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 ALGEBRAIC

FORMS

299

Adding the first and fourth equations we obtain the complete system of equations satisfied by an invariant. The nd - e = 2w, fourth shows that every term of the invariant is of the same weight. Moreover, if we add the first to the fourth we obtain the invariant relation connecting the four numbers n, 9, e, w. dj 2w. a. The satisfaction of the differential equations is not only necessary, but it is also sufficient. To establish this, in the case of inoak 71 k variants let J be any solution of the differential equations, in where 9 is the degree of the invariant; this shows, as we have which the quantities A* are independent variables, and K any before observed, that for an invariant other solution. We obtain 0K 1/„0J t0K ,Xl + -1 2, t-0J t -2)a2=°, w — ^nd. K V 0A, 0A 0^i 0^1 1 0K The second and third are those upon the solution of which the k|^--j^)a = 2 IYk— K2 0/42 C^J 2 0, Kff 0^ 0/q theory of the invariant may be said to depend. An instantaneous ^dJ t0ka „ 0K v 0J t-0K deduction from the relation w = -nd is that forms of uneven orders 1/^=0. 2 K K2 K—-J— 0A, 0Ai 0XJ'Ml ^ K possess only invariants of even degree in the coefficients. The 0K 1_ „0J t0K ^ 1 2 Kf- ■ J^V=0; two operators — Mi + K d/Xo t K2 K5—— 0Mi 0/h/ fi = <Zq—=- + 2a1-=- + ... + na,n-i—=r or, if J = KL, U&l 0C&2 OCtn 0L. 1
 * 0;

Xl+ 0 = nal^+(n- l)a2^-+ oAi 0X i? ^X2_0; 2 1 CCCq CCi-^ CCln— 0L 0L . 0E 0L _ have been much studied by Sylvester, Hammond, Hilbert, and ^i + ^-0’d^1 + dy2^ °Elliott (Elliott, l.c. ch. vi.) It has been established that, if As (A/x) does not vanish, these equations necessitate F(a0, a^.-.an) be any rational, integral, homogeneous, and isobaric function of the coefficients, 0F _ n — 0 0X ^—0 —=0 0x"r ’ 2_ ’ F^Aq, Aj,...A: showing that L is independent of Aj, A2, y-i, /x2. = . w, nd - 2wexp {^0 + [xS^2}F(ao,al,•••an), We may put K = (Ayu)“’ so that J = (A/4)“L a result from which all the important facts concerning invariants and noww putting A1 = 7x2=l, =/t1 = 0, L becomes equal toy, and may be deduced. The analogous partial differential equations satisfied by any J = <a) j ; hence this relation is satisfied by every solution j or the differential equations. If we have several binary quantics we covariant j may be established as follows. We have have similarly the two operations 0 fi_a:2 where J = F{A0, A^...(ayt), (A#)} SV 2‘ y=F(a0, aq, x2) the new variables being ^ = {xy) = ^2*1 ~ Mi*2 2i° 3310*2’ f2 = (Xa3)= — A^ + X^. which cause the vanishing of a covariant and the invariable We then obtain relation 2710 - 2w—e 2(D-^)aT,+=(w+j » k connecting the numbers iq, 01( %, d^.-.w, eThe Evectant Process.—If we have a symbolic product, which 2(d^)£+^d^i)^=o» contains the symbol a only in determinant factors such as {ao), we may write x2, - xx for ax, a2, and thus obtain a product in k which {ab) is replaced by bx, {ac) by cx and so on. In particular, 2/D A — n 3J . when the product denotes an invariant we may transform each ( /xA fc)g^ + - 0^ of the symbols a, b,...to x in succession, and take the sum ot the k resultant products ; we thus obtain a, covariant which is called the first evectant of the original invariant, dhe second evectant is obtained by similarly operating upon all the symbols remaink ing which only occur in determinant factors, and so on for the 2_ 0J 3J. .T higher evectants. 2 2 (7l )A + =(w,+e)J Ex. gr. From (ac) (6c?) (ad)(&c) we obtain "* ® ^ ’ k (bdy(bc)cldx + («c)2(ac0c/4 'SH Afc- 1 0J _ 5J - {bdy(ad)albx - (ac)2(Sc)a^| ^ - 5A* ^1=°> k = (bdy(bc)cxdx the first evectant; 2_ (H _n and thence the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the J square of the original cubic. 2_ 0J 3, .T If 0 be the degree of an invariant j ej=a—0^L l7 n ca0 ocLfi Now, as before, we pass from J toj ; further, we make use of the formula n^j n-l QL , , n =a 2 ida+ai a23a1+"'+a20«„ 3£2 J> and we obtain and, herein transforming from a to x, we obtain the first evectant x Ca-i ’o i n r k k n-k aa0 px=2y~)x^ 0a* daK a0^i + 2aijt+ ...-xJza!2
 * Afc3A* + %i~(M, + e)Jdj + a— dj± + ...+a— ^L07
 * = 0,

s^ GCL-^ (JCL<£ Combinants.—An important class of invariants, of several binary forms of the same order, was discovered by bylvester. The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation —a 07 — 07' 0? — W l^ + + • • • ~ *20^7 3 ’ of the variables. If the forms be ax, bx, cx,... the Aronhold process, given by the operation 5 as between any two of the forms, the complete system.