60 
ALGEBRAIC INVARIANTS 
The right member is the result of operating on 0 with the 
operator obtained by substituting D for d/dv and D\ for — d/d £ 
in 
whose terms are partial derivatives of order co. Hence, if 
the form 
l{x, y) = 2 c rs x r y s 
(r+s = œ) 
becomes X(£, 77) under the transformation T, our right mem 
ber is the result of operating on 0 with \{d/dv, -9/91:). The 
left member is the result of operating on / with 
Hence if T replaces the forms f{x, y), l{x, y) by 0(£, 77), X(£, 77), 
then 
is a consequence of the equations for T, if co is the order of l(x, y). 
Let / and l be covariants of indices m and n of one or more 
binary forms / t with the coefficients c\, C2, .... Under T 
let the transformed forms have the coefficients Ci, C2, ... . 
Then 
/(C; $, 17)= A m f{c; x, y), 1{C; f, ri)=A n l(c; x, y). 
But 0(£, rj) =f(c; x, y), by the earlier notation. Hence 
0(€, r?)=A —/(C; 77), X(f, 77) =A-/(C; 77). 
Inserting these into the formula of the theorem, and mul 
tiplying by A™ +n , we get 
K 
The function in the right member is therefore a covariant of 
index co+m-\-n of the ft. We therefore have the theorem 
of Boole, one of the first known general theorems on covariants: