ON THE SURFACES DIVISIBLE INTO SQUARES &C. 
98 
[502 
be expressed each of them as a function of the parameters h, k, and we have for the 
element of distance between two consecutive points on the surface 
dx- + dy 2 + dz 2 = Adb? -f Cdk 2 , 
where A, C are in general each of them a function of h and Jc. The condition for 
the divisibility into squares is that the quotient A 4- G shall be of the form function h 
-T- function k. 
It was shown by M. Bertrand that, in a triple system of orthotomic isothermal 
surfaces, each surface possesses the property in question of divisibility into squares by 
means of its curves of curvature. But in such a triple system, each surface of the 
system is necessarily a quadric; so that the theorem comes to this, that a quadric 
surface is, by means of its curves of curvature, divisible into squares. The analytical 
verification is at once effected: taking the equation of the surface to be 
x 2 y* z i 
—+X + - = 1> 
a b c 
then the expressions for the coordinates in terms of the parameters li, k of a curve 
of curvature are 
„_a (a 4- h) (a + k) 
X (a — b) (a — c) ’ 
..2 _ h ( h + h ) (ft + &) 
J (6 — c) (6 — a) ’ 
„ c (c + h)(c + k) 
Z '~ (c — a)(c — b) ’ 
and we have 
* /1 n i „ i o\ /j n f hdh 2 kdk 2 ) 
4 (dx- + dy + dz ) - (h - k) j (a + A) (b + h) (c+/ -) - (« + k)(b + k) (c + ¿)j ’ 
so that A ~ C is of the required form. 
But there is nothing to show that the property is confined to quadric surfaces; 
and the question of the determination of the surfaces possessing the property appears 
to be one of considerable difficulty, and which has not hitherto been examined.