§7] 
DEFINITION OF INVARIANTS 
15 
If, for every linear transformation T of determinant A ¿¿0, 
a polynomial K in the coefficients and variables in f is such 
that* 
K(A.Qj . . . , Ap y £, ff) = A K{a 0 , . . . , dpy x, y), 
identically in do, . . . , d p , £, rj, after the ¿4’s have been replaced 
by their values in terms of the d’s, and after x and y have 
been replaced by their values in terms of £ and 77 from T, then 
K is called a covdridnt of index X of /. 
The definitions of invariants and covariants of several 
binary forms are similar. 
These definitions are illustrated by the examples in §§ 4, 5. 
Note that / itself is a covariant of index zero of /; also that 
invariants are covariants of order zero. 
EXERCISES 
1. The Jacobian oi f = ax 2 +2bxy+cy 2 and L=rx+sy is 
/ = 2 {as — hr) x + 2 {bs—cr) y. 
If J is identically zero, / = tL 2 , where t is a constant. How does this 
illustrate the last result in § 5? Next, let J be not identically zero. Let 
k and l be the values of x/y for which f—0;m that for which L = 0 and n 
that for which / = 0. Prove that the cross-ratio (k, m, l, n) = —1. Thus 
the points represented by /= 0 are separated harmonically by those repre 
sented by L — 0, / = 0. 
2. If J is the Jacobian of two binary quadratic forms/and g, the points 
represented by / = 0 separate harmonically those represented by /= 0 
and also those represented by g=0. Thus / = 0 represents the pair of 
double points of the involution defined by the pairs of points represented 
by/=0 and g = 0. 
3. If f{x, y) is a binary form of order n, then (Euler) 
Hint: Prove this for /= ax k y n ~ k and for /=/i+/2. 
4. The Hessian of (ax+by) n is identically zero. 
Hint: It is sufficient to prove this for x n . Why? 
* The factor can be shown to be a power of A if it is merely assumed to be 
a function only of the coefficients of the transformation.