PROFESSOR CAYLEY ON CUBIC SURFACES. 
305 
of the unode) is made up of the lines w= 0, y—z= 0 ; w= 0, z—x — 0 ; w= 0, x—y=0 
(reciprocals of the rays) each twice. 
155. The nodal curve is at once seen to consist of the lines (y=0, z= 0), (2=0, #=0), 
(x=0, y=0), reciprocals of the facultative lines; in fact, in regard to (y, z) conjointly 
y is of the order 1, and § is of the order 2 ; hence every term of the equation is of the 
order 2 in y, z; and the like as to the other two lines: b'=3 as above. 
156. For the cuspidal curve we have 
I 12 , w- 4/3, 4 y |[=0, 
| w— 4/3, 4 y , -12h I 
or say 
48y-(w-4/3) 2 =0, 
36c$-f-y(w — 4/3)=0, 
whence the cuspidal curve is a complete intersection 2x3; c'=G. 
Section XIII=12— B 3 -2C 2 . 
Equation WXZ+Y 2 (Y+X+Z) = 0. Article Nos. 157 to 164. 
157. The diagram of the lines and planes is 
Lines. 
£ 
^ CO 
fcc ■— 
© 
a cjt 
col H- 
to 
1 
W 
X 
X 
x 
X 
X 
XIII=12— B 3 — C 2 . 
if 
if 
05 
ii 
05 
II 
iSI - 
* 
l 
I 
* 
5 
1 
1 
2 
2 X 6 = 12 
1 • 
" 
Biplanes. 
056 
1x12=12 
; 
; ; 
Plane through the three 
. 
• 
axes. 
• 
. 
5 
Planes each through an 
2x 6 = 12 
axis joining the binode 
6 
* 
with a cnicnode. 
| 
• 
Plane through the axis 
34 
lX 4= 4 
• 
joining the two cnic- 
nodes. 
12 
lX 3= 3 
. 
• 
Planes through the bi- 
planar rays. 
0 
lX 2= 2 
. 
, # 
Plane touching along the 

• 
axis which joins the 
8 45 
two cnicnodes. 
fcd 
t> 
> 
1 
1 1 
iplanar ray: 
biplane, ar 
a transvers 
ris joining 
nodes. 
P o 
O OT 
3 O 
S--P" 
g-'sj 
p o 
5* p 
nf p 
o 
2. 
O g. 
2 P* 
O'CTQ 
P r-f- 
o 
CL 
1 1 
§ 
p ^ 
MDCCCLXIX. 
T 
