554 Mr. G. H. Livens on the Electron 



the electrons themselves are also presumed to be too in- 

 frequent to warrant consideration. The actual bearing and 

 import of these assumptions have been elaborately discussed 

 by Dr. Bohr in his dissertation already quoted, and the 

 reader may be referred to this beautiful exposition of the 

 subject for further information on this point. 



The solution of the above differential equation for the 

 function /is easily obtained, and in the most general possible 

 case of the present type of theory it may be written in the 

 form 





t x -t- T 



wherein 



X ~ X ^ +Y B? +Z B? + % + ^ +r 5F + ^' 



and the suffix is taken to indicate the values of the variable 

 functions expressed explicitly in terms of the auxiliary time 

 variable i ls which is introduced to denote the integrations. 

 To effect this interpretation it is first necessary to integrate 

 the six auxiliary differential relations 



d%\ dvi d£i dssi ___ dy 1 dz 1 dt x 



expressed generally in terms of the auxiliary time variable t ± , 

 introducing as the necessary constants of integration the 

 values of the variables at the time t lm On substitution of 

 these integrals in % x we obtain its complete expression as an 

 explicit function of £ x ; the integration for / is then directly 

 effected, maintaining throughout (f, 77, f, a\ y, z, t) all 

 constant. 



There is perhaps one remark necessary respecting the form 

 in which the above solution is written. It is implied in it 

 that T m is not a function of the time, although it is essentially 

 a function of the varying quantity u, velocity of the electron. 

 The slight variation of r m on this account is, however, 

 negligibly small and may safely be left out of account, 

 especially considering that, as the final result verifies , r m 

 merely appears as a factor in the already small correction 

 term in /. This assumption implies of course that the 

 external forces have no direct influence on the collisions 

 themselves, which are presumed to be of a comparatively 

 short duration. 



