124 
ON A SINGULARITY OF SURFACES. 
[402 
touches the conic 
(a, b, c, f, g, h) (p, q, r) 2 = 0, 
viz. A, B, G, F, G, H being the inverse coefficients, A = be —/ 2 , &c., this condition is 
(A, B, C, F, G, H'frXP + gP', \Q + gQ', \R + y,RJ = 0, 
or what is the same thing, the pinch-points are given as the common intersections 
of the nodal line p = 0, q = 0, r = 0 with each of the three surfaces 
(A, B, G, F, G, H)(P, Q, Rf =0, 
(A, B, C, F, G, H)(P, Q, R) (P, Q', R') =0, 
(A, B, G, F, G, II) (P, Q\ RJ = 0, 
these last three equations in fact, adding only a single relation to the relations 
expressed by the equations 
p = 0, q = 0, r = 0. 
If the functions P, Q, R, P', Q', R' are linear functions of the coordinates, then 
the curve (p = 0, q = 0, r = 0) is a cubic curve in space, or skew cubic; and if 
moreover (a, b, c, /, g, h) are constants, then the equation 
(a, b, c, f, g, h\p, q, r)‘ 2 = 0, 
belongs to a quartic surface having the skew cubic for a nodal line: this surface is 
(it may be observed) a ruled surface, or scroll. With a view to ulterior investigations, 
1 propose to study the theory of the pinch-points in regard to this particular surface; 
and to simplify as much as possible, I fix the coordinates as follows: 
Considering the skew cubic as given, let any point 0 on the cubic be taken for 
the origin; let x = 0 be the equation of the osculating plane at O; y = 0 that of 
any other plane through the tangent line at 0; z=0, that of any other plane through 
O, not passing through the tangent line; and w = 0 that of a fourth plane; then 
the equation of the cubic will be 
x, y, z = 0, 
U \\ 
y> 2, w 1' 
or what is the same thing, the values of p, q, r are yio — z 2 , zy — xw, and xz — y- 
respectively. And conversely, the cubic being thus represented, the point (x = 0, y — 0, 
2 = 0) may be considered as standing for any point whatever on the skew cubic; the 
osculating plane at this point being x = 0, and the tangent line being x = 0, y = 0. 
For the purpose of the present investigations, we may without loss of generality write 
*.0 = 1; and for convenience I shall do this ; the values of p, q, r thus become y — z-, 
yz — x, xz — y 2 , and the equation of the surface is 
(a, b, c, f g, K$y- z\ yz - x, xz - y 2 ) 2 = 0.