ON THE REGENT PROGRESS OF ANALYSIS. 273 



The rational transformation in question leads to a continually 

 increasing series of moduli, or is, to use an expression of M. 

 Jacobi, a transformation ' minoris in majorem.' The law con- 

 necting two consecutive moduli is the same as in Lagrange's, 

 which is, as we have seen, an irrational transformation ; so 

 that M. Gauss's method does not afford us a new scale of moduli. 

 Nevertheless, as no rational transformation had I believe been 

 noticed when his tract appeared*, his method is, in a historical 

 point of view, of considerable interest. 



18. In the second volume of Crelle's Journal, p. 101, we 

 find Abel's first memoir on elliptic functions. It was published 

 in the spring of 1827, and therefore before M. Jacobi' s announce- 

 ment in No. 123 of Schumacher's Journal. But it contains 

 nothing which interferes with M. Jacobi's discovery of the 

 general theory of transformation. Abel's researches on this part 

 of the subject appeared in the third volume of Crelle's Journal, 

 p. 160. This second communication is dated, as we are in- 

 formed by an editorial note, the 12th of February, 1828, and 

 though it is announced as a continuation of the former memoir, 

 it is yet in effect distinct from it, as its contents are not 

 mentioned in the general summary prefixed to the first com- 

 munication. 



These details may not be without interest, though it is not 

 often that questions of priority deserve the importance sometimes 

 given to them. There is no doubt that Abel's researches were 

 wholly independent of those of M. Jacobi ; and though the co- 

 incidence of some of their results is therefore interesting, yet 

 the general view which they respectively took of the theory of 

 elliptic functions is essentially different, as different as the style 

 and manner of their writings. 



With M. Jacobi the problem of transformation occupied the 

 first place ; with Abel that of the division of elliptic integrals. 

 Both introduced a notation inverse to that which had previously 

 been used, and as an immediate consequence recognised the double 

 periodicity of elliptic functions. Expressions of these functions 

 in continued products and series were given by both, but those 

 of Abel were deduced by considering the limiting case of the 



* The fundamental formula of his transformation is incidentally mentioned in 

 Legendre's second work (TraiM des Fonct. i. 61). 



18 



