32 
ALGEBRAIC INVARIANTS 
To prove the theorem, apply to / the transformation 
y=ocv- 
We obtain a form with the literal coefficients 
Ao = qq, Ai =a\a, A2—CL20Î 2 , . . . , A p = a v a v . 
Hence if I is of index X, 
I{a 0 , aia, . . . , a p a p )=a x I{a 0 , ai, , a P ), 
identically in a. and the a’s. The term of the left member 
which corresponds to the above term t of I is evidently 
coOo e ° . . . a v e va w . 
Hence w — \. The weight of an invariant of degree d of a 
binary p-ic is thus its index and hence (§ 15) equals \dp. 
17. Weight of an Invariant of any System of Forms. Let 
/1, . . . , f n be forms in the same variables x x , . . . , x a . We 
define the weight of the coefficient of any term of ft to be 
the exponent of x a in that term, and the weight of a product 
of coefficients to be the sum of the weights of the factors. 
For q = 2, this definition is in accord with that in § 16, where 
the coefficient a t of xi p ~ k X2 k was taken to be of weight k. 
Again, in a ternary quadratic form, the coefficients of xi 2 , 
X1X2 and X2 2 are of weight zero, those of aqæs and #2X3 of weight 
unity, and that of X3, 2 of weight 2. 
Under the transformation of determinant a, 
Xl — £1, • • • J —1 = £5—1.) %a =<x ^Qj 
ft becomes a form in which the coefficient c' corresponding 
to a coefficient c of weight k in ft is ca k . If I is an invariant, 
I(c')=a x I{c), identically in a. Hence every term of I is of 
weight X. 
Thus any invariant of a single form is isoharic; any invariant 
of a system of two or more forms is isoharic on the whole, hut 
not necessarily isoharic in the coefficients of each form separately. 
The index equals the weight and is therefore an integer't. 0.