MATHEMATICS: J. LIPKA 
8f 
difference of their geodesic curvatures is equal to the geodesic curvature 
of a curve through that point but in a perpendicular direction of a re- 
lated N family. This is evidently true for every point and in every 
direction. 
Considering our ^ ^ curves as composed oi geodesic curvature ele- 
ments (w, V, v' , v"), and defining corresponding geodesic curvature ele- 
ments on a surface as two elements which have the same initial point 
and the same direction, we may state the following result: 
Given any two isogonal families. If in each direction through each 
point we construct a geodesic curvature element whose geodesic curvature is 
equal to the difference of the geodesic curvatures of corresponding elements of 
the two isogonal families, and then rotate each new element in the same direc- 
tion through a right angle {keeping its geodesic curvature unchanged), the 
^ ^ new elements will form a natural family. 
If (X) and a are the functions determining the two isogonal families, then 
the above transformation leads to the natural family whose characteristic 
function, F, is the exponential of 03 — a. 
According as we subtract the geodesic curvatures of the curves 
from those of the curves or vice versa, we get the N family, F = 
or Fi = e""", so that F = l/F^-, hence 
Two isogonal families give rise, by the above mentioned transformation, 
to two natural families whose characteristic point functions are reciprocals, 
and such that corresponding geodesic curvature elements have their geodesic 
curvatures numerically equal but opposite in sign. 
3. The analytic curvature transformation which changes an / family 
into an N family, is 
(D =«, =-!,.: = _ ''"-[aogVx-a).-(log^/x-.)„z.'] 
v' v'^ 
where a is an arbitrary point function. 
This changes 
/ = (co„ + CO, v') (1 + v'^) {type I) (4) 
into 
v"={ [(co - (logVx).] - [(co-a)t.-|- (log V\)uV} {l+^^'^l {type N) 
(14) 
and by comparison with (2), we have 
log F = CO - or F = e'^-" (15) 
Hence {T) is the analytic statement of the geometric transformation 
described in §2. 
It is interesting to note the results of repeated applications of the 
