125 
402] 
ON A SINGULARITY OF SURFACES. 
At a pinch-point, we have 
(A, B, C, F, G, H) (x, y, zf — 0, 
(A, B, C, F, G, H)(x, y, z) (y, z, 1) = 0, 
(A, B, G, F, G, H) (y, Z) l) 2 = 0, 
and hence the origin will be a pinch-point if (7=0, that is, if ab — h 2 = 0. This however 
appeals moie leadily by remarking, that the equation of the pair of tangent planes 
at the origin is 
or what is the same thing, 
(a, b, c, f g, h\y, - x, 0) 2 = 0, 
(a, h, b^y, -x) 2 = 0; 
the two tangent planes therefore coincide, or there is a pinch-point, if only ab — h 2 = 0. 
By what precedes, it appears that if we wish to study the form of the quartic 
surface, 1 \ in the neighbourhood of an arbitrary point on the nodal line; 2°, in the 
neighbourhood of a pinch-point; it is sufficient in the first case to consider the general 
surface 
(«> b, c, f g, h\y - z 2 , yz - x, xz- y 2 ) 2 = 0, 
in the neighbourhood of the origin; and in the second case, to study the special 
surface for which ab — li 2 = 0, or writing for convenience a = 1, and therefore b = h 2 , the 
surface 
(1, h 2 , c, f g, h\y-z 2 y yz-x, xz - y 2 f = 0, 
in the neighbourhood of the origin. 
Consider first the surface 
(a, b, c, f, g, h) (y - z 2 , yz - x, xz-y-f = 0. 
A plane through the origin is either a plane not passing through the tangent line 
(x = 0, y = 0), and the equation z=0 will serve to represent any such plane; or if it 
pass through the tangent line, then it is either a non-special plane, which may be 
represented by the equation y = 0; or it is a special plane: viz. either the osculating 
plane x = 0 of the nodal line, or else one or the other of the two tangent planes 
(a, h, b\y, — xf = 0 of the surface. I consider therefore the sections of the surface 
by these planes z = 0, y = 0, x = 0, (a, li, b\y, -x) 2 =0 respectively. 
Section by the non-special plane z = 0. 
The equation is 
(a, b, c, f g, h\y, -x, -yj = 0, 
which represents a curve having at the origin an ordinary node, the equations of the 
two tangents being (a, h, b\y, — ¿r) 2 = 0, viz. these are the intersections of the two 
tangent planes by the plane z — 0.