206 
PROFESSOR CAYLEY ON RECIPROCAL SURFACES. 
Explanation of the New Singularities. Article Nos. 13 to 39. 
I proceed to the explanation of the new singularities. 
13. The cnicnode, or singularity C=l, is an ordinary conical point; instead of the 
tangent plane we have a proper qnadricone. 
14. The cnictrope, or reciprocal singularity C'=l, is also a well known one; it is in 
fact the conic of plane contact, or say rather the plane of conic contact, viz. the cnictrope 
is a plane touching a surface, not at a single point, but along a conic. 
15. Consider a surface having the cnicnode C=l, and the reciprocal surface having 
the cnictrope C'=l. There are on the quadricone of the cnicnode six directions of 
closest contact*, and reciprocal thereto we have six tangents of the cnictrope conic, 
touching it at six points. The plane of the cnictrope meets the surface in the conic 
twice, and in a residual curve which touches the conic at each of the six points. It 
would appear that these six contacts are part of the notion of the cnictrope. 
16. We may of course have a surface with a conic of plane contact, but such that the 
residual curve of intersection in the plane of the conic does not touch the conic six times 
or at all ; for instance the general equation of a surface with a conic of plane contact is- 
PM + V 2 N=0, where P = 0 is a plane, V=0 a quadric surface; and here the conic 
P=0, V=0 does not touch the residual curve P=0, N=0. The reciprocal surface 
will in this case have a cnicnode, but there is some special circumstance doing away 
with the six directions of closest contact which in general belong thereto. I do not 
further pursue this inquiry. 
17. For a surface having the cnictrope C'=l, 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=l, 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 
* Taking for greater simplicity coordinates x, y, z, 1, then for a surface having a cnicnode at the origin, the 
equation is U 2 +U 3 + &c. =0, the suffixes showing the degree in the coordinates ; the equation of the quadri- 
cone is U 2 =0, and the six directions are given as the lines of intersection of the two cones U 2 =0, U 3 =0. 
