The collected mathematical papers of Arthur Cayley.

3341

A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES.

[411

11. Forming the combinations 4i+ 6r, 24t - 8q+ 18r (the last of which introduces
on the opposite side the term + 48t), we obtain
4i+6r= c(5n-12)-5y7-18/9+30-2X,
- 24t - 8q + 18r = - (8n - 16) b + (15n - 36)c - 34/3 + 9y + 4j + 90 + 6x,
equations which are used post, No. 53.
12. I remark that if there be on a surface a right line which is such that the
tangent plane is different at different points of the line, the line is said to be
scrolar: the section of the surface by any plane through the line contains the line
once. But if there is at each point of the line one and the same tangent plane,
then the section of the surface by the tangent plane contains the line at least twice;
if it contain it twice only, the line is torsal; if three times the line is oscular;
and the tangent plane containing the torsal or oscular line may in like manner be
termed a torsal or an oscular tangent plane. These epithets, scrolar, torsal and oscular,
will be convenient in the sequel.
Article Nos. 13 to 39. Explanation of the New Singularities.
I proceed to the explanation of the new singularities.
13. The cnicnode, or singularity C = 1, is an ordinary conical point; instead of
the tangent plane we have a proper quadricone.
14. The cnictrope, or reciprocal singularity C' = 1, 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= 1, and the reciprocal surface having
the cnictrope C' =1. There are on the quadricone of the cnicnode six directions of
closest contact(1), 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 + V2N=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.
1 Taking for greater simplicity coordinates x, y, z, 1, then for a surface having a cnicnode at the origin,
the equation is U2 + U3 + &c. = 0, the suffixes showing the degree in the coordinates; the equation of the
quadricone is U2=0, and the six directions are given as the lines of intersection of the two cones
U2=o, U3=0.