CHAP. II.] STANDARD TRINOMIAL FORM. 439 (6.) It will be found an useful check on formulæ of this sort, to consider each beta, in what we have called the Standard Form (1.) of op, as being of the first dimension; for then we may say that & and p' are also of the first dimension, but & and ' of the second, and m of the third; and every formula, into which these symbols enter, will thus be homogeneous: a, a', a", and λ, μ, ν, ρ, being not counted, in this mode of estimating dimensions, but o being treated as of the first dimension, when it is taken as representing φρ. (7.) And although the trinomial form XV. has been seen to be sufficiently general, yet if we choose to take the more expanded form, ΧΧΧΙ. . . φρ = Σβαρ, which gives XXΧΙΙ. . . φ ́ρ = Σαββρ, any number of terms of op, such as βαρ, β'δα'p, &c, being now included in the sum 2, there is no difficulty in proving that the equations VIII. and IX. are satisfied, when we write, and ΧΧΧΙΙΙ. . . ψρ = Σναα ́δβ ́βρ, with XXXIV. . . ψ'ρ = ΣΤββ ́δα'αρ, XXXV... m = Σδαα'α"β"β'β = Σδββ ́β"Sa"a'a. (8.) The important property (2.), that the auxiliary function is changed to its own conjugate, when is changed to o', may be proved without any reference to the form Σββαp of pp, by means of the definitions IV. and XI., of o' and 4, as follows. Whatever four vectors μ, ν, μι, and v1 may be, if we write XXXVI... λί= Vμινι, and XXXVII... ́μν = V. φμφν, adopting here this last equation as a definition of the function ', we may proceed to prove that it is conjugate to 4, by observing that we have the transformations, XXXVIII. . . Sλ'ιψ'λ' = $(Vμινι. V.φμφν) = S.μι (V.νιν.φμφν) = δμιφν. Σνιφμ – Σμιφμ. Σνιφν = δμφ'νι. δνφ'μι – δμφ ́μι . δνφ'νι = S.μ (V. νν.φ ́μιφ'νι) = S (Vμν. V. φ ́μιφ ́νι) = Βλ'ψλι; which establish the relation in question, between & and '. (9.) And the not less important property (3.), that m remains unchanged when we pass from $ to o', may in like manner be proved, without reference to the form XV. or XXXI. of op, by observing that we have by XXXVII., &c. the transformations, ΧΧΧΙΧ... S. φλφμφν = S. φλψ'λ' = βλ'ψφλ = mSx'x = mSμν, because the equations III. and VIII. give, XL. .. ψφρ = mp, whatever vector p may be; so that the value of this scalar constant m may now be derived from the original linear function 4, exactly as it was in X. or X'. from the conjugate function φ'. 348. It is found, then, that the linear and vector equation, Ι... φρ = σ, gives II. . Μρ = ψσ, as its formula of solution; with the general method, above explained, of deducing m and 4 from 4. We have therefore the two identities, III. Μσ = φψσ, mρ = ψφρ ; or briefly and symbolically, III'. . . m = φψ = ψφ; with which, by what has been shown, we may connect these conjugate equations, III". . . m = φ'ψ' = ψ'φ'. Changing then successively u and v to ψ' and 'v, in the equation of definition of the auxiliary function 4, or in the formula, ψѴμν = V.φ ́μφ'ν, we get these two other equations, IV. . . – 4V. νψ'μ = m V. μφ'ν; 347, ΧΙ., V... ψν.ψ ́μψ'ν = m2V μν ; in the former of which the points may be omitted, while in each of them accented may be exchanged with unaccented symbols of operation: and we see that the law of homogeneity (347, (6.)) is preserved. And many other transformations of the same sort may be made, of which the following are a few examples. (1.) Operating on V. by 4-1, or by m ̄1ø, we get this new formula, VI. . . ν.