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



3. In the first edition of the 'Higher Plane Curves' (1852) SALMON, under the 

 subject " Poles and Polars " (of cubics), showed that 



*= 



is the envelope of the polar lines of points on L, touching each tangent to u at the 

 points L = 0, u = in the point harmonic conjugate to its contact with u relatively 

 to its intersections with the other two, and proposed for the locus of s = the 

 designation " Polar conic of the line " L, for which I have ventured to suggest, and 

 use, the shorter name " Poloid " of (u and) L. 



In CAYLEY'S Memoir -" On Curves of the Third Order," 'Phil. Trans.,' 1857, the 

 additional property is given of the polar lines of points on L all passing through a 

 point (viz , the intersection of the two right lines into which s, " the lineo-polar 

 envelope of the line," then breaks up) on the Hessian of u, when L joins corresponding 

 points on that curve. Beside these I have not been able to find any notice of the 

 conic s. 



4. If through any point P in its plane chords are drawn meeting the cubic in O 1; 

 Og, O 8 , and a point is taken on the chord determined by the relation 



3 . ^_ J_ J 

 pn ' PA ~T~ w 



PO " PO! ir PC, ' PO S ' 



the locus of is a straight line, according to a theorem of COTES'S, communicated by 

 his friend, Dr. ROBERT SMITH, Master of Trinity College, Cambridge, to MACLAURIN, 

 after COTES'S lamented death, and proved by MACLAURIN* as a case of a more 

 general theorem which presented itself to his mind when " meditating on this com- 

 munication." For shortness I have called the point O the " CoTES-point " on the 

 chords through P, the locus of which is now well known to be the polar line of the 

 point P relative to the cubic u. 



5. If P describes a line L, the CoTES-points of the polar lines of the points on L 

 regarded as chords of u relatively to their intersections with L, may be considered. 

 SS 28-32. 



Jv 



The locus of the CoTES-points of this complex of lines is shown in the sequel to be 



a nodal cubic (37) and (40), 



v = 0, 



which covariant of u and the line L, I propose to call their " Cotesian." 



6. Considering ( 33-37) more generally the locus of CoTES-points on chords of u 

 subject to the condition of touching s, the result comes out as a concomitant breaking 

 up into two factors, one the cubic v just referred to, the other being the equation to 



* ' De Lin. Geom. Prop. Gen. Theor. IV.,' p. 24, ed. 1748. The Theorem is not given in the ' Harmonia 

 Mensurarnm,' as sometimes erroneously stated, with the subjects of which treatise it has no connexion. 



