convenience now in adopting these last equations V. and I., namely, dfp=2f (p, dp), ƒ(p,p′) = $p'pp; because this function Sp'op, or Spop', occurs frequently in the applications of quaternions to surfaces of the second order, and not always with the coefficient 2. (3.) Retaining then the recent notations, and treating dp as constant, or d2p as null, successive differentiation of fp gives, by IV. and V., the formulæ, VIII. . . d2fp = 2ƒ(dp); d3fp=0; &c; so that the theorem 342, I. is here verified, under the form, or briefly, IX... efp = (1+d+}d2) fp X... εdfp = f(p + dp), =fp+2ƒ (p, dp) + füp ; an equation which by II. is rigorously exact (comp. 339, (4.)), without any supposition whatever being made, respecting any smallness of the tensor, Tdp. 363. Linear and vector functions of vectors, such as those considered in the present Section, although not generally satisfying the condition of self-conjugation, present themselves generally in the differentiation of non-linear but vector functions of vectors. In fact, if we denote for the moment such a non-linear function by w(p), or simply by wp, the general distributive property (326) of differential expressions allows us to write, I. . . dw(p) = $(dp), or briefly, I'...dwp=ødp; where has all the properties hitherto employed, including that of not being generally self-conjugate, as has been just observed. There is, however, as we shall soon see, an extensive and important case, in which the property of self-conjugation exists, for such a function ; namely when the differentiated function, wp, is itself the result v of the differentiation of a scalar function fp of the variable vector p, although not necessarily a function of the second dimension, such as has been recently considered (361); or more fully, when it is the coefficient of dp, under the sign S., in the differential (361, I.) of that scalar function fp, whether it be multiplied or not by any scalar constant (such as n, in the formula last referred to). And generally (comp. 346), the inversion of the linear and vector function in I. corresponds to the differentiation of the inverse (or implicit) function w1; in such a manner that the equation I. or I'. may be written under this other form, (1.) As a very simple example of a non-linear but vector function, let us take the form, III... σ = w(p) = pap, where a is a constant vector. This gives, if dp = p', so that op' and IV...op' = pdp = dwp = p'ap + pap' = 2Vpap'; VI... 'λ = 2Vλpa=2Vapλ, p'p'= 2Vapp' ; 'p' are unequal, and the linear function øp' is not self-conjugate. (2.) To find its self-conjugate part pop', by the method of Art. 361, we are to form the scalar expression, VII... fp' = Sp'¢p' = p22Sap; of which the differential, taken with respect to p', is VIII... dfp' S. pop'dp' = 2SapSp'do', giving IX... Pop' = 2p'Sap; and accordingly this is equal to the semisum of the two expressions, IV. and VI., for op' and its conjugate. (3.) On the other hand, as an example of the self-conjugation of the linear and vector function, X... dv=dwp =ødp, when X'. . . dfp = 2Svdp = 28. wpdp, even if the scalar function fp be of a higher dimension than the second, let this last function have the form, XI... fp=Sqpq'oq′′p, q, q', q′′ being three constant quaternions. Here XII. . . v = wp = §V(qpq'pg′′ + q′pq"pq + q′′paea') ; XIII... dv = pdp = pp' = {V(qp'q'eq′′ + q′pq′′p′q) + ¿V(q′p'q′′pq+q′′pap'q') + {V(q′′p'qpq'+qpq′e̱ʼq′′) ; CHAP. II.] LINear function of a quaternion. and XIV... Sλ¢p'=}S.qpq′′(^qp′ + pʻqλ) + &c. = $p'pλ ; so that p', as asserted. 