OF A SURFACE, AND THEIR C4EOMETRIC SIGNIFICANCE. 
351 
and the equation (II5)' requires 
for fR = Pj, 
01 
10 
dF 
+ 
01 
Qi, Pt^, and 
We thus have 24 equations giving the derivatives of the four quantities 
Pn Qe P'n with respect to E^q, Eqj, Fjq, Egi, G^q, Guj. Each of the four quantities 
is then given by effecting the quadrature 
The results are 
0 
00 
0E 
<^^10 + ■ 
10 
. + 
2WP, = 3 ( - E,,G - EoiF + 2 F,oF) , 
2W’Q, = 3 (E,oF + E„iE - 2V,,E ), 
2V2R, = 0, 
2WS, = {EGio(2Fio - Eoi) + F(EoU - 2Eo,F,o - E,oG,o) + GE.oEoi} r 
+ (P (Eoi" ~ 4 Eo^Fjq 4- 4Fiy) + 2 FEiq(Eoi — 2Fi,3) + ^^ 10 ") 
The solution in question remains a solution when it is multiplied by 2^^"^; denotino- 
this product by k, we have 
K = 2Vbq + a 2 )y + hq^ -f c)\ + rc^ -f- sj^. " 
Similarly we obtain 
X' = 2V-h^^ + cqj .2 + J>q., + cr., + rc^ + .s/', , 
fx' = 2 Vb ^3 + ap )3 + l>q^ + cr^ + re. + sf.^ 
F = 2\+ €12^4, + hq^ + cr.j. + re^, + . 
where 
Pi — 3 ~ F (Eqj 2F;^o) GE^q} 
i, = 3SE(Eo,-2F,„) + FEi„i j 
r, = 0 , 
c, = - EG,„(E„, - 2F,„) + F(E„,» - 2E„,F,„ - E,„G„) + GEi„E„, 
A = E (Eoi - 2F,(,)= + 2FE,„ (E,„ - 2F,„) + GE„ ^ i 
/b — 2 FGjq — 2GEg^ 
q, = - 2 EG 10 + F (Eoi + 2F,o) - GE,, 
7q = E (Eoi - 2F,o) + FE,o 
Co = EGk/ - 2FEoiG,o + GEoU 
_/o = C|, aliove 
