﻿Hypothesis 
  as 
  to 
  the 
  Origin 
  of 
  Series 
  Spectra. 
  1013 
  

   The 
  Steady 
  Motions. 
  

  

  For 
  a 
  steady 
  motion 
  of 
  the 
  negative 
  electron 
  the 
  orbit 
  

   must 
  be 
  a 
  circle 
  having 
  the 
  ^-axis 
  as 
  axis. 
  In 
  general, 
  for 
  

   this 
  to 
  take 
  place 
  the 
  attractive 
  electrostatic 
  force 
  must 
  

   balance 
  the 
  repulsive 
  mechanical 
  force, 
  and 
  so 
  we 
  must 
  have, 
  

   if 
  m 
  be 
  the 
  mass 
  of 
  the 
  electron, 
  

  

  « 
  = 
  2*/m(P/r 
  3 
  + 
  Q) 
  

  

  3a>0r 
  2 
  + 
  ?/ 
  2 
  )P/r 
  5 
  =.(?R/r 
  s 
  . 
  

  

  "We 
  now 
  make 
  use 
  of 
  our 
  assumption 
  that 
  the 
  electric 
  

   density 
  of 
  the 
  interior 
  of 
  the 
  atom 
  is 
  uniform 
  and 
  that 
  the 
  

   electron 
  is 
  moving 
  in 
  this 
  region. 
  On 
  evaluating 
  the 
  inte- 
  

   grals 
  the 
  above 
  equations 
  take 
  the 
  forms 
  

  

  a> 
  = 
  § 
  e/mMY 
  -% 
  Trpflr 
  2 
  . 
  e/m. 
  . 
  . 
  . 
  (l) 
  

  

  K* 
  2 
  +i/>a 
  = 
  c 
  2 
  (2) 
  

  

  where 
  Y 
  = 
  \ 
  kirp'rdr 
  denotes 
  the 
  potential 
  of 
  the 
  atom 
  at 
  

  

  Jo 
  

   its 
  centre. 
  It 
  will 
  be 
  noticed 
  that 
  (2) 
  expresses 
  the 
  fact 
  that 
  

   in 
  every 
  steady 
  motion 
  the 
  angular 
  momentum 
  of 
  the 
  negative 
  

   electron 
  has 
  the 
  same 
  constant 
  value. 
  

  

  It 
  may 
  be 
  remarked 
  that 
  the 
  theory 
  of 
  Bohr* 
  starts 
  with 
  an 
  

   assumption 
  that 
  the 
  angular 
  momentum 
  of 
  an 
  electron 
  about 
  

   the 
  centre 
  of 
  force 
  is 
  constant 
  and 
  equal 
  to 
  7*/27r 
  where 
  h 
  is 
  

   Planck's 
  constant. 
  In 
  our 
  theory 
  the 
  constant 
  is 
  found 
  to 
  

   be 
  (within 
  the 
  limits 
  of 
  experimental 
  error) 
  h/ir. 
  The 
  con- 
  

   nexion 
  between 
  the 
  two 
  theories 
  so 
  very 
  different 
  from 
  one 
  

   another 
  can 
  be 
  still 
  more 
  clearly 
  shown 
  when 
  it 
  is 
  remem- 
  

   bered 
  that 
  the 
  frequency 
  of 
  the 
  radiation 
  emitted 
  by 
  the 
  Bohr 
  

   electron 
  is 
  one 
  half 
  of 
  the 
  frequency 
  of 
  rotation 
  in 
  the 
  

   permanent 
  state. 
  

  

  As 
  regards 
  the 
  stability 
  of 
  these 
  motions, 
  one 
  finds 
  that 
  

   two 
  of 
  the 
  oscillations 
  about 
  the 
  steady 
  motion 
  are 
  apparently 
  

   neutral 
  and 
  the 
  third 
  is 
  stable. 
  

  

  A 
  further 
  question 
  arises, 
  that 
  of 
  the 
  applicability 
  of 
  the 
  

   above 
  formulae 
  for 
  high 
  speeds. 
  The 
  formula? 
  for 
  the 
  mag- 
  

   netic 
  forces 
  are 
  true 
  no 
  matter 
  what 
  magnitude 
  Q 
  has. 
  As 
  

   for 
  the 
  motion 
  of 
  the 
  negative 
  electron, 
  its 
  velocity 
  is 
  of 
  the 
  

   order 
  10" 
  3 
  of 
  the 
  velocity 
  of 
  light, 
  so 
  that 
  its 
  mass 
  m 
  may 
  be 
  

   taken 
  to 
  have 
  its 
  Newtonian 
  value. 
  

  

  * 
  Phil. 
  Mag. 
  July 
  and 
  Sept. 
  1913. 
  

  

  