166 MR. J. J. WALKER ON THE DIAMETERS OP A PLANE CUBIC. 



v = - u (AZ 2 + Bm 2 -|- Cn 2 + 2Fmn + 2Gnl + 2HZm) 



S + ' H &} ; " 

 (AZ + Hm + Gn + (HZ + m + n) | + (Gl + Fm + Cn)| 



2s 



where, as above (7), M U . . . , r% . . . are the contravariants of 



du , du ,dn 



' W = ' W =' 



taken singly and in pairs, these being the coefficients of the poloid, viz., 



s = u n x* + w 22 y 2 

 and A, B, C, F, G, H are the coefficients of its reciprocal, viz., 



4A = 



4F = 



31. The coefficient of ,s in the above expression admits of an important transforma- 

 tion, viz., it may be written 



and this (71 ), 55, is identically equal to 



4P (Ix + my + nz), ........ (38) 



where P is the Pippian, or Cayleyan, of the cubic u. 

 Also (55) 



Ifi j(AZ + Hm + Gn)~ + (Rl + Bm + Fn) || + (Gl + Fm + Cn)| I + 24Ps 



s-{(A'+ 12A) ?;+...+ 2 (F+^F)^ +...}(/*+ my + ), . (39) 



