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



are combined, the coordinates of the point of contact of the double chord with its 

 envelope will be obtained in terms of those (x'y') of the point on L, through which it 

 is drawn, viz., eliminating z between the equations (82), the point of contact is deter- 

 mined by the line 



(Vox' 3 + by' 3 ) y'x (ax' 3 + 2by' 3 )x'y = 0. . ... . . (83) 



The intersection of the double chord with L lies on the line 



y'x - x'y = 0, . . .., ;.,.., . . ; . . . (84) 



and its two intersections with the poloid on the lines 



bi/*x - ax'*y = 0, ... .'"" !,'' : . " . . (85) 

 ax' 2 x by'*y = Q, (86) 



the latter being its CoTES-point, viz., the point of contact with s of the polar line of 

 (*', y', 0)-(61), 47. 



Now these four lines through x = 0, y 0, form a harmonic pencil, since, writing 

 (84) (86) 



y'x x'y=p, 



ax' z x by'~y = q, 

 [giving 



(ax' 3 by 3 ) x= by'-p + x'q, 



(ax' 3 by' 3 ) y = ax^p + y'<l\ 

 then (85) 



bi/*x ax'*y = (a 2 x 6 b 2 y*) p + (%' 3 a %' s ) x'y'q, 

 and (83) 



('2ax' 3 + by*) y'x - (2by' 3 + ax' 3 ) x'y = {(- 2ax* - by'*) bij' 3 + (2by' 3 + ax*) ax' 3 }p 



+ {(2ax 3 + by' 3 ) x'y - ( 2by' 3 + ax' 3 ) x'y'} q 

 = (a-x' & lry' 5 )p + (ax 3 by' 3 ) x'y'q ; 



viz., the later pair are harmonic-conjugate with respect to the former pair_p, q. Thus 

 it appears that the CoTES-point on a double chord, and its intersection with the line L, 

 are harmonic-conjtigate points with respect to its second (or " empty ") intersection with 

 the poloid s and its contact with its envelope w, as stated, 7. This is shown in 

 Plates 7, 8, in the case in which L is the line at infinity, by D being the mid-point 

 of CCi. 



60. Returning to the equation (81) of the quartic w, if y and x be alternately 

 eliminated from either 



