400 Professor Car slaw, The cooling of a solid sphere 



Though the discovery of radioactivity has definitely closed the 

 controversy as to the rehability or otherwise of the results obtained 

 by Kelvin's method, or similar methods, the mathematical problem 

 treated in this paper seems of sufficient interest to justify the 

 pubhcation of a solution on lines which I have found useful in 

 dealing with other questions of .the conduction of heat. 



2. It is more convenient to start with the case of a sphere 

 whose surface is kept at a constant temperature, the initial tem- 

 perature of the whole being zero. 



Let the sphere be of radius 6,. the core being of radius a. The 

 surface r = 6 is kept at a constant temperature Vq. 



Let the temperature, conductivity, specific heat and density of 

 the core from r = to r = a be -yi, K^, c^ and p^ : and the corre- 

 sponding quantities from r ^ a to r = b he v.^, K^, c^ and p^. 



Also let /Ci = KJcip^ and k^ = K^jc^p^. 



Then if we write u^ = v^r and u^ = v^r, we have the equations : 



W^"'^' 0<r<a...(l), _2=«,— /, a<r<b...il), 



^2 = (^2 sin fjia {r — a) + B^ sin /xa (6 — r)) e'"'"-'* 

 satisfy (1) and (!'), when p, = V{'<i/'<2)- 



They also satisfy (4) and (5), provided that 

 Ai sin aa = B2 sin /xa (b — a), 

 Kj^Ai [aa cos aa — sin aa] = K^ {piaa {A.^ — B^ cos p^a (6 — a)) 



— B2 sin p,a (b — a)]. 



Thus we have 



crcosaasiii/Aa(&-a) + sinaacosMa(6-a) -i sin aa sin «,a(& -a) 



sin pa [b- a) 

 where K^ = p^aK^. 



Tj _ sm aa 



sm pua (0 — a) ^ 



