312 



28 



which is a complete solution of equation (73) for F = 0, we may according to § 1 

 calculate the variables which are canonically conjugated to /j, by means of 



the formulae 



ÔS ÔS dS 



""^-JT,' "'^=^,' '"-^^di,- 



The coordinates and momenta of the electron considered as functions of the /'s 

 and w's are periodic in each of the ly's with period 1. The rectangular coordinates 

 X, y, z of the electron may therefore be expressed by trigonometric series of the 

 form 2'Cti, T„ r3e2'r'(^>«'i + ^>«'« + '^««»\ where the coefficients Ct-i.t,, rg depend on the /'s 

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

 entire values of the r's. The values of tlie C's may be calculated by means of the 

 general method exposed in § 1. We will, however, not enter on these calculations 

 because they are entirely analogous to those performed in § 2 and in § 3 and be- 

 cause the result may be directly deduced from formula (42). They give that the 

 trigonometric series for z and a- + iy are of the form 



Z = ID. cos 2 TT (z^ 1 w, + w. ), I 



„ ^_ (77) 



where the summations are to be extended over all positive and negative entire 

 values of r, and where the coefficients Dr, Dr, Dr with neglect of small quantities 

 of the same order of Å are given by the expressions 



Dr = —xP ^ { (1 4- e') Jr^x {re) — (1 — c' ) Jr+, (re) ] , D„ = | ef, xP, 



= -^/^ ^-^^'{(l -^OJr-i(r£)-Ml--cO./r+i(rs)j, D',, = | ^(1 + f.');^ 



Dr = -y.P ^-^f' { ( 1 - t')Jr-i ire) — {\ + b') ./r+i (re) } , = J e (1 - // ) xP 

 where 



^_2_i^3^ e=l/l- 



■'2 I ^3 



(78) 



(79) 



while Jp{x) represents the value of the Bessel coefficient of argument x and of order 

 p. The formulae (77) and (78) are actually seen to coincide with the formulae (42). 

 deduced in connection with the problem of the influence of a small magnetic force 

 parallel to the z-axis, if in these formulae we replace w^t, m.^t, w^t by Wj, w^, w.^ 

 respectively, A simple consideration would show that this is just what must be expected. 



As long as we assume that F = 0, i. e. that we have to do with the system 

 in its undisturbed state, the motion of the electron is directly given by (77) if we 

 consider the i's as constants and for Wj. w^, substitute their expressions as linear 

 functions of the time by means of the formulae 



