MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 195 



which has been proved ( 71) to be the chord of contact with s of the two tangents 



referred to 



(by + 2& s z) 2 = 0. 



76. Generally then, when the transversal L meets the cubic u in three real points, 

 w the envelope of the double chords of pencils of lines through points in L, is a 

 tricuspidal quartic having triple contact with s, the poloid of L, at the points in which 

 it is touched by the three tangents to u at the points L = 0, = ; the cuspidal 

 tangents being the s-polars of those three points; and the fourth pair of points 

 common to w and s lie on the line L. 



In Plate 8 the envelope w is figured for the case of L being the line at infinity 

 and the cubic u a modification of the Cissoid, as more particularly described 

 below ( 91). 



The property of the double chord touching w in a point which is harmonic to its 

 second intersection with the poloid * relatively to its CoTES-point and intersection 

 with L, has been proved in 59 by the use of other lines of reference, with the reality 

 or imaginariness of which it is, plainly, unconnected. 



77. When L touches w, it and the tangent at the point where it again meets the 

 cubic being taken as 



2 = 0, y=0, 



and their chord of contact with the poloid s as 



x= 0, 

 the equation of the poloid (10) is reduced, since 1 = 0, m = 0, to 



(06, - a s 8 ) a 2 + (6a 3 + a s &3 - 2\e) yz = 0, 



the conditions for which are (ib.) 



60, -&, = <>, ........ (i.) 



, 8 6 3 e 2 =0, ........ (ii.) 



a&s + A 2<V = 0, ....... (iii.) 



ab a 2 &! = ......... (iv.) 



But since y = 0, z = is on u, 



a=0; 

 therefore, by (iv.), 



&i = 0, 



since o 2 = is excluded by the form of s ; 



2c2 



