318 
PROFESSOR CAYLEY OJN CUBIC SURFACES. 
The equation is thus 
(y 2 +&wx + 4w) 2 — 6 4lW 2 £cz = 0, 
or in an irrational form 
wx+%s/‘ wz=0. 
The section by the plane w=0 (reciprocal of B 4 ) is w=0, y=0 (reciprocal of edge) 
4 times. 
The section by the plane 2=0 (reciprocal of C 2 =C) is z= 0, y 2 -\-iwx =0 (reciprocal 
of nodal cone) twice; and similarly for the section by x=Q (reciprocal of C 2 =A). 
Nodal curve. Writing the equation in the form 
y * + 8 wy 2 (z x) -f- 1 6w 2 (;r — z ) 2 = 0 , 
we have a nodal line y— 0, x—z=0, reciprocal of the mere line : 
And writing the equation in the form 
we have y= 0, w = 0 (reciprocal of edge), a tacnodal line counting as two lines ; V— 3. 
There is no cuspidal curve; c'=0. 
Section XIX=12-B 6 -C 2 . 
Equation WXZ+Y 2 Z+X 3 =0. Article Nos. 190 to 193. 
190. The diagram of the lines and planes is 
X 
ft 
II 
B' 
o 
© 
N 
KJ 
p 
© 
II 
II 
o 
O 
U)| — 
xix=: 
12— B 6 — C 2 . 
X 
X 
IT 
II 
Planes are 
15 
27 
to 
Z=0 
... 
1x15=15 
* 
Oscular biplane. 
x=o 
• 
1 x 30=30 
Ordinary biplane. 
2 45 
ft 
<JQ 
O 
o 
c-h s’. 
p" s* 
. 
o 
g* 
| f 
p 5‘ 
o 
where the equations of the lines and planes are shown in the margins. 
191. The axis is a facultative line counting 3 times (as will appear from the reciprocal 
surface); §'=b'=o, i'=l. 
