EussELL — Geometry of Surfaces derived from Cuhics. 473 



13. Let us now recall a property of the Hessian and Cayleyan of a 

 plane cubic curre. 



If the line joining a pair of corresponding points PP' on the 

 Hessian meet it again in IT, the polar conies of P and P' are pairs of 

 lines touching the Cayleyan, and the four points of contact lie on a line 

 passing through ZT", which we may call a companion line ; the two 

 lines form a polar conic, and this companion line touches the Cayleyan 

 in a point whose polar line with regard to the cubic touches the 

 Hessian at V the correspondent of U. 



If the line joining a pair of corresponding points on the Hessian 

 of a cubic surface meet it again in U, U', the polar quadrics of P and 

 P' are cones touching C and the twelve points of contact lie on a 

 twisted cubic passing through U, V , which we may call the com- 

 panion curve ; the line PP' and this curve determine two polar cones, 

 and this companion curve touches C in ten points whose polar planes 

 with regard to the [cubic are tangent planes to the Hessian at points 

 V, V, the correspondents of U, V . 



14. The Class of C. — In any plane there are three lines joining 

 corresponding points, and for the plane PP'Vixoxn. Art. 3, we see 

 that of these lines two coincide with PP' and the third is TIV. If, 

 therefore, we require to know how many tangent planes to C can be 

 drawn through" UV we have at once the solution. The six lines PP' 

 which can be drawn through U, and the six through F (see Art. 7) 

 when joined to UV give twelve tangent planes ; but UV being a 

 double tangent line to C four more planes (two coincident pairs) have 

 to be added to^the twelve. The Class is therefore 16 as was before 

 determined. 



15. The sections of the Cubic and Sessian hj the li-planes 



X Ja = yjh = z^c = vjd = tvje. 



If X, y, z, V, IV be the coordinates of any point on the Hessian the 

 equation of the tangent plane at the corresponding point is 



«^-X+ hfY^ cz^Z^ dv^V+ ew'JV= ; 



this plane will pass through the double point 0, 0, 0, 1, - 1, if 



dv~ - ew^ = 0. 



We see, therefore, that the plane vJd- wje = intersects the 



E.I.A. PEOC, SEE. III., VOL. V. 2 1 



