MR. J. J. WALKER ON THE DIAMETERS OF A PLANE CUBIC. 1G7 



6A', . . . 6F', . . . being the second differential coefficients with respect to /, m, n of 

 the reciprocal (v) of u, 6W . . . ; while, otherwise, (21) 21, 



v = 4 (A/ 2 + Bm 8 + Cn* + 2Fron + 2Gid + 2HJw). 



With these substitutions the equation to the locus of the CoTE8-points on the polar 

 lines of points in L, or Ix + my + , takes the form 



2(F'+ 16F)- + . . .- 8PL* = 0, . (40) 



a cubic, degenerating into a conic when the line L touches the cubic u, which will be 

 traced further on ( 50-53) by referring it to the line L and the two tangents to s at 

 the points L = 0, s = 0. But previously it will be of interest to show the signifi- 

 cance of the part 



. . (41) 



in the equation to v; and to add a few remarks on the relations (38) (39). 



32. It is not easy to verify these relations by means of the invariants of (12) 20, 

 because the variables which enter into them are perfectly general except satisfying 

 ax + /3y + yz constant. They are verifiable, the former with slight, the latter with 

 moderate labour by means of the canonical form of the equation to the cubic ; but 

 much more readily by means of the simultaneous forms of u and s, to which every 

 form of u and its concomitant are reducible, referred to in 10. The verification is 

 therefore deferred until the reduction to those forms is explained in the sequel, and 

 will be found in the paragraphs cited. 



33. The tangent to u at any one of the three points in which it is met by the 

 transversal L, being the polar line of that point, also touches the poloid s, the point 

 of contact with s being its CoTES-point viz., it is that determined on it by the polar 

 with respect to * of the point in which it meets L ; or, otherwise, as the point in 

 which it is cut by the coincident tangent. 



The general equation to the three tangents, at the points u = 0, L = 0, may be 

 obtained without the difficulties which would attend the direct investigation hitherto 

 unattempted, at least successfully through the property of their touching the poloid 

 also, as follows : the point of contact of one of the tangents being (x'y'z), its equation 

 is 



or, u being a cubic, 



