MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 171 



4cam8/ + 4 &" S - 32(a6c + 2e)^mn - 6(abc 

 + cam 8 / + a&n 8 /) + 32(a6c + 2e*)ePmn -f 8(abc + 



whence, 



Zn2 ) x ( 5 1 ) 



From (oO) (51) there results, as the equation of the satellite chord of L, 



(abc + %<?)S (bcl* caw 8 / ZabiM 6aem 2 n 2 )ar; .... (52) 



which, ifa=6 = c=l, and e, I, m, n be replaced by m, a, ft, y respectively, agrees 

 exactly with the form given in the " Memoir on Curves of the Third Order" ('Phil. 

 Trans.,' 1857, p. 439). 



37. The values of the second differential coefficients of tr and v used in the 

 preceding verification may be calculated from the formulae (10) given, 19. Making 

 all the coefficients of u having suffixes vanish, u . . . ig ... for the present form of M 

 are found ; taid thence 2d i o-/3 2 = ^u^u^ Wgj 8 . . . obtained. Next v, expressed 

 in terms of I, m, n as line coordinates, is determined as 



viz., therefrom it is found that 



v = 26W - 2 (abc + IGe 3 ) 2am 3 n 3 - 24e 2 /mn2 (bcP) - 24 (ale + 26 s ) eWn. 



Finally, the second differential coefficients of v may be found. And the value of v so 

 found affords an independent verification of the relation (i.), 21, 



v = 4cr, 



when in <r and v f, 77, have been replaced by /, m, n. 



I may remark, in conclusion, that I first obtained the general form of v by considering 

 the question mentioned in 35, determining in the resulting form (47) which of the 

 two factors represented v, which m, by examination of the special forms when the line 

 L and the tangents to * at the points L = 0, * = 0, were the lines of reference. 

 Subsequently, I observed the method of arriving at the general form of v given 

 30, 31, independently ; which completed the general proof in as simple a manner as 

 could be expected. 



38. The triad of tangents m = reappear as a part of the complete solution of a 



2 2 



