112 ON THE DELINEATION OE A CUBIC SCROLL. [328 
Hence, starting from the basic cubic and the line Kk, we have a construction for the 
point Q the locus whereof is the cubic, the projection of a section of the scroll; for 
the projections of the parallel sections, we have only to vary the length Kk. By what 
precedes, the construction gives for the locus of Q a cubic having a node at 0, and 
having its asymptotes parallel to those of the basic cubic. As P moves up to K, 
the distances Pr, rK become indefinitely small; but their ratio is finite, hence the 
cubic, the locus of Q, does not pass through the point K. The construction shows, 
however, that it does pass through the points L, M, which are the other two inter 
sections of Kk with the basic cubic; these points L, M are in fact the feet of the 
generators parallel to the nodal directrix. 
The general conclusion is, that a series of cubics having each of them at one 
and the same given point a node—having their asymptotes parallel—and besides 
passing through the same two given points—may be considered as the projections of 
a series of parallel sections of a cubic scroll; and such a series of cubics will thus 
afford a delineation of the scroll. 
2, Stone Buildings, W.C., April 15, 1863.