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



contact C'] of another double ordinate C'D', of the diameter D'D' , D' being the centre 

 of C'C\. It will be observed that D , D' are the mean points of the diameters DD , 

 D'D' , D being harmonic conjugate to D with respect to DjDo, the other twx> real 

 intersections with v ( 58). 



86. In Plate (8) is shown an "alien" ordinate EE of DD , having its mean point 

 EQ on that diameter ( 61), but being an ordinate of the other tangent to s which 

 might be drawn through E . The parabola, which is the envelope of these " alien " 

 ordinates through the different points on DD (J , is also shown, touching DD at the 

 point D , and having the connector of that point with the centre of the centroid as 

 diameter (88), this connector being now the representative of the 



y'x + x'y = Q 



of the equation just cited, the connector of the CoTES-point on the polar line of 

 (x'y'O) with the pole of the transversal L, now become the line at infinity. 



87. The asymptotes of the centroid offer themselves as an unique pair of Cotesian 

 axes to which the cubic may be frequently referred with advantage, its equation being 

 then 



u = ax 3 + bf -f- 6exy + Sc^x + 3c 2 y + c = 0, . . . . . (i.) 



and the centroid, with changed sign which is now immaterial 



e~=-0. . , ....... (ii.) 



For many discussions also it is convenient to define the diameter by the coordinates 

 (x"y") of its point of contact with s. 



The equation of its double ordinate will now be 



ax"(x - x") + e(y- y"} = ; . . ... . (iii.) 



and that of its " alien " ordinate at the point x 1 y l 



x x^+ey ] (y yJ=Q. . ,;.-.-,. . (iv.) 



It will be sufficient to indicate the steps by which these may be verified indepen- 

 dently of the general formulae given in the earlier part of this Memoir. 

 The diameter is the tangent to (ii.) at x"y" or 



y'x + x"y - 2x"y" = 0, . ,, ,. -,,-*. ,., . fa (v.) 



the coordinates of the mean point of which may be found as one-third of the sums of 

 those of its intersections with u. The polar of this mean point is (iii.) ; and that 

 of the mean point of the second tangent to s through ,r l y l on (v.) is (iv.) 



