ON THE RECENT PROGRESS OF ANALYSIS. 289 



Trans, for 1831. His design is to give in a simple form 

 M. Jacobi's theorem for transformation. The demonstration 

 is essentially the same as that in the Fundamenta Nova. 



But Mr Ivory does not set out with assuming y = -=^ , U and 



F being integral functions of x, but with assuming it equal to 

 the continued product of a number of elliptic functions (whose 

 arguments are in arithmetical progression), multiplied by a con- 

 stant factor. This is one of M. Jacobi's transcendental expres- 

 sions for y, and the two assumptions are therefore perfectly 

 equivalent in the transformations of odd orders ; but in those of 

 even orders, or where the continued product consists of an even 

 number of factors, Mr Ivory's amounts to making y equal to an 

 irrational function of x. Transformations by irrational substitu- 

 tions, though long the only kind known (since Lagrange's be- 

 longs to this class), had not of late been considered in detail. 

 Abel indeed remarked in the beginning of the general investi- 

 gation contained in Schumacher's Journal (No. 138), that the 

 existence of an irrational transformation implied that of a rational 

 one leading to an integral with the same modulus as the other. 

 He was, therefore, in seeking for the most general modular 

 transformation, exempted from considering irrational substitu- 

 tions ; but in a historical point of view it is interesting to see 

 the connection between Lagrange's transformation and those 

 which have been more recently discovered*. 



If y = (i+&)a A /_LL^_, where 6 a +c*=i, 

 V i - c 2 ^ 3 



then 

 dx 



This is Lagrange's direct transformation. The corresponding rational transforma- 

 tion is 



which satisfies the same differential equation as before. 

 . dy i+c dx 



where 



is satisfied by if = i + ex 9 - V(i - x 2 ) (i - c 2 x 2 ), 



19 



