GENERAL EQUATION OE A CUBIC SURFACE. 
45 
Then the equation (A) takes the form 
xyzu = (x — aT) (y — bT) (z — cT) {u — c^T) .... (D), 
and it represents, besides the plane T, the cubic surface passing through the twelve 
straight lines, which are represented in the annexed ligure, as well as three other 
straight lines which are not represented in the figure. 
£ 
The equations of the lines may be written as follows 
and 
y 
X 
z 
X 
u 
X 
= 6T 
= 0 
= cT 
= 0 
= dT 
= 0 
( 6 ) 
.( 8 ) 
( 10 ) 
X 
y 
= aT 
= 0 
y 
u 
y 
= cT 
= 0 
= dT 
= 0 
•(^) 
( 9 ) 
( 11 ) 
0-/ 
~a ~d 
a; = aT 
z 
y 
0 
hT 
2=0 
u 
z 
= dT 
= 0 
•( 2 ) 
(4) 
(12) 
-f + - -- T 
K13), 
i 
J 
X 
u 
y 
u 
z 
u 
= aT 
= 0 
hT 
0 
cT 
= 0 
•( 3 ) 
•( 5 ) 
•( 7 ) 
which meets (3), (4), (9), and (10); 
