ALGEBRAIC In general for a form in n variables the Hessian is 0y ^x_ 32/_ dx cx-fix^ "'dxidxn 02/ 02/ 02/ dx1dx2 dxl ’ dx2dxn = H^'2)=H; _02/_ _0y C2/ 0x13iC„ dx2dxn ca;^ and there is a remarkable theorem which states that if H = 0 and n = 2, 3, or 4 the original form can be exhibited as a form in 1, 2, 3 variables respectively. Ex. gr. If (a&)2a™-2i™_2 = 0, for the binary form a™, the theorem states that the form is a simple mth power (a^ + and therefore, by linear transformation, depends only on a single variable. It may be verified that, if ax =/, i(an/)i2H ^—J-J ==/./!-(/y y y. 2; and now assuming/ to have the form a£.</>£ where ax is a linear form and <px a form of order a ; p + <T=m, fy=max
a <t>x r
y ' maP^x
a
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297
co variants is connected with the form /+ X0, where/and 0 are of the same order in the variables and X is an arbitrary constant. If the invariants and covariants of this composite quantic be formed we obtain functions of X such that the coefficients of the -various powers of X are simultaneous invariants of / and 0. In particular, when 0 is a covariant of /, we obtain in this manner covariants of/. Consider, for example, the cubic f=al> and take for 0 its cubic covariant Q = {ab)cb)c%a,x. If J be any covariant of /, denote by J/+xQ the corresponding covariant of/+XQ. There are four fundamental covariants of /, viz. /=4. A = {f,f')2=(ab)2axbx, Q = (/, A)1 = (ab)2(cb)c%ax, B, = (A, A')2=(ab)2(cd)2(ac)(bd). To find the fundamental system of/+XQ we have //+xq=/+xQ> Ay+X2Q = (/+ XQ,2 / +2XQ)2, 2 = (/,/') + 2X(/,Q) + X (Q,Q') . To reduce this expression take the first polar of A = A|, 2 A* Ay = (ab)2axbv + (ab)2aybx,
m{m — 1)/^=p(p ~ 1 ) S y Px + %P x ay<Px fiy or /
P , o’ — 2 0^,
,2 + <r((r-l)a^ tkxk.y = {ab)2axby. leading to Herein, A . A^ being symbols equivalent to A^ A2, substitute 0 = m2(m-l){/./2-(/J) } A ,A 12 for -x2,xi and multiply by £:>y, so that = {m/)(p-l)-(7n,-l)p2}a2p 2a^0 {tk)^y={ab)'1^ )byAy. 2p 1 a + 2pcra " o2/^. <£ + { m(r((r - l- (m - l)<r (0y) } x 5 Now • and, since each term on the right must vanish separately, m — p, Ql=(f,A)' = (aA)a2xAx, (r=0, which proves the theorem. SQ?Qy=(aA)(ct%.Ay+2(la£lyAx) , It has been established above that every symbolic product can be expressed by means of transvectants. We have now to show = (uA) "I 2>CL^Ay "b 2^/b/A^. *— dXAy) ^ that every invariant can be expressed by means of symbolic products. It suffices to prove this for invariants as will appear. = 3(aA)a%Ay + 2{xy){aAfax; When an invariant is expressed in terms of the umbrse a/a2; Z>j, b2 ; Cj, c2 ;... it is homogeneous and of weight wc in the senes of and, since it is easy to see that (/, A)2= (aA)2®* vanishes identisymbols «!,&!, Cj,... and also in the series a2, &2> 2>• • • • Let then cally, an invariant j of weight w be written symbolically Qx^y~(a^ax^y » j = (Si*! -(- S.2bl + SgCj + ... + hc2 + • • • )W > and, herein writing b2, - bx for xi, x2, transforming, by linear substitution, J = (p.)wj ; or (bQ)2Qy = (aA)(ab)2Av; J= + s2bk + s3cK+... + «3V +••■)"’ or (bQ)2bxQx=(aA)(ab)2bxAx, = {(sjQ!] + s26j + SgC! + .. .)Xi + (S]a2 + s2b2 + s3c2 + • • • )X2} w C i a C = (AA)AXAX=(A> A)1; x {(<iai + <2^1 + ^3 l+ •••)/ l + (^l '2 + ^2 + ^3 2+ , , , sW, ww 2 (/, Q) =o. —+7*2^2) (^iMi+ Again, since where = s^i + s2bi + S3C1 + ..., QxQy=(aA')axAy’ p2—sla2 + s2b2 + s3c2 + ..., g-y—- "l" ^2^1 d" ^3^1 "b •. • , substituting Q2, - Qj for xv x2, g2 = ty(X2 -f- t2b2 -f- ^3^2 "b ... 5 and (QQ)2QyQy=(aA)(aQ)2AyQy. But = Syt2{ab) + Syt3(ac) + ... + s2t3(bc) + ... ; (/, Q)22 = (aQ)2a*Qx=0, . •. (aQ) (aa:Qy + a.yQ*) = 0, = W now, since (X/a) y, if we operate with and 2 2 2 3 0 (aQ) (^Qy - «yQx) = (aQ)3(*2/) ; . •., by addition, dd/J.2 0X20^i 1, 3(a;y) ; we obtain (aQ)%a:Qy = ^(aQ) W w-1 w (pq)pK w-1 = w(w + l)(X/i) V> . •. (aA)(aQ)2A^ = {aq?A2y=. A2, .w-1. / V ww-1 — 1 w-1 w ■ = (X/i) j ; since it is quite easy to show that (aQ)3 = (A, A')2, w + 1 (PV)Pk 9V again operating (Q, Q')2=2r-a; W-l. ,2 w-2 w-2 x . w +!^{pq)Pk ^ =( w and finally and continuing to operate we arrive at V+ A.Q = A + X2 • 2R - A' 1 w Similarly J = w + (pq) , Q/4-A.Q = ( /+ ^Q; A + ^RA)1, = (s^ab) + s/ (ac) + ... + s ( (bc) +
3
ffa
FORMS
23
= (/ A)1 + X(Q, A/r^RC/, A^ + ^RCQ, A) , establishing that the invariant j is expressible as a sum of sym1 1 bolic products, each product involving w determinant factors. The Formf+<t>.—An important method for the formation of wherein (/, A) = Q, and we have to reduce (Q, A)S.. I. — 38
