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



so that the connectors of the pole of L (with respect to s) with this point and the point 

 of contact of v ivith s are harmonic conjugates relatively to the tangents to s from that 

 pole. 



53. The polar line of P (xfy'o), a point on L, meets the Cotesian v in three points 

 (say, D its CoTES-point) and two other points (say, D lf D 2 ), while it touches the 

 poloid s in a point D. Now D is also the CoTES-point on this polar line relatively to 

 the point D and the cubic v : to prove this 



The coordinates of D are, (59), 45, 



x : y : z = bey 1 ' 2 : aex' 2 : abx'y', . . . ..... (72) 



and 



. t ..{'.. (73) 

 becomes, for those values of the coordinates of D, 



3,y 82 



' : Sax'* 



viz., the u-polar line of D is 



3 (ax' 3 + fy' 3 ) (y'x + x'y) - x'y' (ax'*x + by' 2 y + Zex'y'z) = ; . . (75) 



but 



00;'% + by' 2 y + Zcx'y'z = ........ (70) 



is the w-polar line of (x'y'o), and 



y'x + x'y =0, . . . .tl.^j aaiJ", / B .iJ x*' ( 77 ) 



the s-polar of the same point, cuts it in its CoTES-point, D (35), 28. Thus the 

 three lines (75) (76) (77) meet in one point (D ), and this is the -CoTES-point 

 relatively to P, and the v-CoTES-point to D of (76) ; whence 



3 ' + ' 



I 



DD DD n DD, 

 or 



2 1 1 



DD ~ DD, " 



as verified very exactly in figs. 2, 3, i.e., the CoTEB-point on the polar line of a point 

 in L is harmonic-conjugate to its point of contact with the poloid of L and u relatively 



