§24] 
ALTERNANTS 
41 
3. Find the covariant of the binary cubic / whose leader is 
a 0 2 a 3 —3o 0 uiO2+2ai 3 , the only seminvariant of weight 3 and degree 3. It 
is the Jacobian of / and its Hessian. 
4. A covariant of two or more binary forms is annihilated by 
Xn-y—, 
d* 
20—*- 
dy 
5. Find a seminvariant of weight 2 and partial degrees 1, 1 of a binary 
quadratic and cubic. Show that it is the leader of the covariant 
(flo^2— 2iilbl -\-<l2bo)X 2&1&2 T#2^l)y. 
24. Alternants. Consider the annihilators 
0 = 2 jaj-i— = 2 (^Tl)#*— , 
7=1 dUj k= 0 Odt + v 
0 = 1 (p-j+l)a,^- = 2(p-k)a t+1 ^- 
j= i ddj-i * = o 9®» 
of invariants of a binary form. We have 
itO = 2 ja, -, 10 -7+1)—^- + 2 (/>- *)a.+i-~- 
y=i [ 9«7-i fc=o odjoa* 
00 = 2 (/>— (£ + l)—— jaj-i 
k =0 
9&t+i i=1 
d°kdcij 
The terms involving second derivatives are identical. Hence 
£20 — 0£2 = 2 {i-\-\){p — t)cLi—— 2 i{p — l)flf-^" 
i =o 9^< *=i 9^ 
p p) 
= 2 (p-2i)ai--, 
i =0 9®< 
since the first sum is the first sum in £20 with j replaced by 
i-\-1, and the second is the first sum in 0£2 with k replaced 
by i — 1. 
If S is a homogeneous function of ao, . 
d and hence a sum of terms 
cao e °ui®i . . . a v e v (£o+£i~K 
we readily verify Euler’s theorem: 
I Z*=dS. 
i= o 9Vi 
, a p of total degree 
,-\-e p =d),