ON THE RECENT PROGRESS OF ANALYSIS. 281 



limite, and are therefore analogous to the expression of sin $ in 

 terms of < which we have already mentioned. 



The second contains the development of what had already 

 been pointed out with respect to the lemniscate, so far as relates 

 to the division of its perimeter by any prime number of the form 

 4m + 1. In an interesting note which M. Liouville communicated 

 to the Institute in 1844, and which is published in the eighth 

 volume of his Journal, p. 507, he has proved generally that 

 the division of the perimeter of this curve can always be 

 effected whether the divisor be a composite or prime number, 

 real or complex (that is, 01 the form p + V q, p and q being 

 integers). In order to do this, it was only requisite to follow 

 m . m . , the reasoning by which Abel has shown that the equa- 

 tion which presents itself in the problem of the division of 

 the circumference of the circle is always resoluble. Thus, as 

 M. Liouville has remarked, his analysis is implicitly contained 

 in Abel's. 



This memoir also contains Abel's theorem for the transfor- 

 mation of elliptic integrals of the first kind. It is equivalent 

 to that of M. Jacobi ; nor is the demonstration, though presented 

 in quite a different form, altogether unlike M. Jacobi's. 



Abel begins by considering the sum of a certain series of <f> 

 functions whose arguments are in arithmetical progression. He 

 shows that the sum of this series is a rational function of its first 

 term. If we call this sum (multiplied by a certain constant) y, 

 and the first term #, then y is such a function of x as to satisfy 

 the differential equation already mentioned, viz. 



d dx A 



or rather an equation of equivalent form. In fact y is m . m . the 

 same function of x that it is in M. Jacobi's theorem. Thus the 

 sum of the series of elliptic functions is itself, when multiplied 

 by a constant, a new elliptic function, having a new modulus, 

 and whose argument bears a constant ratio to that of the first 

 term of the series. It appears also that for the sum of the 

 elliptic functions we may, duly altering the constant factor, sub- 

 stitute their continued product. Thus, beside the algebraical 

 expression of y, there are two transcendental expressions of it, 

 both of which are given by M. Jacobi in the Fundamenta 



