E-ussELL — Geometry of Surfaces derived from Cubics. 465 

 and on substituting the coordinates of V in it, we obtain 



ax {ax^ - ^o) 



= 



the line VV is therefore a double tangent to the Hessian, as may 

 indeed be seen otherwise. 



The tangent planes to the Hessian at P and P' obviously pass 

 through the line VV . 



4. The converse of this theorem is also true. If the line joining 

 ^1) y\i %) ^1) ^^1 to x-i, ^21 ^2) ^2; ^2 touch tlic Hcssiau at these points, 

 the line joining their correspondents intersects the Hessian in a pair of 

 corresponding points. 



The first condition is satisfied, if 



1 



reduces to A.-/X- = 



a (A«i + /x^2) 



and a little reduction shows that this is equivalent to 

 ^ ^1 





0, 



ax-r 



Now let 6, 4>, p he any three quantities, and put 

 {6xi + Xn) {cfiXi + ^2) - paxi~x^ = ^ 1 



(%i + ^2) {^yi + '1/2) - phi^!/2~ = B 



(6zi + So) (^Si + Z2) - pczy%" = C 

 {6vy + Vo) (^Vi + Vo) - pdv^-vi = B 

 {Ow^ + tOi) (cfiWi + tvs) - pew^wi = E 



(6) 



then 



and 



2 -A^ = 6,^5 — + (^ + </.):§ — + S A 



ax^x. axi ax^ ax{' 



2^.0 



(XOC-\pCi 







but ^, <^, p may plainly be determined, so that A = B = C = ; and 

 therefore, from the last two equations, it follows that B = 0, E=0 

 at the same time. 



