136 Mr. A. R. McLeod on the Lags of 



satisfied is : 



~du 2 /B 2 w , 2 "bit 



-dt 



9 /d m . z d«\ /,x 



the temperature being supposed to depend only on r. 

 The value of a 2 is K//)cr, where K is the thermal conduc- 

 tivity, p the density, and a the specific heat of the substance 

 in the sphere. 



Writing v=ru, we get : 



at " a a-- 2 



"« 2 i- 2 (2) 



Since z/=£0 when r = 0, we must have v = for r = 0. 

 To get the temperature as a function of r and £, we require 

 a solution of (2) which satisfies the conditions 



v = for t = 0, 

 r = for r= 0, 



r = 0(£J for r— c = radius of sphere. 



Such a solution is (Weber, Partielle Differential- Gleich- 

 ungen, vol. ii. § 48) : 



^ (~iy n sin — J ^-' 1 W-)</)(t)^ 



where 



A 2 _ 7T 2 K 

 C 2 ^^O" 



Suppose 4>(t) = Gct, where G is a constant. Then, on 

 tegration, 



The mean temperature is given by 



" n=l l J 



Since % l/n 2 = 7r 2 /6 and 2 l/n 4 = 7r 4 /90, this becomes 



