316 



32 



ivl ^ ^)sin 27: i— CO, -i- w,)t. (/f = 1, 2, 3) (88i 



2?: eil \ / 112' 



The frequency o given by (86) is easily seen to represent the frequency of the slow 

 rotation of the Keplerian orbit which the electron at any moment may be considered 

 lo describe (compare page 10). The appearance of this small frequency o in the 

 denominators in the expressions on the right side of (87) and (88) may physically 

 be interpreted by observing that the deviation of the undisturbed orbit from a perio- 

 dic orbit, which is characterised by this frequency, is small, so that even a small 

 external force is sufficient to produce large changes in the character of these deviations. 



In order to find now for the perturbed motion the expressions for the coordi- 

 nates X, y, z of the electron as a function of the time, with the approximation 

 mentioned on page 26, we shall put 



X = .r" - a-', ij = y" + ii', z = z" + z' , (89) 



where x^', and z" represent the values of these functions for F = 0, while x^, ?/' 

 and z^ are small quantities of the order fjÅ. From (77) and (84) we find for x", y" 

 and -" 



z" = 2'Dt cos 2 ;r(r — 1 (u^ ^ (Wg) ' ' 

 The quantities x\ and z' will be given by 



(90) 



(91) 



where the summations are to be extended over k = 1, 2, 3 and where the /' 's and 



iv^'s are the functions of t given by (87) and (88), while the quantities ( tt-f-) , ( ^ " I , 

 /ô(x-\-iu)\ /ô(x-\-iii)\ \ö^t/o V^'^^A/ü 



\ ~~a T ] > ( ^ I functions of / obtained by first ditTerentiating the 



\ dl,, \ dwk /q 



expressions for z and x -^iy given by (77), and by replacing in the expressions thus 

 obtained the /'s by the constants ll and the iv's by wl = co^ t. 



It is seen from (91) that for z' and + l y' we obtain expressions in the form 

 of trigonometric series. While in the series for z" the frequencies corresponding to 

 the single terms were of the form r — lojy^-^-œ^ they will for r' be of the form 

 (:" — 1 + '«^2) dz ( — (»1 + «>2) I > so that there aj)pear, owing to the perturbing force, 

 new frequencies in the motion of the electron parallel to the direction of the electric 

 force, the amplitudes of which are of the order f,Å, and the frequencies of which 

 are of the form aaj^, a — 2 aji~\-2 co.^ and a -f 2 co^ — 2 w.,, where a is a positive integer. 

 As regards the motion jjerpendicular to the direction of the perturbing field, we see 

 that, wliile x" + ly" contained only terms of frequencies r — \ cd^-^- w., and 

 \t-\- \ iu^ — 2^2 + a>3 , .t' -f- 'y' contains terms of frequencies | (r — 1 lo^ -\- (o^) 



