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OX THE DELINEATION OF A CUBIC SCHOLL.
Ill
)LL.
-530.]
2d by lines each
(double) line on
e on the surface,
a cubic passing
the point on the
; on the single
the cubic curve,
s by a series of
id the section by
section projected
a may be spoken
ibic has a node
this point four
three points at
points at infinity
tersection of the
these points are
points of infer
tile basic cubic,
> generating lines
)ic in nine fixed
lity, and the two
nvs that if P=0
ir cubic meeting
f ‘ the cubic ’ is
• to the position
(in the series of parallel planes) of the plane of the section. Suppose that the basic
cubic U = 0 is given, and suppose for a moment that the cubic V = 0 is also given,
these two cubics having the above-mentioned relations, viz. they have a common node
and parallel asymptotes: the cubic U + X,]^ = 0 might be constructed by drawing
through the node (say 0) a radius vector meeting the cubics in P, P' respectively,
and taking on this radius vector a point Q such that PQ= —~ PP\ or, what is
Qjp \QP'
the same thing, OQ = —^ —-; the locus of the point Q will then be the cubic
U + AP = 0. And we may even suppose the cubic F=0 to break up into a line and
a conic (hyperbola), and then (disregarding the line) use the hyperbola in the con
struction. In fact, if the hyperbola is determined by the following five conditions,
viz. to pass through the node and through the feet of the two generators parallel to
the nodal directrix, and to have its asymptotes parallel to two of the asymptotes of
the basic cubic, and if the line be taken to be a line through the node parallel to
the third asymptote of the basic cubic; then the hyperbola and line form together a
cubic curve meeting the basic cubic in the nine points, and therefore satisfying the
conditions assumed in regard to the cubic V — 0. And it is to be noticed that as in
general the cubic 1^=0 is the projection of some section of the scroll, so the
hyperbola and line are the projection of a section of the scroll, viz. the section through
one of the generating lines (there are three such lines) parallel to the basic plane.
But it is better to construct ‘the cubic’ by a different method (using only the basic
cubic U = 0) which results more immediately from the geometrical theory. Taking
the basic plane as the plane of the figure, let O be the node, or foot of the nodal
directrix, K the foot of the single directrix, Kk the projection of the single directrix,
k being the projection of the point in which the single directrix meets the plane of
the section. Drawing through 0 any radius vector meeting the basic cubic in P, and
the line Kk in r, and producing it to a properly determined point Q, then OPrQ will
be the projection of the generating line which meets the nodal directrix, the basic cubic,
the single directrix, and the section in the points the projections whereof are O, P, r, Q
respectively: and the consideration of the solid figure shows easily that the condition
for the determination of the point Q is
Pr
PQ = Kk .-jr.