246 PEOEESSOE CAYLEY ON CUBIC SUEEACES. 
38. It has been mentioned that the number of double-sixers was =36, these are as 
follows : — 
1, 
2 
" , 
3 , 
4, 
5, 
6 
Assumed primitive 
. 1 
r, 
2', 
S', 
4', 
5', 
6' 
i, 
1', 
23, 
24, 
25, 
26 
Like arrangements 
. 15 
2, 
2', 
13, 
14, 
15, 
16 
i, 
9 
U 5 
3, 
56, 
46, 
45 
Like arrangements 
. 20 
23, 
13, 
12, 
4, 
5, 
6 
36 
where, if we take any column of two lines, we have the complete number 216 of pairs 
of non-intersecting lines (each line meets 10 lines, there are therefore 27 — 1 — 10, =16, 
which it does not meet, and the number of non-intersecting pairs is thus -^27 . 16=216). 
39. We can out of the 45 planes select, and that in 120 ways, a trihedral-pair, that 
is, two triads of planes, such that the planes of the one triad, intersecting those of the 
other triad, give 9 of the 27 lines. Analytically if X=0, Y=0, Z=0 and U=0, V=0, 
W=0 are the equations of the six planes, then the equation of the cubic surface is 
XYZ +MJVW=0. See as to this post, No. 44. 
The trihedral plane pairs are — 
12', 23', 
31' 
1'2, 2'3, 
3'1 
No. is =20 
12', 34', 
14.23.56 
2'3, 4'1, 
12.34.56 
= 90 
14.25.36, 
35,16.24, 
26.34.15 
14.35.26, 
25.16.34, 
36.24.15 =10 
120 
The construction of the last set is most easily effected by the diagram 
1 2 3 x 4 5 6 
3 1 2 5 6 4 
2 3 1 6 4 5 
v ' 
II 
14 25 36 
35 16 24 
26 34 15 
It is immaterial how the two component triads 123 and 456 are arranged, we obtain 
always the same trihedral pair. 
