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



i.e., at the points determined by it, and 



= 0, ....... '. . (106) 



the polars of those two points of the three common to L and u. 

 For these points are determined (93) by (z = 0) and 



67/2=0, 

 or _ 



x:y:z = b: \/ 3ba. 2 : 0, 



the polars of which, with respect to (94) 



a<?x* + ajyf a 3 6 3 2 2 = 0, 



are _ 



a 2 bx b \/ Sbag/ = 0, 



or 



2 = 0. 



72. The equation of the double chord ( 41) of the pencil through x'y'z on L being 



generally (56) 



^ Bs 1 3s ck 3s 3jA _ 



^" u> 



is, in the present case, from the values of the differential coefficients given (98), 65, 

 (a^-b^)(y'x-x'y) + (3l^ 2 -a^)x'z = 0, . . . (107) 

 or, arranged as a binary cubic in x'y', 



3M x 'y" 2 - ^ = o. . 



The discriminant of this last form gives the envelope of the double chord as the 

 point x'y' describes the line L, or z = ; viz., it is 



3w = 



- {afx* +3 (a 2 y + a 3 ?) (by + 3b s z)} (Bba^ + (by + 3b 3 z)*} . (108) 

 or 



3w = 36a 2 (agx 3 + 36y 2 ) 2 



+ 2a 2 (a<pr + 3% 2 ) (2by 3& 3 z) 2 - 3 (2a^ - a 3 z)(2% - 36 3 z) 3 , ( 1 09) 

 in vktue of the relation (92) 



