300 



16 



ciliar lo llie r-axis. Then the coordinates .r, ;/, z defining the position of the electron 

 in space w ill be connected with ^ and rj by means of the formulae 



z = çcost/, -r + iy = {çsin -T i-^)e-~^'"'' -"''^' . (41) 



Now, according to (37) and (38j, the dependency on the time of the quantities ^ and 

 is expressed by 



where the summation has lo be extended over all positive and negative values of 

 r, except r = 0, and where for simplicity we have taken the quantities and d.^ in 

 (38) equal to zero, what is easily seen not to restrict the generality of the consider- 

 ations. By means of this formula we get from (41), denoting cos by a. and sin by 



z = l^ne,Pe^^'(-">+'"^)t_,^,p v^^ j (1 . _ £')J^^i(rs)_(l _ .')J^^i(,£)|cos 2;7(r- 1 to.^oj.^t 



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

 values of r except r = 0. It is seen that the motion of the electron may be regarded 

 as a superposition of linear harmonic vibrations parallel to the axis and of fre- 

 quencies It — 1 coy-\-w^\, and of circular harmonic rotations perpendicular to this 

 axis and of frequencies \t — 1 w^-]- \ and \t: -\- \ w^ — 2(0^^ (u^ \. In the expres- 

 sions, given by (42), for the amplitudes of these vibrations small quantities of the 

 same order as "^/c^ are neglected, just as in (37), while from the above calculation 

 it is seen that the magnetic field, at any rale in first approximation, does not aftecl 

 Ihe values of these amplitudes. 



3. Hydrogen atom under the influence of a strong homogeneous electric 



field of force. 



In lliis chapter we shall consider a mechanical system, consisting of an elec- 

 Iron of charge — e and mass m, whicli is subject to the attraction of a nucleus of 

 charge Ne and of infinite mass as well as lo Ihe influence of a homogeneous electric 

 field of intensity F, assuming that the motion of the electron is governed by 

 Ihe laws of Newtonian mechanics. We shall assume that the force eF is small 

 compared wilh the force which the nucleus exerts at any moment on the electron, 

 and it will be our purpose to solve the equations of motion by means of trigono- 

 metric series of the type (12), in such a way that we shall neglect in the calcula- 

 tion of the coefficienLs C small quantities which are proportional to the first power 



(42) 



