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 ALGEBRAIC

296 1

the operator, for the k * transvectant, becomes effectively

FORMS

Now for yi, y2 write C2, — Ci and multiply by cx.

l, 4.}’; dc + k and disregard the factor g~. 'We can transform the operator by expressing, by means of one or if a—b%—c% =/=/ =/, this is of the fundamental identities, (bg), (eg), ... in terms of (ag) and determinant factors which are free from g. Thus (/>/)v'Tn{(/./)vT= -(ag)( da + b~ db + cdc i+~) a V a~ Now (ai+bTb+4c+-y-m/’ where m = m1 + m2 + m3+... ; therefore the first transvectant is i ~(ag)mfgn - hlLoT'g11 {(ab) an important form, because it shows that the transvectant can be broken up into two portions, in only one of which does the symbol g occur in the determinant factors. Conversely, if we have a form involving m + 1 symbols in the determinant factors, in one of which g occurs only once, we may exhibit it by means of the transvectant of a form not involving g upon the original form, and of a form whose determinant factors involve only m symbols and do not involve g. This theorem is of great significance in the proof of Gordan’s theorem concerning the finite number of covariants of a given form, and we must further generalize it. The operator for the second transvectant is which may be written either as d + b,.

'db

dc

}]

da dc Taking the first form we find the second transvectant to be w(m - i^O/2/. g11-2 together with a number of terms in which (ag) occurs atthmost once. Hence a form, containing m + 1 symbols and the m + l symbol g only in the form (ag)2, can be exhibited by means of the second transvectant of a form in m symbols over the original form and of a form which involves (ag) at most once ; and now, combining the former result, we find that a form in +1 symbols which involves the m + lth symbol g only in the form (ag)2, can be expressed by means of forms involving m symbols, and by first and second transvectants of such a form upon the original form. The second form of operator above written leads to the same conclusion in regard to the factor (ag)(bg). Similarly it is proved that a form in m + 1 symbols which involves the determinant factors (ag), (bg), ... to the order k is expressible by means of forms in m symbols and 1st, 2nd, ... kth transvectants of such forms upon the original forms. Every symbolic product is expressible as a sum of transvectants. Ex. gr. Gordan takes as an example g>=(abf(acf(bc)2axbxcx; substituting for Cj, c2 the cogredient variables — y3, we obtain (xy)(ab) a^zfiyby which is a member of the fourth polar of (ab)2a%bl multiplied by (xy) ; putting (a6)2ay|=G® = (a6)2F® we find 2 UxbaPyby = (^x)y + 5(^2/) {(ab)a%bl} y > . •. (abfa^ybl=(G® )* + l(xy)(H.*yy, where

= (ab)3a2b2.

In general it suffices to say that we transform one set of umbral symbols to a new set of variables, and make use as above of the properties of polars. The theorem establishes that transvectants are as inclusive as symbolic products in general. First and Second Transvectants.—A few words must be said about the first two transvectants as they are of exceptional interest. Since, iff=a%, 4> — b^., j 1/0/00 0/ 00 ... m-1 m-1 T (/, <*>) -—< mn OX} ^cx2 = dx2 dx1 (ab)ax bx =J, the first transvectant differs but by a numerical factor from the Jacobian or functional determinant, of the two forms. We can find an expression for the first transvectant of (/ 0)1 over another p form c. For (m + ?i)(f, 0)i = nf. 0i + m/i.0, and 7YI — "1.— 11 t / /•, 1 y^x^y ~ My^x)® ^y' X bx =(xy)(f,(t>) ; fv'y't'$ (P'vtPy 71 n Put m - 1 for m, n-1 for n, and multiply through by (ab); then {(/»0)}:i=(a^)a™ _ 2aybx ~1 + m “ t+ n to - 1 = (ab)ax b„ b„V - m+n -r4,xy){f,4>) Multiply by and for y2 write c2, - c1; then the righthand side becomes / tp-l —1 c t, ( 2 (ab)(bc)a bx cx +, inTO x + n_2 x(f> P) 1 of which the first term, writing cp=f/, is p-2, . O'x ®x 1 m-2,n-2 Cxp-2 x-{ (befal + (abfcl - (acfbl) — - ^ bx -c' ,n, .2 m-2 p-2) 1 faxvi,,^2.n-2v-2 p.a^) , ,2am-2,n-2 — 2l 'bc) bx cx +, cx( bx — bx(ac) ax cx j x =

if,'Pf-'f'-(/1)2.0} ;

and, if (/,0)1 = £™+'l~2, {(/^^crwr^vrV and this, on writing c2, - c1 for yi, y2, becomes (kc)k.m+n-3 p-l_ and thence it appears that the first transvectant of (/ 0)1 over 0 is always expressible by means of forms of lower degree in the coefficients wherever each of the forms /, 0, 0 is of higher degree than the first in x1( x2. The second transvectant of a form over itself is called the Hessian of the form. It is / 7A2 m-2,to-2 tt2to-4 tt. {f,f)=(ab)a bx =H =H; x X unsymbolically it is a numerical multiple of the determinant 02y 02^ / 02y 2 dxfdx2~dxdx.) ' ^ is als0 the first transvectant of the differential coefficients of the form with regard to the variables, viz. ( ua< ra c 0^"’ 0^") ’ ^'°r l ^ ti it is the discriminant (ab)2 and for the cubic the quadratic covariant (ab)2axhv.