Page:EB1911 - Volume 01.djvu/673

 and this, on writing c2, − c1 For y1, y2, becomes

(kc)K X 'T 3C X 1＝ (ƒ,0 1 ', G 1; ï¿½

∴1{F,O}1 M 1＝1 M 2 0`,4)) (ƒ,2).+ (0,0 2 .F '

and thence it appears that the first transvectant of (ƒ, 1 over ) is always expressible by means of forms of lower degree in the coefficients wherever each of the forms F, 0, 4, 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

(ƒ,ƒ′)2 = (ab)2 a n-2 r7 2 =Hx - =H;

unsymbolically it is a numerical multiple of the determinant ∂2ƒ a2f (32 f) It is also the first transvectant of the differxi ax axa x 2 ential coefficients of the form with regard to the variables, viz. (L, _f_)'. For the quadratic it is the discriminant (ab) 2 and for ax2 the cubic the quadratic covariant (ab) 2 axbx.

In general for a form in n variables the Hessian is 3 2 f 3 2 f a2f ax i ax n ax 2 ax " ï¿½ï¿½ ' axn and there is a remarkable theorem which states that if H =o and n=2, 3, or 4 the original form can be exhibited as a form in 1, 2, 3 variables respectively.

The Form ƒ+. - An important method for the formation of covariants is connected with the form ƒ+, where ƒ and 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 f and 4). In particular, when 4) is a covariant of f, we obtain in this manner covariants of f. The Partial Differential Equations.--It will be shown later that covariants may be studied by restricting attention to the leading coefficient, viz. that affecting xi where e is the order of the covariant.

An important fact, discovered by Cayley, is that these coefficients, and also the complete covariants, satisfy certain partial differential equations which suffice to determine them, and to ascertain many of their properties. These equations can be arrived at in many ways; the method here given is due to Gordan. X1, X 2, u1, /22 being as usual the coefficients of substitution, let x1a ? + X 2 - = D, X 1 -' j +X 2 =D 2 AA' ?2 / 2 1 3 - 5 -, =112 87,2 = ?1a a + ?2a a =Dï¿½ï¿½, 1 be linear operators. Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.

Operation upon J results as follows D AA J = wJ; D A J=0; D ï¿½A J =0;D ï¿½ï¿½ J = wJ.

The first and fourth of these indicate that (a 2) w is a homogeneous function of X i, X2, and of /u1, ï¿½ 2 separately, and the second and third arise from the fact that (X / 1) is caused to vanish by both Da ï¿½ and Dï¿½A. Since J= F(A0,A11...Ak,ï¿½..), where A k= we find that the results are equivalent to. aJ - ., _ A aJ ï¿½. k (DwAk) Ak 0; (D (ï¿½ A k) Ak =wJ.

k k According to the well-known law for the changes of independent variables. Now D A xA k = (n - k) A k; Aï¿½ A k = k A?1; D ï¿½A A k = (n - k) A k+1;D mï¿½ A k = kA k; (n - k)A ka - w Ak - 1 aA k = O; a _ J (n - k) A k +l A k = O; kA k Ak = wJ; equations which are valid when X 1, X 2, ï¿½ 1, ï¿½2 have arbitrary values, and therefore when the values are such that J =j, A k =akï¿½ Hence Â°a-do +(n -1)71 (a2aa-+... =wj, - aj aj - aj a Â°aa1 +2a 1aa2 +3a 2aa3 +... =0, - aj aj aj nal aao +(n-1)a2 at i -} (n - 2)a 3aa2+... =0, a 1 a ? +2a 2 a? +3a 3 a +... = wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant. The fourth shows that every term of the invariant is of the same weight. Moreover, if we add the first to the fourth we obtain aj 2w ak = 7 1=6, j, =0j, where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant w= - n0. The second and third are those upon the solution of which the theory of the invariant may be said to depend. An instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the coefficients. The two operators - a a - a = a Â°aa 1 +2 a 1aa2 +... +na" -laan -a a O = na laao + (n 1)a 2aa1 +ï¿½.. +a"aa"-1 have been much studied by Sylvester, Hammond, Hilbert and Elliott (Elliott, Algebra of Quantics, ch. vi.). An important reference is “The Differential Equations satisfied by Concomitants of Quantics,” by A. R. Forsyth, Proc. Lond. Math. Soc. vol. xix.

The Evectant Process.—If we have a symbolic product, which contains the symbol a only in determinant factors such as (ab), we may write x 2 ,-x 1 for a 1, a 2, and thus obtain a product in which (ab) is replaced by b x, (ac) by c x and so on. In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant. The second evectant is obtained by similarly operating upon all the symbols remaining which only occur in determinant factors, and so on for the higher evectants.

Ex. gr. From (ac) 2 (bd) 2 (ad)(bc) we obtain (bd) 2 (bc) cyd x +(ac) 2 (ad) c xdx - (bd) 2 (ad)axb x - (ac)2(bc)axbx =4(bd) 2 (bc)c 2. d x the first evectant; and thence 4cxdi the second evectant; in fact the two evectants are to numerical factors pres, the cubic covariant Q, and the square of the original cubic.

If be the degree of an invariant j

aj aj a; oj =a Â° a a o +al aa l +... +anaan naj n.-1 aj naj =a l aa Â° +a 1 a2c3a1...+a2aan

and, herein transforming from a to x, we obtain the first evectant

(-) k, x1x2 aak k

Combinants. - An important class of invariants, of several binary forms of the same order, was discovered by Sylvester. The invariants in question are invariants quâ linear transformation of the forms themselves as well as quâ linear transformation of the variables. If the forms be ax, b2, cy,... The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish. Thus it has annihilators of the forms

a0 db - 0 +al d 1+a2d 22+... Â°c - iao l a12da2+'..

and Gordan, in fact, takes the satisfaction of these conditions as defining those invariants which Sylvester termed " combinants." The existence of such forms seems to have been brought to Sylvester's notice by observation of the fact that the resultant of of and b must be a factor of the resultant of Xax+ 12 by and X'a +tA2 for a common factor of the first pair must be also a common factor