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

THE LINES UPON THE CUBIC SURFACE 75
The plane of BCD (x = 0) was chosen for the base of the tetrahedron,
since it contains three of the lines. The lines (0) and (1) are the
edges BD and AD respectively; the line (2') passes through D and
is parallel to BC; the line (3') also passes through D and bisects the
edge BC. The line (12') passes through the middle point of AD and
is parallel to BC; the line (13') passes through the middle point of
BC and is parallel to AD.
There is a binode at D and the edge is torsal. The tangent plane
coincides with one of the biplanes; we have thus an ordinary biplane,
and an oscular biplane (Plate 8).
31. Species VIII = 12- 3C2
The equation of the surface is
y3+ y (x + z + w) + 4axzw = 0,
where we set (m - 1)2= 4am.
Consequently ml + m2 = 2 + 4a, mm2 = 1.
1 1
Putting a =, we obtain mn = 2, = 2.
8 2i
The equations of the twelve distinct lines upon the surface are as
follows:
(7): = 0, =O0,
(8): x=0, Y=0,
(9): z=0, y=0,
(7): y+z+x=O, w=0,
(8) y + x + tv= 0, ^ = 0,
(s): yx+Zw=0, z=0,
(9): + + +w, x = O,
(1): 2y=2x -z,
(2): 2/:=- -=2z,
(3): 2y =-w= 2x,
(4): 2y = 2w=- x,
(5): 2y= 2z - w,
(6): 2y=-z = 2w.
The lines (7), (8) and (9) are the edges CA, CD and AD,
respectively. The lines (7), (8) and (9) are the lines at infinity in the
planes ABC, ABD, BCD, respectively. The lines (1) and (2) pass
through the point D, and are parallel to the internal bisectors of the
angles C and A, respectively, lying in the plane ABC. The lines (3)
and (4) pass through the point C, and are parallel to the internal
bisectors of the angles D and A, respectively, lying in the plane ABD.