172 



total moment of momentum of electron and molecule together; it 

 determines the rotation of the system as a whole. 



If ^I\ = 0, the motion of the electron is determined by tlie function : 

 1 / &' W' \ ¥^' 



It will now be assumed that it is possible to find solutions of 

 the problem characterized by the Ham. function (4) (in this problem 

 there is no disturbing influence of the rotation), and that these 

 solutions are of the following form : the coordinates and momenta 

 can be expressed as periodic functions (with period 2ji) of three 

 variables q^, q^, q^, which depend linearlj- on the time (so-called 

 "angular variables")^)- H' the canonical momenta p.^, p^, p^, cor- 

 responding to these variables, are introduced '*), the original 

 coordinates and momenta r, v)-, \p^, K, O, W^ can be expressed as 

 functions of q-^ q^q^ Px p^ Pz- This transformation of the variables 

 possesses the property of conserving the canonical (Hamiltonian) 

 form of the equations of motion. ') 



To find solutions of the problem given by (3) {W^=\=Q), it may 

 be considered as a problem of disturbed motion, and instead of 

 the original coordinates and momenta the p and q may be introduced 

 as new variables. The equations of motion of the q and p then are 

 the Hamiltonian equations, derived from the function /i (^, /;), which 

 is obtained if in (3) the original coordinates and momenta are 

 replaced by their expressions as functions of the q and p. This 

 function has the form : 



K{q,p):=:H, L_^ + -^ .= 



= A{PiP,Px) +2/ J 



T-L 





Sin 



(5)^] 



') Solutions of this kind are — as is known — of frequent use in Astronomy, 

 especially for the treatment of problems of disturbed motion. In the most common 

 cases they have the form of trigonometric expansions according to sines and 

 cosines of combinations of the q. — (The" expression found for 4»i has a slightly 

 different form, as this variable can increase indefinitely; for instance j|»i may be 

 found to be equal to q^ plus a periodic function of qi qo gg). 



In the theory of quanta K. Schwarzschild was the first to introduce solutions 

 of that nature (Sitz. Ber. Berl. Akad p. 548,1916) Compare also : J. M. Buegees, 

 these Proceedings p. 163. 



*) These momenta Jh P-2 P& ^^'^ constant. 



3) Comp. Whittaker, Anal. Dynamics (Cambridge 1904) p. 297, 396. 



*) £' denotes a summation over all positive and negative values of the m, with 

 the exception of simultaneous zero values. 



