§23] ANNIHILATORS OF COVARIANTS 39 
Consider a covariant A homogeneous and of total order ca in the variables 
x lf . . . , x Q of two or more forms /». As in § 15, A need not be homo 
geneous in the coefficients of each form separately, but is a sum of covariants 
homogeneous in the coefficients of each. Let such a A be of degree d t in 
the coefficients of f t , of order pi. As in §21, SpA— u = q\. The total 
weight of A is w+X. 
For example, if pi —pi—q — 2, 
fi = aox 2 +2(hxy+a 2 y 2 , f 2 =b 0 x 2 +2bay+b 2 y 2 . 
The Jacobian of /i and / 2 is 4A, where 
A = (a A—aib ü )x 2 +(a A—a 2 b 0 )xy+(aA—aA)y 2 . 
di = d 2 =l, u — 2, X= 1, and A is of weight 3. 
Here 
23. Annihilators of Covariants K of a Binary Form. Pro 
ceeding as in § 20, we have instead of (2) 
d v(A a . t \ v dK d A i,d K 9£ , dK dr, 
9n j= 0 dn 9 c on dv 9n 
I 9 K 9 K 
and obtain the following result: K is covariant with respect 
to every transformation x=^+nrj, y = rj, if and only if it is 
annihilated by * 
(1) 
C —ao~-+- • .-\-pcL p - 
v 9a 1 дПр 
The binary form is unaltered if we interchange x and y, 
at and a p -i for ¿ = 0, 1, . . . , p. Hence A is covariant with 
respect to every transformation x= £, y = r]-\-n£, if and only 
if it is annihilated by 
(2) O-xio=pai~-+(p-l)a 2 ~-+. . 
9 y \ 9«o 9a 1 9a p -i/ 
Denote a covariant of order co of the binary ^?-ic by 
A=Ax"-f5ix“ _1 y+. . .-h5o,y". 
* For another derivation, see the corollary in §47.