290 
MATHEMATICS: L. P. EISENHART 
TRANSFORMATIONS OF CONJUGATE SYSTEMS WITH EQUAL 
INVARIANTS 
By Luther Pfahler Eisenhart 
DEPARTMENT OF MATHEMATICS, PRINCETON UNIVERSITY 
Presented to the Academy, March 25, 1915 
Recently we considered at length two types of transformations of 
conjugate systems of curves on surfaces, which we called^'^ transforma- 
tions K and transformations i^. It is the purpose of this note to show 
that there is a fundamental relation connecting these transformations. 
If X, y, z are the cartesian coordinates of a surface S, referred to a 
conjugate system with equal point invariants, it is necessary and suffi- 
cient that X, y, z are solutions of an equation of the form 
m ^ blogVp ^ _ ^ 
bubv bv bu bu bv 
where p is a function of u and v in general. If is any solution of (1) 
and Xi is the function defined by the consistent equations 
b\i bdi b\i bdi 
bu bu bv bv 
the equations 
b , . /bdi bx\ b , . (bdi bx\ 
-(X.xO = -p(a;--..-j, -(X.x0=p(x--^x-j (3) 
are consistent, and Xi satisfies an equation of the form (1) when p is 
replaced by pi, given by 
Pi = 
(4) 
The three functions Xi, yi, Zi, given by (3) and similar equations in 
the y's and z's, are the cartesian coordinates of a surface 5i, which 
by definition is in the relation of a transformation K with the surface 
S. If M and Mi are corresponding points on these surfaces, the 
developables of the congruence G of the fines MMi cut S and Si in the 
parametric curves and the focal points on the line MMi are harmonic to 
M and Mi, Conversely, when two surfaces S and 6*1 are so related 
that the congruence of lines joining corresponding points meet 5 and Si 
in conjugate systems and the points of S and Si are harmonic to the 
focal points of the congruence, they are in the relation of a transforma- 
tion K, as defined analytically by (3) [cf. Mi, p. 403]. 
