466 Proceedings of the Royal Irish Academy. 



Witli these values of 6, <^, p, we have 



= 0, 2 . =0; 



and therefore 



ax^X2 ' 3j/iy2 ' ■ ■ 

 are points in the Hessian on the line joining 



1 1 



1 





1 1 



1 







to 







ax^^ bi/i' 



etvi 





ffijPa' ii/o^ 



ew'i 



From (6) we see that they are corresponding points. 



5. The coordinates of any point on the line PFmay be written 

 , ixx txy 



p^ = ^ + t£—^^ pv = y + 



This will he on the Hessian if 



which, since 



ax {ax- + /x - ^i) 

 ax'^ - 6i p qx 



ax {ax"- + iJ- — 0]) ax ax- + fx - Oi 



X 



is equivalent to 2 — jr • 



ax- + fji - Ui 



We therefore see iinally that PF touches the Hessian at F, passes 

 through V, and also through the point for which jx = Oi - Oo, and 

 therefore having for its coordinates 



ax'' - e. hf - e. 



ax- - di by- - Ui 



In a similar manner we obtain the remaining points in which P'V 

 PV\ and P'F" meet the Hessian. 



