86 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
In order to obtain the most general solution which simultaneously satisfies 
Ag / = 0 and 0, it wiU be sufficient to obtain the irreducible functional com- 
bmations of Ui, u^, . . . , Ug, which satisfy Aj/” = 0. Now, 
= 0, AjWo = 0, A^w^ = 0, A^^^g = 0 ; 
and 
A 1^3 = '^i^, 
= — 2u-^u^, 
A 1 W 7 = 
= ~ 6^1% ; 
so that 
Ai (wi'^^6 + = 0, 
Ai — 2u^) = 0, 
Ai + BWgW^) = 0 . 
Hence, the most general simultaneous solution of Ag/^ 0 , and = 0 , can be 
expressed as a functional combination of 
= UlU^ +2%'i^3, 
Vr. — UlUrj — 2u^, 
llg = fi- 
In order to obtain the most general simultaneous solution of Ag/ = 0 , A^f — 0 , 
An/ = 0 , it will be sufficient to obtain the irreducible functional combinations of 
Uq, V-, Vq, Vj, Vg which satisfy An/ = 0. Now, it is easy to show that 
~ b; ^2^^^ ~ b, AnVg = 0, ^-2^8 ~ b 5 
and 
— G'^o, 
A^v? — 2u^u^ I 
so that 
A.2 {u^u- ■§■ wy) = 0, 
Ag ("^7 “■ ”^2^%) —■ b. 
Hence, the most general simultaneous solution of Ag/ = 0, Ai/=: 0, An/= 0 can be 
expressed as a functional combination of 
Ul, U^, Vg, V^, 
^5 = 
tVj = v,^ — v.^u^ = UiU^ — — 2u^. 
