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

28

THE TRIHEDRAL PAIR CONFIGURATION

45 32
there are 45 32240 triheders; but since these fall into pairs such
2.3
that one triheder being given the other is uniquely determined, there
are in all 120 trihedral pairs.
13. Enumeration of Trihedral Pairs.
It is of interest to form, actually, the 120 trihedral pairs in terms
of the planes which constitute them. Remembering that every trihedral pair determines nine straight lines lying wholly upon the
surface, we must choose for combination those triheders that fulfil this
condition.
It is sufficiently obvious, in view of the unique notation already
adopted, that the faces of the trihedral pair, of the type
(12) (23) (31) + (13) (32) (21),
intersect the surface in the nine lines a,, a,, a; b,, b2, b3; c12, c23, C31.
These nine lines may be arranged in the following form:
a,1 2 C12 )
a b3 C23,
a3 bl C31 J
where each horizontal line represents three co-planar lines, and each
vertical column represents three non-intersecting lines-as indicated.
Bearing in mind the order of the letters, as indicated by the diagram,
we may represent all examples of this type in the abbreviated notation
F j k jk i, =, 2,...6 (i k).
k i ki
Fixing our attention on the first column, it is obvious that the number
of this type is clearly 6C, = 20.
Consider next the two similar types:
(35) (64) (12. 34. 56) + (34) (65) (12. 35. 64);
(53) (46) (12. 34. 56) + (43) (56) (12. 35. 64).
The faces of the first pair cut the surface in the nine lines a3, a,;
b4, b5; C12, C34, C56,, C, 46. Also the faces of the second pair cut the
surface in the nine lines a4, a,; b3, b6; c,1, c34, C6, c35, C46.
These two sets of nine lines may be arranged in the following
forms:
f a3 b5 c,5, a4 b3 C34
a6 b4 C46 s a5 b6 C56
C34 C56 C12 C46 C35 C12