77 



361 



by the coiulilions (117), and thai the sine of the angle which lliis plane makes with 



the axis will be equal to " . ') The ditt'erent possible slalionary stales of Ihe per- 

 "2 



turbed system are a-priori equally probable and are obtained by lelling /j, assume 

 the values 0, 1, 2, . . ., /j, the values 1, 2, ;i, ... and n the values 1, 2, . . ., n... That 

 no stationary states exist in which /»., would be equal to zero follows, as mentioned 

 on page 68, from the fact that the motion in these stales would not be physically 

 realisable since the electron would collide with the nucleus. In stales for which n., 

 would be different from zero, but in which n would be equal to zero, the mechani- 

 cal motion of the electron would not show singularities, but as pointed out in 

 BoHH S paper it is possible to conclude, from the principle of the invariance of the 

 a-priori probability of the stationary slates for continuous transformations, lhal 

 these slates cannot represent stationary stales since it would be possible lo Irans- 

 form them continuously into physically unrealisable states.-) 



From the calculations in ,^ 4 it follows thai, with neglect of small quantities 

 proportional to the square and higher powers of F, the total energy of the perturbed 

 system may be expressed in terms of I^, by the same function as that by 

 which in (76) the quantity «j is expressed in terms of /j, 7^, L, (compare page 31). 

 Introducing (124) it will therefore be seen that the total energy of the stationary 

 states of the perturbed system with this approximation will depend on /», and n., 

 only, and will be given by the same formula as that holding for the energy in 

 the stationary states of the undisturbed system, which was given by (118). '') 



') In Bohr's paper (Part I, page 35; Part II. page 55) it has been mentioned that quite generally 

 we must expect that one of tlie conditions whicii fix tiie stationary states of an atomic system which 

 possesses a fixed axis of symmetry will claim that the total angular momentum of the system round 

 this axis is equal to an entire multiple of ^' 2- Starting from this result it may directly be proved 

 that the conditions (124 1 are in concordance with the principle of the mechanical transformability of 

 the stationary states. In fact, it can be proved that during a slow increase of the intensity of the elec- 

 tric field the mean values of the quantities /[, /., and /.. , taken over the motion of the perturbed 

 system, with neglect of small quantities projiortional to ' will remain the same. Since now, according 

 to the calculations in § 4 (see page 30), the quantities 7,', and /!,' appearing in the conditions (124) 

 just represent the mean values of /,, I., and /.p it will therefore be seen that, if we start from a sta- 

 tionär}' motion of the undisturbed atom which satisfies the additional condition of the angular momen- 

 tum, the atom will during a slow establishment of the perturbing electric field pass mechanically into 

 a state which satisfies the conditions il24). 



If, for instance, we imagine that the intensity of the electric force increases to values which 

 arc so large that the relativity modifications may be neglected we would obtain the system considered 

 in 5 3 and § 0, and the states in question would be continuouslj* transformed into the corresponding 

 states of the latter system, the motion in which, as mentioned on page 50, involves an essential sin- 

 gularity. Compare Uohk, loc. cit. Part II. page 5(1. 



\) The fact that in the present case the alteration in the total energy of the system due to the 

 presence of the external forces, i. e. what Hohk calls the "additional energy ' of the perturbed system, is 

 e([ual to zero as far as small quantities proportional to F arc concerned may be directly deduced from 

 a general theorem which states that if a conditionally periodic system is perturbed by a constant small 

 external field of force the value of the additional energy in the stationary states of the perturbed system 

 is, with neglect of small cpiantities proportional to the s(|uare of the external forces, simply rt|ual lo 



