374 
A MEMOIR ON CUBIC SURFACES. 
[412 
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 
1', 
2', 
3', 
4', 
5', 
6' 
1, 
r, 
23, 
24, 
25, 
26 
Like arrangements 
15 
2, 
2', 
13, 
14, 
15, 
16 
1, 
2, 
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 
4.27.16 = 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 X YZ + k TJVW = 0. See as to this post, No. 44. 
The trihedral plane pairs are : 
12', 23', 31' 
1'2, 2'3, 31 No. is = 20 
12', 34', 14.23.56 
2'3, 41, 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 
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.