VI. On the Diameters of a Plane Cubic. 

 By J. J. WALKEE, M.A., F.R.S. 



Received June 16, Read June 16, 1887. 

 Revised February 9, 1888. 



[PLATES 6-8.] 



I. ABSTRACT. 



1. THE object of this Memoir is to develop relations which subsist between a cubic () 

 and the complex of lines, in its plane, which are the polars with respect to it of the 

 points on any transversal (L). This complex becomes the system of NEWTONIAN 

 DIAMETERS of the cubic, when the points on the plane are projected on a second plane 

 parallel to that containing the vertex of projection and the line L (Ix -f- my + nz = 0). 

 This development involves frequent reference to the envelope of the complex in 

 question, the conic 



which, in analogy with the " pole " of a line in the theory of conies, I propose to call 

 the " POLOID " of the cubic u and the line L ; and, in particular, when the line L is at 

 infinity, the " CENTROID " of u. 



2. HESSE* first appears to have used the equation to s in the theory of the ternary 

 cubic form but without any recognition of its geometrical significance to obtain 

 the equation to the cubic in " line-coordinates:" viz., in the form of the resultant of 



the system 



3s 3 8s . 



a* : ay : az = ' : m ' "- 



with 



Ix -f- my -f nz = 0, 



and this resultant will plainly be, to a factor, the equation to s itself in line- 

 coordinates , 17, , with /, m, n substituted for f, rj, respectively. 



* ' CRKLI.K, Jonrn. Math.,' vol. 41, p. 285 . . . , under date January, 1850. 



18.688 



