62 
MATHEMATICS: L. P. EISENHART 
TRANSFORMATION OF SURFACES 
By L. p. Eisenhart 
DEPARTMENT OF MATHEMATICS. PRINCETON UNIVERSITY 
Presented to the Academy, December 28, 1914. 
When any surface 6* is referred to a conjugate system with equal 
point invariants, its cartesian coordinates y, z, are solutions of an 
equation of the form 
m _^dlogv^ dg_^ dlogVp d^_Q 
bubv dv bu bu bv 
If di is any solution of this equation, linearly independent of x, y and z, 
the surface Si whose cartesian coordinates, Xi, yi, Zi, are given by equa- 
tions of the form 
ou \ bu bul bv \ bv 
|), (2) 
where \i is given by the quadratures 
^=-p^ ^' = p^ (3) 
bu bu bv bv 
is referred to a conjugate system with equal point invariants, and cor- 
responding points M and Mi on S and Si are harmonic with respect 
to the focal points of the line MMi for the congruence of these lines. 
We say that Si is obtained from 5 by a transformation K. We have 
studied these transformations at length in a recent memoir. ^ In the 
present note we consider the case where the lines MMi form a normal 
congruence. In this case there exists a solution t of equation (1) such 
that x^ -\- y'^ -{- z^ — t^ also is a solution. Thus the parametric conju- 
gate system is 2 O in the sense of Guichard, and / is the complementary 
function. The surface Si has the same properties. 
By definition a surface C is one possessing a conjugate system 2 O 
with equal point invariants. When this system is parametric, the first 
fundamental coefiicients of C have the form 
E=m\l, F=^^, G=(^y+1, (4) 
\bu/ p bu bv \bv/ p 
and this property is characteristic. 
It can be shown that when a surface C is referred to the system 2 0 with 
equal point invariants, there can be found without quadratures two sur- 
