336 
ON THE GAUSSIAN THEORY OF SURFACES. 
[767 
We thus obtain 
E: - F; = A (( - | FG, + IGEJ E' + (- IGE, + FF, - iFF) F’} 
+ A{( IFF- EF 1 + iEE,)ff + ( \FG,-FF, +) S EG % )E'), 
F1-G;=~{( ire,- FF + iEG s )F' + ( -iEG, + \FE,)G'\ 
+ A ((- i GE, + FF, - iFE,) G' + (- i GG, + GF, - \FQ,) E'}; 
or, finally, 
Ei - F; = i {(- ire, + GE, - FF, + iEG,) E' 
+ (- GE, + 2FF, - FF,) F'+(iFE, - EF, + iEE,) G'\, 
f: - G,' = ^5 ((— i GG,+GF,—ire,) E' 
+ <re, - 2iy,+re,) i”+(- i G£,+ff, - re, + i re,) G'j, 
which are the required formulae; and which may, I think, be regarded as new formulae 
in the Gaussian theory of surfaces. 
Writing herein as before, the first of these becomes 
that is, 
or finally 
(VK), + (FT),= y, fun, VK), = V.K, 
V 2 K+ VK 2 + V*T 1 + 2VV 1 T= V 2 K ; 
VT 1 + 2TV 1 + K 2 =0, 
which is Bour’s third equation. And the second equation becomes 
— ( FT), - ( m = Y* {- i F* (F’), F/f + (F a ), (- FT) - (V s ), V H), 
= - V’-V,K - 2 FF.r - 2 F=F,if. 
that is, 
- V*T 2 -2VV 2 T- V*H 1 -3V*V 1 H=-V*V 1 K-2VV 2 T-2V*V 1 H; 
or finally 
T. 2 + VH 1 + (H—K) V 1 = 0, 
which is Bour’s second equation.