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



question of considerable interest in connexion with the subject of this Memoir : what, if 

 any, pairs of polar lines of points on L are conjugate, in the sense of having a common 

 CoTES-point ? 



The polar line of the point x'y'z' on L, viz., 



meets L in the point 



, N , . . . 9?*' du' , 3tt' du' 3w' , du' 



(^(y)'(^ = n^-m^:l^--n^:m^-l^, . ,. . (52) 



and the condition plainly is that the polar line of (x) (y) (z) should pass through the 

 original point x'y'z', i.e., that 



when for (a:) (y) (z) in (du/dx) . . . the values just given in terms of x'y'z' (52) are 

 substituted. The result is at once obtained by means of the general theorem, 24 



(and 27) ; viz., it is simply 



u's = 0, 

 with 



lx' + my' + nz = 0. 



Now, at the three points u' = 0, lx' -f my' + nz' = 0, [() (y) (z)] plainly coincides 

 with (x'y'z') ; thus each tangent to u at those points is a double conjugate line, which 

 accounts for the factor u' in the result above. 



39. The points 



s' =0, Id + my 1 + nz' = 0, 



determine the unique pair of distinct or proper conjugate polar lines; viz., these are 

 the tangents to the poloid at the points in which it is met by the transversal L ; which 

 will be real only when L meets s in real points. 



These two conj ugate tangents to s and their chord of contact L form, as before 

 remarked, an unique triad of lines, to which the cubic u and its system may be con- 

 veniently referred, in questions involving *, v, and certain other concomitants of L and 

 u particularly, as will appear in the sequel. Plainly the equations thus referred will 

 only be symmetric as regards two coordinates, however. 



40. Through any point (x"y"z") in the plane of the cubic u and the transversal L two 

 chords of u can, in general, be drawn having it as their CoTES-point, relatively to their 

 intersections with L ; viz., the connectors of (x"y"z") with the points (x'y'z'} (...) in 

 which its polar conic meets L. 



If (xyz) is any other point on one of these chords, then the coordinates of the 

 point in which it meets L, or 



lx -{- my + nz = 0, 



