402] 
123 
402. 
ON A SINGULARITY OF SURFACES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), 
pp. 332—338.] 
A SURFACE having a nodal line has in general on this nodal line points where 
the two tangent planes coincide, or as I propose to term them “pinch-points.” Thus, 
if the nodal line be the curve of complete intersection of any two surfaces P = 0, 
Q = 0, then the equation of the general surface having this curve for a nodal line is 
(a, h, c#P, Qy = 0 (where a, b, c are any functions of the coordinates), and the pinch- 
points are given as the intersections of the nodal line P = 0, Q = 0 with the surface 
ac — b 2 = 0. Consider the case where the nodal curve is a curve of partial intersection 
represented by the equations P, Q, R = 0, or say by the equations p= 0, q = 0, 
P, Qf, R' 
r = 0 (viz. p, q, r denote the functions QR' — Q'R, RP' — R'P, PQ' — P'Q respectively), 
and consequently we have identically 
(P, Q, R^ip, q, r)= 0, 
(P, Q', R'\p, q, r)= 0, 
or what is the same thing, (A, p) being arbitrary, 
(AP + pP', \Q + pQ', \R + pR'][p, q, r) = 0. 
The general surface having the curve in question for its nodal line is represented by 
the equation 
(a, b, c, f, g, hftp, q, r) 2 = 0, 
(where (a, b, c, f, g, h) are any functions of the coordinates), and it is easy to see that 
the condition for a pinch-point is the same as that which (considering p, q, r as 
coordinates and all the other quantities as constants), expresses that the line 
(AP + pP', AQ + pQ', AR + pR’\p, q, r) = 0, 
16—2