MR. A. R. FORSYTH ON A CLASS OF FUNCTIONAL INVARIANTS. 
99 
©^'U^ = 0, < 
and 
= 0 ; = 0, = 0 ; @. 2^5 — 0 5 
©2^2 = 
@ 2^2 = i. 
Hence 
©3V3 = — 3 u^. 
©2 {UiU'2 — U2U{) = 0, 
and therefore, if 
©2 + 2 up>^) = 0 ; 
P = upi^ — iipL\ — qt' — qt, 
Q = 3 upi^ + 2 w^ 7 ’ 3 , 
the irreducible combinations which satisfy = 0 = ©gi// = ©^xjj, are u^, u\, P, 
4 > Q- 
It is now necessary to obtain the irreducible combinations of these seven quantities 
which satisfy ©^r// — 0. We have 
so that 
II 
0 
11 
0 
Again, 
= u^. 
©4P — p>t' — p't + 2 qs — 2 q's. 
Now, 
q {pt' — qj't) z= (P + q't) — p)'qt = p>V + Urju., 
so that 
and hence 
©4P = 3 {pt' — p/t) + 2r’3 
= ^ (pP + upi^) + 2 v ^; 
Again, 
^^^©^P — 3 pP = Q. 
but 
\ ©4Q = 35^^5 + pv^ + qBpj.^ ; 
= qr' — ps — {qr — p/s) ; 
qr' - ps' = i {qS' - 2 p>qs + pH') + ^ {qs - pt') 
qr - 
^ U^U'/ ^ ~ > 
1 1)^X1^ 
- ps — {u 4 W 4 + 2 w — U 2^5 ) + , 
0 2 
and 
