352 
PROFESSOR A. R. FORSYTH ON THE DIFFEREN'HAL INVARIANTS 
P?j 
''01 
J'i = 
FGqi — G ( 2Foi — G-io) 
-FX4„i + F(2Foi + GjG-2GE 
- 2EG,, + 2FE,, 
EGjqGqj + 1 ( EjjjGq^ GEqj (2F| 
e.2, above 
01 
- Gio) 
Pa. 
= 0 
r, = 
e, = 
h = 
3 jFG„ -G(2Fo,-Gio)j 
3 ^ — EGoi + F (2Foj — G^o) \ 
EGo;^ - 2FGoi (2Foi - Gio) + G (2Foi - G^o)^ 
63, above 
Other four solutions are S’iven bv 
K = 1 \ 4 - + P^\ + cr/’i ^ 
— ^ 'To + «'|io + />'(/o + e'ro + pTo + cr/o I 
/x" — P\ \’g + a !+ h'fp 4- cVg 4- peg 4- cr/g j 
v" — 2Vb4 + 4- pe,^ 4- cr/l ' 
and there is a last solution i^iveii bv 
V = E {(Eqi — 2h Gm 4 - Gj,-|~} 
■E ^ '(^loGoi E|-)j (2hQ3 4“ Gjq) 4“ 2F^g (2 Fqj — G^q)] 
4“ G {Eqj- E^q (2Fq^ Gjq)]- 
— 2V-'(E(,o — 2Fj^ 4 " Goq). 
C'onsetjiiently, it IoIIoavs that evei'y snnnltaneous solution ot the fourteen ec|uations 
made up of the eight (Illi) • • • (Hlg) and of the six (IIi) . . . (llg), is a functional 
combination of the fifteen quantities 
o, />, c, a\ //, r', 
k', y, p\ v, k \ X", p", v’\ 
V, 
and ot the quantities derivatives with regard to wdiich have not occurred in those 
fourteen equations, viz., E, F, G, L, M, N, P, Q, K, S, a, A y, S, e, <f>,„ <^^ 1 , i/^io, 4n 
making 34 aiguments in all AVhat now is required is the algebraically independent 
functional combinations of these 34 arguments satisfying the 
remaining four differential equations (I,)... (I,). 
