The Roots of the Equation u = tan u

Abstract

IN many treatises on optics it is stated that the roots of the equation, u = tan u, were calculated by Schwerd. Verdet (“Oeuvres,” t. v. p. 266), says:—“These roots have been calculated by Schwerd, who arrived at the following values: u₁/=1.4303, u₂/π=2·4590,…” up to u₇/π. Preston (“Light,” p. 255, second edition) says:—“The values of u corresponding to the maximum values of the illumination have been given by Schwerd as follows:—“The values given are precisely the same as Verdet's. Rayleigh (“Encyc Brit.,” vol. xxiv. p. 430, art., Wave Theory) gives a method for calculating the roots of the equation, and remarks that they were obtained in another manner by Schwerd. (There is a misprint in Rayleigh's value for u₄). Other references might be added.