ON THE RECENT PROGRESS OF ANALYSIS. 257 



this method originated, as we have seen, with Lagrange. It is 

 developed at great length by Legendre, with a special reference 

 to the modifications required in applying it to the different 

 species of integrals. As Lagrange had shown, the series of 

 transformed integrals extending indefinitely both ways conducts 

 us, in whichever direction we follow it, towards a transcendent 

 of a lower kind than an elliptic integral, or in other words, 

 towards a logarithmic or circular integral. There are thus tw r o 

 modes of approximation, one of which depends on a series of 

 integrals with increasing moduli, and the other on a series 

 whose moduli decrease. Thus for the three species of integrals 

 there will be in all six approximative processes to be considered. 

 In the case of the elliptic integral of the third kind, we have 

 to determine the law of formation of the successive parameters 

 , n l , &c. 



. Keductions of transcendents not contained in the general 



formula (e.g. \ } to elliptic integrals. 



77. Lastly, applications to various mechanical and geometri- 

 cal problems. 



This analysis, however slight, will give an idea of the 

 contents of that part of the Exercices de Calcul Integral which 

 relates to elliptic functions. In the third volume there are 

 tables for facilitating the calculation of integrals of the first and 

 second kind : they are accompanied with an explanation of the 

 manner in which they were constructed. The ninth table is 

 one with double entry, the two arguments being the angle of 

 the modulus and the amplitude. 



13. In 1825 Legendre presented to the Academic des 

 Sciences the first volume of his Traite des Fonctions Ellip- 

 tiques. A great part of this work is precisely the same as 

 the Exercices de Calcul Integral. By far the most important 

 addition to the theory of elliptic functions consists in the dis- 

 covery of a new system of successive transformations quite dis- 

 tinct from that of Lagrange. 



In the earlier work Legendre had shown that a certain 

 transcendent might be expressed in two ways by means of 

 elliptic integrals of the first kind. Comparing the two results, 

 he obtained a very simple relation between the two elliptic 



17 



