518 
MATHEMATICS: G. M. GREEN 
calls the axis of the point y. The totality of axes, which correspond 
to all the points y of the surface Sy, constitute a congruence, the axis 
congruence. 
We may write the first of equations (2) in the form 
yuu - hu ~dy = ay,, + cy,. 
The left-hand member represents a point in the osculating plane to the 
curve V = const., and the right-hand member a point in the osculating 
plane to the curve u = const., at y. Therefore, since the coordinates 
are homogeneous, the point 
a 
lies on the line of intersection of the two osculating planes, and the 
line yz is the axis of the point y. 
We may determine the developables of the axis congruence as fol- 
lows. If the point y moves to the point y + dy, the point z moves to 
z + dz, where dy = y^^ du % dv and dz = z^^ du z, dv. We wish 
the line yz to generate a developable. This will happen if and only if 
the four points y, z, y + dy, z -\- dz He in a plane, or what is the same 
thing, if the points y, z, y^ du + % dv, z^ du -j- z, dv are coplanar. We 
have on differentiation of equations (2) 
Jvvv =amy^, + _^ ^(03)3;^ + 5(03)^^ 
where in particular 
^(12) = ^'^ ^(12)^5/2 + ^/^ yi2) = 5V4-c: + ^^,' 
«(03) =:y-t-^\oga, ^m = i (y^/ + 5; _ + , (5) 
a dv a 
ym = l[b'c-{.c'{c'-b)+c'u-c,-d], 
a 
so that on using these and equations (2) we find 
Zu = yuvv + -ym + (-) % 
a \a/u 
Zv = yvvv + -yvv+ (-) % 
a \a/ V 
= (b'~~ log a) y„ + iS^'^^^y, + |^7'°* + (^) j % + ( )y, 
