146 
SURFACES DIVISIBLE INTO SQUARES BY THEIR CURVES OF CURVATURE. 
[505 
and if in the notation of Gauss we write 
x? + yf + zf = E, 
x£ + y 2 + z£ = G, 
then adding the equations multiplied by x ly y ly z 1 respectively, and also adding the 
equations multiplied by x 2y y. 2y z 2 respectively, we find 
A = 
L 1 dE 
2 Ë dq ’ 
B = 
1 1 dG 
2 G dq 
and the equations thus become 
1 dE 1 dG 
2jOC a ~~~ T-j -, CG\ ^ 002 — U. 
Edq G dq 
&c. 
&c. 
&c., 
which, in fact, agree with the equations (10 bis) in Lamés “Leçons sur les coordonnées 
curvilignes,” Paris (1859), p. 89. The surface will be divisible into squares if only 
E : G is the quotient of a function of p by a function of q, or say if 
P = ©P, G = ®Q, 
where © is any function of (p, q), but P and Q are functions of p and q respectively; 
we then have 
1 dE 1 d© 1 dG 1 d© 
E dq © dq ’ G dp © dp ’ 
and the equations for x, y, z are 
n 1 d<0 Id© . 
AiOG4 -7=r 7 y X2 — Oj 
© dq © dp 
&c. &c. 
&c., 
viz. x, y, z being functions of p, q such that x x x 2 + y x y 2 + z x z 2 — 0, and which besides 
satisfy these equations, or say which each of them satisfy the equation 
„ 1 d© 
ZM4 ~ pr j Ui 
<$)dq 
1 d© 
© dp 
u 2 = 0, 
then the values of x, y, z in terms of (p, q) determine a surface which has the 
property in question.