310 
PROFESSOR CAYLEY ON CUBIC SURFACES. 
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 2 (X^+Z;s)+X(YZ-wX) 2 , 
or, as this may be written, 
(3w 2 , — 2 \yw, y 2 -\-ixw, 12 zwJX, Z) 3 . 
The equation is in the first instance obtained in the form 
108 wV 
— 32 iv 4 y 3 z 
+ 3Qw 4 yz(y 2 -{-4:Xw) 
+ w 2 («/ 2 + 4xw) 3 
— w 2 y 2 {y 2 + 4 xwf = 0 ; 
but the last two terms being together =4:W 3 x(y 2 -\-4:Xtv) 2 , the whole divides by 4w 3 , and it 
then becomes 
27 wV 
— 8 wy 3 z 
+ 2wyz(y 2 -\-4:Xw) 
+ xiyf + 4 ot ) 2 = 0 ; 
or, expanding, it is 
w 3 . 27 z 2 
+ w 2 . 2>§xyz-\- IQx 3 
. y 3 z 4- 8 x*y 2 
+ xy*=0. 
The section by the plane w = 0 (reciprocal of B 5 ) is w=0, y—§ (reciprocal of edge) 4 
times, together with w=0, #=0 (reciprocal of biplanar ray). 
The section by the plane z — 0 (reciprocal of C 2 ) is x(y 2 -}-4xw) 2 =0, viz. this is 
z=0, y 2 -\-ixw=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 vicinity is y=— t\X — gyjwf, showing that the line is a cusp-nodal line count- 
ing once in the nodal and once in the cuspidal curve : wherefore 5'=1. 
