112 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
0 A„j _ 2 
^ dp 
. 0A,„_3 
dq 
. 0 A,„ _ 2 
op 
. 0 A,„_o 
= 0 
1 
0 A 
I 
I’ ; 
m — 2 
d 'p j 
d<i 
02 
- = 0 
J 
From these it follows that, if neither I nor m be zero, 
J/, m — 
{1 d~ 1) “k 2) (qJi-i,m — A/_ 
dp , 
) + (w 
+ 1 ) ('^A-f- 2 ) 
m—X “k A„j_ 
ah 
^ 0p / j 
li 
s 
<1 
— (^d“ l)(^d“ 2 ) + A^_ 
0 A„A 
02 j 
— ('To 
+ 1)(to+2)(2A, 
— A 
n—\ -^m- 
ay ; 
dq j j 
Ag J m — 
b - A^J 
J 
and 
• 
11 
o 
< 
— 2 q (to + 2 ) A„, -f (to + 2 ) 
( to -f- 
l)(gj 
4- — A ^1 
dp ”^-7 
i 
1 
A2 J 0, i,i — 
22) {m + 2 ) A,„ — {m -f 2 ) {m 
+ 1 ) 
(^^ J 0 , nt 
-1 0^ 
!>• 
i 
A3 J 0, M — 
0 = A4J 0, m 
J 
Connexion with Theory of Binary Forms. 
34. In connexion with the fact (§ 15) that the irreducible invariants proper to the 
rank 3 are expressible in terms of the simultaneous concomitants of Ag and A^, viewed 
as binary forms (quadratic and cubic) in q and — p as variables, it is important to 
remark that the equations = 0 and = 0 are in fact the difierential equations 
satisfied by all concomitants of binary forms which have q and — p for their variables, 
and have 
r, s, t ; 
a, h, c, d; 
e, f g, h, i ; 
for their coefficients, that is, of Aq, A^, Aj, . . . , viewed as binary forms. Each form 
of a concomitant-system satisfies the differential equations characteristic of its con- 
