ON THE RECENT PROGRESS OF ANALYSIS. 275 



of transcendents in the manner already described ; the other to 

 the solution of the equations presented by the problem of the 

 division of elliptic integrals. The second of these applications 

 is contained in the memoir published in the second volume of 

 Crelle's Journal. 



He begins by introducing an inverse notation <> (u) corre- 

 sponding to the function denoted in the Fundamenta Nova 

 by sin am M, while f(u) and F(u) correspond respectively to 

 cos am u and A am u. This notation has the defect of appro- 

 priating three symbols which we cannot well spare. On the 

 other hand it is certainly more concise than M. Jacobi's. 



He then verifies the fundamental formulae for the addition of 

 the new functions, and goes on to show that they are doubly 

 periodic*. He next considers the expressions of <wa, &c. in 

 <>a, &c., and proceeds to prove the important proposition that 

 the equation of the problem of the division of elliptic integrals 

 of the first kind is always algebraically soluble. 



In order to illustrate this, which is one of the most remark- 

 able theorems in the whole subject, it may be observed, that as 

 any circular function of a multiple arc can be algebraically ex- 

 pressed in terms of circular functions of the simple arc, so may 

 (pna^fna, Fno. be algebraically expressed by means of ^a,^, Fa.. 



Conversely, as the determination (to take a particular 

 function) of sin a in terms of sin no. requires the solution of 

 an algebraical equation, so does that of <j>a in terms of <wa. The 

 equation which presents itself in the former case is, as we know, 

 of the nth or of the (2w)th degree as n is odd or even. But the 

 equation for determining <a rises to the (n 2 )ih degree in the 

 former case, and in the latter to the (2w*)th. We may however 

 confine ourselves to the case in which n is a prime number; 



* The formulae in question differ from those already given, only because Abel's 

 form of the elliptic integral is I . , which becomes the same as 



;V('-<**) ('+**) 



Legendre's on making e 2 = - i. The double periodicity of the functions is ex- 

 pressed by the formula 



00=0 {(-i 



with similar formulae for /and F. The quantities m and n are integral, and 

 f dx p 



W=2 I ? 57=2 I = 



182 



