390 Lord Rayleigh : Problems in 



So far we have been considering instantaneous sources. 

 As in II., the effect of constant sources may be deduced by 

 integration, although the result is often more readily obtained 

 otherwise. A comparison will, however, give the value of a 

 definite integral. Let us apply this process to (33) repre- 

 senting the effect of a cylindrical source. 



The required solution, being independent of t, is obtained 

 at once from (1). We have inside the cylinder 



v = Ap n cos nO, 

 and outside 



v = Bp~ n cos n6, 



with Aa n = Ba~ n . The intensity of the source is represented 

 by the difference in the values of dv/dp just inside and just 

 outside the cylindrical surface. Thus 



a' cos n0=n cos nO (Bar"- 1 -|- Aa 71 - 1 ) , 

 whence 



Aa n = Ba- n = a'a/2n, 



a' cos 7i0 being the constant time rate. Accordingly, within 

 the cylinder 



and without the cylinder 



'^&'™ n0 («) 



These values are applicable when n is any positive integer. 

 When n is zero, there is no permanent distribution of tem- 

 perature possible. 



These solutions should coincide with the value obtained 

 from (33) by putting a — cr'dt and integrating with 

 respect to t from to go . Or 



the + sign in the ambiguity being taken when p<a, 

 and the — sign when p>a. I have not confirmed (50) 

 independently. 



In like manner we may treat a constant source distributed 

 over a sphere. If the rate per unit time and per unit of 



