80 MATHEMATICS: J. LIPKA 
If we have any other simple system of curves 
v' = tan a (u, v), (5) 
the differential equation of its isogonals is similarly given by 
v" = {a, + a, v') (1 + {type /J (6) 
For these families, «• ^ curves pass through each point on the surface, 
one in each direction. 
The geodesic curvature for any curve on the surface is 
1 v'' -[ (log VX). - (log VX )„ t;'] [ 1 + v"'] 
- = 3 (7) 
^ \/y(l + z;'2)2 
If we apply ( 7 ) to the unrelated /^^ and /q, curves, we have, for a 
curve in any direction v' , 
[ o)u - (log V'\)v ] + [ CO. + ( log Vx" )u ] 
[;],= 
(8) 
(9) 
VX (1 + V'') 
rn [an - { log VT ). ] + [ + ( log V\ )u ] v' ^ 
L p J " Vx(l + 2;'2) 
and hence 
r 1 1 r 1 1 ( COM - au) + ( - ) 
Ui-Ui= vfTTT^^ 
On the other hand, if we apply .(7) to the iV curves, we have, for a 
curve in any direction d' ^ 
_ (log F\ ~ (log F\ v' xj^x 
pJ^ Vx(l + 2;'2) 
and for a curve in a direction^— i^, i.e., in a direction perpendicular 
to the direction v' , 
rn _ (log F).H-(log F\v' ,j2) 
LpJ^ Vx(i+.'2) 
Comparing (12) and (10) we see that the right members of these 
equations will coincide if 
log F = CO - a, or F = . (13) 
Thus if we have two distinct I famiHes and we choose one curve of 
each family passing through the same point in the same direction, the 
