MATHEMATICS: G. M. GREEN 
519 
in which the coefficients of y do not concern us. Consequently, 
+ 
Now, if the points y, z, dy, dz are to be coplanar, the determinant of the 
coefficients of y^, % in the expressions for s, dy, dz must vanish; 
on expansion this determinant yields the quadratic in du: dv, 
a pi2) + du'' -^dudv-a ^^''^ dv' = 0, (6) 
where, on using (4), we find 
^ = + ab'' - + b'c + be' + ab^ - C (7) 
The quadratic (6) determines the direction in which y must move, 
in order that the axis yz may trace out a developable; there are two 
such directions at each point of Sy. We may regard (6) as a differen- 
tial equation defining a net of curves on Sy having the property that if 
the point y traces out a curve of this net, the corresponding axis gen- 
erates a developable surface. We call the two curves of the net which 
pass through the point y the axis curves of the point y. 
In like manner, we may determine the developables of the ray con- 
gruence, i.e., the net of curves on having the property, that if the 
point y traces out a curve of the net, the corresponding ray traces out 
a developable of the ray congruence. The differential equation defin- 
ing this net of curves, which we call the ray curves, is without difficulty 
found to be 
aB. du'-^dudv~K dv'' = 0, (8) 
where is given by (7), and 
}i = d' + b'c'-b:„ -K = d' + bW-c: (9) 
are the Laplace-Darboux invariants of the given conjugate net. 
If we use (9), we find from (5) that 
a /3(03) = H -f - b„ y^^'^ + (-) =K + 2c', + {-) , 
the latter of which becomes, on use of (4), 
'(12^ -ff-) =Y. + 2b',-b,-~^\oga. 
\a/u 
du dv 
