80 



can be solved by tbe method of separation of variables, we obtain, by fixing the 

 stationary states by means of the conditions (22), a relation between the total energy 

 of the atom in the presence of a magnetic field and the fundamental frequencies 

 characterising the motion of the electron, which is exactly the same as that holding 

 between the energy and frequencies in the stationary states of an ordinary con- 

 ditionally periodic system. By a procedure analogous to that applied by Burgers 

 in bis proof of the mechanical invariance of the relations (22) for slow changes of 

 the external conditions, mentioned in Part I on page 21, it may further be proved 

 that also in the presence of a magnetic field these relations are invariant, when 

 regard is taken to the effect of the induced electric forces which, according to the 

 ordinary theory of electrodynamics, will accompany a variation of the magnetic 

 field. In the following, however, we shall not treat the problem of the influence of 

 an external magnetic field on the hydrogen spectrum by means of the method of 

 separation of variables, but in analogy to the treatment of the problems of the fine 

 structure and of the Stark effect of the hydrogen lines, given in the preceding 

 sections, we shall treat the problem from the point of view of the theory of per- 

 turbed periodic systems. Before entering on the detailed discussion of the necessary 

 modifications to be introduced in the general considerations in § 2, in order that 

 they may be applied also to the problem oT the fixation of the stationary states of 

 the atom in the presence of external magnetic forces, we shall for the sake of 

 illustration first show how it is possible in certain cases to treat the problem of 

 the effect of a homogeneous magnetic field on the hydrogen spectrum in a simple 

 way, which will be seen to present a close formal analogy with the theory originally 

 devised by Lorentz on the basis of the classical theory of electrons. 



In these considerations we shall make use of a well known theorem of Larmor, 

 which states that, if we look apart from small quantities proportional to the 

 square of the intensity of the magnetic field, the motion of a system of electrons 

 moving in a conservative field of force possessing axial symmetry round a fixed 

 axis will, in the presence of an external homogeneous magnetic field parallel to 

 this axis, differ from a mechanically possible motion of the system without field, 

 only by a superposed uniform rotation of the entire system round the axis, the 

 frequency of which is given by 



OH = T~^H, (79) 



4: TT m c 



where H is the intensity of the magnetic field and c the velocity of fight, while — e 

 and m represent the charge and the mass of an electron. ') If the magnetic field 



') J. Larmor, Aether & Matter, Cambridge 1900, p. 3-H. This tlieorem, wliicli was establislied in 

 connection with an attempt to develop a general theory of the Zeeman effect based on the ordinarN* theorj' 

 of electrodynamics, is directly proved bj' observing that, with the degree of approximation in question, 

 the accellerations of the electrons due to the presence of the magnetic field are equal to the changes in 

 the accellerations of the particles due to the superposed rotation of the system. 



