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



conjugate ; to which lines the other two asymptotes will form a conjugate harmonic 

 pair. These axes are the special form which those of the self-conjugate triangle of 

 reference employed, 63-76, take ; and all the equations there deduced will be 

 adapted to the above Cotesian axes simply by substituting unity for z. 



90. When the cubic u is parabolic, i.e., touches the line at infinity, its "centroid" s 

 also touches that line, viz., it becomes a parabola, which may then be referred to the 

 tangent parallel to the finite asymptote of u, and the diameter through its point of 

 contact. 



The Cotesian v will then be made up of the line at infinity and a parabola, and the 

 chords of u having their mean points on a diameter of which they are not ordinates 

 will envelope another parabola, while the double chord will envelope a third. 



In this case all the equations and formulae of 77-81 are applicable by simply 

 making 2=1. 



It has not been thought necessary to add a figure in illustration of this case, the 

 curves being all of familiar character. 



VI. DESCRIPTION OF PLATES. 



91. The two types of cubic which have been drawn to illustrate results arrived at 

 in this Memoir have been constructed geometrically with great accuracy as follows : 



(i.) Those in Plate 6 and Plate 8 from two conjugate diameters of a hyperbola, as 

 the well-known Cissoid of DIOCLES from two rectangular diameters of a circle ; viz., 

 from a vertex of one diameter a pencil of lines was drawn, each to the extremity of an 

 ordinate parallel to the other diameter, and then its intersection with the equidistant 

 ordinate on the other side of that diameter determined a point the locus of which 

 gave the cubic u as represented, with one cuspidal branch and two hyperbolic. For 

 the figure in Plate 6 the line L was drawn arbitrarily, cutting u in three real points 

 A, A', A", and the tangents to u at these points traced, the accuracy of their directions 

 being vouched by their tangential points proving to range in a right line KKjK 2 . 

 The poloid s was next inscribed in the triangle formed by the triad of tangents, 

 touching them at points A lf A 2 , A 3 , harmonic-conjugate severally to A, A', A" with 

 respect to the corners of the triangle. Another arbitrary line OD having been drawn 

 touching the conic in D and meeting the cubic in three real points, its CoTES-point O 

 was found by actual calculation from the measured lengths of the segments between L 

 and the cubic u. The polar of O, being the double chord of the pencil through the 

 point on L of which OD was the polar line, of course determined that point (x'}. 

 [Otherwise x might have been taken arbitrarily and the CoTES-point on L relatively 

 to it have been found by measurement ; and so the polar-line of x' with the double 

 chord have been arrived at.] The accuracy of the figure so far was tested by 

 examining the agreement of the CoTES-points of L itself and another chord with 

 their intersections by the polar line at O', O". For the figure in Plate 8, corre- 



