TowNSEND — Geometrical Properties of the Atriphthaloid. 69 



tial chord of the curve, and the essentially real and positive root to 

 the pair of inflexional chords PQ (see fig.) of the conchoidal branches ; 

 ■which, equidistant in opposite directions from the axis of y, lie neces- 

 sarily somewhere between it and the parallel tangents at the vertices C 

 of the branches. 



The four remaining roots, with the four pairs of inflexional chords 

 reflexions of each other with respect to the axis of y to which they cor- 

 respond in the curve, may be all real, or all imaginary, or two real and 

 two imaginary, according to the particular value of the parametric ratio 

 h ^ k on which alone they depend. When the two ovals are real and 

 finite, as they are for all values of the ratio for which 4/i^ > 271^, the 

 entire number of pairs of inflexional chords intersecting them at pairs 

 of real points is necessarily even; but none of the four pairs^ even 

 when themselves real as regards their positions, need intersect them 

 necessarily at real points at all. So that the ovals may be, and in 

 fact often if not always are, as represented in the figure, concave 

 to their interiors throughout the entire cii'cuits of their perimeters. 



The application of Sturm's theorem to equation (14) gives us, for 

 all numerical values of the parametric ratio, the exact numbers of 

 corresponding pairs of inflexional chords which occupy real positions 

 within any two assigned limits of distance from the centre of the 

 curve ; and, as the corresponding values of OA and OJB can also be 

 determined for all such values to any degree of approximation, from 

 equations (1) or (4), by Horner's and other methods of numerical solu- 

 tion, the exact numbers of real pairs of inflexional chords lying within 

 the intervals AP, and therefore intersecting the ovals at real points, 

 can consequently be determined by its aid for all numerical values of 

 the ratio. Its applications, however, are in general laborious, and 

 in the present instance uninstructive except for such values of the 

 ratio. 



By its application to the equation for the two particular cases 

 when h -^ k = 1 and 2 respectively, which correspond, the former to 

 an imaginary and the latter to a real pair of ovals ; we find, with 

 comparatively little trouble in either case, aiising from the circum- 

 stance of the quadratic functions having imaginary roots in both, and 

 therefore dispensing with the necessity of proceeding any ftu'ther with 

 the process in either, that the equation, in addition to its evanescent, 

 has for each of them but a single real root, that, viz., corresponding to 

 the pair of inflexional chords of the in finite branches which are always 

 real. And, by a similar application, attended with a little more trouble 

 arising from the reality of the roots of the quadratic functions in each 

 case, we arrive at the same result for the two cases when 4h^ = 26k^ 

 and 28li? respectively, which correspond again, the former to an imagi- 

 nary and the latter to a real pair of ovals ; the intermediate case for 

 which 4A^ - 21]<? being that for which they pass through evanescence 

 from their real to their imaginary state, and conversely. 



In the particular case when 4/?^ = 21 B, for which the ovals contract 

 into points, the five finite roots of equation (14) are on the contrary 