ψ ́μψ'ν = mφѴμν; comparing which with the lately cited definition of 4, we see that we may change to, if we at the same time change to mp, and therefore also m to m2; ' being then changed to ', and ' to mp'. (2.) For example, we may thus pass from IV. and V. to the formulæ, VII. . . – φννφ'μ = Vμψ'ν, and VIII... V. φ ́μφ ́ν = mIVμν; în which we see that the lately cited law of homogeneity is still observed. (3.) The equation VII. might have been otherwise obtained, by interchanging μ and v in IV., and operating with - m-1, or with-4-1; and the formula VIII. may be at once deduced from the equation of definition of 4, by operating on it with 4. In fact, our rule of inversion, of the linear function $, may be said to be contained in the formula, ΙΧ. . . φ-μν =m-1V.φ'μφ'ν; where m is a scalar constant, as above. (4.) By similar operations and substitutions, Χ. . . φεν. φ ́μφ ́ν = mφѴμν = V.ψ ́μψ'ν; CHAP. II.] SECOND FUNCTIONS, QUATERNION CONSTANTS. 441 (5.) But we have also, XIV. . . δ. λφέρ = S. φρφ ́λ = S. ρφ ́ελ, so that the second functions 2 and 2 are conjugate (compare 347, IV.); hence, by XIII., 42 is formed from $2, as from ; and generally it will be found, that if n be any whole number, and if we change to o", we change at the same time p' to φ'η, ψ το ψη, ψ' to 'n, and m to mr. (6.) It may also be remarked that the changes (1.) conduct to the equation, XV... (8. φλφμφν)2 = Suv8. ψλψμψν; and to many other analogous formulæ. 349. The expressions, Χ ́φλ + μ'φμ + ν'φν, λ'ψλ + μψμ + ύψν with the significations 347, XX. of λ', μ', ν', and others of the same type, are easily proved to vanish when λ, μ, v are complanar, and therefore to be divisible by Sauv, since each such expression involves each of the three auxiliary vectors λ, μ, ν in the first degree only; the quotients of such divisions being therefore certain constant quaternions, independent of λ, μ, ν, and depending only on the particular form of 4, or on the (scalar or vector, but real) constants, which enter into the composition of that given function. Writing, then, and Ι... 91 = (λ'φλ + μ'φμ + ν'φν) : Sλμν, ΙΙ. . . 92 = (λ'ψλ + μψμ + ν'ψν) : Sλμν, we shall find it useful to consider separately the scalar and vector parts of these two quaternion constants, q1 and q2; which constants are, respectively, of the first and second dimensions, in a sense lately explained. (1.) Since Vλ'φλ = μνφλ – νλφ'μ, &c., it follows that the vector parts of qi and q2 change signs, when is changed to o', and therefore to ψ'. On the other hand, we may change the arbitrary vectors λ, μ, ν το λ ́, μ', ν', if we at the same time change d' to Vμ'ν', or to – λάλμν, &c., and Sauv, or Sad', to – (Suv) ; dividing then by – Sauv, we find these new expressions, ΙΙΙ. . . 91 = (λφλ' + μφμ' + νφν') : Sλμr, operating on which by S, we return to the scalars of the expressions I. and II., with and changed to d' and '. (2.) Hence the conjugate quaternion constants, Kqı and Kq2, are obtained by passing to the conjugate linear functions; and thus we may write, ν... Και = (λ'φ'λ + μ'φ' μ + ν'φ'ν) : Βλμν; VI. . . Κq2 = (λ'ψ'λ + μ' ψ'μ + ν'ψ'ν): 8μν; or, interchanging A with λ', &c., in the dividends, VII. . . Και = (λφ'λ' + μφ' μ' + νφ'ν') : δμν; where λ' = Vμν, &c., as before. (3.) Operating with V.p on Vq1, and observing that V. ρνλ'φλ = φ(λλ'ρ) – λ ́Βλφ ́ρ, &c., the vector φρ – φ'ρ, if it do not vanish, must be a line perpendicular to p, and therefore of the form, ΧΙ. . . φρ – Φ ́ρ = 27 γρ, in which y is some constant vector; so that we may write, ΧΙΙ. . . φρ = φορ + γρ, Φ ́ρ = φορ - ν γρ, where the function pop is its own conjugate, or is the common self-conjugate part of φρ and φ'ρ; namely the part, ΧΙΙΙ. . . φορ = (φρ + Φ ́ρ). And we see that, with this signification of y, XIV... V(λ'φλ + μ'φμ + ν'φν) = - 2ySuv, or XIV... Vq1 = - 2y; while we have, in like manner, if XV... V(λ'ψλ. + μ ́ψμ + ν'ψν) = 28λμ, or XV... Vq2 = -28, As a confirmation, the part po of o has by (1.) no effect on Vq1; and if we change φλ τo Vyd, &c., in the first member of XIV., we have thus, (λόγλ ́ + μδγμ' + νγν ́) – γδ (λλ' + μμ' + νν') = γSμν – 3y8μν. (5.) Since Vλ'ψ' = – φѴλφλ', &c., by 348, VII., while we may write, by (1.), (2.), and (4.), or and XVII. . . V(λφλ' + μφμ' + νφν') = – 2γδμν, we have this relation between the two new vector constants, ΧΧΙ. . . δ = – φγ = - Φ ́γ = - Φογ ; for ø, φ', and po have all the same effect, on this particular vector, γ. (6.) We may add that the vector constant y is of the first dimension, and that d is of the second dimension, with respect to the betas of the standard form; in fact, with that form, 347, XV., of op, we have the expressions, CHAP. II.] SYMBOLIC AND CUBIC EQUATION. and δ: XXII... y = (βα + β ́α' + β"α"), XXIII. . . d = (νβ'β".να ́α" + Vβ"β.να"α + νββ ́.Ναα'). 443 (7.) If we denote by yo and mo, what & and m become when is changed to $ο, we easily find that ΧΧΙV. . . ψρ = ψορ - γδγρ + δρ; XXV... ψ'ρ = ψορ – γδγρ – Ѵδρ; so that the self-conjugate part of up contains a term, -ySyp, which involves the vector y, but only in the second degree; and in like manner, XXVI... m = mo + Syd = mo - Sypy; y again entering only in an even degree, because m remains unchanged, when we pass from to p', or from y to - y (8.) It is evident that we have the relations, XXVII... mo = φοψο = ψοφο; and that, in a sense already explained, φο, ψο, and mo are of the first, second, and third dimensions, respectively. 350. After thus considering the vector parts of the two quaternion constants, q1 and q2, we proceed to consider their scalar parts; which will introduce two new scalar constants, m" and m', and will lead to the employment of two new conjugate auxiliary functions, xp and χρ; whence also will result the establishment of a certain Symbolic and Cubic Equation, which is satisfied by the Linear Symbol of Operation, d, and is of great importance in this whole Theory of Linear Func tions. (1.) Writing, then, IL... m"=Sqı, and III. m' = Sq21 we see first that neither of these two new constants changes value, when we pass from to φ', or from y to - y; because, in such a passage, it has been seen that we only change qı and 92 to Kqı and Kq2. Accordingly, if we denote by m', and m", what m' and m" become, when is changed to do, we easily find the expressions, IV...m" =m"; and V... m' = m'o - y2. (2.) It may be noted that m", or m"o, is of the first dimension, but that m' and m', are of the second, with respect to the standard form of ; and accordingly, with that form we have, and and VI... m" = Saẞ + Sa'ẞ + Sa"β"; VII... m' = S(Va'a".Vẞ"β ́+Va"a.Vββ" + Vaa'.νβ ́β). (3.) If we introduce two new linear functions, χρ and χ'ρ, such that VIII. . . xVμν = V (μφ'ν – νφ'μ), |