ON THE RECENT PROGRESS OF ANALYSIS. 285 



The commensurability of the transcendental functions - 1 , - 



is therefore a necessary condition, in order that an integral with 

 modulus c can be transformed into one with modulus 7c, the 

 regulator a being real and c indeterminate. And it may be 

 shown that this condition is not only necessary but sufficient. 

 Similar considerations apply to the case in which a is impossible. 

 Simple as this view is, it leads to many consequences of great 

 interest. The function ^, of which we have already spoken 



cs 



(p. 270), is merely- 6 w , an& as we know for the first real trans- 



nty 



formation of the nth order, it becomes e~"~^~ . Hence in this 

 case we have ( j =n( j according to the general law. It 



may be well to remark, that if k = c we have a m~=m (an 



integer). Hence in multiplying an integral, the multiplier must 

 be an integer, if y is rational in #, except for particular values 

 of c. 



In the paper of which we are speaking Abel has applied 

 precisely similar considerations to the case in which x and y 

 are connected by any algebraical equation. 



Passing over one or two shorter papers, one of which has 

 been already referred to at p. 276, we come to a Pr&cis of the 

 theory of elliptic functions, published in the fourth volume of 

 Crelle's Journal, p. 236. The work of which it was designed to 

 be an extract was never written, and the Precis itself is left 

 unfinished. A general summary was prefixed to it, from which 

 we learn that the work was to be divided into two parts. In the 

 first elliptic integrals are considered irrespectively of the limits 

 of integration, and their moduli may have any values, real or 

 imaginary. Abel proposes the general problem of determining 

 all the cases in which a linear relation may exist among elliptic 

 integrals and logarithmic and algebraical functions in virtue of 

 algebraical relations existing among the variables*. 



His first step is to apply his general method for the com- 

 parison of transcendents to elliptic integrals, which may be 



* In the assumed relation, the amplitude, or rather the sine of the amplitude 

 of each elliptic integal, is to be one of the variables, and not a function of one 

 or more of them. 



