MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
101 
But, if B = 0‘, then 
SC-^= - = - nu,0^-' ; 
so that 
S = — n'u-C'‘'^^ ; 
and therefore 
2w5"C"'^‘’ = 2RC%5^ = ~ n{n 3) 
= — n {n + 3 ) 
whence 
n = ~ 1 or — 2 , 
First, taking n= — 1 , we have 
= 20u\ — 2C~hi^, 
and therefore 
Tw. = CV4 + 2w4C"h 
Hence we may take as one irreducible solution 
A . 2u^Jr CV^ 
X = -+BC\.,- 
Second, taking n = — 2 , we have 
= 2u\ — 2C“X i 
id therefore 
Twg = 2Qu\ + G~hi^. 
Hence we may take as another irreducible solution 
Y — ^ 4- 2BCt< — + 2Ch/.h 
y _ C3 4 
And it follows from the method of derivation, and by an application of the theory 
of linear partial differential equations, that every simultaneous solution of the 
equations ©^i/; = 0 = @ 2 ^ = @ 31 // = © 4 . 1 // which involves no quantity of order higher 
than r, s, t, r, s', t' can be expressed as a functional combination of ; u^, u'^ ; X, Y. 
23. It is now necessary to consider the index equations. We have for u-^ (= J) 
= 3J, 
<j)^J = 3J ; 
so that J is an invariant of index unity. For u^{=- Aq) we have 
