140 Royal Irish Academy, 



Let P„_i, P;,; P n+ i be corresponding points on the (ra — l)th, nth, 

 and (w + l)th curves of the positive series respectively, and V their 

 common vertex, which is also that of the hyperbola, then will 



arc VP„_! + right line P w _! P„ m l!Lzl arc V P»j.i, 



2«+l 



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 Bernoulli's lemniscate, or the function F 



{ *2> P}> an d 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 hyperbolic arc and the portion of the tangent 

 applied at its extremity, 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 w , S n +i, denoting the quadrants of the rath, and (ra+l)th 

 curves, the following very remarkable relation exists between them : 



S„ Src+i = (2 ra+1)— . 



The curves of the negative series enjoy analogous properties. 



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 in- 

 volve 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 demonstration of the 

 equation which results from the application of Lagrange's celebrated 

 scale of reduction to elliptic functions of the second kind, and which 

 is nothing more than the analytical expression of Landen's theorem. 



William Roberts, Esq., F.T.C.D., read a paper on a class of sphe- 

 rical curves, the arcs of which represent the three species of elliptic 

 transcendents. 



A cone of the second order, whose vertex is upon the surface of 

 a sphere, and one of whose principal axes is a diameter, will inter- 

 sect the sphere along a curve which admits of several varieties, ac- 

 cording to the nature of the sections of the cone parallel to its prin- 

 cipal planes, and the position of its internal axis. This curve may 

 be made to furnish, by means of its arc, a geometrical representation 

 of the three species of elliptic trancendents, including the two cases 

 of the third. 



In the course of the investigations alluded to, Mr. Roberts was 

 also led to consider two species of the curve called the spherical 

 conic, which appear to possess many remarkable analogies to the 



