MATHEMATICS: G. M. GREEN 
517 
tion is of great interest. In the present note, we propose to give a 
new and simple geometric characterization of isothermaily conjugate 
nets which i^ entirely different from Wilczynski's. 
Let y y'-\ y'^\ y'^\ be the homogeneous coordinates of a point in 
space, and let the four functions 
(u,v) (^ = 1,2,3,4) (1) 
define a surface Sy on which the curves u = const., v = const, form a 
conjugate net. Then the y'^^^s satisfy a completely integrable system 
of two partial differential equations of the form^ 
yuu = (^yvv + by^ + cy, + dy, . 
yu.=b'y, + c'y, + dy. 
The second of these is of the familiar Laplace type, characteristic of 
conjugate nets; the first shows that the conjugate net defined by equa- 
tions (1) is isothermaily conjugate if and only if 
-^log a = 0. (3) 
ou ov 
The coefficients in equations (2) are not arbitrary, but are subjected 
to certain integrability conditions. One of the relations yielded by 
these conditions is that^ 
A(6 + 2c')=|-(2&'-i-Alog a), 
OV Ou\ a ov / 
or 
+ 24 = 2bi-('-)- log a. (4) 
\a/u ouov 
The minus first and first Laplace transforms of the point y are re- 
spectively 
p=yu~c'y, <y=yv-'b'y, 
which represent covariant points on the tangents at y to the curves 
of the net passing through y. The surface Sp is the second focal sheet 
of the congruence of tangents to the curves v = const, on Sy, and 
is the second focal sheet of the congruence of tangents to the curves 
u = const, on Sy. Let us, with Wilczynski,^ call the line pa correspond- 
ing to the point y the ray of the point y, and the totality of rays, which 
form a congruence, the ray congruence. 
The osculating planes of the two curves u = const, and v = const, 
at a point y meet in a line which passes through y arid which Wilczynski 
