=Xi(77 2 f3 — ^3^2)+^2(l?3fl — +^3(^1 ^2 —7?2fl), 
§67] 
CONCOMITANTS OF TERNARY FORMS 
95 
A concomitant of the forms ji(x\, X2, xs) is evidently a 
covariant of the enlarged system of forms /< and u x . We may 
therefore restrict attention to covariants. In the proof of the 
corresponding theorem for binary forms, we used the operator 
(1), § 42. Here we employ an operator V composed of six terms 
each a partial differentiation of the third order: 
a 
0£i 
J3_ 
a £2 
_a_ 
a& 
a 
a 
a 
a 3 
a^i 
dv2 
dvs 
aaa^afs 
a 
ah 
_a_ 
af 2 
_a_ 
afs 
the determinant being symbolic. It may be shown as in 
§ 43 that 
F(^r)” = w(w+l)(w+2)(^r)”" 1 . 
As in § 44, the result of applying V r to a product of k factors 
of the type l factors of the type /3,, and m factors of the 
type 7 f , is a sum of terms each containing k—r factors a 
l — r factors p v , m—r factors 7$-, and r factors of the type (a/3y). 
For the case of an invariant /, the theorem can be proved 
without a device. In the notations of § 65, we have 
i{A)=dvifm. 
Each A is a product of factors a^, a n , a$. Hence I {A) equals 
a sum of terms each with X factors of the type a^, X of type 
a v , and X of type a^. Operate on each member of the equation 
with F x . The left member becomes a sum of terms each a 
product of a constant and factors of type (afiy). The right 
member becomes the product of 1(a) by a number not zero. 
Hence I equals a polynomial in the (afiy). 
For a covariant K, we have, by definition, 
K(A,X) = (MYK(a, x). 
Solving the equations of our transformation T in § 65, 
we get