Mr. A. Cayley on the Delineation of a Cubic Scrole. 529 



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 scrole ; and 

 these points are therefore the same for each of the two curves. 

 The remaining two points of intersection 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 genera- 

 ting 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 = is the equation of the basic cubic, V = the equa- 

 tion of some other cubic meeting the basic cubic in the nine 

 points in question, then the equation of ' the cubic ' is U + XV = 0, 

 \ being a parameter the value of which varies according to the 

 position (in the series of parallel planes) of the plane of the sec- 

 tion. Suppose that the basic cubic U = 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 TJ-f\V = 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 



1 + A, 



is the same thing, 0Q= — - ' — ; the locus of the point Q 



1 + A 



will then be the cubic U + \Y = 0. And we may even suppose 

 the cubic V = 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 follow- 

 ing 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 con- 

 ditions assumed in regard to the cubic V = 0. And it is to be 

 noticed that as in general the cubic V = is the projection of 

 some section of the scrole, so the hyperbola and line are the 

 projection of a section of the scrole, viz. the section through one 

 of the generating lines (there are three such lines) parallel to 

 the basic plane. 13ut it is better to construct ' the cubic ' by. a dif- 

 ferent method, using only the basic cubic U = 0, and which re- 

 sults more immediately from the geometrical theory. Taking the 



