PEOFESSOE CAYLEY ON EECIPEOCAL SUEFACES. 
211 
cuspidal curve], such that the point of contact counts 4 times in the intersection of the 
cuspidal curve with the curve of section. At a close-point the form of the curve of 
section is altered ; viz. the point of contact is here in the nature of a quadruple point 
with two distinct branches, one of them a triple branch of the form y 3 =x*, but such 
that the tangent thereof, y— 0, is not the tangent of the cuspidal curve; the other 
of them a simple branch, the tangent of which is also distinct from the tangent of the 
cuspidal branch: the point of contact counts 3 + 1 times, that is 4 times, as before, in 
the intersection of the cuspidal curve and the curve of section. The tangent to the 
simple branch may conveniently be termed the cotangent at the close-point ; that of the 
other branch the cotriple tangent. 
34. We may look at the question differently thus : to fix the ideas, let the cus- 
pidal curve be a complete intersection P = 0, Q=0; the equation of the surface is 
(A, B, CX?, Q) 2 =0, where AC— B 2 =0, in virtue of the equations P=0, Q=0 of the 
cuspidal curve, that is, AC— B 2 is =MP+NQ suppose. We have (as in the investiga- 
tion regarding the pinch-point) a critic surface AC — B 2 =0, this meets the surface in the 
cuspidal curve and in a residual curve of intersection ; the residual curve by its intersec- 
tion with the cuspidal curve determines the close-points ; the tangent at the close-point 
is I believe the tangent of the residual curve. Analytically the close-points are given 
by the equations P — 0, Q=0, (A, B, CJN, — M) 2 =0. It is proper to remark that if 
besides the cuspidal curve there be a nodal curve, only such of the points so determined 
as do not lie on the nodal curve are the close-points. 
35. I take as an example a surface which is substantially the same as one which pre- 
sents itself in the Memoir on Cubic Curves, viz. the surface (1, w, xy$w 2 —xy, z) 2 = 0, 
having the cuspidal conic w 2 —xy~ 0, 2=0. Since in the present case AC— B 2 =P, we 
have M=l, N=0, and the close-points are given by P=0, Q=0, C=0; that is, they 
are the points (2 = 0, w = 0, #=0) and ( 2 = 0 , w= 0, y= 0). 
36. I first however consider an ordinary point on the cuspidal curve, or conic w 2 —xy=0, 
2 = 0; the coordinates of any point on the conic are given by x: y: z: w=l : Q 2 : 0 : 0, 
where 0 is an arbitrary parameter ; we at once find 0 2 x-\-y—0(z-\-'2w) = O for the equation 
of the tangent plane of the surface or cuspidal tangent plane at the point (1, 0 2 , 0, 0). 
Proceeding to find the intersection of this plane with the surface, the elimination of z 
gives 
(0 2 , 0w, xyjw 2 —xy, 0 2 x-\-y , — 24w) 2 =0, 
which is of course the cone, vertex (a?=0, y= 0, w = 0), which passes through the required 
curve of intersection. In place of the coordinates x, y take the new coordinates 
8 2 x— y=2jq, and 0 2 x-j-y— 2Qw=2q; we have 
6 2 x= Qw-\-jp—q, 
—y=—Qw+p—q, 
— (jrxy =p 2 — (q + kv) 2 =p 2 —q 2 — 2Qqw — Q 2 w 2 , 
6 2 {w 2 —xy) =]) 2 —q 2 —2Qqiv, 
2 g 2 
and thence 
