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V. On the Conduction of Heat in Ellipsoids of Revolution. 
By C. Niven, M.A., Professor of Mathematics in Queens College, Cork. 
Communicated by J. W. L. Glaisher, F.R.S. 
Received May 7—Read May 29, 1879. 
The object of the present paper is to investigate the expressions which present 
themselves 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 Lame, the corresponding problem of conduction has not been so successfully dealt 
with. M. Mathieu, indeed, in his ‘ Caurs 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 <^> of the 
meridional section through it and the parameters a and fi 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~ K "' il cos m<£ 3j(/3)n,f(a), in which k is 
determined by an equation whose roots are infinite in number. 
The function 3 is expanded in what Mr. Todhunter, translating Heine’s term, 
calls associated functions of cos /3, and we shall also follow Heine in denoting by 
m 
Pthe expression (/P—1 )2(/W w — &c.). The language of harmonic analysis has 
been greatly benefited by Professor Maxwell’s introduction of the words type and 
degree into the specification of a tesseral harmonic, though we prefer to replace the 
term type by order. We shall therefore call the product Pp(/r) cos w</> the tesseral 
harmonic of the m th order and n ih degree, and the factor P „i‘(f) the associated function 
of the m th order and n th degree. The expansion of the function 3 will then consist of a 
series of associated functions of the m th order. 
It is shown that the roots of the equation in k fall into two classes, and that the 
corresponding expressions for 3 take different forms, for one of which the difference 
between the degree and order of the associated functions involved is an even number 
