28 



we shall further see, that this connection leads to certain general considerations 

 about the probability of transition between any two stationary states and about 

 the nature of the accompanying radiation, which are found to be supported by 

 observations. In order to discuss this question we shall first deduce a general ex- 

 pression for the energy difference between two neighbouring states of a conditionally 

 periodic system, which can be simply obtained by a calculation analogous to that 

 used in § 2 in the deduction of the relation (8). 



Consider some motion of a conditionally periodic system which allows of 

 separation of variables in a certain set of coordinates q^, . . . qs, and let us assume 

 that at the time t = ä the configuration of the system will to a close approxima- 

 tion be the same as at the time f = 0. By taking ê large enough we can make 

 this approximation as close as we like. If next we consider some conditionally 

 periodic motion, obtained by a small variation of the first motion, and which allows 

 of separation of variables in a set of coordinates q\, ... q'^ which may differ 

 slightly from the set q^, . . . qs, we get by means of Hamilton's equations (4), using 

 the coordinates q\, . . . q' , 



dEdt 





SE ,eE, 



)'it =^ ^\%Sp^~p',^dq,)dt. 



By partial integration of the second term in the bracket this gives: 





Now we have for the unvaried motion 



s 



« = >5i 



t = ü 



Pkikdt =2:. ^ich' 



(27) 



where h is defined by (21) and where Nj, is the number of oscillations performed 

 by qk in the time interval ä. For the varied motion we have on the other hand: 



2^P'k<i'kdt =\ ^p'.dql =^X/, 



+ 



f = o 



^P>^<l'k 

 1 



' = .51 



t = o 



where the f's correspond to the conditionally periodic motion in the coordinates 

 q[ ■ ■■ q'^, and the dq"s which enter in the last term are the same as those in (27). 

 Writing I'k — Ik = à Ik, we get therefore from the latter equation 



1.5 . 



dEdt = S^Nkdh. 



(28) 



