Q8 Proceedings of the Royal Irish Academy. 



double ordinates to the curve "which are real as regards both position and 

 magnitude, viz., those MN oi the ovals when real (see fig., page 63). 



In the particular case when 4/j^ - 21W, that is when the two ovals 

 contract into points, the three roots of equation (6) are, respectively, 



3? = -K', «^' = H (^^ - 1)^'^ and ;r- = - ^(V3 + \)li\ 



the first only of which gives a pair of chords real as regards both posi- 

 tion and magnitude of the curve. 



EKminating 'j[? between equations (3) and (12), we get, after a few 

 ordinary reductions, for the actual maxima and minima values of y", 

 the cubic equation 



42/S - 8Ay + 4A (A3 _ <^W) tf + ^ (4^3 _ 21 B) - 0, (13) 



which has always one real positive root corresponding to the real 

 negative root of equation (12), and belonging consequently to no 

 chords real even as regards position of the curve; and whose remain- 

 ing roots will be real and have opposite signs when 4/i^ - 21h^ is posi- 

 tive, that is, when the two ovals are real and finite. 



In the particular case when Ali^ = 27X-^, that is, when the ovals 

 contract into points, one root of equation (13) is evanescent, and cor- 

 responds to the evanescent ovals ; and the remaining two are respec- 



2 - 



tively = 7r (1 t- V3); which correspond respectively, the former to the 



o 



real negative root of equation (12) and therefore to no real chords even 

 as regards position of the curve, and the latter to the pair of chords 

 between the evanescent ovals and the infinite branches, which, though 

 real as regards their positions, have no real intersections with the 

 curve. 



Differentiating with respect to x the value of -j- given by equa- 



tion (11), having first substituted in it for y its value in x given by 

 equation (3), and equating the result to 0, we get, after a few ordi- 

 nary reductions, for the values of a?" for which -z-^ = 0, that is, for the 



dx~ 



several points of inflexion, real or imaginary, of the curve, the equa- 

 tion of the sixth degree, 



x~ [h-x'"' - UhTc's? + U^ {2¥ + 5k^) x^ - ISh'^k'x^ 



+ 18MV--6P] = 0, (14) 



of whose roots one is obviously evanescent, one essentially real and 

 positive, and the remaining four when real essentially positive also. 



The evanescent root corresponds of course to the axis of y, which 

 is consequently a doubly inflexional as well as asymptotically tangen- 



