298 



14 



The product of the first two factors of the integrand may be written as a sum 

 of terms a^^e-^'9> a^^e- '<P -\- a,^ -\- a, e^9 a^e^''^. Remembering that 



-^^e-'''<^+'P^"'</'d(P ^ J„(p), (/2 integer) (34) 



where Jnip) is the Besse! coefficient of argument /> and order n, we see therefore 

 that may be written as a sum of Bessel coefficients of different orders and of 

 argument re, each muHiplied by a certain factor. Performing the necessary calcula- 

 tions and contracting terms by means of the well known formula 



Jn-i{p}-\-Jn + l{p)-=^Jn{p), (35) 



we finally get the result 



= {(l + c-')Jr-i(r=-)-(l-£')Jr + i(r£)|. (36) 



This expression becomes undetermined for r = 0. By introducing, however, this 



3 



value for r directly in (33) we get = ^ s. For the expansion of x-\-iij in a tri- 

 gonometric series we therefore get from (31), (32) and (36) 



Q if I 



x^iy^'^exP e^^i(-"'M - P 1' ^ _ |(1 + £')-^r i ire) — ( 1— s') Jr+i {re) | e2-«-(^-i (37) 



where the summation is to be extended over all positive and negative entire values 

 of r except r = 0, and where the factor x P, as mentioned, is equal to the half 

 major axis of the orbit of the electron. 



The values of the coefficients are, as mentioned above, calculated witli neglect 

 of small terms containing the square and higher powers of i/c; it will, however, be 

 observed that, also if these terms were taken into account, there would in the 

 expansion for x -\- iy only occur terms of the form e^"'^~-^'"' + ^'\ due to the symme- 

 try of the system. 



The expressions for and w,^ as linear functions of the time are given by 



W, = Wj/4-Oi, H>2 = (O^t^U.,, (38) 



where, according to (11), = ^2 = ' "1 representing the total energy 



of the system as given by (24), and where Oj and are constants. We thus see 

 that the motion of the electron may be considered as a superposition of an infinite 

 number of circular harmonic vibrations, the frequencies of which are given by the 

 numerical values of r — lu>^-\-oj.^, where r may assume all positive and negative 

 entire values, and the amplitudes of which are directly given by (37). 



The values of co^ and oj.^ differ only by small quantities of the order "•/c-, their 

 difference being equal to the frequency mentioned on page 10, and become equal when 

 the relativity modifications are neglected (c = 00). In this case the expression (37) gives 



