436 Mr. J. H. Jeans on the 



Substituting these values for F« and G» in equations (13) 

 and (14), and putting rc = x> , we find as the equivalent of 

 equations (13) to (16) the two equations 



(<rp 2 -W)Y=0, (22) 



{o- i? «-47rp 2 (l + ? 1 )-47ra 2 £ J }Z = 0. . . . (23) 



§ 18. These equations are not difficult to interpret. Cor- 

 responding to any value of n, the displacement-potential at r 

 of the shell of radius r' contains the factor r n /r /n+l or 

 r ' n / n+1 y according as r< or >r' . This shows that in the case 

 of n = co, the potential created by the displacement of any 

 shell will vanish everywhere except in the immediate neigh- 

 bourhood of that shell. It is therefore clear that the normal 

 vibrations for which n = co w T ill split up into the normal 

 vibrations of the separate shells. 



This is the meaning of equations (22) and (23). Corre- 

 sponding to a given value of p 2 ) the second factors of equations 

 (19) or (20) can only vanish over a single isolated shell of 

 ions ; hence the first factor must vanish everywhere except 

 over this shell. The corresponding displacement and displace- 

 ment-potential are limited to the shell in question. 



Corresponding to any specified shell, equations (22) and 

 (23) show that there will be two vibrations of order n = oo . 

 For the first of these Z = and <rp 2 =W. For the second, 

 Y=0 and <7p 2 = 47r{p 2 (i + f 1 ) + <7 2 (l + £,)}. The displacement 

 of the shell is therefore in the former case purely radial ; in 

 the latter purely tangential. 



§ 19. This concludes the investigation of the frequency- 

 equation for the case of n = co . Corresponding to each par- 

 ticular solution for the case of n = oo , there will be a general 

 solution for all values of n from to go . Hence the number 

 of spectrum-series will be equal to twice the number of shells 

 in the atom. The frequencies of the heads of the series are, 

 as we have seen, given by 



W 

 P 2 =^, (24) 



^=^{p>(l+fc) +«*&}. .... (25) 



§ 20. It is easily verified that all possible degrees of freedom 

 have been accounted for. The (2ra-l-l) degrees represented 

 by the nth line of a spectrum-series may be associated w r ith 

 the (2n + l) independent spherical surface-harmonics of 

 order n ; so that a complete spectrum-series may be asso- 

 ciated with the most general function of position on any 



