MB. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 153 



the three tangents to u at the points in which it is met by L, now first obtained in 

 a general form. In the Memoir referred to ('Phil. Trans.,' 1857, p. 439) CAYLEY 

 obtained, with his peculiar skill, the equation of the three tangents in question 

 for HESSE'S canonical form of u ; viz., 



ax 3 + &/ 4 v? + Qexyz (2) 



finding for the " satellite line " of L, or line in which the tangents again meet u, the 

 equation that, further on ( 36) will be shown to verify the general form at which I 

 have arrived ; and in terms of which (if it be taken as a fundamental covariant of u 

 and L) and of 6- the equation of the nodal cubic r may be expressed. 



7. Whereas through any point in the plane of the cubic u and the line L there can 

 be drawn in general ( 40-43) two chords having it as their COTES- point relatively to 

 those in which they meet L, viz., those determined by the polar-conic of the first 

 point ; if, however, that point be taken on s the two chords coincide, and thus a 

 complex of double chords is obtained, which are the polars with regard to s ( 4'2) of 

 the CoTES-points of the polar lines of the points of L, and are shown ( 58-60) to 

 have as their envelope a tricuspidal quartic (81) 



w= 0, 



the equation of which canuot be found explicitly except for specifically assigned forms 

 of u. 



The point in which a double chord meets s being its CoTES-point, that in which it 

 touches its envelope is shown to be harmonic conjugate to the former with respect to 

 the intersections of the chord with the line L and with the conic s again ; and the 

 two intersections of the double chord with s are thereby discriminated ( 59). 



8. Again, considering ( 61) the points of a line having its pole on L, the chords of 

 which those are CoTES-points constitute two groups : viz., one a pencil through the 

 pole of the polar line, the other having as its envelope a conic ( 88) touching the line 

 L as well as the polar line in question (in virtue of its being the line of the latter 

 complex through its own CoTES-point) and the double chord through its point of 

 contact with s, which is both a ray of the pencil and a line of the complex. 



This system of conies ( 62) has as its envelope the sides of the quadrilateral formed 

 by the transversal L and the tangents to the cubic at the points in which L meets it. 



9. A question of some interest is considered ( 38-39) : what, if any, of the complex 

 of polar lines of points on the transversal L are conjugate, in the sense of their inter- 

 section being the CoTEs-point on either ? Discarding the tangents to u at the points 

 in which it is met by the transversal L, each of which is conjugate to itself, the o.dy 

 distinct conjugate polar lines of points on L are the two tangents to the poloid (s) of L 

 from the pole of that line with respect to the poloid. 



10. As the chord of contact with the poloid of the two tangents through its pole 



MDCCCLXXXVIII. A. X 



