the Conduction of Heat. 339 



ifp=rrpL Now (< Theory of Sound/ § 334) 



£^^^ d ^^\/{j^n + h (p), . (40) 



whence 



J_" P„ O) ef" dp = 2i" +J v /( J) J„ H (-ip), (41) 

 or, as it may also be written by (27), 



-VGp^W < 42 ) 



Substituting in (2) 



q = 27rc 2 F n (fx)dfi, (43) 



we now get for the value of v at time t, and at the point for 

 which p — r } /a=1, 



It may be verified by trial that (44) is a solution of (38). 

 When fju is not restricted to the value unity, the only change 

 required in (44) is the introduction of the factor F„(/jl). 



When n — 0, P n (/x) = l, and we fall back upon the case of 

 uniform distribution. We have 



J iW=V / (i) sin * , > -...(«) 



or J i( -i-) = ^~~ (46) 



Using this in (44), we obtain a result in accordance with (6), 

 in which Q, representing the integrated magnitude of the 

 source, is equal to Aire 2 in our present reckoning. 

 When ra = l, Pi (/*)=/*, and 



J ^V©{^~ — }; ■ • («> 



and whatever integral value » may assume J, +4 is express- 

 lble in nnite terms. 



We have supposed that the rate of distribution is repre- 

 sented by a Legendre's function V n (ji). In the more 

 genera case it is evident that we have merely to multiply 

 the right-hand member of (44) by S„, instead of Pn. 



Phil. Mag. S. 6. Vol. 22. No. 129. Sept. 1911. 2 D 



