PEOFESSOE CAYLEY ON CUBIC SUEFACES. 
315 
Section XVII=12 — 2B 3 — C 2 . 
Equation WXZ+XY 2 +Y 3 =0. Article Nos. 181 to 185. 
181. The diagram of the lines and planes is 
N 
HJ 
M 
II 
II 
|| 
|| 
II 
£1 
o 
p 
O 
O 
o 
a* 
M 
M 
N 
Kj 
p 
+ 
+ 
|j 
|| 
II 
3 
Kj 
Kj 
o 
o 
O 
|| 
11 
o 
O 
« 
H- 
o 
XVII 
Planes are 
= 12- 
-2B 3 — C 2 . 
enl U> 
X 
ssi « 
X 
05 
X 
ca 
II 
CD 
X=0 
0 
IX 6= 6 
• • • 
Common biplane, through 
the axis joining the two 
binodes. 
Z =0 
13 
‘ * 
O 
fs 
24 
2x 6=12 
; 
Remaining biplanes, one 
for each binode. 
Y=0 
012 
1X18=18 
* * 
• 
Plane through the three 
axes. 
X+Y=0 
034 
lX 9= 9 
5 45 
• • 
• 
Plane through the axis 
joining the two binodes. 
w 
<x> C* 
m 
o ** 
pj o 
cd a 
Axes each through 
the cnienode and 
a binode. 
Axis joining the 
two binodes. 
where the equations of the lines and planes are shown in the margins. 
182. There is no facultative line ; 6'=^ = 0, t'=0. 
183. The Hessian surface is 
X(WXZ+ 3YZW+XY 2 )=0, 
viz. this breaks up into X=0 (the common biplane), and into a cubic surface. 
The complete intersection with the cubic surface is made up of X=0, Y=0 (BB-axis) 
4 times, of Y=0, Z=0 and Y=0, W=0 (CB-axes) each three times; and of a residual 
conic, which is the spinode curve; a'= 2. The equations of the spinode curve are 
Y 2 — 3ZW=0, 4X+3Y=0; viz. it lies in a plane passing through the BB-axis; since 
there is no facultative line, (3'=0. 
2 u 2 
