[328 
328] 
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.