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

PRELIMINARY THEOREMS

11

every plane II corresponds one straight line I lying entirely on the
surface, and there are twenty-seven* (n = 3) double tangent planes to
the tangent cone, vertex P, therefore there are twenty-seven straight
lines I upon the cubic surface+.
3. Triple Tangent Planes.
By properly determining the plane passed through any straight
line I upon the cubic surface, the conic C (~ 2) will degenerate into a
pair of straight lines. Here the plane intersects the surface in three
intersecting straight lines (a degenerate curve of the third order having
three double points) and the points of intersection of the lines taken
in pairs are the points of contact of the plane with the surface. Now,
through each of the three lines in the plane there may be drawn,
besides the given plane, four other triple tangent planes. For these
twelve new planes give rise to twenty-four lines upon the surface,
making up, with the former three lines, twenty-seven lines upon the
surface. It is clear that there can be no lines upon the surface besides
the twenty-seven. For since the three lines upon the triple tangent
plane are the complete intersection of this plane with the surface,
every other line upon the surface meets the triple tangent plane in a
point upon one of the three lines, and must therefore lie in a plane
passing through one of these lines, such plane (since it meets the
surface in two lines, and therefore in a third line) being obviously a
triple tangent plane. Hence the whole number of lines upon the
surface is twenty-seven. Every straight line on the surface is met by
ten others.
If all the twenty-seven intersect in pairs, there would be 351 points
of intersection. But since each line is met by ten other lines, there
remain sixteen lines by which it is not met. Therefore there are
27 x 16
-27 6 = 216 pairs of lines that do not mutually intersect. Consequently
there are 135 points of intersection. Since these 135 points, by threes,
determine the triple tangent planes, there are forty-five triple tangent
planes.
* Salmon (Geometry of Three Dimensions, 4th edition, ~ 286) gives
n (n - 1) (n - 2) (,13 - n2 + - 12)
2
as the number of double tangent planes, drawn through a point P to a surface of
the nth degree.
t For other proofs, cf. for example, R. Sturm, Flichen dritter Ordnung, Kap. 2,
~ '0; Cayley, Coll. Mtath. Papers, Vol. i. No. 76, pp. 445-456.