100 
MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
so 
that 
+ ’iiA _!% _ Atj („3 + „/) _ p. 
r*' 1 
u.^u 
1 
Hence 
Hence 
pv^ + 2®4^3 = 2pv3 + 3l ^^'4 - % - -V 2 P - ;7 ( 2 M 3 + w's)- 
w • 
u\ 
— 'iq 
upif^ 
M 1 
p 
/ 0 
U J 
I^p, 
_ “5Q 
up' 
“1 
,£ 2 _ 
2 
If now we write 
? = C, 
= B, 
Q 
= A, 
the three results can be put into the forms 
u 
^t 7 @^B = AC-^ 
m 7 @,A = 2CSi\ - 2 C-J u, - 2 BCX' 
And, further, we have 
©4^4 = 0 > ®4'^^4' = 0 , @4^5 = 0 - 
Since the result of operating with on B gives a quantity into which A enters 
linearly, and the result of operating with @4 on A gives another quantity into which B 
enters linearly, we are led to assume that the irreducible solution (or solutions) of 
z= 0 are of the foi'm 
RA + SB + T, 
where R, S, T are independent of A and B. If this be a solution, we have 
w7©^T = R ( 2 CV, - 20-H^ - + SAC"" + ; 
and we suppose R and S so determined that 
SC-^ = - idu©^R, 
