183 



(n—iy'\ n"', and («+ 1)"' curves of the positive series respec- 

 tively, and V their common vertex, vrhich is also that of the 

 hyperbola, then will 



arc vp„_i + right hue p„_i p„ = -— arc vp„+i. 



Tin -f- i 



Mr. Roberts states that he has demonstrated the property 

 in a manner purely geometrical. 



This equation shows that the arcs of all the curves of an 

 odd order will depend only on that of Bernouilli's lemniscate, 

 or the function f { v^^, ^ \ , and those of an even order only 

 on the arc of the second of the series. This latter arc is 

 three times the difference between the corresponding hyper- 

 bolic arc and the portion of the tangent applied at its extre- 

 mity, which is intercepted between the point of contact and 

 the perpendicular dropped upon it from the centre : and the 

 entire quadrant is three times the difference between the 

 infinite hyperbolic arc and its asymptot. 



Also, s„, Sn^.], denoting the quadrants of the «*, and 

 (w+ 1)*'' curves, the following very remarkable relation exists 

 between them, 



s„ s„+i=: (2w4- 1)|. 



The curves of the negative series enjoy analogous pro- 

 perties. 



Lastly, let the base curve be a circle, the origin being 

 within it: and it appears that the rectification of the curves 

 of both series, which are of an even order, can be effected 

 by the arcs of circles; and that those of an odd order, which 

 belong to the positive series, will involve elliptic integrals of 

 the first and second kinds in their arcs. The negative curves 

 of an odd order contain a term depending on a function of 

 the third kind, which is however reducible to a function of 

 the first kind and a logarithm. 



By the particular consideration of the first negative curve 

 in this case, Mr. Roberts was led to a very simple demon- 



