Theory of Spectrum Emission. 49 



and the kinetic energy, 



T = £m'a 2 (cosh 2 f — cos 2 rf) . (f 2 +^ 2 J 



+ hn'a? sinh 2 f . sin 2 y . cf> 2 , . (5) 



where <£ is the angle between the variable and an arbitrarily 

 fixed meridian plane. The moments corresponding to the 

 canonical coordinates f, ??,(/> (which are the derivatives of T 

 with respect to f, ?;, <£) are 



]j l = m'a 2 (cosh 2 !j~ cos 2 rj)$j, p 2 =m l a 2 (idem) ?; 5 1 • ,v 



p 3 =m'a 2 sinh 2 £ . sin 2 ?; . <£. 



Since the energy does not contain c£, we have, as one of the 

 -canonical equations, dp 3 /dt = Q, i. e. 



p 3 = const,,- 



an obvious result. Thus, one of the quantizing equations 

 will simply be 



p z d$ = 27rp 3 = hi 3 , .... (7) 



J 



o 



where n 3 is an integer. And since there is with f, tj what is 

 familiar as u separation of variables/' the other two equations 

 will be, according to the principles used by Sommerfeld, 

 Epstein, and others, 



jVidf— n^, §p 2 drj=n 2 h, ... (8) 



where n i9 n x are two more integers. (For integration limits 

 cf. infra.) Using the well-known method of separation of 

 variables, and denoting by TT'the constant value of — V— T, 

 we have at once the two first integrals, which are Jacobi's 

 integrals, 



2h = av/^^.V/Q-cosl^^ + ^coshf-P-Vsinh 2 ^ 



p 2 =. a\/'lm! W . \/ — /3 + cos 2 r\ — P 2 /sin 2 y, 



\ 



p 2 Wn 3 . 



where F~ — -,- , \yTfr = Tr^> — ^~2Tf-5 an d 6 is an integration 

 zm a 2, \\ bir-m tr II ^ ^ 



constant. 



TVith the values (9) of jD 1; p 2 we have in (8) two equations 

 for the purpose of quantizing f$ and W. Thus the problem 

 of finding the required 



W= W(n u n 2 , n 3 ) 

 corresponding to the stationary orbits, and thence the spectrum 

 Phil Mag. S. 6. Vol. 39. No. 229. Jan. 1920. E 



