the Conduction of Heat. 391 



area of surface be S„, we find, as above, for inside the 

 sphere (c) 



»=2^-lG)" S »' • • • • • (51) 

 and outside the sphere 



•-arTi©*" 8 " • • ' • (52) 



and these forms are applicable to any integral n, zero included. 

 Comparing with (44), we see that 



. n+ iC"dt _r?±£! T / irc\ 2 /r\±(»+*) 



(53) 



which does not differ from (50), if in the latter we suppose 

 ra = integer -f^. 



The solution for a time-periodic simple point-source has 

 already been quoted from Kelvin (IV.). Though derivable 

 as a particular case from (4), it is. more readily obtained 

 from the differential equation (1) taking here the form — see 

 (38) withn=0— 



d*(rv) d 2 (rv) , 

 dt ~ dr 2 



or if v is assumed proportional to e !pt , 



d 2 (rv)ldr 2 -ip(rv)=Q, . . . . (54) 

 giving 



rv = Ae i t >t e-V r , ...... (55) 



as the symbolical solution applicable to a source situated at 

 r = 0. Denoting by q the magnitude of the source,, as in (5), 

 we get to determine A, 



r-47rr 2 ^] =qe^ = ^irA, 

 L drJ r=Q 



so that v= JLy& e -$p\ ...... (5G) 



If from (56) we discard the imaginary part, we have 



^47/~ fVWcos ^^VW)},. • (57) 



corresponding to the source q cos pt. 



From (56) it is possible to build up by integration solutions 

 relating to various distributions of periodic sources over lines 

 or surfaces, but an independent treatment is usually simpler. 



2I>2 F 