487 (4.) In general, if d be used as a second and independent symbol of differentiation, we may write (comp. 345, IV.), XV... ddfq=dd fq, where fq may denote any function of a quaternion; in fact, each member is, by the principles of the present Chapter (comp. 344, I., and 345, IX.), an expression for the limit,* XVI. . . lim. nn'{ƒ(q + n ̄1dq+n'−1dq) − f(q+n−1dq) − f(q+n ́ ̄1ôq) +ƒq} . (5.) As another statement of the same theorem, we may remark that a first differentiation of fq, with each symbol separately taken, gives results of the forms, XVII... dfq=f (q, dq), _dƒq=ƒ (q, dq) ; and then the assertion is, that if we differentiate the first of these with d, and the second with d, operating only on q with each, and not on dq nor on dq, we obtain equal results, of these other forms, For example, if XVIII. . . ddfq=f(g, dq, dq) = ƒ (q, dq, dq) = đôfq. XIX... fq=qcq, where c is a constant quaternion, the common value of these last expressions is, and XX... ddfq=ddfq=dq.c. dq + dq.c.dq. (6.) Writing then, by X., XXI... dfp = 2Swpdp, &fp=2Swpop, we have the general equation, XXIII... S(dp.pdp) = $(dp.pdp), in which dp and dp may represent any two vectors; the linear and vector function, p, which is thus derived from a scalar function fp by differentiation, is therefore (as above asserted and exemplified) always self-conjugate. (7.) The equation XXIII. may be thus briefly written, XXIV... Sdpdv = Sdpdv; and it will be found to be virtually equivalent to the following system of three known equations, in the calculus of partial differential coefficients, XXV... D2Dy = DyDx, DyD2 = D2Dy, D-D2 = D2D.. 364. At the commencement of the present Section, we reduced (in 347) the problem of the inversion (346) of a linear (or distributive) quaternion function of a quaternion, to the * We may also say that each of the two symbols XV. represents the coefficient of ly1, in the development of ƒ (g+xdq+yồq) according to ascending powers of x and y, when such development is possible. corresponding problem for vectors; and, under this reduced or simplified form, have resolved it. Yet it may be interesting, and it will now be easy, to resume the linear and quaternion equation, I... fq=r, with II...f(q + q)=fq +ƒq', and to assign a quaternion expression for the solution of that equation, or for the inverse quaternion function, with the aid of notations already employed, and of results already established. (1.) The conjugate of the linear and quaternion function fq being defined (comp. 347, IV.) by the equation, IV... Spfq=Sqfp, in which P and q are arbitrary quaternions, if we set out (comp. 347, XXXI.) with the form, V... fq=tqs+t'qs' + . Etqs, in which s, s',... and t, t', ... are arbitrary but constant quaternions, and which is more than sufficiently general, we shall have (comp. 347, XXXII.) the conjugate form, whence VI...fp= spt + s'pt' + ... = Σspt; VII... ƒ1 = Σts, and VIII... ƒ'1 = Est; it is then possible, for each given particular form of the linear function fq, to assign one scalar constant e, and two vector constants, e, e', such that IX. . .ƒ1 = e + ɛ, ƒ'l=e+ε'; and then we shall have the general transformations (comp. 347, I.): and X... Sfq=S.qƒ'1 = eSq + Se'q ; XI... Vfq=eSq + V.fVq = &Sq +¢Vq; XII... fq= (e + ε) Sq + Sɛ'q + ¢Vq; in which Se'q=S. 'Vq, and Vq or VfVq is a linear and vector function of Vq, of the kind already considered in this Section; being also such that, with the form V. of fq, we have XIII. . . øp = ΣVtps. (2.) As regards the number of independent and scalar constants which enter, at least implicitly, into the composition of the quaternion function fq, it may in various ways be shown to be sixteen; and accordingly, in the expression XII., the scalar e is one; the two vectors, ɛ and e', count each as three; and the linear and vector function, Vq, counts as nine (comp. 347, (1.)). (3.) Since we already know (347, &c.) how to invert a function of this last kind , we may in general write, XIV. . . r = Sr + Vr = Sr+pp, where XV... p=4-1Vr=m21&Vr; the scalar constant, m, and the auxiliary linear and vector function, V, being deduced |