A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
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411] 
it appears that there are also the three cuspidal conics y 3 — 2x 3 = 0, a? 2 — zw = 0. Reducing 
by means of these two equations, the equation of the second polar is at first obtained 
in the form 
(4, 6x, &x 2 + zw\Sy 2 Ay — §x 2 Ax, 2xAx — zAco — wAz) 2 = 0 ; 
but further reducing by the same equations and writing for this purpose y = tox (eo 3 = 2), 
the equation becomes 
(4, 6x, 9x 2 ^x 2 (3a> 2 Ay — 9 Ax), lx Ax — zAw — wAz 2 ) 2 — 0, 
that is 
x 2 [2x (Sco 2 Ay — 6A#) + 3 (2xAx — zAw — wA^)] 2 = 0, 
and we have thus the off-points ac 2 = 0, y 3 — 2x 3 = 0, x 2 — zw = 0, in fact the before- 
mentioned two points each 6 times; and the complete value of 0 is 0 = (4 + 12=) 16; 
viz. the off-points are the points (sc = 0, y = 0, z = 0), (# = (), y = 0, w — 0) each 8 times. 
On account of this union of points the singularity is really one of a higher order, but 
equivalent to 0 = 16. 
I am not at present able to explain the off-plane or reciprocal singularity 0' = 1. 
33. As to the close-point or singularity %=1. I remark that at an ordinary point 
of the cuspidal curve the section by the tangent plane touches, at the point of contact, 
the cuspidal curve: the point of contact is on the curve of section a singular point 
■in the nature of a triple point, viz. taking the point of contact as origin, the form 
of the branch in the vicinity thereof is y s — x i = 0, where y = 0 is the equation of the 
tangent to the 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 A , 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 cuspidal 
curve be a complete intersection P = 0, Q = 0; the equation of the surface is 
(A, B, C^P, 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 + JS r Q 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 
intersection 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, CQN, — 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. 
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