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ALGEBRAIC
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3 3 to , • s v-TO TO e 1111(1i X dk) ~Xl^~ 23| ’ Up0n ax ^ ^Sm' ^ / 3 3 TO / . o v—TO TO—1/ 'Wl =m(Sm “} 'Xa' ^ Calling the operation 3 1 ^ X2 ~ oi} ^•0fs Tc m-k when performed upon a™, P, we have P a™—(xa) a~. (m-Ic) Sin . M U(X , 3 X 3 Vk is, in fact, ^- WJ ’ ml the operator being expanded symbolically. Similarly 3 .to / vm-k k k(™X (-)' d£J Xa =xa) Hence (4)’ has the effect, practically, of converting a™ into (xa)m; this can be otherwise accomplished by writing -~x2,xl in place of x1,x,2, and changing a into a. (xa)m is similarly converted into os™, only here a is changed into a. If the operand be os™ k‘(xa)^ it becomes converted by the operation into (xa)m ka%., The covariants of the series (£ca)fca™ k are all reducible except os™ and (a;a)os™_1, as may be seen from the identities . znwf£caJos r k m-k +fa:a) , / k+2am-k-2 sm x x / sk m-k-2 ( ■ 2 wos Z +, (a:aj , j = {xa) ax |sm x / kam-k-2xg ; —aa{xa) x which exhibits the reduction of (a?a)fc+2os™-fc~2. Next consider the operation of 32 32 dx^x dx2c%2 upon os™ and (xa)m. We find f S2 +, 32 ka xm_ Vil akaxm-2k
dxxdh dx&J ’ ~ {m-2ky °>
/ 32 32 ml m-2k (xa) = (m - 2k) !l°a (xa) . BcrjSfi dx2c£2 2 2 k .' (j) ff 3 3 to — 1/{xa) (m —1)! ic to — 2k — 1/™— {Xa)
dxjd^x dx2d^2
=(m-2k-iya°-ax Before proceeding to give examples of these processes it must be observed that the transvectant / to , o■Ln'.ko (a x x) is always reducible where k2 is even. This arises from the symbolic identity sin2 w(os&)2 + osj3=a0&£ ; for, thence, / TO,bt»,2 , . o / to ,un2,0 / m-2/, (a )(bpb7n-2 ), x x)’ +sm-u(ax ,bx) ' =(o,aax x ( TO ,7i,4 ,0-02 / to ,tc,2 , . 44 / to ,m,0 WA; +2 sm + sm ,bx) f 2 2,w-4 ~aa.ax JyJfPx ), with similar identities which establish the theorem. -Sc. gr- Consider transvectants of a,x over b% (o.l,b$)^ = (ab)a%bl, (> 2^x)° ’1 — ospct’xbx 2') which, for the single cubic osx gives (a — 2os5 cos u + b )^ + 4 {ab — (osc + b2) cos w + bcx?y + 2 {osc + 252 - (ad + 5&c) cos w + 6c? + 2c2} a;2;/2 + 4{6c-(6c? + c2) cos w +2 cc?} ccy3 2 4 + (c - 2cc? cos w + c? )y , which is unchanged when os, 6, c, c?, a:, y are changed into d, c, b, a, y, x respectively. (al,bl)2,2=(ab)2axbx, (ax>bfy1’1— (ab)apaxbx, which vanishes for the single cubic, (ax, 6|)3’0=(a6)3, G4> ^x)2’1=(«6)2«/3 ; which for a single cubic, is 2 {(osc - 62) - (ad - be) cos w + (6c? - c2,}.
FORMS 1,2
(a|, 6|) = («6)a|; which vanishes for a single cubic, but, for two cubics, is ab' - ba' + 2(6c'— c6') + (cdr - dc’) — 2(ac' — cos' + 6c?' - c?6') cos w + (ad' - da’ + 6c' - c6') cos2 w. 3 (c4,6|)°’ =os£2 which, for a single cubic, is os + 362 2+ 3c2 + c?22 - 6(os6 + 26c + cd) cos w + 6(osc + 6 + 6c? + c ) cos2 co - 2(osc? + 36c) cos3 w. Again the linear covariant osaax is (os - 26 cos w + c)x + (6 - 2c cos w + d)y ; and the linear covariant na(xa) is {6 + o?- (os + 3c) cos w + 26cos2o;}as - {os + c- (36 + c?)cos w + 2ccos2a>}2/. An invariant of the cubic which is of the fourth degree in the coefficients is u c a b ll y8 which is derived from 2 (ab) (cd)2(ac)(bd) by an obvious method of universal application. Generally, from any covariant given by its symbolic product in the theory of the general linear substitution, we may derive a conjugate form by writing mv for (mn) and (xy) for mx, and we are permitted to do this whether or no the former ultimately vanishes ; the conjugate cannot vanish. In the general Boolian theory, in any symbolic product, we can pass to a conjugate form by writing mv for (mn), (mn) for mv, (xy) for mx,mx for (xy). A form, containing only a,a or x^, will have no conjugate, and may be called a non-conjugate form. Other forms may have vanishing conjugates. The differential equation, satisfied by Boolian covariants of the binary form ax, has been shown, by Sylvester, to be cos
"(“‘s; - a42)+(“4; - *4,)+L=0 0 0 where L = 0„ - 0,, - cos w i nan^=- + (n- 2)a1-=r ^ OCf'Q + (n - 4)a2^ + ...+(»- 2)aH_1—+ nanjL } ; da2 8an_i oan J if then a covariant be operation gives 2[(e-2£)coswCfcaq ^x^ + C^^e - k)x{ k 1a32+1 +^-^.0^ = 0; leading to — (& + l)Cj:+i + {L + (e — 2k) cos w} C* + (e — ?;+ l)Ci_i = 0, and thence to the series of relations — Cx+ (L + ecos w) Cq — 2C2 + {L + (e — 2) cos w} 0^ +e Cq = 0, — 3O3 + {L + (e — 4) cos C2 + (e — l)Cj:= = 0, — 4C4 + {L + (6 — 6) cos w} C3 + (c— 2)C2 0, which may be written C1 = (L + ecos w)C0, 202= {L2 +2sin2w + (e- 1)(...)}C0, 6C3= {L(L + 4 sinV) + (e - 2)(...)}C0 24C4= {(L2+ sin2 w)(L2+ 9 sin2 w) + (e - 3)(...)}C0. Hence for leading coefficients of covariants of orders 2s and 2s +1 we have annihilators 2 2 2 2 L(L +222 sin w)(L +242 sin w).. .(L2 2 + 4s22 sin2 w), 2 2 2 2 2 (L +1 sin w)(L + 3 sin w)... {L + (4s + 4s +1) sin w} respectively. These conditions are necessary but plainly not sufficient; for every leading coefficient of a covariant, of even order ^25, satisfies the first operator and, of uneven order ^2s +1, the second operator. We may say, however, that, if the order be 2s, either (L2 + 4s2 sin2 w)C0 = 0, 2 2 2 or else (L + 4s sin w)C0 is the leading coefficient of a covariant of even order ^ 2s - 2. Similarly, for uneven order 2s +1, either {L2 + (2s + l)2 sin2w} C0 = O, 2 2 2 or {L + (2s + l) sin w}C0, as the leading coefficient of a covariant of uneven order ^25 - 1.
