290 ON THE REGENT PROGRESS OF ANALYSIS. 



The question presents itself, what is the connection between 

 the irrational transformation (that of which Lagrange's is a 

 particular case) and the rational transformation of even orders ? 

 Perhaps the simplest answer to it (though every question of the 

 kind is included in the general investigations contained in Abel's 

 Precis) is found in a paper by M. Sanio in the fourteenth 

 volume of Crelle's Journal, p. 1. The aim of this paper is to 

 develope more fully than Mr Ivory has done the theory of trans- 

 formations of even orders, and particularly of the irrational 

 transformations, which M. Sanio considers more truly analogous 

 to the rational transformations of odd orders than the rational 

 transformations of even orders ; and also to discuss the multipli- 

 cation of elliptic integrals by even numbers, a subject intimately 

 connected with the other. We have already mentioned the ex- 

 istence of what are called complementary transformations, each 

 of which may be derived from the other by an irrational sub- 

 stitution, by which two new variables are introduced. In the 

 case of transformations of odd orders, the original transformation 

 and the complementary one are both rational, and are both in- 

 cluded in the general formula given by M. Jacobi's theorem ; 

 but to the rational transformation of any even order corresponds 

 as its complement the irrational transformation of the same order. 

 This remark, which, as far as I am aware, had not before been 

 made, sets the subject in a clear light*. 



which may be called Lagrange's inverse transformation, Tc being now the same func- 

 tion of c, that c was before of Jc. The corresponding rational transformation is 



_(!+)* 



y i+cx*> 



which is M. Gauss's, and is termed in M. Jacobi's nomenclature the rational trans- 

 formation of the second order. It satisfies the equation 



dx 



* Lagrange's transformation being 



where, as before, 1c= 



J + c 



then we find that tf 



while the differential equation becomes 



dy r 



" 6) 



where 



