OF A SURFACE, AND THEIR GEOMETRIC SIGNIFICANCE. 
355 
Similarly, we write 
k" = k', r 
aad we find 
= l\ jj." — ’yVct = rn', V 
" ~ i i' 
V,k' =(}, 
V,I' =k', 
Vjwfi 2l', 
V,7d = Sm\ 
V,k' = Si' 
VJ' = 2 m' ! 
- I 
V.^m' = n' I 
~ I 
V.m' = U I 
These quantities k, I, m, n, k', I', in', n' replace k', v', fx", v" ; moreover, 
V is a simultaneous solution of the equations. What we require are the functional 
combinations of the quantities 
a, h, c, a', ly, c', 
k, I, in, n, k', I', m', n', 
E, F, G, L, M, N, P, Q, R, S, a, /3, y, 8, c, 
making 33 arguments in all. 
For this purpose, we transfoi’in the equations, so that these 33 arguments may 
become the independent valuables. The process would be laborious but not 
intrinsically difficult, were it not that the effect of the operators Vj and Vo upon the 
various arguments has already been obtained ; and the results are 
aG ' ^ 0F 
^ aN ^ an 
+ 31! ® ' + 2Q If, + V y 
CD ah oQ 
¥ , 3/ 
¥' , 3/ 
I a/’ of 
_L ■'>]/ _i_ a' 
^ " ae' ^ " a// 
+-imy + n ^ iM 
an cin at 
+ ^\o 
+ ’Aio 
At 
At 
= 0 
(I3)', 
2 z 2 
