Part I. 



Examination of the trigonometric series representing the 

 motion of the electron in the hydrogen atom. 



1. General method applicable to conditionally periodic systems. 



Consider a mechanical system of s degrees of freedom, the equations of motion 

 of wliich are given by the set of canonical equations 



, ik=l,2....s) (2) 



dl (it cpk 



where q^, ... qg is a set of generalised coordinates by means of which the positions 

 In space of the particles of which the system consists are uniquely determined, 

 while p^, ■ ■ ■ ps are the canonically conjugated momenta, and where E is the energy 

 of Ihe system, which is assumed to be a function of the p's and 7's only. The so 

 called Hamilton-Jacobi partial ditrerenlial equation is then obtained by writing 

 d S 



Pi = 2 where S is a function of the q's, and bv putting /s, considered as a func- 



lion of the q's and „ 's, equal to a constant a, ; 



' cq ' 



A complete solution of this equation will contain, besides an additional conslant C, 

 s — 1 other integration constants .... Us- Now it may happen thai, for a suitable 

 choice of orthogonal generalised positional coordinates r;, , .... (/s, il is possible to 

 write a complete solution of equation (3) in the form 



S = l\%{qk; r/,) : C, (4) 



where depends on the as and on 7/- only. If this is Ihe case it is said thai Ihe 

 equation (3) allows of "separation of variables'" for the special choice of coordinates 

 under consideration, or briefly, that the system allows of separation of variables. 

 For such a system '^^ , as seen from (4), will depend on llu- corresponding r//,. only ; 



moreover remembering that in Newtonian, as well as In relalivislic mechanics, E 



rS 



contains the p's in the form of a sum of squares, ^ must necessarily be Ihe sipiare 

 root of a one-valued function of 7/.. Hence, denoting tiiis one-Mihied I'lniclion by 

 F/;, we see lhat 5 mav be written in the form 



