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 ALGEBRAIC so that 0ia, possess, each of them, the invariant property. There are thus six invariant symbolic factors, viz., (ab), aa, {pea), ax, xg; and from these types we are able to form invariant symbolic products. We must note the results— (£a) = (xa)', a£ = sin2wax; (a/3) = sin2 w(a6) (xif) = cos w(a;2 - x^); (aa) = cos - a2). Since sin w is the discriminant of we may regard and (a/3) as reducible, and we take as the general form of invariant— (abyiacji2(jbcf‘3.. ■ ftj,3- ■■a^aifby3... x (pcaf^xpy^xy)1*3...43...xf, which should be contrasted with the corresponding expression for orthogonal invariants. If this be of order e and appertain to an nic + 2Z + — e, the pair of symbols a, a must together appear n times ; so also for 5, p ; c, 7 ; ... We have the symbolic identities 2

313

FORMS

0 I m =a»m-Tca..k i(m-k)[ '—(t 0 ,-y2W ^— v’toi dk7 (m - k) / cm-kr(ya)k, (xa) =(xa) + 2/2 ! k)( 0 0 m_ m-k k (t?i- Z:)! m 1 V7?10^ + ,?20^/ _ax ay> (m - A;)! (3 , 9 ^7 t m-k/ k 7?1 0^ + 1?20r / a:a =^a;a' ^ ^> (m-k) k)f ■ 22a;f( 2/ 8—-y dyj| kam=am-ki(ya)-Jc , y-^|sm 1£ 2s x x k (m-k) k)f 0 ^ ^r m ( ,m-k •y> m (m-k) k)f 0 d kaxm axm-kr{y )k T~ ( ^ ~ ’ (m — k) f • o / 0 0 nto / V,m-k_k. ’’y> sHsm "(’’■sr’'*®;)) y£=xv> aabp - apba = (ap)(ab) — sin2 ufab)* ;

and, as regards (xa)m, only 0 3,00 y 'S^+^ and 5,105 ■ ’ ciaxg - (xa f — axa^ = sin2w a|; and we have the processes (ab)ca + (bc)aa + (ca)ba = 0 ; (m-k^-k^.f 0 0 0 JLfc2 m and many others, derivable from these, which are of assistance m!Ti V^'^dxJ a* in the reduction of symbolic products. We may give here some = a™-kl~hlaj(ya)hl ■ simple examples of Boolian invariants. fc Ex. gr. For the linear form ax—bz —, we have 0 (m - Aq - k2)fA ni _00 2_00 fci/ i 0 m! 0a:2y v^~y*Kj ( } (i.) ax =, = (ax)m ~kl~ ^(ay^Oy1. (ii.) £C^=a3^+2cos wjc1a;2 + a:|, 2 (iii.) sin w, (xa)m~k(ya)lc, satisfy the partial differential The two forms equation (iv.) (xa) = (»] - a0 cos u>)x1 - (a0 - ax cos w)a;2, 02ii1 02M'1 _ -0; (v.) ap = al-2,a0a1 cos w+ al. 0*102/2 ?*232/i For two linear forms (v.) yields the simultaneous invariant and the two forms a™ k(ya)k,(xa)m the partial differential equation a0b0 + a1b1 - cos w(a0&1 + a^g). 2 2 0 «9 0 Mo — 0; cxy:y1 + 3aj20?/5 For the quadratic form a2 = &!=..., we have the system to verify this statement recall that (i.) «|» (ya) = (ya), an = sin2 • (ii.) xg, 1: l wr Taking, as operand, any form b y we write (iii.) sin2w, 2 0 = (m - k)k cxpdy 0x00i/ (iv.) ^(ab)3=a0a2-a{, 2 2 rj1 2 0 ^ (m - A)Ax / 30?7 dx32dri J ’ (v.) (xa)ax = («! - a0 cos w)a:f + (a2 - a0)x1x2 + (a2 cos w - a^x%, 1

1

2

2

which is the Jacobian of a2 and xt; (vi.) a.a=a0-2coswa1 + a2;

and we can establish the relations

and it may be shown that all other covariants are rational integral functions of these six forms. Again, take the quadratic a| = Sx=... and the linear form a'x=b'x=...; besides the forms, appertaining to the linear form and quadratic separately, which have been already given, we have four additional forms involving the coefficients of both forms, viz. :— 2 2 / (aa’)(ab’) = a{}al - 2a1a[|a1 + ajiQ,

d^O^iypr - (- )ab)}a^ " fcl" %p)n ~ kl ~ ^ Ox^' Xa^by - (- ^M)^£U)m - fcl - X ~ ^ A (f2 cosec2 W)fcjO^(*a)m(^)U = (ab)klaf(xa)m ~ kl ~ %pf ~ fcl" ^ From these relations, by putting 2/=:e, we derive the processes of transvection, and we may write 0™, O*1 ’ ^ = iabW^ ~ h ~ Snx ~ k' ~ h, {a™, (xp?}1*1' k2=a}(ab)™“k'"H”~kl~k2, {(xa)m, 6X }fcl;l2=a$(ab)xa)m~kl~ { (xaf1, (xpf } > ^ = (ab)kla^(xa)m ' " xp)n ~ kl ~ h. Of order k=k1 + k2 there are A + l transvectants, and k may have any value not greater than the least of the numbers m, n. The process is practically equivalent to the performance of the differential operation 00 _ 00fci/_0/ dtp df 0

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