110 
[328 
328. 
ON THE DELINEATION OF A CUBIC SCROLL. 
[From the Philosophical Magazine, vol. xxv. (1863), pp. 528—530.] 
Imagine a cubic scroll (skew surface of the third order) generated by lines each 
of which meets two given directrix lines. One of these is a nodal (double) line on 
the surface, and I call it the nodal directrix; the other is a single line on the surface, 
and I call it the single directrix. The section by any plane is a cubic passing 
through the points in which the plane meets the directrix lines; i.e. the point on the 
nodal directrix is a node (double point) of the curve, the point on the single 
directrix a single point on the curve; the two directrix lines, and the cubic curve, 
the section by any plane, determine the scroll. Consider the sections by a series of 
parallel planes. Let one of these planes be called the basic plane, and the section by 
this plane the basic section or basic cubic; and imagine any other section projected 
on the basic plane by lines parallel to the nodal directrix: such section may be spoken 
of simply as £ the section,’ and its projection as ‘ the cubic.’ The cubic has a node 
at the node of the basic cubic; that is, the two curves have at this point four 
points in common. The two curves have, moreover, in common the three points at 
infinity (or, in other words, their asymptotes are parallel); in fact the points at infinity 
of either curve are the points in which the line at infinity, the intersection of the 
basic plane and the plane of the section, meets the scroll; and these points are 
therefore the same for each of the two curves. The remaining two points of inter 
section of the cubic with the basic cubic are also fixed points on the basic cubic, 
i.e. they are the points of intersection of the basic plane by the two generating lines 
parallel to the nodal directrix. Hence the cubic meets the basic cubic in nine fixed 
points, viz. the node counting as four points, the three points at infinity, and the two 
points the feet of the generators parallel to the nodal directrix. It follows that if U = 0 
is the equation of the basic cubic, V = 0 the equation of some other cubic meeting 
the basic cubic in the nine points in question, then the equation of ‘ the cubic ’ is 
U + \ V = 0, A, being a parameter the value of which varies according to the position