S IS = 
1 
[501 
number 
M 
times the number 
proportional parts 
iredths (instead of 
of the last figure, 
example of the 
to balance that 
502] 
97 
502. 
ON THE SURFACES DIVISIBLE INTO SQUARES BY THEIR 
CURVES OF CURVATURE. 
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873), 
pp. 8, 9. Read December 14, 1871.] 
Geometrically, the question is as follows:—Consider any two curves of curvature 
AB, CD of one set, and any two AC, BD of the other set, as shown by the continuous 
lines of the figure: drawing the consecutive curves as shown by dotted lines, the curve 
consecutive to AB at an arbitrary (infinitesimal) distance from AB, the other three 
curves may be drawn at such distances that the elements at A, B, and C shall be 
each of them a square; but this being so, the element at D will not be in general 
a square, and it is only for certain surfaces that it is so. But if (whatever the carves 
of curvature AB, CD, AD, BC may be) the element at D is a square, then it is clear 
that the whole surface can be, by means of its curves of curvature, divided into 
infinitesimal squares. 
Analytically, if for a given surface the equations of its curves of curvature are 
expressed in the form h=f{x, y, z), k = <j>{x, y, z); then the coordinates x, y, z can 
C. VIII. 13