PROFESSOR CAYLEY ON CUBIC SURFACES. 
235 
those of the third binode Z = 0, W=0, and that (except in the case XII=12 — U 6 ) the 
uniplane is X=0. For example, in the surface XVII=12 — 2B 3 — C 2 , as represented 
by its equation WXZ+Y 2 Z+X 3 =0, we have a B 3 at D, the biplanes being X=0, Z=0, 
a B 3 at C, the biplanes being X=0, W=0 (therefore X=0 the common biplane), and 
a C 2 at A. 
13. It will be convenient (anticipating the results of the investigations contained in 
the present Memoir) to give at once the following Table of Singularities ; the several 
symbols have of course the significations explained in the former Memoir. 
o 
d* 
O 
cq 
cf 
d* 
cq 
pf 
o 
1 
1 
1 
1 
1 
cf 
1 
pf 
gpf 
1 1 
pf 
pf 
CO 
ffl pf 
1 1 
p 
1 
pf 
i 
pf 
1 
p 
1 
o 
rtl 
pf 
1 
pf 
pf 
cq 
1 
p” 
i 
p 
CO 
1 
E 
oi 
Cq 
cq 
cq cq 
cq 
cq 
cq 
cq cq 
cq 
cq 
cq 
cq 
cq 
cq 
cq 
cq 
cq 
§ 5 " 
II 
II 
ii 
ii ii 
ii 
ii 
ii 
ti ii 
ii 
ii 
'll' 
7 i 
il 
'll 
"ii 
Tl 
1 
ll 
II 
i— i 
M 
M 
M 
M 
VIII 
M 
w 
K 
h 
> 
£ 
X 
P 
X 
3 
& 
X 
M 
M 
HH 
M f> 
> 
> 
M M 
M 
M 
X 
X 
X 
X 
M 
X 
X 
n 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
3 
~3 
a 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
4 
8 
0 
1 
0 
2 
1 
3 
0 
2 
4 
1 
0 
0 
K 
6 
6 
7 
6 
7 
6 
8 
7 
6 
8 
9 
3 
b 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
lc 
0 
0 
0 
0 
0 
0 
0 
O 
0 
0 
0 
0 
t 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
q 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
P 
0 
0 
0 
0 
0 
0 
0 
0 
y 
0 
0 
1 
i 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
2 
c 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
h 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
r 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
a 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
e 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
X 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
P 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
y 
y 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
i 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
C 
0 
1 
0 
2 
1 
3 
0 
2 
4 
1 
y 
0 
B 
0 
0 
1 
0 
1 
0 
2 
0 
2 
3 
0 
I 
II 
in 
IV v 
VI VII 
VIII X XI 
XII 
IX 
XIII XIV XV 
XVI XVIII XIX 
xvn xx 
XXI 
XX3I XXIII 
n' 
12 
10 
9 
8 
7 
6 
6 
5 
4 
4 
3 
3 
a' 
6 
6 
6 
6 
6 
6 
6 
6 
6 
0 
6 
4 
8' 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
k' 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
3 
V 
27 
15 
9 
7 
3 
3 
0 
l 
3 
0 
0 
I 
V 
216 
60 
18 
12 
3 
0 
0 
0 
0 
0 
0 
0 
( f 
45 
15 
6 
3 
0 
1 
0 
1 
0 
y 
0 
‘s' 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
P 
27 
15 
9 
7 
3 
3 
0 
1 
3 
0 
0 
1 
/ 
0 
0 
0 
1 
0 
3 
0 
1 
6 
0 
0 
2 
o' 
24 
18 
16 
12 
10 
6 
8 
4 
0 
2 
0 
0 
hi 
180 
96 
72 
38 
24 
6 
12 
2 
0 
0 
0 
0 
r' 
30 
24 
42 
17 
24 
9 
32 
5 
0 
2 
0 
0 
a' 
12 
12 
12 
10 
9 
6 
8 
4 
0 
2 
0 
0 
6 ' 
0 
0 
16 
0 
8 
0 
16 
0 
0 
0 
0 
0 
x' 
0 
0 
0 
0 
1 
0 
0 
2 
0 
2 
0 
0 
F 
54 
30 
18 
13 
6 
3 
0 
1 
0 
0 
0 
0 
y\ 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
C' 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
B' 
0 
0 
0 
0 
0 
0 
1 
0 
0 
1 
3 
0 
2 k 2 
n 
8 
b 
k 
t 
S 
P 
3 
e 
k 
r 
a 
9 
X 
P 
7 
i 
C 
B 
%’ 
a' 
8' 
b> 
k' 
if 
S' 
P 
3 
<f 
hi 
o' 
O' 
x' 
F 
V 
C' 
B' 
