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

THE TRIHEDRAL PAIR CONFIGURATION

29

That there are forty-five of each of these types follows from the
fact that, if we keep 12 fixed, for example, then there are three ways
in which the c-triangles may be written:
( 35 ( f 36 ( 34
46, 45, 56
34.56.12 35.46.12 36.45.12
Hence there are fifteen such sets.
All examples of this type may be represented in the abbreviated
notation
( i J l }
ji' j' i'j' ij, k, 1, 2,...6 (i j k1),
[ ij' i'j li
the affixes denoting that a different choice of numerals must be made
for the letters in the second line to those in the first line.
Finally, there is a type:
(14. 25. 36) (35. 16. 24) (26. 34. 15)
+ (14. 35. 26) (25. 16.34) (36. 24.15).
The faces of this pair cut the surface in the nine lines ca1, ci5, c16, c24,
C25, C26, C34, C35, C36. These nine lines may be arranged in the following
form:
C14 C15 16 )
C24 C25 C26 L
C34 C35 C36 j
Obviously such a form arises from the two forms:
1 1 1 r 4 5 6
2 2 2, 4 5 6
3 3 3 4 5 6
Hence the number of such forms is I (6C3)= 10.
All examples of this type may be represented in the abbreviated
notation
j ik il ]
i'j i'k i'l,j, k, 1= 1, 2,...6 (i jokl1),
' i"j k il J
the affixes denoting that a different choice of numerals must be made
for the letters for each line.
Hence we have enumerated all the different types, the total number
of trihedral pairs being 120 = 20 + 90 + 10.
Below are listed the 120 trihedral pairs, according to the rules just
enumerated.