59^ 



ticles for such a system. Since moreover, in complete analogy to llie conditions 

 (22), the stationary states of the perturbed system are characterised l)y 



/ = nh, Zsk = u/c/i , [k = 1, . . . .s ~- 1) (67) 



we see consequently that, for sufficiently small intensity of the external forces, we 

 obtain in the region of large values of n and of the it's a connection between the 

 frequencies of the components of the spectral lines, determined on the quantum theory 

 by means of relation (1), and those to be expected on ordinary electrodynamics, 

 which is of exactly the same type as the analogous connection, discussed in Part I, 

 in case of ordinary conditionally periodic systems which allow of separation of 

 variables. In perfect analogy with the general considerations in Part I, we are 

 therefore led directly to certain simple conclusions as regards the intensities and 

 polarisations of the components into which the lines of the spectrum of the undis- 

 turbed periodic system are split up in the presence of the external field. Thus we 

 shall expect that there will exist an intimate connection between the probability of 

 spontaneous transition between two stationary states of the perturbed system, for 

 which n = n', u^. = uj. and n = n", n,. = n"_ respectively, and the values in these 

 states of the coefficient Cr, tj, . . t^^j in the expressions for the displacements of the 

 particles, for which r = n — n" and t,^ ^ n^. — uj^. If for instance, for a certain set 

 of values of r and tj, ... tj_j, the coefficient Cr, ti, .. . tj_i in the expressions for the 

 displacements in every direction will be equal to zero for all motions of the per- 

 turbed system, we shall expect that the corresponding transitions between two 

 stationary states will be impossible in the presence of the given external field; and 

 if this coefficient is zero for the displacements of the particles in a certain direc- 

 tion only, we shall expect that the corresponding transitions will give rise to the 

 emission of a radiation which is polarised in a plane perpendicular to this direction. 

 With a characteristic example of these considerations we meet in the case of 

 the spectrum of a hydrogen atom exposed to an external field of force which pos- 

 sesses axial symmetry round an axis through the nucleus. In analogy with the 

 resolution of the motion of an ordinary conditionally periodic system which pos- 

 sesses an axis of symmetry in its constituent harmonic vibrations, discussed in 

 Part I on page 33, it follows from the discussion of the general character of the 

 secular perturbations on page 54 that the motion of the electron in the perturbed atom 

 in this case can be resolved in a number of linear harmonic vibrations parallel to 

 the axis with frequencies | tiup +tiDi and in a number of circular harmonic rotations 

 perpendicular to the axis with frequencies rcop + t^ Oj + i\ . In complete analogy with 

 the considerations in Part I, we are therefore led to conclude that in the present 

 case only two types of transitions between the stationai'y states of the perturbed 

 atom are possible. In the transitions of the first type n., will remain unaltered and 

 the emitted radiation will give rise to components of the hydrogen lines which will 

 show linear polarisation parallel to the axis. In the transitions of the second type 

 n^ will change by one unit and the emitted radiation will show circular polarisa- 

 tion when viewed in the direction of the axis. Remembering Ihal, according lo 



