188 MR. J. .T. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 



the term y? must disappear from u, or 



a=0, 

 which (91) necessitates either 



a. 2 or ?>i = ; 



but the supposition . a., = would imply a second of the tangents to u at the points 

 u = 0, 2=0 passing through the point y = 0, 2 = 0, so that 



6, = 0, 

 and therefore, by the second of the three conditions (91) 



e = ; 

 whence, by the first, 



b:a 2 = -& 3 : 3 ....... ( 92 ) 



64. Thus the equation to the cubic is reduced to the terms 



by* + C2 s + sc^fy + SagO^z + 3b<y*z + Sc^x + 3c 2 2 2 y =0, . . (93) 

 with the relation (92) ; and (90) the poloid to 



+ 3 & 3 2 2 = 0, 



or ....... (94) 



the reciprocal of which is 

 where 



A = 6a 2 a 3 6 3 n 4 ' = a 3 2 6 3 2 n*, 

 B = a 2 2 a 3 & 3 n 4 = ba 2 a^n*, 



C = - bafn*, 

 or 



A : B : C = a s b 3 : a 3 2 : a^. 



(95) 



65. For the above forms (93) (94) of u and s, the values of their first differential 

 coefficients, with the coordinates (x'y'O) of a point on L substituted for xyz, are 



9s 9s 9a 



S'Jy'S- 



8tt OU OU , , 7 /n , I* '9 I 7 ,'2 . 



a? : 97 : a? = 2asa;y : &y + a ^ : a '** + & ' 



