334 
ON THE GAUSSIAN THEORY OF SURFACES. 
[767 
In fact, writing herein E = 1, F= 0, and therefore the differential coefficients of E 
and F each = 0, the equation becomes 
4 (E'G' — F' 2 ) = Gi — 26rCr u , 
which is 
4 V 4 (.HK - T 2 ) = (2 VVy - 2 V 2 (2 b? + 2 VV n ), = - 4 V s V u ; 
or finally it is 
F u = V (T 2 - HK). 
The other two of Bour’s equations are derived from equations which give 
respectively the values of E* - F ' and F.! - Gi; viz. starting from the equations 
E' = Aa + B/3 + Cy , 
F' = Ad + B{3' + Cy , 
G' = Aa" + B/3" + Cy", 
we see at once that E 2 ' and Fi contain, E 2 the terms Aa 2 + B/3 2 + Cy 2 , and Fi the 
terms Aai 4- H/3/ + C'y/, which are equal to each other (a, = a/ since a and d are 
the differential coefficients x n , x 12 of x, and so ¡3, = /3/ and y 2 — yi). Hence 
E'- F' = A 2 ol + B 2 p + C 2 y - Ay - By - Cy ; 
and similarly 
f: - g;=Ay + B.y + ay - Ay - By - cy. 
Here, from the values of A, B, C, we have 
A = bc’-cbA^/30' -yb' +by' -c,S'; A 2 = ¡3'c'- y'b' +by" - cj3"; 
B = cd — ac'; B 1 = yd — ac' + ca' — ay'; B 2 = yd — a'c + ca" — ay"; 
C = ah' - ba!; C, = a6' - /3a' + a/3' - 6a'; C 2 = zb' - ¡3'd + a/3" - 6a"; 
and, substituting, we find 
E 2 — Fi = 2d ad + aa"a, 
F 2 ' -Gi = -2ada"-da"a, 
if, for shortness, dad denotes the determinant 
a', a, a' , 
v, ¡3, /3' 
o', y, y 
and so for the other like symbols. Observe that, with 
а, a, a, d, a" , 
б, 6', /3, /3', /3" 
C, c', y, y, y"