MB. J. J. WALKKll ON THK Dl \\IKTK1JS OF A PLANE CUBIC. 199 



ttjg = 0, that the line at infinity should be cut harmonically by the conies 3,u, 

 3jtt . . . . 



The condition that the cubic u should touch the line at infinity in the standard 

 form (b*c* 4-...)a 8 -f... = is equal to four times that for the centroid being 

 parabolic and touching the finite asymptote of u ; and 



83. When the cubic meets the line at infinity in three real points, or the three 

 asymptotes are real, the centroid is an ellipse inscribed in the triangle formed by 

 these lines, so as to touch them at their middle points (Plate 8). 



But if the cubic u has only one real intersection with the line at infinity, then the 

 centroid is a hyperbola, having the single real asymptote as a tangent ; and 



The asymptotes of the hyperbolic centroid are the only real pair of conjugate 

 diameters of the cubic u, viz., each cuts every chord parallel to the other in its mean 

 point, and in particular divides the tangential chord of the cubic parallel to the other, 

 or the parallel nodal chord if the cubic is nodal, in the ratio of 2 : 1. Thus in Plate 7 

 the chords BBj, B'B'j, B' 2 , parallel to one asymptote of the centroid, are divided in 

 such wise by the other asymptote in the points Bg, B' . 



84. The mean point on any diameter of u regarded as a chord of that cubic is the 

 point in which it is met by the diameter of the centroid conjugate in direction to its 

 chords, the "double" one ( 7) of which is the s-polar 42 of that mean point. 



The locus of the mean points of diameters of the cubic is the nodal cubic v, the 

 Cotesian of the line at infinity, having as its nodal tangents the asymptote* of the 

 centroid, and being therefore acnodal when the three asymptotes of the cubic are all 

 real (Plate 8), the acuode, or conjugate point, being the centre of the elliptic centroid. 



The asymptotes of the cubic v meeting, two and two, on the diameters of s through 

 its points of triple contact with v, are parallel to those of the primitive cubic u ; and 

 the line at infinity is its inflexional axis, these inflexions being at the points in which 

 the asymptotes of meet that line. 



In Plate 8 the Cotesian is represented with three hyperbolic branches, each touching 

 the centroid ; in Plate 7, with a real loop touching the centroid, and a single real 

 asymptote parallel to that of the cubic u. 



85. The envelope of the " double ordinates " of the Newtonian Diameters is repre- 

 sented in Plate 8 as a quartic (w) with three real cusps, the three diameters of the 

 centroid conjugate to the asymptotes of u being the cuspidal tangents. This quartic w t 

 as well as the Cotesian v, has triple contact with the centroid at the points of contact 

 of the asymptotes of u. In Plate 7, the cubic u having only one real asymptote, the 

 quartic w has only one real cusp ; also it and v have only one real contact with * at 

 the point of contact of the real asymptote with *. Within the limits of the diagram 

 the only parts of w visible are (i.) that adjoining the contact referred to, terminating 

 on one side with the point of contact (C 7 ) of the double ordinate CD of the diameter 

 DD in this case (of L being at infinity) the point on CD conjugate to D ( 59) 

 being at infinity, D is the centre of the segment CC' and (ii.) the cusp of w with the 



