474 



Hessian in the line t? = 0, 

 may be ■written 



Proceedings of the Royal Irish Academy. 



= 0, and a cubic curve whose equation 



1 1 

 ax hy 



X + y 



c% dv 



+ «; 1 + 



Ji 



(8) 







and that the section of the cubic surface by the same plane is 



ax^ + ly'^ + c-3^ + dv' i 1 -} 

 X ■{ y + % + v\l + 





= 0, 



= 



(9) 



From the above "we easily infer — 



(1.) The curve (8) is the Hessian of the curve (9). 



(2.) All points of the Hessian surface on the curve (8) have their 

 correspondents on the same curve, and that these pairs of points are 

 correspondents on the curve (8) in the sense in -which the -word is used 

 in the geometry of plane cubic curves. 



(3.) The line joining a pair of corresponding points on the curve 

 (8) -which -we kno-w touches the Cayleyan of (9) (see Salmon's " Higher 

 jplane Curves") must touch the bitangent surface in the same point; 

 hence this Cayleyan is part of the section of C by the bi-plane 



vjd - wje = 0. 

 (4.) The tangent cones from the node to the Hessian are cubics 

 ■whose curves of contacts are 



''i ! 



do 



1 



1 



1 



— + ^ + — 

 ax by cz 



X + y + %-^ v\l + 



Je. 



> 



(8) 



d 



111 J^ n 



— + J- + — + — = 0, 



ax by cz dv 



X ■'r y + Z ■{- v{ 1-"^ 



>, 



Xs 



(10) 



