296 



12 



where the first integral is to be extended twice between the two roots of the inte- 

 grand. Evaluating these integrals and expressing the «'s as functions of the 7's, we 

 get, expanding after powers of l/c^ neglecting terms containing squares and 



higher powers of this quantity, 



In the expression for the energy a^, the term which does not contain i 'c^ gives the 

 value of the energy for an unrelativistic motion, while the terms containing i c= are, 

 as will be seen in Part II, determinative for the fine structure of the hydrogen 

 lines. We may, however, neglect these terms in the following since, for the purpose 

 of the present paper, it will only be necessary to calculate the values of the 

 coefficients C in (21) to the first approximation, /. e. with neglect of quantities 

 containing I'c^ and higher powers of l/c-. 



Introducing the above values for «j and u., in (22) we find in this way for S 

 the expression 



where we have introduced the abbreviations 



X - , ^^T > ' ^ = ^ + (26) 



It is easily shown that x I - will be equal to the half major axis of the orbit des- 

 cribed by the electron. 



According to (8) the angle variables and w.^ will be defined by 



o o 1 



2 TT = 2 ;r 



' ^ if' 



^ o /^S dS\ , , r dr , 



2 7z(w,, — u;, ) = 2 ;r ^ tt-t = — x I L \ — , -+- w. 



(27) 



Introducing now the abbreviations 



= ^, £ = 1/1 (28) 



where s may be simply shown to be equal to the eccentricity of the orbit, and 

 introducing (compare (19)) a new variable (p by means of 



r = xPi\ +>cosyA), (29) 



it is easily seen that 



