PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
FEBRUARY 15. 1915 * Number 2 
Volume 1 
CONJUGATE SYSTEMS OF SPACE CURVES WITH EQUAL 
LAPLACE-DARBOUX INVARIANTS . . 
By E. J. Wilczynski 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF CHICAGO ' • ' 
Readbeforethe Academy, December 7, 1914. Received. December 16; 1914 . 
It is the object of this note to provide a new geometrical interpre- 
tation for the condition that the Laplace-Darboux invariants of an 
equation of the form 
+ap + b^ + c. = 0 • ■ (1) 
dxdy bx by 
be equaL These invariants are 
h = + ab — Cj k = hy ah — c, (2) 
where we have put 
da ^ d6 
bx ^ by ' 
a notation which we shall employ throughout this paper. 
Let z^^\ z^'^\ z^^\ z^^^ be any four linearly independent solutions of 
(1), functionally independent in the sense that no two of their three 
ratios can be expressed as a function of the third. If. z''^\ . . z^^} be 
interpreted as the homogeneous coordinates of a point P of space, vari- 
ation of the parameters x and y will cause P to generate a non-degen- 
erate surface S which is not a plane. Let us assume further thatjS^ 
is not developable. Then 2^^^ z^^^ will satisfy, besides (1), just 
one other linear homogeneous differential equation of the second order, 
of the form 
= mz^x -f- nz^ -f pZy + qz, ' (4) 
59 
