Mechanism of Radiation. 433 



equation so obtained, and equation (11), may be written 



JT?m rl(V n 



(<rp*-W)cK n =p~^ r +<r-j—, .... (13) 



trpV(«+m 7 ™) = -n(» + l) (pF™ + crGff), - (14) 

 while we have (cf. equation (4), p. 430) 



^=-^T^{£,(r'V^) + r'Ym + hny':)y„dr', . (15) 



g:= ~ 2S1 f{i ( r ' VcC) + r ' v (< <3 "» + "»7'») } >»*•'• ( 16 > 



Equations (13) to (16) must hold for all values of?', and 

 contain only the four variables a™, f3£+imrff, F™, and G», each 

 equation being homogeneous and of the first degree in these 

 four variables. It is easily seen that there is no solution of 

 these equations other than 



«T=0, /8T +ifi*7r=0 l r: = 0, G?=0, . . (17) 



except only when a special relation is satisfied by the various 

 constants of the equations. Now m has entirely disappeared, 

 so that any such relation may be written in the form 



Af,n)=0, (18) 



where the coefficients which determine the functional form/ 

 are constants o£ the atom aud of the potential-function. 



Hence the normal vibrations fall into two classes, given by 

 the two equations (17) and (18) respectively. In either class 

 the displacement exists only for single values of n and m. 



For a vibration of the first class ««=(), and 0%, 7™ may 

 have any values such that 



ftn+hny™ = 0. 



The displacement is purely tangential, and no forces of 

 restitution are called into play. Hence the frequency is in 

 every case p = (as is otherwise obvious from equations (9) 

 and (10)), and, strictly speaking, the corresponding normal 

 degrees of freedom do not give rise to vibrations at all. 



For a vibration of the second class, the frequency is a root 

 of equation (18), and the functions c£, fflt + invfi, F?, G? are 

 determined by equations (13) to (16). These functions do 

 not contain m ; to determine ft™ and y„ separately we must 

 give to m any one of the (2n -j- 1) integral values between n 

 and — ??, and determine ft™, y™ from equations (9) and (10), 



