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A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
335 
17. For a surface having the cnictrope C'= 1, the Hessian surface passes through 
the conic, which is thus thrown off from the spinode curve; or there is a reduction 
= 2 in the order of the curve, which agrees with a foregoing result. 
18. The binode, or singularity B — 1, is a biplanar node, where instead of the proper 
quadricone we have two planes; these may be called the biplanes, and their line of 
intersection, the edge of the binode. The biplanes form a plane-pair. 
19. The bitrope, or reciprocal singularity B' = l, is the plane of point-pair contact; 
but this needs explanation. 
20. Consider a surface having a binode, and the reciprocal surface having a bitrope. 
We have the bitrope, a plane the reciprocal of the binode; in this plane a line, the 
reciprocal of the edge; in the line two points, or say a point-pair, the reciprocal of the 
biplanes: these points may be called the bipoints. There are in each biplane three 
directions of closest contact; the reciprocals of these are in the bitrope three directions 
through each of the two points. The section of the reciprocal surface by the bitrope 
is made up of the line counting three times (or the line is oscular), and of a curve 
passing in the three directions (having therefore a triple point) through each of the 
two bipoints. The bitrope contains thus an oscular line ; but it is part of the notion 
that there are on this line two points each a triple point on the residual curve of 
intersection. 
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 Mx + Ny 3 = 0; 
the intersections of the line x = 0, y = 0 with the curve x = 0, N = 0 w T ill be all of 
them ordinary points. The reciprocal surface will have a binode, but there will be 
some special circumstance 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' = 1, 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 j = 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 (H, B, CQP, Q) 2 = 0 
(A, B, G functions of the coordinates); we have a surface AG— 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, [402]. 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