The twenty-seven lines upon the cubic surface ... by Archibald Henderson.

42

THE TRIHEDRAL PAIR CONFIGURATION

3 4 34 3 1
6 2 62 1 5
32 64 15 13
{2 4 24 1 3
5 3 53 4
23 54 16 j 31
3 5 35 (r 1
4 2 42, 6
32 45 16 J 13
2 5 25 4
3 4 34 \ 5
24 35 16 J 41

6 16 2 5 25
3 3 1 4 1 41
56 24 21 45 36
6 36 1 5 15 j
1 41 6 2 62
46 25 12 65 34
4 14 r2 6 26 1
3 63 5 1 51
64 25 J 21 56 34

6 46 1
1 51 6
56 23 J 12

3 13
2 62
63 45

4 3 43
5 2 52,
42 53 16

1 5 15
6 4 64,
[ 14 65 23

2 6 26
3 1 31
21 36 45 J
r1 4 14
6 2 62
12 64 35

4 2 5 25 2 3
4 3 43, 5
23 45 16 31
3 3 4 34 1
5 2 52 1 J 6
32 54 16 13

6 36
1 51
56 24

5 15 (, 2 6 26
3 63, 4 1 41
65 24 J 21 46 35

15. The Cubic Surface referred to a Pair of Triheders.
It follows from the preceding discussion, in connection with ~4,
that the equation of the cubic surface, when written in the form
U F V -- x YZ= 0,
may be said to be referred to a pair of triheders. Since there are
precisely one hundred and twenty trihedral pairs, it follows that the
equation of our fundamental cubic surface may be reduced to the form
UVW- XXYZ= 0
in one hundred and twenty ways: That is to say, we have arrived at
a geometrical solution of the algebraic problem: In how many ways
may the general equation of the third degree (x, y, z, w)= 0 be
reduced to the form
UVW- XXYZ= 0,
where U, V, WY; X, Y, Z are linear polynomes in x,, z, w?