ON THE RECENT PROGRESS OF ANALYSIS. 253 



researches relates to the transcendents which are commonly 

 called elliptic functions or elliptic integrals. For a reason 

 which will be mentioned hereafter the latter name seems pre- 

 ferable, and it is sanctioned by the authority of M. Jacobi, 

 though the former was used by Legendre. Elliptic integrals 

 then may be defined as those whose differentials are irrational 

 in consequence of involving a radical of the form 



ex* 



But it may perhaps be more correct to say that all such integrals 

 may be reduced to three standard integrals, to which the name 

 of elliptic integrals has been given. 



In the Turin Memoirs for 1784 and 1785, p. 218, Lagrange 

 considered, as has been already mentioned, the theory of these 

 transcendents. He showed that the integration of every func- 



tion irrational in consequence of containing a square root may 



p 

 be made to depend on that of a function of the form -^ , P being 



rational, and R the radical in question ; and that if under the 

 sign of the square root x does not rise above the fourth degree, 

 it may ultimately be made to depend on that of 



Ndx 



where Nis rational in x\ He thus laid the foundation of that 

 part of the theory of elliptic transcendents in which a proposed 

 integral is reduced to certain canonical or standard forms, or 

 to the simplest combination of such forms of which the case 

 admits. In Legendre's earliest writings on elliptic functions 

 there is nothing relating to this part of the subject. Having 

 thus, in the simple manner which distinguishes his analysis, 

 reduced the general case to that which admits of the application 

 of his method, Lagrange proceeded to prove that if we intro- 

 duce a new variable whose ratio to x is the subduplicate of the 

 ratio of 1 p*x* to 1 jV, the last written integral is made to 

 depend on another of similar form, but in which p and q are 

 replaced by new quantities p 1 and q l . If p is greater than q, 

 p 1 will be greater than p, and q l less than q, and thus by 

 successive similar transformations we ultimately come to an 

 integral in which q is so small that the factor 1 + q l x* may be 



