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



V. NEWTONIAN DIAMETERS. 



82. If L is the line at infinity, then 



1 : m : n = sin A : sin B : sin C, 



= a. : j8 : y\ 



and to the CoTES-point of a chord through a given point on L (x'y'z') corresponds 

 the " mean point " on a chord parallel to a given line, the " direction coordinates " of 

 which are, 44, 



m-ftl *-y ft-i\ ..... (122) 



the line being written 



& + yy + & - ; 



viz., the quantities (122) may be considered as the coordinates of the point in which 

 this line, and all parallel to it, meet the line at infinity. 



To every equation in the preceding part of this Paper involving as parameters 

 (x'y'z') the coordinates of a point on L, a finite line, now corresponds one in which 

 x'y'z' are replaced by 



X : /* : v = yr? - ft : at, - y - : ft - ay. ' ." . . (123) 

 Thus, for the polar line of x'y'z' (7) now appears D 2 ?< (4), 17, i.e., 



which is, plainly, the locus of a point on a chord parallel to r + . . . = 0, and 

 meeting the cubic u in the points O 1( O 2 , O 3 , such that 



OO, + O0 2 + 00 3 = ; '. . !' " . . . (125) 



viz., O is the " mean point " on the chord relatively to the triad 0^ 0.;, O 3 . 



The line (124) is the Newtonian Diameter of the system of chords (V 1 ') '> an< ^ ^ 

 envelope, since (123) 



X + /8/x + yv = . .. . .. ^ ( . . .. . (1 20) 



the " centroid," i.e., the " poloid " of the line at infinity, is 



or 



u n y? 4- ... + Uy$)Z + . . . = 0, 



where ,, = is now the condition that d,u should be a parabola . . . , 



