317 



-_x^ ( — + w.^) and (r 1 -~ 2(t»2 + c^) -r ( — "^i -\- a^i) > tl^at I'lis motion 

 there appear new frequencies of the form a —li co^^ w^-h w^, a -|- 2 Wj — — co^, 

 «et»! — + aw, +Wo--£f3, a-\~2ü}j — 'dwi-fcu^ and a — 2 w, -3a>2 — cus, where a 

 is again a positive integer. Since the quantities m.^ and w., appearing in the cal- 

 culations do not differ from each other as far as small quantities proportional to F 

 are concerned, we see that the motion of the perturbed system under consideration 

 with the approximation in question may still be represented as a sum of harmonic 

 vibrations the frequencies of which are built up of only two fundamental frequen- 

 cies and w,. In the undisturbed system appeared only frequcMicies a — Ito^-l-w^, 

 a -\- 1 (Vi~ a>.^; in the perturbed system appear the new frequencies a — 2 1- 2^2, 

 aw^, a -f 2 — 2^2- The new frequencies are seen to be equal to the sum or to 

 the difference of two of the frequencies appearing in the motion of the undisturbed 

 system, and correspond to the sum-tones and ditTerence-tones in acoustics. [That to 

 the first approximation only sums and differences and not other linear combinations 

 of the original frequencies appear, lies in the circumstance that the perturbing field 

 is homogeneous, so that its potential is a linear function of the Cartesian coordinates 

 of the electron.') In fact, as a consequence of this the quantities /] and iv[. appearing 

 in (91) contain according to (81) only frequencies which appear also in the undis- 

 turbed motion of the electron. The same is the case for the quantities ( n^,- ) , . . . 

 in (91), so that the new frequencies in the perturbed motion can only be sums or 

 (iilTerences of frequencies occurring in the undisturbed motion.] 



Although with the approximation mentioned oj., and w.j do not differ from each 

 other, the fact that they are not identical will nevertheless be essential for the des- 

 cription of the motion over a time interval of the same order as ^-'p (compare note 

 on page 79). For sake of the applications in § 7 we shall therefore keep oj.^ and w., 

 separated in the following formulæ, what, as it will be discussed more closely in the 

 paper mentioned on page 26, will he justified on account of the special character of 

 the system under consideration. 



Let us now proceed to the explicit calculation of the trigonometric series for 

 r' and for .c' -f ly'. For we gel from (91), (88), (87) and (77), omitting in the 

 calculations for the sake of simplicity, here as well as in the following, the index (°1 

 in and in D?, 



D„ cos 2 r (r — 1 w, -t- w,) t cos 'Ijzi — co. - - <u.,) t 



[ SI 



^ - ' //, {"0) + //. ) ' - 2"'- ». 



cos2r(r- - 2 



Compare Bohh, lue. cit l'art I. p. Mi. 



II. K. D. Vldeusk. Selsk. Skr . niitiirvldensk. nj; ninlhrm, Vfil.. S. Hii-kkr. III. .t. 



COS 2r 7(0, 1 . 



(92) 



