250 Proceedings of the Royal Irish Academy. 



the curve at a, then chords of the cubic meeting a' will also meet V and be 

 divided harmonically by these lines and by the cubic." (Cremona, Crelle, 

 vol. 58.) 



Fig. 1. 



Theoeem II. — " If P = and ^ = be osculating planes, the locus 

 of the poles of planes through their intersection with respect to conic sections 

 of the developable is a hyperboloid of one sheet." 



(It is usual in this connexion to speak of the pole of a plane as meaning 

 the pole of the intersection of the plane with the plane of the conic.) 



Theorem III. — " If P = and ^ = be osculating planes, then the 

 locus of poles with respect to conic sections of the developable of the 

 plane P + X(? = is a conic in the plane F -\Q = 0." 



5. I shall also have occasion to use a property of the tricusped plane 

 quartic which can be inferred from Theorem III, paragraph 4. 



" The fourth harmonic to the points in which a tangent to a tricusped 

 quartic meets the curve again, and the point in which it meets the bitangent, 

 lie on a conic." 



I append a direct proof of this theorem, with a view to making a useful 

 extension. 



Writing the quartic x-^ + y~^ + z'i = 0, 



the bitangent is x + y + z = 0. 



