Russell — Geometry of Surfaces derived from Cubics. 467 



In the figure, all the points marked, with, the exception of T and 

 jT', lie on the Hessian, and in each case the " x coordinate " is given. 



From these values the pairs that are correspondents are easily seen, 

 "We may also verify that the line joining the points 



ax" - Vi 

 ax {ax^ - 6z) 



ax^ - 6i 



touches the Hessian at Z7; for any point on this line has for its 

 coordinates 



{ax" - B) {ax' - $ ,) ^ {If - ^) (^/ - ^0 



^^~ ax{ax'-e^) ' ^'^ hy{hf-6-,) ' '" 



and if this lies on the Hessian, we have 



X {ax' - 62) 



= 0: 



{ax~ - 6) {ax- - ^1) 

 or, replacing by partial fractions, and making use of the identity 



_ X 



this equation becomes 



ax' - Ox 

 -6-, x 



0, 



0, 



V - tfi ax" - U 

 and therefore two values of 6 are each equal to 62. 



