160 MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 



= 211^.3 + 2U 2 W t - VjV, ........ (26) 



Hence (22), 



8A' 6 D.'(s) = (tt,u>, + UjW! - 21;^) (2U 1 W 1 + 2U 1 W 1 - V, V a ) 



- 2t; 1 v ! 



= 2A' 6 P 3 , . . ,. ... ...... ...... ...... (27) 



where, 



P = (U ltfl + . -)/A 3 = (V,^ + . .)/A 3 = 



viz., P is the condition that L should be cut in involution by 



w x = 0, % = 0, u 3 = 0, 



which condition, multiplied by A 3 , has been shown (' London Math. Soc. Proc.,' vol. 0, 

 p. 233) to be 



?< 1 (D 2 D 2 ?< 3 D 3 D% 2 ) + w 2 (D 3 DX Di^D-Mg) + M 8 (DM 1 D 8 t/ 3 DM 8 DX) = 0. 



Thus it is proved (27) that four times the discriminant of s is equal to the square of 

 the Caleyan of u ...................... (ii) 



23. If the line Ix + my + nz = joins corresponding points on the Hessian of u its 

 coefficients satisfy the Cayleyan 



P=0; 



hence its poloid s breaks up into two right lines through the intersection of which the 

 tangents of s all pass ; and the locus of this Intel-section, as the line L varies in 

 position, is the Hessian itself. This is the property mentioned in the Introduction 

 ( 3) as proved by CAYLEY, ' Phil. Trans.,' 1857, p. 432. 



24. A theorem on the result of certain substitutions, which will be of use in 

 subsequent investigations, may be conveniently considered before entering upon 

 them : 



If <j> is any ternary form of order p, and /> a quadric say 



^ = ax 2 + by 2 + cz 2 + Zfyz + 2g 2 a; + 2hoy ...... (28) 



