1904.] BROOKS— ORTHIC CURVES. 318 



ingly, the foci of the cubics will lie on the curves which are the 

 maps in the x-plane of two concentric circles in the s-plane. The 

 centre of these circles maps into the triad common to all the cubics, 

 and the circles themselves map into two cassinoids of the sixth 

 order, about the triad, as M. Lucas^ has shown. Each of the circles 

 goes through one of the branch points, and, therefore, each of the 

 cassinoids must have a node. If the point which corresponds to 

 the triad ajSy is equidistant from the branch points, the two circles 

 and also the two cassinoids coincide. In this case the cassinoid 

 has two double points. 



The lines which correspond to the cubics are all perpendicular to 

 the circles which correspond to the cassinoids, and so, by the prin- 

 ciple of orthogonality, the ovals are orthogonal trajectories of the 

 cubics of the pencil. 



XVII. The Position of the OrtJiic Cubic in Projective Geo??tetry. 



I shall close this study of the metrical properties of the orthic 

 curve of the third order by showing that from the point of view of 

 projective geometry the orthic cubic is really a general cubic. Any 

 proper plane curve of the third order can be projected into an 

 orthic curve. 



We know that the points of contact of three of the six tangents 

 to a cubic curve from any point of its Hessian lie in a line. Now 

 these three points, considered as a binary cubic, have a Hessian 

 pair. If this pair of -points be projected to the circular points at 

 infinity, the three tangents become equally inclined asymptotes, 

 and continue to meet in a point. The cubic curve is then orthic 

 and the transformation is accomplished. This projection requires 

 two points to go into given points, and can, therefore, always be 

 made. J?i projective geometry the orthic cubic is any proper plane 

 cubic. 



As an illustration of the way in which information about the 

 orthic cubic applies to cubic curves in general, let us see what the 

 characteristic property that the asymptotes are concurrent and 

 equally inclined means. The circular points / and y are a pair of 

 points apolar with the curve. Their join, the line at infinity, meets 

 the curve in three points such that the tangents at these points meet 



1 Felix Lucas, Geometric des Polynomes, Journal de P Ecote Polyiechnique, 

 XXVIII, p. 5. 



