424 
A MEMOIR ON CUBIC SURFACES. 
[412 
Reciprocal Surface. 
130. The equation is obtained by means of the binary cubic 
X (xX + yY) 2 + 4zw (a, b, c, d\X, F) 3 , 
viz. calling this (*$X, Y) 3 the coefficients are 
(3# 2 + 12aztu, 2xy 4-12bzw, y 2 + 12czw, 12dzw). 
The equation is found to be 
432 (a 2 d 2 — Qabcd + 4ac 3 + 4 b 3 d — 36 2 c 2 ) zhu 3 
4-216 [(ad 2 — 3bed 4- 2c 3 ) x 2 + (— 2acd + 4>b 2 d — 26c 2 ) xy 4- (— abd + 2ac 2 — b 2 c) y 2 \ z 2 iu 2 
+ 9 [3d 2 a? — 12cdic 3 y 4- (106d + 8c 2 ) x 2 y 2 — (4ad 4- 8be) xy 3 4- (4ac — b 2 ) y*\ zw 
— y 3 (da? — 3 cx 2 y + 3 bxy 2 — ay 3 ) = 0. 
The section by the plane %u — 0 (reciprocal of B 3 = D) is the line w = 0, y = 0 (reciprocal 
of edge) three times, and the lines w = 0, da? — 3cx 2 y 4- obxy 2 — ay 3 = 0 (reciprocals of the 
biplanar rays). And similarly for the section by the plane z — 0 (reciprocal of B 3 = G). 
The section by the plane y = 0 is made up of the lines (y = 0, ^ = 0), (y = 0, w = 0) 
each once, and of two conics, y = 0, 
16 (a 2 d 2 — Gabcd + 4ac 3 + 4b 3 d — 3b 2 c 2 ) z 2 w 2 
4- 8 (ad 2 — Sbcd + 2c 3 ) x 2 zw 
4- d 2 a? = 0. 
131. There is not any nodal curve; b' = 0. 
132. Cuspidal curve. The equations may be written 
ox 2 4-12a^w, Ixy 4- 12bziv, y 2 + 12czw | =0. 
2 xy + 12 bzw, y 2 + 12 eziu, 12 dziu 
Forming the equations 
(bd — c 2 ) . 144^ 2 w 2 4- 2 (dxy — cy 2 ). 12zw —y i = 0, 
(ad — be). 144^ 2 w 2 4- (3dx 2 — 2cxy — by 2 ). 12zw — 2xy 3 = 0, 
— y 3 \ on the first surface it is a nodal line, the one tangent plane being 
these are two quartic surfaces having in common the lines (y = 0, w = 0), (y = 0, z = 0) : 
attending to the line (y = 0, 0 = 0), this is on the second surface an oscular line, 
1 
2 ISdajw 
6 (bd — c 2 )w. z + dxz . y = 0, the other tangent plane being z = 0, but the line being in 
regard to this sheet an oscular line, 2 = — yHence in the intersection of the 
two surfaces the line counts (14-3=) 4 times; similarly the line y = 0, w = 0 counts 
(1 + 3=) four times; and there is a residual intersection of the order (16 — 4 — 4=) 8, 
which is the cuspidal curve ; c' = 8.