530 Mr. A. Cay ley on the Delineation of a Cubic Scrole, 



basic plane as the plane of the figure, 

 let be the node, or foot of the nodal 

 directrix, K the foot of the single 

 directrix, K& the projection of the 

 single directrix, k being the projection 

 of the point in which the single direc- 

 trix meets the plane of the section. 

 Drawing through any radius vector 

 meeting the basic cubic in P, and the 

 line K k in r, and producing it to a pro- 

 perly determined point Q, P r Q 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 0, P, r, Q respectively. And the consi- 

 deration of the solid figure shows easily that the condition for 

 the determination of the point Q is 



Pr 



Hence, starting from the basic cubic and the line K k } we have a 

 construction for the point Q the locus whereof is the cubic, the 

 projection of a section of the scrole ; for the projections of the 

 parallel sections, we have only to vary the length K k. By what 

 precedes, the construction gives for the locus of Q a cubic having 

 a node at O, and having its asymptotes parallel to those of the 

 basic cubic. As P moves up to K, the distances P r, r K 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 intersections of K k 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 scrole ; and such a series of cubics 

 will thus afford a delineation of the scrole. 



2 Stone Buildings, W.C., 

 April 15, 1863. 



