320 Lord Rayleigh on Conduction of Heat in a Spherical . 



When the value of n is greater than zero, the first of these 

 conditions gives C = and the second D = 0; so that 



6=-s/cc=(mr)-»J nH (mr) .Y n , . . . (34) 



and s + <x0 = O. Accordingly these terms contribute nothing 

 to the pressure. It is further required that 



J n+i (ma)=0, (35) 



by which the admissible values of m are determined. The 



roots of (35) are discussed in ' Theory of Sound,' § 206 . . . ; 



but it is not necessary to go further into the matter here, as 



interest centres rather upon the case n = 0. 



. If we assume symmetry with respect to the centre of the 



1 d* 

 sphere, we may replace V 2 in (8) by - y^ r, thus obtaining 



^|fr«" 0) +™Mfl-C)=0, . . . (36) 



of which the general solution is 



n fi . cos mr ^ sin mr 

 u = \u -f A f- r> 



mr mr 



But for the present purpose the term r~ l cos mr is excluded, 

 so that we may write 



= Q + B™2*, - S /«=D + B™', . (37) 

 mr mr 



giving 



5 + a0 = a(C--D) = H. . . . {37 bis) 



The first special condition gives 



7waC + Bsinma = (38) 



The second, that the mean value of s shall vanish, gives on 

 integration 



im 3 tt 3 D 4- B(sin ma — ma cos ma) = 0. . . (39) 



Equations (18), derived from (9) and (37 his), giving C 

 and D in terms of H, hold good as before. Thus 



D _ g-h _ a/Bq-vm 9 - 



C ~ hafi ~ a/3{qivm 2 ) { } 



Equating this ratio to that derived from (38), (39), we find 



3 ma cos ma — sin ma _ vni 2 — aftq 

 m?a* - sin ma a/3 (vm 2 + q) ' ^ ' 



