70 Proceedings of the Royal Irish Academy. 



all real, and correspond in consequence to five pairs of chords all real 

 as regards their positions ; one of which intersects, as in all cases, the 

 conchoidal branches, and, of the remaining four, three coincide at the 

 evanescent ovals, and the fourth intersects the curve at imaginary- 

 points. 



For, equation (14) for x^ is easily seen to be equivalent in that 

 case to 



A^ x" ix" - ^ h^ix' - ^ li'x'- + V^ h^\\ = 0, (15) 



4 



which, besides its single root = 0, has evidently three roots each - h^, 



and two others equal respectively to 



9 V V3, 



which correspond respectively, the first three to the evanescent ovals, 

 the latter with the lower sign to the conchoidal branches, and the 

 latter with the upper sign to no real points at all on the curve. 



Of the three pairs of chords coinciding at the evanescent ovals in 

 this case, two however correspond to the evanescent radii of curvature 



at their vertices ; the function - y^ — , which multiplied by a;^° is mani- 

 festly equivalent to the quantity within the brackets at the left side of 

 equation (14), representing in all cases, as is well known, at the several 

 vertices of any curve symmetrical with respect to the axis of x, the 

 squares of the corresponding radii of curvature at the vertices, and 

 being consequently evanescent at the two coincident vertices of every 

 acnodal double point on the axis of the curve. 



As regards the third pair in the same case. Taking it in con- 

 nexion with the pair intersecting the curve at imaginary points, and 

 conceiving both pairs to change position together with the gradual 

 and continued increase of the parametric ratio from its critical to every 

 higher value, and the consequent accompanying dilatation of the ovals 

 from their evanescent to every greater magnitude ; they are to be re- 

 garded, while real, as two variable pairs of inflexional chords inter- 

 secting the expanding ovals at pairs of imaginary points, and after 

 coming together in the course of their variation, as the above particu- 

 lar cases show they do very rapidly with the increase of the ratio, then 

 passing through coincidence from their real to their imaginary state, 

 beyond which the particulars above stated supply no clue to follow 

 them. 



N.B. — The questions, as to whether for any values of the parame- 

 tric ratio the ovals have ever real points of inflexion, and as to the 

 critical values (if any) of the ratio for which they cease (if ever) to 

 be, as represented in the figure, concave to their interiors throughout 

 the entire circuits of their perimeters, have, it will be observed, been 

 left undecided in the above investigation. 



