PBOFESSOE CAYLEY ON EECIPEOCAL SUEEACES. 
207 
there are on this line two points each a triple point on the residual curve of inter- 
section. 
21. We may however have on a surface an oscular line without upon it two or any triple 
points of the residual curve of intersection. Such a surface is M^r+Ny 3 =0 ; the inter- 
sections of the line #=0, y = 0 with the curve #=0, N=0 will be all of them ordinary 
points. The reciprocal surface will have a binode, but there will be some special cir- 
cumstance doing away with the existence of the directions of closest contact in the two 
biplanes respectively. I do not at present pursue the question. 
22. For a surface having a bitrope B'=l, it appears from what precedes, that the 
oscular line must count 4 times in the intersection of the surface with the Hessian ; for 
only in this way can the reduction 4 in the order of the spinode curve arise. 
23. The pinch -point, or singularity^^ 1, is in fact mentioned in Salmon ; it is a point 
on the nodal curve such that the two tangent planes coincide, or say it is a cuspidal 
point on the nodal curve. If, to fix the ideas, we take the nodal curve to be a complete 
intersection P=0, Q=0, then the equation of the surface is (A, B, CT£T, Q) 2 =0 
(A, B, C functions of the coordinates) ; we have a surface AC— B 2 =0, which may be 
called the critic surface, intersecting the nodal curve in the points P = 0, Q=0, 
AC — B 2 =0, which are the pinch-points thereof; or if there be a cuspidal curve, then 
such of these points as are not situate on the cuspidal curve are the pinch-points : see my 
paper “ On a Singularity of Surfaces,” Quart. Math. Journ. vol. ix. (1868) pp. 332-338. 
The single tangent plane at the pinch-point meets the surface (see p. 338) in a curve 
having at the pinch-point a triple point, = cusp + 2 nodes, viz. there is a cuspidal branch 
the tangent to which coincides with that of the nodal curve ; and there is a simple 
branch the tangent to which may be called the cotangent at the pinch-point. In the 
particular case where the nodal curve is a right line the section is the line twice (repre- 
senting the cuspidal branch), and a residual curve of the order n — 2, the tangent to 
which is the cotangent. 
24. The pinch-plane, or reciprocal singularity /=1, is in fact a torsal plane touching 
the surface along a line, or meeting it in the line twice and in a residual curve. Let 
the line and curve meet in a point P ; for the reason that the section by the plane is 
the line twice and the residual curve, the section has at P two coincident nodes ; that is, 
the plane is a node-couple plane with two coincident nodes. The plane meets the con- 
secutive node-couple plane in a line ^ passing through P and touching at this point the 
residual curve. Considering now the reciprocal figure, the reciprocal of the pinch-plane 
is thus a point of the nodal curve, and is a pinch-point ; the tangent plane at the pinch- 
point is the reciprocal of the point P ; the tangent to the nodal curve is the reciprocal 
of the line that is, of the tangent at P to the residual curve ; and the cotangent at the 
pinch-point is the reciprocal of the torsal line. 
25. There is in this theory the difficulty that for a surface of the order n , the torsal 
plane meets the residual curve of intersection in(w— 2) points P, and if each of these be 
a point on the node-couple curve, then in the reciprocal figure the pinch-point would be 
