505] 
145 
505. 
ON THE SURFACES DIVISIBLE INTO SQUARES BY THEIR 
CURVES OF CURVATURE. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 120, 121. Read June 13, 1872.] 
Professor Cayley gave an account of an investigation recently communicated by 
him to the Academy of Sciences at Paris. The fundamental theorem is that, if the 
coordinates x, y, z of a point on a surface are expressed as functions of two parameters 
p, q (such expressions, of course replacing the equation of the surface); and if these 
parameters are such that p = const., q = const, are the equations of the two sets of 
curves of curvature respectively; then (writing for shortness 
dx dx di l x d 2 x 
dp ~ Xl> dq~ X2 ’ df X *' dpdq Xi ’ dq- 
and the like for y, z), the coordinates x, y, z, considered always as functions of p, q, 
satisfy the equations 
XjX 2 + y$ 2 + z x z 2 = 0, 
®1, Vi, = 0. 
*®4 > Vi’ ^4 
The last equation is equivalent to 
Xi + Ax x + Bx 2 = 0, 
y 4 + Ay 1 + By2 = 0, 
s 4 + Az 1 + Bz 2 = 0; 
C. VIII. 
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