340 
A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES. 
[411 
35. I take as an example a surface which is substantially the same as one which 
presents itself in the Memoir on Cubic Surfaces, viz. the surface (1, w, xy\w 2 — xy, z) 2 = 0, 
having the cuspidal conic w 2 — xy = 0, z = 0. Since in the present case AC — B 2 = P, we 
have M = 1, N = 0, and the close-points are given by P = 0, Q = 0, C = 0 ; that is, they 
are the points (z = 0, w = 0, x = 0) and (z = 0, w = 0, y = 0). 
36. I first however consider an ordinary point on the cuspidal curve, or conic w 2 —xy = 0, 
z = 0 ; the coordinates of any point on the conic are given by x : y : z : w = 1 : 9 2 : 0 : 0, 
where 0 is an arbitrary parameter; we at once find 6 2 x + y — 9 (z + 2w) = 0 for the 
equation of the tangent plane of the surface or cuspidal tangent plane at the point 
(1, 0-, 0, 6). Proceeding to find the intersection of this plane with the surface, the 
elimination of z gives 
(0-, 9w, xy'^iu 2 — xy, 0-x + y, — 20iv) 2 = 0, 
which is of course the cone, vertex (x = 0, y = 0, w = 0), which passes through the 
required curve of intersection. In place of the coordinates x, y take the new coordinates 
9 2 x — y = 2p, and 0' 2 x + y — 20w =2q\ we have 
0 2 x= 0io-\-p — q, 
-y = -0w+p — q, 
and thence 
— 0' 2 xy —pi 2 — (q + 9wf - p 2 — q 2 — 20qw — 0Hv 2 , 
9 2 (w‘ 2 — xy) = p 2 — q 2 — 29qw, 
and the equation thus is 
(0 4 , 9\v, —p 1 + q 2 + 29qw + 9 % vr\p 2 — q 2 — 29qw, 29 2 q) = 0, 
or, what is the same thing, 
(1, 9w, —p 2 + q 2 + 29qw + 9 2 w 2 \p 2 — q 2 — 29qiv, 2q} 2 = 0 ; 
viz. this is 
(.V 2 - <f ~ 20qw) 2 + 4<0qw (p 2 - q 2 - 29qw) + 4q 2 (- p 2 + q 2 + 29qw + 0hu 2 ) = 0 ; 
or reducing, it is 
(p 2 — q 2 ) (p 2 — 5q 2 ) + 80q 3 w = 0, 
the equation of the section in terms of the coordinates p, q, w. The equation is 
satisfied by the values £>=0, q= 0 which belong to the assumed point (1, 0' 2 , 0, 0) of 
the conic, and in the vicinity of this point we have p 4 + 80q 3 w = 0, which is a triple 
branch of the form y 3 = a?, the tangent q = 0 being, it will be observed, the tangent 
of the conic. But at the close-points, or when 0=0 or 0 = oo, the transformation 
fails; and these points must be considered separately. 
37. At the first of these, viz. the point. z=0, w= 0, x = 0, the tangent plane of the 
surface or cuspidal tangent plane is x = 0, and this meets the surface in the curve x = 0,