588 
MATHEMATICS: G. M. GREEN 
The two points defined by the equations 
P = yu - py, (T = yv - fiy, « (4) 
where m and v are functions of u, v, He on the tangents to the parametric 
curves. Let us denote by / the Hne joining p and cr; we have thus asso- 
ciated with each point y of the surface a definite Une / lying in the cor- 
responding tangent plane. The congruence of lines / we shall denote 
by r. 
Again, if /x and v are any functions of u, v, the point 
z = yuv — fxyu — vyt (5) 
does not lie in the tangent plane to S at y^ and the line V joining y and 
z therefore protrudes from the surface. The Hnes I' constitute a con- 
gruence r'. 
If, now, the functions \x and v are the same in equations (4) and (5), 
the lines / and /' are in a certain characteristic geometric relation, which 
may be described as follows: The ruled surface formed by the tan- 
gents to the curves v = constant along a fixed curve u = constant is 
skew, since the parametric net is non-conjugate. This ruled surface 
is touched, in the point p defined by (4) , by the plane determined by 
the line V and the tangent y yu. Similarly the point a is the point in 
which the plane determined by and the other parametric tangent 
y > is tangent to the ruled surface i^^"^ formed by the tangents to the 
curves u = constant along a fixed curve v = constant. The line / is 
therefore determined uniquely by the Hne l\ and, conversely, if the line 
I is given, or in other words the points p and cr, the Hne /'is defined as 
the Hne of intersection of the tangent planes to the ruled surfaces R^^^ 
and R^^^ constructed at the points p and a. The relation between the 
Hnes / and or between the congruences r and r', is a reciprocal one, 
which for want of a better name I shall call the relation R. It is deter- 
mined of course by the particular parametric net to which the surface 
is referred. 
The developables of the congruence r' cut the surface 5 in a net of 
curves whose differential equation is 
+ { -C'y-y^^y (flr (12) _ _ ^) _ [ ^(21) _ _ ^ (^(21) _ _ ^) ] } ^ludv (6) 
The quantities with double upper indices are coefficients of equations 
of the form (3). Likewise, the developables of the congruence T corre- 
spond to a net of curves on S defined by the differentia] equation 
