Royal Irish Academy. 1 39 



which has been given by Mr. Talbot, in a letter addressed to 3VI. 

 Gergonne, and inserted in the Annates des MatMmatiques, torn. xiv. 

 p. 380. 



A function of the third kind, with a circular parameter — 1 + £ + , 

 where b is the semiaxis minor of the ellipse, its semiaxis major 

 being unity, and the modulus of which is the eccentricity, enters 

 into the arcs of all the positive curves ; and their general rectification 

 depends only on that of the ellipse, and of the first derived, both 

 positive and negative. 



The quadrants of the ellipse, and of the first two curves, positive 



and negative, are connected by the following relation : — 



(S^ + SJS.., = (3S-S_ 2 )(2S-S 2 ). 



*/~5 I 



It is worthy of notice, that if the eccentricity be 1 the 



functions of the third kind disappear, and the rectification of both 

 series depends only on that of the ellipse and of the first negative 

 curve. 



If the base curve be a hyperbola, whose centre is the origin, the 

 arcs of all the curves of the negative series will depend only on el- 

 liptic functions of the first and second kinds. But the general ex- 

 pression for the arc in the positive series contains a function of the 

 third kind, the parameter of which is alternately circular and loga- 

 rithmic ; the curves of an odd order involving the same function 

 of the circular kind, and those of an even order the same of the lo- 

 garithmic kind, if the real axis of the base-hyperbola be greater than 

 the imaginary, and vice versd. 



Mr. Roberts also shows, that besides the case of the equilateral 

 hyperbola, in which the first positive curve is the lemniscate of 

 Bernoulli, and which has been the only one hitherto noticed, at least 

 as far as he is aware, there are two others, in which the arc of the 

 first positive curve can be expressed by a function of the first kind, 

 with the addition of a circular arc in one case, and of a logarithm 

 in the other. The first of these occurs when the imaginary semiaxis 



is equal to — (the distance between The centre and focus being 



unity), and this fraction is the modulus of the function. The other 

 case is furnished by the conjugate hyperbola, and the modulus is 

 complementary. In both these cases functions of the third kind dis- 

 appear from the arcs of the positive curves. 



If the hyperbola be equilateral, and its semiaxis be supposed equal 

 to unity, the general equation of the derived curves of both series 

 may be presented under the form 

 2 



+ 2n-l 



r +M = cos 



\ + 2« — 1/ 



The successive curves represented by this equation are very curiously 

 related to each other. The following property appears worthy of 

 remark : — 



