§21] 
HOMOGENEITY OF COYARIANTS 
37 
of the first degree in the o’s and first degree in the b’s is a multiple of 
Oo&2+O'>6o 2,d\h\. 
5. A binary quadratic and quartic have no such lineo-linear invariant. 
6. Find the invariant of partial degrees 2, 1 of a binary linear and 
a quadratic form. 
7. Find the invariant of partial degrees 1, 2 of a binary quadratic and a 
cubic form. 
8. The first two properties in the theorem of § 20 imply that 7 is homo 
geneous. For, under replacement (1), any term ca 0 e o. . . a p e v of 7, of 
weight w=ei+2e 2 + ■ • • +pe P , implies a term ±ca 0 ep ai e p-i. . . a v e o 
of weight w=e P -i+2e p -i+ . . . + (p — l)ex+ pe 0 . Adding the two 
expressions for w, show that the degree d = e 0 +ei+ . . . +e p is the constant 
2 w/p. 
21. Homogeneity of Covariants. A covariant which is not 
homogeneous in the variables is a sum of covariants each homo 
geneous in the variables. 
For, if a, b, . . . are the coefficients of the forms, and K 
is a covariant, 
K(A, B, . . . ; Ç, v, • • .)=A x K(a, b, . . .; x, y, . . .). 
When x, y, . . . are replaced by their linear expressions in 
£, 77, . . , , the terms of order co in x, y, . . .on the right (and 
only such terms) give rise to terms of order co in £, ??,... on 
the left. Hence, if K\ is the sum of all of the terms of order 
co of K, 
Kx{A, B, . . . ; $, v, • • .)=A x ATi(a, b, . . . ; x, y, . . .), 
and K\ is a covariant. In this way, K = Ki-\-K2-\-. . . . 
Henceforth, we shall restrict attention to covariants which 
are homogeneous in the variables, and hence of constant order. 
A covariant K of constant order œ of a single form f is homo 
geneous in the coefficients, and hence of constant degree d. 
For, let / have the coefficients a, b, . . . and order p, and 
apply the transformation x=a%, y=ar\, .... The coefficients 
of the resulting form are A =a p a, B=a p b, .... Thus 
K{a p a, a p b,... ; a- 1 *, a~ l y,. . .) = {a.*YK{a, b, . . .;x,y,...) } 
identically in a, a, b, , x, y, ... , since the left member