GENERAL EQUATION OF A CUBIC SURFACE. 
53 
are simultaneously true if X/u, = (f)^. It follows, therefore, that each of the pair of 
lines 16, 17 cuts one or other of the pair 24, 25. 
In an exactly similar manner we can prove that each line of one pair intersects 
one or other of the lines of the second pair in the case of each of the sets of pairs 
ii., iv,, and hi., vi. 
§ 9. We have thus shown that any one of the original twelve lines cuts ten others; 
the line 1, for instance, cuts 2, 3, 6, 9, 11, 15, 16, 17, 18, and 19. 
Also we have shown that any one of the last twelve lines cuts nine others ; 16, for 
instance, cuts 1, 4, 7, 10, and one from each of the pairs ii., hi., iv., v., vi. It must, 
therefore, cut one more, and that must be one from the group 13, 14, 15. since these 
three form a triangle. 
The equations of 14 are 
f+ 7-T=«^ 
X 
- + - - d' = 0 
a 0 J 
and the equations of 16 or 17 are 
X — aT = \ij 
s — cT = 
where 
and 
T — ax + /3y + 
abXfj. -f" (na-j- ^(3 — l)/r — c8 
ydKjx + (cy + dS — l) X — = 0 j 
If the lines intersect, the first five equations must be simultaneously true. Hence, 
eliminating x and s, we see that the equations 
are simultaneously true. 
Hence 
f+^-T=0 
-y -Jr-u-^-T = 0 
a. c 
(1 — «a — cy) T = (aX + ^) y + (y/r + 8) ll 
1/5 
l/d 
kja 
file 
X + ^ 
yp, + 8 
or 
- y \ \ I — 1 
7-- ¥ +-te- 
1 
aa + cy — 1 
cy + clB — 1 
= 0, 
cal 
