316 Lord Rayleigh on Conduction of Heat in a Spherical 



In order to find the normal modes into which the most 

 general subsidence may be analysed, we are to assume that s 

 and 6 are functions of the time solely through the factor e~ w . 

 Since p is uniform, s-\-a0 must by (1) be of the form H e~ h \ 

 where H is some constant ; so that if for brevity the factor 

 e~ M be dropped, 



s + «<9=H; ....... (6) 



while from (5) 



vS7 2 0+{h-q)0 = hi3s (7) 



Eliminating s between (5) and (7), we get 



V s + m 2 (0-C)=.O, ..... (8) 

 where 



hy — q n 7</3H 





, ' 0-J^. • • • • 0) 



These equations are applicable in the general case, but when 

 radiation and conduction are both operative the equation by 

 which m is determined becomes rather complicated. If there 

 be no conduction, v = 0. The solution is then very simple, 

 and may be worth a moment's attention. 

 Equations (6) and (7) give 



0= kpR , = (H)H, . . . (10) 



hy — q hy — q v ' 



Now the mean value of s throughout the mass, which does 

 not change with the time, must be zero ; so that from (10) we 

 obtain the alternatives 



(i.) h = q, (ii.) H = 0. 



Corresponding to (i.) we have with restoration of the time- 

 factor 



0=(H/a>-««, s=0. .... (11) 



In this solution the temperature is uniform and the condensa- 

 tion zero throughout the mass. By means of it any initial 

 mean temperature may be provided for, so that in the 

 remaining solutions the mean temperature may be considered 

 to be zero. 



In the second alternative H = 0, so that s=—a6. Using 

 this in (7) with v evanescent, we get 



(hy-q)d=0 (12) 



The second solution is accordingly 



0=<j>{x,y,z)e-«% s=-x<l>fay,z)e-Mr, . (13) 

 where <£ denotes a function arbitrary throughout the mass, 

 except for the restriction that its mean value must be zero. 



