288 



4 



the fine structure and in that of the Stark effect of the hydrogen lines, and to 

 compare the result of the calculations with the observations. 

 The paper is divided in two Parts. 



Part I deals with the problem of the determination of the values of the ampli- 

 tudes of the harmonic vibrations in which the motion of certain mechanical systems 

 may be resolved, and is divided in four chapters. 



In § 1 a short account will be given of the theory of mechanical systems for 

 which the Hamilton-Jacobi partial differential equation may be solved by means 

 of separation of variables, and it will be shown how it is possible to reduce the 

 calculation of the amplitudes of the harmonic vibrations, in which the motion of 

 these systems may be resolved, to the evaluation of simple definite integrals. 



In § 2 the method exposed in § 1 will be applied to the model of a hydrogen 

 atom which is uninfluenced by external forces, assuming that the motion is governed 

 by the laws of relativistic mechanics. 



In § 3 the same method will be appUed to the model of a hydrogen atom, 

 which is subject to the influence of an external homogeneous electric field of force, 

 the intensity of which is so large that it is possible with a high degree of approx- 

 imation to determine the motion by means of ordinary Newtonian mechanics. 



In § 4 the perturbing influence is considered which a very weak homogeneous 

 electric field of force will have on the motion of the system considered in § 2. 



Part II deals with the application of the calculations given in Part I to the 

 problem of the intensities of spectral lines, and is divided in four chapters. 



§ 5 contains, besides a brief exposure of the theory of stationary states of 

 systems which allow of separation of variables, an account of Bohr's theory of the 

 connection between the polarisation and intensities of spectral lines emitted by an 

 atomic system and the amplitudes of the harmonic vibrations in which the motion 

 of such a system may be resolved. 



In § 6 a discussion is given of the application of the theory to the relative 

 intensities of the components in which Ihe hydrogen lines are split u]) in case of 

 the Stark effect, on the basis of the formulae deduced in § 3. 



§ 7 contains a discussion of the relative intensities w ith which the components 

 of the fine structure of the hydrogen lines appear, based on the formulae deduced 

 in § 2 and § 4. 



In § 8 a brief discussion will be given of certain questions which stand in 

 connection with the application of the theory to the problem of the Zeeman effect 

 of the hydrogen lines. 



Finally I wish to express my best thanks to professor N. Bohr, the creator 

 of the beautiful theory underlying the present paper, for his kind interest and 

 encouragement during the achievement of the work. 



