OF A SUEFACE, AND THEIE GEOMETRIC SIGNIFICANCE. 
347 
rq — 2V'<^3 q ( 2 Foq 
- Eyy)!- - 
?,q = 2 V^^oi ~ 
iq = 2 Y'(p^., — 
GiP’ 
Ey.,? 
iq = 2 V 2 (/,y 3 - 
Cry^jl’ ( 2 Fy 3 
- 
iq = 2 V^i/; 3 o - ( 2 F 2 y 
— En)p — 
Eoycr 
Vo = 2 VVoi - 
CFiP “ 
Eno- 
iq = 2 W> 3 . - 
Giip — 
EoW 
i\ = 2 'V'i/;Qg 
^0-2P ( 2 Fyn 
- G,,)a 
^ — ^03 ~ -f- 
where V“ = EG — F^, and 
E(^oi - r(/.io = Ei/zoi - Ft/z^y = pi 
^4*10 ~ ^4*0'i — f El//y, = O- j 
Any functional combination of these nine (quantities will satisfy the set of eight 
equations which have been considered, as will also any functional combination of the 
derivatives of (f) and xff of orders lower than 3, of the derivatives of E, F, G of 
orders lower than 2, and of L, M, N, P, Q, R, S, a, y, S, e. We therefore have 
to find the functional combinations which will satisfy the remaining- equations. 
16. For this purpose, we make iq, u,, u,, iq, iq, v,, v,, v„ d; E, E^y, Ey^; 
F, bio? Fyy; G, Gyy, Gq^ ; (f)yQ, the variables ; 
and rve transform the set of e(piatlons (1T|). . . (Ily), so that the derivatives of / are 
taken witli regard to these varialiles. Denoting J' with the new variable byy’ for a 
moment, we have 
I v ^^6/ I V j_ h/' 
Of 0»: ^ - 0 ,,^ 0 ^^ + 0 ^ 0 ^^ ’ 
for all the quantities ^ in the original equations ; the magnitude 0//0^ is zero if f be 
not one of the new variables. 
The result of tlie transformation is to reqilace the set of equations (TTj). . . (Tly) 
by tlie set 
2E + F ¥ . j , , vf 0/’ 
3E,o ^ aF,y + -f- ^10 + (Gio- 2b ,y) 
¥ 
+ (-^01 - GFiy)r - 5 E,y.S'] + q^[ 2 WVji - Cf.yr 
J- (jU a 
2E„,.] 
+ + (2E„, - G¥„)f, ~ 5E,„<r] + ^ - G,„p - 2E„,cr] = 0 . (II,)', 
2 Y 2 
¥ rr.-xrs. 
