412] 
A MEMOIR ON CUBIC SURFACES. 
425 
133. The cuspidal curve is a system of four conics; in fact from the preceding 
equations written in the forms 
(bd — c 2 , 2 (dxy — cy 2 ), — y A \\2zw, l) 2 = 0, 
{ad — be, Sdx 2 — 2cxy — by 2 , — 2xy‘ i ’^l 2zw, l) 2 = 0, 
eliminating zw, we obtain 
' 3 {bd- c 2 ), 5 
2 (— ad 2 — 3 bed + 4c 3 ), 
6 (acd + b 2 d — 25c 2 ), (£ x, yf = 0, 
6 {be — ad) b, 
a 2 d — b 3 , 
which shows that the cuspidal curve lies in four planes, and it hence consists of four 
conics; these are of course the reciprocals of the quadric cones which touch the cubic 
surface along the four conics which make up the spinode curve. 
134. The equation of the surface, attending only to the terms of the second order 
in y, z, w, is 27d 2 x A zw = 0 ; it thus appears that the point y — 0, z=0, w — 0 (reciprocal 
of the plane X = 0) (which is oscular along the axis joining the two binodes, or 
BB-axis) is a binode on the reciprocal surface, the biplanes being z = 0, w = 0, viz. these 
are the planes reciprocal to the binodes {X=0, Y= 0, W= 0) and (X = 0, 7 = 0, Z= 0) 
of the cubic surface; we have thus B'= 1. 
It is proper to remark that the binode y — 0, z=0, w = 0 is not on the cuspidal 
curve, as its being so would probably imply a higher singularity. 
135. A simple case, presenting the same singularities as the general one, is when 
a= d, b = c = 0: to diminish the numerical coefficients assume a — d = ^, the cubic 
surface is thus \2XZW + X 3 + Y 3 = 0, and the equation of the reciprocal surface, 
multiplying it by 4, becomes 
z 3 w 3 
+ 6x 2 zhv 2 
+ (9 ¿r 4 — 12# 3 y) zw 
— 4y 3 {x 3 — y 3 ) = 0, 
viz. this is the surface 
4 y ti 
— ty 3 x {x 2 + 3zw) 
+ zw (3# 2 + zw) 2 = 0 
considered in the Memoir “ On the Theory of Reciprocal Surfaces.” The cuspidal curve 
is, as there shown, composed of the four conics y = 0, Sx 2 +zw = 0 and y 3 - 2cc A - 0, 
X 2 - zw = 0; and it is there shown that the two points {x = 0, y = 0,z = 0), {x = 0, y = 0, w=0), 
each reckoned eight times, are to be considered as off-points of the reciprocal surface. 
C. VI. 54