1887.] 



On the Diameters of Plane Cubic s. 



335 



so that the centroid cuts the line at infinity in two real, coincident, 

 or imaginary points, according as two of the three intersections of 

 the cubic with that line are unreal, coincident or real. 



The discriminant of the centroid is, similarly, equal to minus one- 

 fourth of the square of the " Cayleyan " of the cubic, with the same 

 substitutions. 



A diameter is the locus of mean points of a system of parallel 

 -chords, which may be called " its " chords ; but through any point 

 pass two chords which have that as mean point. Considering the 

 points then on a given diameter, its own chords through those points 

 are all parallel to the polar of its own mean point with respect to the 

 "centroid," which polar is itself a double chord; the other system of 

 chords touch a parabola, which is touched by the diameter itself at its 

 •own mean point ; viz., for that point the diameter is itself the chord 

 •of the second system ; and the connector of that point w T ith. the centre 

 of the " centroid " is a diameter of the parabola. To every diameter 

 of the cubic corresponding to a parabola, the envelope of all these 

 parabolas is a quartic curve ; while the double chords, which are 

 otherwise distinguished as those having their mean points on the 

 " centroid," envelope a second cuspidal quartic. 



The locus of the mean points of the diameters of the cubic is a 

 second cubic, having a node at the centre of the " centroid," and its 

 asymptotes as its nodal tangents. Every diameter cuts this cubic in 

 its own mean point, and the mean points of two other diameters; 

 and this latter pair of points are harmonic conjugates with respect to 

 the first and the point of contact of the diameter with the "centroid." 



If the " centroid " is an ellipse its centre is merely an acnode, or 

 conjugate point, on the locus of mean points of diameters. 



The elliptic cubic — that which has only one real point on the line 

 at infinity — has two real conjugate diameters, i.e., diameters each of 

 which is the locus of mean points of chords parallel to the other, viz., 

 the asymptotes of the (then hyperbolic) " centroid." 



In the case of the parabolic cubic, the " centroid" being a parabola, 

 the preceding statements require some modification : viz., the en- 

 velopes of the double chords and of the parabolas connected each 

 with a diameter of the cubic in the manner above described, degenerate 

 from quartics into parabolas ; and on each diameter there lies only the 

 (finite) mean point of one other diameter, this being the mid-point of 

 the segment between the mean point of the diameter itself and its 

 point of contact with the " centroid." 



The above are fundamental properties of the diameters of cubics. 

 Some of them might have been stated more generally of the second 

 polars of points lying in any right line ; but it has been thought 

 proper in this notice to limit the statements to the diameters only, 

 viz., the second polars of points lying on the line at infinity. 



