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

THE DOUBLE SIX CONFIGURATION

17

respectively, and we understand by (ij) the determinant
ai, bi, ci, di
ai, Pi, 7i,, i
j bj, bj, cdj
aj, Pj, j,
is as follows:
a,= 0, (12), (13), (14), (15)_ O0.
(21), 0, (23), (24), (25)
(31), (32), 0, (34), (35)
(41), (42), (43), 0, (45)
(51), (52), (53), (54), 0
The five lines 1', 3', 4', 5', 6' are co-tractorial, since the equations
identically satisfy A, = 0, as may be shown on trial. Moreover these
five lines do not mutually intersect, since in forming the determinants
(ij), no one of them is found to vanish. A difficulty arises in the
event of the hyperboloid through any three of these five collinear lines
touching a fourth, that is to say, that certain four of the lines might
have a double tractor t. That such is not the case will appear in the
sequel.
Determining now the common tractor, 2, of these five lines, we find
it to have the equations:
2:. 7yB ('x - a'w) - 8'A (y'y - z') = 0
1yB (s& - aw) - A (yy - P) = '
Now, in general, four given lines have a pair of tractors. Since the
five lines 1', 3', 4', 5', 6' already have a single tractor 2, they have, in
sets of four, five more tractors thus: the lines 1, 3, 4, 5, 6 are tractors
of the sets (3', 4', 5', 6'), (1', 4', 5', 6'), (1', 3', 5', 6'), (1', 3', 4', 6'),
(1', 3', 4', 5') respectively.
Let us proceed to find the equations of the five lines 1, 3, 4, 5, 6.
Recalling the values of A, B, C and D above, it is obvious by inspection that the equations of lines 1 and 3, meeting the quadruples
(3', 4', 5', 6') and (1', 4', 5', 6') respectively, are
1: x=0, wz=0,
3: y=0, z=0.
* Sylvester, " Note sur l'involution de six lignes dans l'espace," Comptes Rendus,
Vol. LII. (1861), pp. 815-817.
t Cayley, "On the Six Co-ordinates of a Line," Trans. Camb. Philos. Soc.
Vol. xi. Part 11. (1869), pp. 290-323.

H.

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