412] 
A MEMOIR ON CUBIC SURFACES. 
399 
(a) 
(b) 
(0) 
(/) 
(?) 
(A) 
fi 
-¥ 
0 
0 
0 
y 
(!') 
AT - F = 0, yF + f x lF= 0 
f 2 
-V 
0 
0 
0 
y 
(2') 
f 3 
-¥ 
0 
0 
0 
y 
(3') 
55 
f 4 
0 
0 
0 
y 
(4') 
55 
8 
/1 i\ 
1 
/1 1\ 
y 
1 
/1 1\ 
1 /1 l\ 
(12. 3'4')* 
¥2 
8 
U + tJ 
-y 
(l + fj 
“p4 
55 1 
v5 + 5) 
~5f s (5 + f 4 ) 
s 
fl l \ 
1 
/1 1\ 
y 
1 
/1 1\ 
(13.2'4') „ 
~¥s 
8 
-y 
(5 + 5) 
“ f~f 
1 2 i 4 
p 
l5 + fj 
_ 55 (5 + 5) 
s 
¥4 
8 
r-H |ej-? 
+ 
1—1 It+i 1 
1 
-y 
(H) 
y 
f 2 f 3 
55 
:hj 
-f.f.Gus) 
(14.2'3') „ 
8 
/1 1\ 
1 
/1 1\ 
V 
1 
a i\ 
(23. 1'4') „ 
8 
U + fj 
-y 
(fl + fj 
55* 
vf 2 + f 3 ) 
~p,(5 + 5) 
8 
fl 1\ 
1 
fl 1\ 
y 
1 
a i\ 
1 /1 1\ 
(24. 1'3') „ 
8 
U + fj 
-y 
(fl + V 
~ TJ 
1 1 I 3 
fiv 
vf 2 + f) 
~ 55 (5 + 5) 
8 
/1 1\ 
1 
/1 1\ 
V 
1 
a i\ 
(34.1'2') „ 
~ ¥4 
8 
-y 
(fl + f 2 ) 
-55 
p 
A + f.) 
”55(5 + f.) 
^equations are 
s {X - (f, + 5) Y] - u,Z = 0, y (X - (f, + f 4 ) Y] -f,f 4 W= 0, 
[and similarly for each of the remaining five lines]. 
76. To verify the equations of the line 12.3'4', observe that the two equations give 
+ 8 IT = 7 8 jx (A + J.) _ F(1 + 1 +1 + J)}, 
ZW = fffj (X - (f, + Q YX - (f, + f 4 ) Y\: 
the equation of the surface, multiplying by Z and observing that — yS = afif 2 f 3 f 4 , 
becomes 
X‘ZW + X Y> (yZ + S W) + yS Y> - f , (X - f, Y) (X - f. Y) (X — f, Y) (X - f 4 Y) = 0; 
n h b M 
and substituting the values just obtained, this is 
X 2 [X - (f> + f 2 ) F] [X - (f 3 + f 4 ) F] + IP [Z (fif 2 + f 3 f 4 ) - F(fifsfs + f x f 2 f 4 + fif 3 f 4 + f 2 f 3 f 4 )] 
+ f 4 f 2 f 3 f 4 F 4 - (Z - fjF) (Z- f 2 F) (Z - f 3 F) (Z - f 4 F) = 0, 
which is in fact an identity. 
77. The facultative lines are the transversal and the six mere lines; b'= p =7; 
t' = 3.