<98 



Prof. C. Niven on the 



[May 



May 29, 1879. 



THE PRESIDENT in the Chair. 



The Presents received were laid on the table, and thanks ordered for 

 them. 



The following Papers were read : — 



X ki On the Conduction of Heat in Ellipsoids of Revolution." By 

 C. NlVEN, M.A., Professor of Mathematics in Queen's 

 College, Cork. Communicated by J. W. L. Glaisher. 

 F.R.S. Received May 7, 1879. 



(Abstract.) 



The object of the present paper is to investigate the expressions 

 which present themselves in the mathematical treatment of the problem 

 of the conduction of heat in an ellipsoid of revolution. The results 

 obtained constitute a generalisation of the corresponding solution for 

 the sphere, and are found, in the first instance, for an ellipsoid whose 

 major axis is the axis of revolution, but a slight alteration, will render 

 them applicable also to a planetary ellipsoid. We may effect the 

 transformation to ellipsoidal co-ordinates of the general equation of 

 conduction as follows : the semi-axes of the ellipsoid and hyperboloid 

 through any point conf ocal to the surface being denoted by c cosha, 

 c sinha, and c cos /3, c sin /3, the co-ordinates of the point may be expressed 

 by c cos sinha sin /3, c sin sinha sin ft, c cosha cos ft, and the general 

 equation of conduction becomes 



+ + coth* ■ — +cot/3 — -f( — — + — | — r 



da? d/P dec H dp Vsin 2 /3 sinh 2 « J d^ 



= — (cosh 2 * — cos 2 fi) 



h at 



V being the temperature at any point. 

 We may satisfy this equation by 



V=(cosm0 or sinm0)e~ (/3)Q^(a), 



where 



d?0 . . „d& m 2 .co o „ „ „ 



