V. On the conduction of heat in ellipsoids of revolution

Abstract

The object of the present paper is to investigate the expressions which present them selves in the solution of the problem of the conduction of heat in an ellipsoid of revolution. For although the question of the stationary temperature of ellipsoids in general has been completely solved by means of the functions introduced by Green and Lamé, the corresponding problem of conduction has not been so successfully dealt with. M. Mathieu, indeed, in his ‘Cours de Physique Mathematique,’ has shown how to reduce the solution to ordinary differential equations, and for the special case of an ellipsoid of revolution has shown how to approximate to their solutions. His method, which is novel and remarkable, enables him to calculate a few terms of the expressions, but does not afford a view of their general constitution and properties. In the present paper the question is treated in a more direct and general manner. Choosing with M. Mathieu, as coordinates of a point, the azimuth ϕ of the meridional section through it and the parameters α and β of the ellipsoid and hyper-boloid confocal to the surface which intersect in the point, it is first shown how to transform the general equation of conduction to these coordinates. This equation is then satisfied by a series of terms of the form e ¯λ 2 ft cos mϕ ϑ m k ( β ) Ω m k ( α ), in which k is determined by an equation whose roots are infinite in number.