MR. J. J. WA1.KKR ON THE DIAMETERS OF A PLANE CUBIC. 203 



spondingly, the " centroid " was inscribed in the triaugle formed by the asymptotes of 

 u so as to bisect the sides. The Cotesian v and the quartic w were laid down from 

 their equations without much difficulty, their triple contacts with s, and the parallelism 

 of the asymptotes of v to those of u, combined with the simply defined positions of the 

 cusps, and directions of the cuspidal tangents of IP, enabling them to be traced with 

 great accuracy. The parabolic envelope shown in connexion with the diameter DD , 

 touching that line at its mean point and having an axis parallel to the connector of 

 that point with the centre of the centroid, as well as touching the asymptotes, was 

 readily constructed geometrically from these data, and its accuracy tested by a 

 tangent drawn arbitrarily, meeting u in three real points E, E t , E 2 and cutting DD 

 in Ey, proving to have the last as its mean point, as determined by actual measure- 

 ment of segments, with great exactness. 



(ii.) For the second illustration, Plate 7, the figure of a cubic u having only a single 

 real asymptote was obtained by constructing a Cissoid from two conjugate diameters of 

 an ellipse, precisely as that of DIOCLES from the circle, of which in fact this, as well as the 

 figure previously described, may be regarded as projections. The hyperbolic centroid 

 was then constructed from its equation referred to the two conjugate diameters of the 

 generating ellipse, which are also conjugate diameters of the centroid, and to one of 

 which the real asymptote of u is parallel. The Cotesian v with its real loop and 

 single real asymptote, parallel to that of u, having also the asymptotes of the centroid 

 as nodal tangents ( 50), was constructed from its equation, the loop touching the 

 centroid at the contact with it of the real asymptote of u. It will be remarked in 

 this as in the case preceding where they are all three real, how soon the cubic v 

 approaches its asymptote, and its curvature becomes inappreciable. As regards the 

 little of w visible within the limits of the figure some remarks have already been 

 made, 60. Two diameters, besides the conjugate pair which the asymptotes of * 

 form, 83, have been introduced, DD touching one branch of s and having three real 

 intersections with v ; D'D' touching the other branch of s and meeting v in only one, 

 its mean point, D' . The " double " chords of these diameters, polars of their mean 

 points, touching their envelope w at the points Cj, C' lf equidistant from D, D' 

 respectively as C, C', are typical of their kind. It will be remembered that they are 

 " double " chords of u, with which, however, in this figure, they have only one real 

 intersection. The mean point of the chord AA'A", drawn parallel to the double 

 chord CD, as determined by actual measurement of segments, coincides with Ay, its 

 intersection with the diameter DD , to a nicety. Some remarks have already been 

 made, 83, on the chords BBj, B'B',8'., parallel to one asymptote of s and cut by the 

 other in their mean points B , B ' with great exactness. It has not been thought 

 desirable to introduce the parabolic envelope connected with either of the diameters 

 drawn, into this figure. 



2 D 2 



