16 
ALGEBRAIC INVARIANTS 
5. Conversely,if the Hessian of a binary form/(x, y) of order n is iden 
tically zero, / is the »th power of a linear function. 
Hints: The Hessian of / is the Jacobian of df/dx, df/dy• By the 
last result in § 5, these derivatives are dependent: 
tg-M-m 0, 
dx dy 
where a and b are constants. Solving this with Euler’s relation in Ex. 3, 
we get 
9/ 0f 
(ax+by) — = naf, (ax+bv) — = nhf, 
dx ' 3 y 
9 log/_ na 9 log/_ nb 
dx ax+by dy ax+by‘ 
Integrating, 
log;—» log (ax+by) = <t>(y)=t(x). 
Hence — = constant, say log c. Thus f=c(ax+by) n . 
8. Invariants of Covariants. The binary cubic form 
(1) f(x, y) = ax 3 -\-2bx 2 y+3cxy 2 +dy 3 
has as a covariant of index 2 its Hessian 36 Jr. 
(2) h = rx 2 -\-2sxy J rty 2 , r — ac — b 2 , 2s = ad —be, t = bd—c 2 . 
Under any linear transformation of determinant A, let/ become 
(3) ^ = ^ 3 +3££ 2 77+3C£r7 2 +£b, 3 . 
Let H denote the Hessian of F. Then the covariance of h gives 
(4) H = R!t 2 +2Sitr l +Tr ] 2 = A 2 h, R = AC — B 2 , . . . 
Hence A 2 r, 2A 2 s, A 2 t are the coefficients of a binary quadratic 
form which our transformation replaces by one with the coeffi 
cients R, 2S, T. Since the discriminant of a binary quad 
ratic form is an invariant of index 2, 
RT-S 2 = A 2 \A 2 r-A 2 t-(A 2 s) 2 }=A 6 (rt-s 2 ). 
Hence rt—s 2 is an invariant of index 6 of /, 
A like method of proof shows that any invariant of a covariant 
of a system of forms is an invariant of the forms.