28 



307 



3 3 



10 = 2 ^'^'^'l'23. '^(1 1 2 ^^"'2 «13' (66) 



while for all other combinations of z^ and for which r = î"i + + 1 = the 

 coefficients become zero. From (61), (65) and (66) we see therefore that the expan- 

 sion of X 4- J y in a trigonometric series may be written in the form 



X + ly = I X P («1 «23 "'^ + '2*13^^ 



(67) 



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

 of the r's, with the exception of such combinations for which r = -|- + 1 = 0. 



From (60) and (67) we obtain directly the expressions for z and x -\- iy as 

 functions of the time by introducing for the id's their expressions as linear func- 

 tions of the time. According to (12) we have 



"^1 = <Oit -\- à-i, = a»2^ + <Î2' "'s ^ û^s' + ^^a? (68) 



where by means of the expression (46) for a-^ the ry's are, with neglect of small 

 quantities proportional to found to be equal to 



Û>j = («3 -(- 0/. , — <«3 0/. , I 



_ 4 n^W^ m 3F(J, — I^ _ 3 FI (69) 



'"s— p ^ STT'Nem ' SK'Nem' \ 



while d^, d.^ and d.^ are constants. 



Introducing (68) in (60) and (67), and taking lor simplicity J,, d.^ and o., equal 

 to zero, we get 



2 



/(/, - /. 



x+iif = 2 ^/^(j,f,,e-'-'(-t'A + ^»,)' i f2£,.,e2'r"- <«. + '".)') 



X P 2:\{c,^t.,^Jr,{Ta,)JTjTa.^ — il i^Jz, ,(r<T,)Jr, + i(rff.,)}e2T'(r.«'. + r,a;, + r«. /. 



(70) 



It would be easy to write the series for r as a series of cosine terms with real 

 coefficients, but the form given above is more symmetrical. The formulae show 

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

 number of linear harmonic vibrations parallel to the direction of the electric force 

 with frequencies jico^-l- r^co^i, and of an infinite number of circular harmonic 

 rotations perpendicular to this direction and with frequencies t^w^ ~t Zito^ oj^ . It 

 may once more be remembered that, in the above expressions for the amplitudes 

 of these frequencies, small quantities proportional to F and to higher powers of 

 F are neglected. 



