THE INTEGRAL TABLES OF THE ELLIPTIC FUNCTIONS BY LEGENDRE AND ITS APPLICATION BY POISSONN (Mathematical aspects of nonlinear waves and their applications)

Abstract

We introduce Legendre's elliptic functions with his theory and construction of the Tables in origin of 1825-26 [1]. Legendre emphasizes his theory's superiority to Euler's integral theory, (as you know, this is entitled with'Eulerlian integral'on the total title, of theses huge volumes 1-3 of books), above all, the concept of echelon and the complete functions on which we discuss specially. Succedingly, we introduce Poisson's applications of Legendre's elliptic functions to the capillary model owing to the theory and table by Legendre, which Poisson gives up the self made theories based on the same elliptic functions including tables. Poisson discusses the heat problems, in which he also applies the elliptic functions and tables in the Earth's science, ln 1835. Both application may be the first orthodox applications of Legendre's theory to the nonlinear differential equations. Hence, all these topics are the translations from the original by me.