## EMS: Ince lecture

Dr E L Ince lectured on

*Lamé Functions*. He gave first the derivation of Lamé's differential equation in its algebraic and Jacobian forms by the consideration of Laplace's equation in confocal coordinates. Considering the jacobian form, he then discussed in order

(i) the solutions which arise in degenerate cases

(ii) solutions which are polynomials in the Jacobian elliptic functions

(iii) Lamé functions in general.

He showed that the most appropriate notation for the Lamé polynomials is based on (a) their degree, and (b) the number of their zeros in a multiple period, since this classification (unlike that of Laplace) is also appropriate for Lamé functions in general. Solutions in the form of power series were discussed: sometimes very slow convergence of these series had troubled the lecturer in his work of numerical tabulation of Lamé functions; but he had very recently found out that in such cases a vast improvement was obtained by using Fourier series solutions of Lamé's equation in a trigonometric form. Finally the lecturer showed that those particular cases of Lamé's equation for which two periodic solutions coexist could be detected as points of intersection of the graphs of the associated characteristic numbers.
(ii) solutions which are polynomials in the Jacobian elliptic functions

(iii) Lamé functions in general.

This lucid exposition, based largely on Dr Ince's own discoveries, was very warmly received, and was followed by a lively discussion in which these took part Prof E T Whittaker, Dr A Erdélyi, and others.

JOC/EFR May 2016

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Extras/ems_Ince.html