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



55. What has preceded is, substantially, a verification of the relation (39) 31 ; 

 but, to make it more clear, observing that the only one of the coefficients A ... 

 F ... of the reciprocal of s which does not disappear from the sinister of (39) is C, in 



the term 



*\ 



96C P = - 24a 2 6 2 n 5 (3 



and adding to this 



144Ps = 



that sinister becomes 

 72 nz 



Again, the dexter is (L being now nz) 



nz {(12o6 8 c 1 n*) 6ax + (12a 2 &c 2 n 4 ) 6% + (30a 2 6 2 /i 4 - I8a^ 2 *) 6 (op + c. 2 y '+ cz) 



+ 2 ( 24a6e 2 n* + 

 or, identically, also 



- nz {UtaWn^CiX + cjy) + (72a 2 6 2 n 4 c + 144a&e 2 n*e)z}. 



Lastly, 



o w 



the only term in the sinister of (38), 



= 4aben*z = (71) 51, 4PL. 



56. It has been shown that v and u meet the transversal L in the same three 



points ( 52), 



aa? + by 3 = 0, z = 0, 



of which one only is real when the line L meets its poloid s in real points, the condi- 

 tion of the lines of reference x = 0, y = 0, at present used, being real. That this 

 would be the case appeared from the fact that the discriminants of the cubic (12) and 

 quadratic (13) 20, whose roots are proportional to the intercepts of L between a 

 certain point on it and its intersections with u and s respectively, are of opposite 

 signs. 



57. The tangents to v at the points in which it is met by L are plainly obtained 

 from those to u at the same points, by changing e into e/3 ; viz., they are (66) 



I 



