§15] 
HOMOGENEITY OF INVARIANTS 
31 
Let I be an invariant of r forms fi, . . . ,f T of orders pi, 
. . . , p r in the same q variables X\, . . . , x q . Let a particular 
term t of / be of degree d\ in the coefficients of fi, of degree 
d2 in the coefficients of /2, etc. Apply the special transformation 
X2=a^2, • • • , X t =a£ a , 
of determinant A =a q . Then f t is transformed into a form whose 
coefficients are the products of those of ft by o? 1 . Hence in 
the function I of the transformed coefficients, the term cor 
responding to t equals the product of t by 
{a Vl ) dl . . . («^) d r=a 2 ^'. 
This factor therefore equals A x , if X is the index of the invariant. 
Thus 
T 
S dipi = \q. 
»=1 
Hence Zdipt is constant for all the terms of the invariant. 
For the above two quadratic forms, r = pi~pi = 2. For invariant d, 
we have di~2, d 2 = 0, hdipi=4 = 2X. For s, we have di = d 2 =l, 'Ldipi—\. 
Again, the discriminant (§8) of the binary cubic form is of constant degree 
4 and index X=6; we have ILdipi — 4-3 = 2X. 
If, as in the last example, we take r = 1, we see that an 
invariant of index X of a single q-ary form of order p is of 
constant degree d, where dp = \q, and hence is homogeneous. 
16. Weight of an Invariant / of a Binary Form f. Give to 
/ and / the notations in § 7. Let 
t = cao e °#i ei • • • a v ev 
be any term of I, and call 
w = ei+2g2+3es + . . .-\-pe v 
the weight of t. Thus w is the sum of the subscripts of the 
factors a t each repeated as often as its exponent indicates. 
We shall prove that the various terms of an invariant of a binary 
form are of constant weight, and hence call the invariant isoharic. 
For example, aox 2 +2a\xy+a2y 2 has the invariant «0^2— «i 2 , 
each of whose terms is of weight 2.