PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 3 OCTOBER 15. 1917 Number 10 
ON THE GENERAL THEORY OF CURVED SURFACES AND 
RECTILINEAR CONGRUENCES 
By Gabriel M. Green 
DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY' 
Communicated by W. F. Osgood, August 15, 1917 
During the past two or three years, I have presented several com- 
munications to the American Mathematical Society concerning the 
general theory of curved surfaces and rectilinear congruences. I have 
been unable to find the leisure to w^rite out in full my results, and ta,ke 
the present opportunity to gather together a few of the more important, 
in the hope that I may soon be in position to publish an extended 
treatment elsewhere. 
Let the non-developable surface 5 be referred to a non-conjugate 
parametric net {u, v). Its equations in homogeneous coordinates may 
be taken in the form 
yik) = yik) {u, V) {k = 1, 2, 3, 4), (1) 
where the determinant \yuv, yhi, yv, y\ is nowhere zero, and the four 
functions y^^^ will then be a fundamental system of solutions of a com- 
pletely integrable system of partial differential equations^ 
yuu = a yuv + b yu -\- c > + d y, . . 
y'vv = a'yuv + b'yu + c'yv d'y. 
The coefficients in these equations are functions of u, v which satisfy 
certain conditions of complete integrability, which v/e shall not write 
out here. Suffice it to say that if the integrability conditions are iden- 
tically satisfied, any derivative of y is expressible, and in only one way, as 
a linear combination of yuv, yu, >, y, 
= a'^'bu, + + c<^"y. + d'^'V (3) 
587 
