282 ON THE RECENT PROGRESS OF ANALYSIS. 



Nova. At the close of the memoir Abel compares his result 

 with the one in Schumacher's Journal, No. 123, and mentions 

 that he had not met with the latter until his own paper was 

 terminated. 



19. In the 138th number of this journal, Abel resumed the 

 problem of transformation, and treated it in a more general and 

 direct manner than had yet been done. This memoir appeared 

 in June 1828. M. Jacobi, in a letter to Legendre, has spoken in 

 the highest terms of Abel's demonstration of the formulae of 

 transformation : he says, " Elle est au-dessus de mes eloges, 

 comme elle est au-dessus de mes travaux." An addition to this 

 memoir, establishing the real transformations by an independent 

 method, appeared in Number 148 of the same journal. These 

 two papers are printed consecutively in the first volume of 

 Abel's Works, pp. 253, 275. 



In the first of these two remarkable essays, Abel makes use 

 of the periodicity of the function $0, or, as he here denotes it, 

 X#, to determine h priori what rational function of x, y must be 

 in order that the differential equation 



= . g 



may be satisfied. [I have altered his notation for the sake of 

 uniformity.] Let ^x be the function sought, then considering 

 y = tyx as an equation determining x in terms of y, he shows 

 that certain relations necessarily exist among its roots. Let \Q 

 be one of them and \& another, it will readily be seen that we 

 may put 



dO' = d0, 

 since each is equal to 



Hence & = 6 + a, 



a being the constant of integration, or, which is the same thing, 

 being independent of y. Hence \0 being one root, every other 

 root is necessarily of the form \(0 + a). Again, we see from 

 hence that 



which is to be true for all values of 6, and which therefore 



