MATHEMATICS: J. LIPKA 
79 
field of force, our natural family may be: (1) a system of trajectories 
arising from the principle of least action, where F = \^ W h; (2) a 
system of hrachistochrones or curves of quickest descent, where F = 
\/ (W + h)~^; (3) a system oi general catenaries or positions of equilibrium 
of homogeneous, flexible, inextensible strings, where F = (W+h). 
Again, if F is the variable index of refraction in an isotropic medium, 
the paths of light in such a medium form a natural family. The conformal 
representation of the geodesies of any surface upon certain other sur- 
faces is also a natural family. Natural families of curves have been 
geometrically characterized by Kasner for a plane and for space of 
three dimensions,^ and by me for any surface and any curved space 
of n dimensions. 2 
If we have any set of ^ ^ curves on a surface, the system of °c i curves 
which cut every curve of this set at a constant angle, a, form a system 
of isogonal trajectories of the original set ; there are ^ ^ such systems for 
varying values of the parameter a, and these form the complete fam- 
ily of isogonal trajectories. Isogonal families of curves have been 
geometrically characterized by Kasner for the plane, ^ and by me^ for 
any surface. 
In this paper, §2 gives a very general geometric transformation by 
which a family of isogonals may be transformed into a natural family; 
§3 gives the analytic representation of the geometric transformation 
of §2, and exhibits the interchange of the two families through repeated 
application of this transformation; §4 gives the relations existing be- 
tween the point functions which characterize dual (natural-isogonal) 
families. 
2. If we take an isothermal system of curves as parameter curves 
on the surface, we can write the element of arc length in the form 
ds^ = X {u, v) [du^ + dv^]. 
The variation problem 
y* F (u,v) ds = minimum (1) 
then leads to a family of ^ ^ curves whose differential equation is given 
by 
v" = [ (log FV\\ - (log F^y\), v'] [1 -f v"^] {type N) (2) 
and the problem of finding the isogonals of a simple system of curves 
v' = tan CO (u, v) (3) 
leads to a family of ^ ^ curves whose differential equation is given by^ 
v'' = (co„ + CO, v') (1 -f v'-') (type /J (4) 
