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

AUXILIARY THEOREMS

25

V (2). Five non-intersecting straight lines a1, a2, as, a4, C56, not
belonging to a double six, are met by two straight lines b,, b6. There
is no straight line by which no one of the five given straight lines is met.
Either set of five non-intersecting straight lines is called a "quintuple."
VI. Finally, such a set as six non-intersecting straight lines al, a2,
a3, a4, as, a6 is called a "sextuple."
On the basis of the preceding, it is easy to determine immediately
the number of doubles, triples, etc. in the configuration of the twentyseven lines.
27.16
Number of doubles - 1 = 216;
1. 2
27.16.10,, triples = 720;
1.2.3
27.16.10.6,,,, quadruples = 12 4 =1080;
1.2.3.4
27.16.10.6.3,,,, quintuples = = 648;
s 27.16.10.6.2.1 7,,,, sextuples = 1.2.3.4.5.6 72.
A word must be said about the quintuples, which are of two types.
Every quadruple a1, a, a3, a4 gives (1) one quintuple with two intersecting lines; and (2) two quintuples with one intersecting line each.
That is, we have the three quintuples:
la2c3a4,aa with one intersector b6;
a,1a2a3,a4a6 A,,,d );
aa2a3a4c56,, two intersectors bs, b6.
Thus the quintuples fall into two groups; and there are twice as many
in one group as in the other. Since the total number is 648, it follows
that there are 432 of the type having only one intersector, and 216 of
the type having two intersectors. This explains the derivation of the
number of sextuples, since two quintuples out of every set of three
belong to a double six.
A large number of theorems upon special portions of the configuration of the twenty-seven lines is given by Steiner*, Sturmt, Taylor+,
and others; and to these the reader is referred.
* Crelle's Journ. Vol. LIII. (1857), pp. 133-141.
t Math. Ann. Bd. xxIII. (1884), pp. 289-310. Also see Sturm's work
Synthetische Untersuchungen ilber Fldchen dritter Ordnung.
+ Philos. Trans. Royal Soc. Part I. A (1894), pp. 37-69.