438 
A MEMOIR ON CUBIC SURFACES. 
[412 
168. I say that the spinode curve is made up of the edge X = 0, Z = 0 once, 
and of the cubic curve; and therefore ar = 4. 
In fact in the reciprocal surface the cuspidal curve is made up of the skew cubic, 
and of. a line the reciprocal of the axis, being a cusp-nodal line, and so counting 
once as part of the cuspidal curve: the pencil of planes through the line is thus 
part of the cuspidal torse; and reverting to the original cubic surface, we have the 
axis as part of the spinode curve : I assume that it counts once. 
The edge is a single tangent of the spinode curve; /3'= 1. 
Reciprocal Surface. 
169. The equation is obtained by means of the binary cubic 
4wZ- (Xx + Zz) + X (YZ — wX)-, 
or, as this may be written, 
(3w\ —2yw, y- + 4>ociv, 12zw\X, Z) 3 . 
The equation is in the first instance obtained in the form 
\03w 3 z 2 
— 32 vfy 3 z 
4- 3 Ovfiyz {y 2 4- 4ami) 
4- w- (y 2 + 4 xwf 
— vj-y 2 (y 2 + 4>xw) 2 = 0 ; 
but the last two terms being together = 4w 3 x (y 2 4- 4xw) 2 , the whole divides by 4w 3 , 
and it then becomes 
27 w 3 z 2 
— Swy 3 z 
4- 9 wyz (y 2 4- 4xw) 
4- x (;y 2 + 4xw) 2 = 0 ; 
or, expanding, it is 
w 3 .27^ 2 
+ w 2 .3 Qxyz + lfiic 3 
+ w . y 3 z + 8 x 2 y 2 
+ xy 4 = 0. 
The section by the plane w = 0 (reciprocal of B 5 ) is w = 0, y = 0 (reciprocal of 
edge) four times, together with w = 0, x = 0 (reciprocal of biplanar ray). 
The section by the plane z = 0 (reciprocal of C. 2 ) is x (y- + kxw) 3 — 0, viz. this is 
z = 0, y 2 4- 4xw = 0 (reciprocal of nodal cone) twice, together with z = 0, x = 0 (reciprocal 
of nodal ray). 
170. Nodal curve. This is the line w = 0, y= 0, reciprocal of edge. The equation 
in the 
showing that the line is a cusp-nodal 
line counting once in the nodal and once in the cuspidal curve: wherefore b' = 1.