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



since 6 = or a = is excluded by the form of s, now reduced to 



s/n*= -e*z*+abxy, ... ...... (64) 



and u is reduced to the terms 



cz 



3 



Gexyz ...... (65) 



49. The line z = meets u in the points 



x : y = 6* : - a*, ' x : y = 3&* : - S'a*. x : y = $'& : - 

 (, A' being the imaginary cube roots of 1, so that 



a + 3'=l, A'=l, S 3 =-.', y s =-A, ^ + ^ 

 and the tangents to u at these points are 



o*6*y 2ez, 



( 66 ) 



the common equation to the three being, by actual multiplication, 



a*ba? + a&V SeV + 6a6ea/2 = 0, ...... (67) 



or, restoring lapsed factors, 



w = o?Wn*u - { SaWcjn*x + SaWcjfy + (a 2 6 2 cw* + Babe V)z] nV =0. . (68) 



Further, the same equation may be thrown into the form 



ra = a 2 6 2 n 6 (oua? + &/ - 2ryz) + 8a&en 3 ( e 2 n 2 z 2 + a6n 2 a;?/) 712 = 0. . (69) 



50. The tangential equation of s is now simply 



= 0, 



where 



4C = - a 2 6 2 n 4 , 



4H = 2abe*n*. 



