Open AccessThe third elliptic integral and the ellipsotomic problem.
Author(s) -
A. G. Greenhill
Publication year - 1904
Publication title -
proceedings of the royal society of london
Language(s) - English
Resource type - Journals
eISSN - 2053-9126
pISSN - 0370-1662
DOI - 10.1098/rspl.1904.0001
Subject(s) - elliptic function , elliptic integral , jacobian matrix and determinant , mathematics , expression (computer science) , jacobi elliptic functions , mathematical analysis , series (stratigraphy) , pure mathematics , computer science , paleontology , biology , programming language
The elliptic integral of the third kind, which makes its appearance in a dynamical problem, is of the circular form in Legendre’s classification, and thus the Jacobian parameter is a fraction of the imaginary period, so that the expression by-means of the theta function can no longer be considered as reducing the variable elements from three to two. Burkhardt has given a series rapidly convergent for the numerical calculation of such cases; but the object of this memoir is to develop the exact expression by means of an idea of Abel, given in the first volume of ‘Crelle’s Journal,’ 1826, “Ueber die Integration der Differential-Formel (1)ρdx /√R, wenn R andρ ganze Functionen sind."