ON THE RECENT PROGRESS OF ANALYSIS. 287 



in which the moduli are real and less than unity ; of this how- 

 ever only the summary exists. Abel proposed to introduce three 

 new function's, the first corresponding to that which he had 

 previously designated by <f)9*. He now denotes it by \0. The 

 second and third functions are apparently what the second and 

 third kind of elliptic integrals respectively become, when, in- 

 stead of x, we introduce the new variable x and being 

 of course connected by the equation x = \0. The double period- 

 icity of the function X and its other fundamental properties 

 having been established, it was his intention to proceed to more 

 profound researches. Some of his principal results are briefly 

 stated. I may mention one, that all the roots of the modular 

 equation may be expressed rationally in terms of two of them f. 



One of the last paragraphs of the summary relates to func- 

 tions very nearly identical with those which M. Jacobi discusses at 

 the close of the Fundamenta Nova, and which he has desig- 

 nated by the symbols H and . 



The second volume of Abel's collected works consists of 

 papers not published during his life. Two or three of these 

 relate to elliptic functions. The longest contains a new and 



very general investigation for the reduction of the general trans- 



p 



cendent, whose differential is of the form -j= , P being, as usual, 



V R 



rational and E a polynomial of the fourth degree ; together with 

 transformations with respect to the parameter of integrals of the 

 third kind. 



20. Having now given some account of the revolution 

 which the discoveries of Abel and Jacobi produced in the 

 theory of elliptic functions, I shall mention some of the prin- 

 cipal contributions which have been made towards the further 

 development of the subject since the publication of the Funda- 



* In the Precis Abel has adopted the canonical form of the integral of the 

 first kind made use of by Legendre and M. Jacobi ; so that the quantity under 

 the radical is (i-se 2 ) (r-c 2 * 2 ). It is worth remarking, that in his first paper 

 in Schumacher's Nctchricliten this quantity is (i e 2 e 2 ) (i c 2 ce 2 ), while in the 

 second it is the same as in the Precis. To this form he appears latterly to 

 have adhered. 



t It is not clear whether by roots of the modular equation we are to under- 

 stand the transformed moduli themselves, or their fourth roots, i. e. in M. Jacobi's 

 notation X or v. Vide supra, p. 263. 



