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



L forms a coincident, pair of "double chords" (of the cubic) ; and these three lines, 

 viz., the two tangents and their chord of contact, form a triad of lines of reference by 

 means of which the properties of the complexes of lines here considered may be 

 deduced with far greater facility than through the use of the canonical form (2). 



The two tangents to s from the pole of L with respect to s are the nodal tangents 

 of the Cotesian v ( 50), the line L being its inflexional axis ; a circumstance which 

 explains the unique character of this triad of lines, and marks them out as the best 

 system of lines of reference for the discussion of properties connected with this con- 

 comitant of the primitive cubic u ( 48-62). 



11. But, whereas these tangents to s, the poloid of L, are only real when L cuts s 

 in real points (which is shown to occur only when it meets the cubic u in but a single 

 real point), three other triads of lines exist, of which one at least is always real, con- 

 venient as lines of reference in many of the questions which arise : viz., the sides of 

 one of the three triangles whose corners are the pole of L, with respect to s; one of 

 the three points in which L meets u ; and the pole of the connector of these two 

 points. Each of these being self-conjugate triangles in respect of s, the equation of 

 that conic is reduced to a trinomial form ; and since the cuspidal tangents of the 

 envelope w ( 76) all pass through the pole of L, the equation of that curve is of a 

 comparatively simple character for one of these triangles of reference. In Plate 6, 

 ABC is one of these triangles ( 63-76). 



12. If the line L touches u it touches its poloid s also ; and consequently other lines 

 of reference have to be looked for. These are found in the line L, the tangent to u 

 at the point where L cuts it (itself a tangent to s also, it will be remembered), and the 

 chord of contact of these lines with s. 



In this case the cubic v degenerates into the line L, and a conic having double 

 contact with s; while the envelope w also degenerates into a conic having double 

 contact with s and the Cotesian conic at the same points ( 77-81). 



13. When L becomes the line at infinity the pencil of chords through any point on 

 it is to be replaced by a system parallel to a given line, and the polar line by the 

 diameter which is the locus of the mean point on any chord of the system relatively 

 to its intersections with the cubic. 



The envelope of these diameters I have called the "Centroid" of the cubic, from 

 its evident analogy with the centre of conies, apprehending no confusion in such a 

 connexion with the sense of a " mass-centre," which it sometimes bears. 



The consideration of the Centroid and associated curves occupies the concluding 

 part of this Memoir. 



14. The method of treatment of the discussions, an abstract of which has just been 

 given, is uniformly analytical, trilinear coordinates being employed. The results will be 

 found to be arrived at without much difficulty, or tedious calculation, considering the 

 great generality of most of them. With a view to simplifying three important discus- 



