MATHEMATICS: J. LIPKA 
83 
jugate harmonic to i.e., if co + ^* is a function of u i v, then F = 
X~^e^ . Now for our parameter system, which is any isothermal sys- 
tem, CO = 0 and therefore H = 0 and F = = Fo, and we may write 
F = Foe^. Hence 
// Fo is the characteristic function corresponding to the isogonals of an iso- 
thermal system, an N family can he identified with an I family when and 
only when its characteristic function is the product of Fo and the exponential 
of a harmonic function.'^ 
If we have two isothermal systems, v' = tan co and = tan a, and 
w and a are conjugate harmonic, they form the base systems of two 
N—I families whose characteristic functions are 
F^ = X-*^" and Fa = \-^e'' 
respectively; hence 
Fee ^ ^co-« (21) 
On the other hand, the isogonals of the system v' = tan co are trans- 
formed by (T) into the N family whose characteristic function is F = 
e"~". Comparing this with (21) we may write 
F = ^ (22) 
If 0} -i- i a is a function of the complex variable u i v, it determines 
two isothermal systems, v' = tan cc and v' = tan a, which are the base sys- 
tems of two N — I families. Either of these families may be transformed 
by means of (T) and the remaining family into an N family whose charac- 
teristic function is the ratio of the characteristic functions of the two given 
families. ■ 
iKasner, E., Trans. Amer. Math. Soc, New York, 10, 1909, (201-219). 
2Lipka, J., lUd., 13, 1912, (77-95). 
3Lipka, J., Ann. Math., Princeton, N. J., 15, 1913, 
* Throughout this paper, primes refer to total derivatives with respect to u, and literal 
subscripts to partial derivatives. 
^The condition that the system = tan/3 be isothermal is + = 0- 
^The condition for a developable surface is (log X)«« + (log X)pj, = 0; cf. Note^. 
^ Compare with Kasner, E., New York, Bull. Amer. Math. Soc, 14, 1908, (169-172). 
