520 
MATHEMATICS: G. M. GREEN 
The differential equation (6) of the axis curves may therefore be written 
du' -^dudv~{B.-}- IV^ - dv"^ = 0. (10) 
du dv 
The differential equation of the asymptotic curves is 
adu' + dv''^ = 0 (11) 
The pair of asymptotic tangents at y is of course harmonically sepa- 
rated by the tangents to the curves of our conjugate net. The differ- 
ential equation 
adu"" ~dv'' = 0 (12) 
defines a new net of curves. It evidently has the property, that the 
tangents to the two curves of the net at the point y separate harmoni- 
cally both the pair of asymptotic tangents and the tangents to the two 
curves of our conjugate net. It is moreover the only net which has 
this property; since it also is a conjugate net, we call it the associate 
conjugate net. 
We shall define another net of curves which will be of importance 
in our geometric interpretation. The quadratic 
a K du"" ^ du dv ~ Kdv'' = 0 (13) 
has for its roots the negatives of the roots of (8). It therefore defines 
a net such that the tangents to the two curves thereof at the point y 
are the harmonic conjugates of the two ray tangents (the tangents to 
the ray curves) with respect to the original conjugate tangents (the 
tangents to the curves of the original conjugate net). For convenience, 
let us call the curves defined by (13) the anti-ray curves, and the two 
tangents to the anti-ray curves at the point y the anti-ray tangents of 
the point y. 
Let us now fix our attention upon a point y of the surface Sy, and let 
us regard equations (10), (12), and (13) as binary quadratics whose 
roots give respectively the pairs of axis tangetits, associated conjugate 
tangents, and anti-ray tangents of the point y. The Jacobian of the 
forms (10) and (12) is 
a^du'' + 2a(B.-K + log a) dudv-^-"^ dv'' = 0, (14) 
\ du c)v / 
and its roots give the pair of lines through y which separate harmonically 
both the pair of axis tangents and the pair of associated conjugate 
tangents of y. The Jacobian of the forms (12) and (13) is 
