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

88 ON SOME CONFIGURATIONS ASSOCIATED WITH THE

four lines, and on each line lie three points, as detailed in the foregoing.
Of the fifteen points, nine, viz. the points LP, LQ, LR; MP, MQ,
MR; NP, NQ, NR are, as appeared above, points on two of the six
lines 1, 2, 3, 4, 5, 6; the remaining points are MN, NL, LM; QR,
RP, PQ. These are Brianchon Points:
MN of the hexad 162435,
NL,,,,,, 152634,
LM,,,,,, 142536,
QR,,,,,, 152436,
P,,,,,, 142635,
PQ,,,,, 162534,
for the point MN is the meet of the lines MNP, MNQ, IMNR _ lP,
NP; MQ, NQ; MR, NR - 35, 26; 16, 34; 24, 15; that is, MN is
the Brianchon point of the hexad 162435; and similar reasoning
verifies the above statements for the remainder of the six points.
To summarize, we have two sets of three hexads, such that the
Brianchon points of each set lie in linea; and the two lines so obtained,
together with the eighteen lines through the six Brianchon points, form
a system of twenty lines passing by fours through fifteen points
(Fig. 2).
46. Derivation of the Pascalian Configuration by the
Projection of the Straight Lines lying upon the Cubic
Surface of the Second Species.
For the cubic surface with one conical point (~ 25), the tangent
planes of the type [12. 34.56] are fifteen in number, viz.
[12.34.56], [14.23.56], [15.23.46],
[12.36.45], [13.24.56], [15.24.36],
[12.35.46], [16.23.45], [13.25.46],
[16.25.34], [13.26.45], [14.26.35],
[15.26.34], [16.24.35], [14.25.36].
In terms of these fifteen planes, the equation of the surface may be
written in the form
UVW+ kXYZ= 0
in ten different ways (~ 13, third table of trihedral pairs).