the Conduction of Heat. 393 



In the above formulae z is supposed to be positive. On 

 the other side or* the source, where z itself is negative, the 

 signs must be changed so that the terms containing z may 

 remain negative in character. 



When periodic sources are distributed over the surface of 

 a sphere (radius = c), we may suppose that v is proportional 

 to the spherical surface harmonic S„. As a function of r 

 and t, v is then subject to (38) ; and when we introduce the 

 further supposition that as dependent on t, v is proportional 

 to e lpt , we have 



d 2 (rv) n{n + l) , . . ' 



—J^r- — ^— (rv)-tp(rv)=0. . . (69) 



When 7i = 0, that is in the case of symmetry round the pole, 

 this equation takes the same form as for one dimension; but 

 we have to distinguish between the inside and the outside of 

 the sphere. 



On the inside the constants must be so chosen that v 

 remains finite at the pole (r = 0). Hence 



rw = Ae^(« r %)- e-r'JVP))) . . . (70) 



or in real form 



rv = A e r V(^/2) cos {pt + r \/(p/2)} 



-A e-r V(*/2) C os \pt-r</(p\~2)}. . (71) 



Outside the sphere the condition is that rv must vanish at 

 infinity. In this case 



ry=B && e~ r Wp), (72) 



or in real form 



rv = B e-r «(pP) cos {j>t-r y/(p/2)\. . . (73) 



When n is not zero, the solution of (69) may be obtained as 

 in Stokes's treatment of the corresponding acoustical problem 

 ('Theory of Sound/ ch. xvii.). Writing r \/(ip) —z, and 

 assuming 



rv=Ae z + Be~ z , (74) 



where A and B are functions of z, we find for 13 



<PB rfB nfn + 1) n „ 



