﻿and 
  the 
  Constitution 
  of 
  the 
  Atom, 
  299 
  

  

  secondary 
  circle 
  to 
  re-establish 
  a 
  normal 
  circuit, 
  the 
  dif- 
  

   ference 
  of 
  energy 
  of 
  the 
  whole 
  system 
  (electron 
  + 
  circuit) 
  

   before 
  and 
  after 
  recombination 
  is 
  radiated 
  in 
  a 
  single 
  

   quantum. 
  

  

  Let 
  the 
  effective 
  charge 
  of 
  the 
  nucleus 
  be 
  +N# 
  and 
  the 
  

  

  angular 
  momentum 
  of 
  each 
  electron 
  t 
  =- 
  ; 
  then, 
  as 
  shown 
  

  

  by 
  Bohr 
  and 
  Debye, 
  the 
  total 
  energy 
  of 
  a 
  ring 
  of 
  p 
  electrons 
  

   will 
  be 
  

  

  E= 
  c-V*=M 
  

  

  where 
  C 
  is 
  a 
  constant, 
  and 
  as 
  we 
  have 
  only 
  to 
  deal 
  with 
  

   differences 
  we 
  put 
  

  

  W=MR^=M 
  2 
  , 
  (7) 
  

  

  and 
  we 
  take 
  simply 
  — 
  W 
  as 
  the 
  total 
  energy. 
  R 
  is 
  Rydberg's 
  

   universal 
  frequency. 
  Debye 
  puts 
  R 
  = 
  2*7337 
  . 
  10 
  15 
  ^-, 
  or 
  in 
  

   wave-number 
  per 
  cm. 
  

  

  Further, 
  

  

  R 
  = 
  109740 
  

  

  i—p— 
  1 
  ^ 
  

  

  -i2 
  

  

  . 
  .7T 
  

  

  »=i 
  sini 
  — 
  

   P 
  

  

  Let 
  the 
  total 
  energy 
  of 
  the 
  restored 
  ring 
  be 
  — 
  Wi, 
  that 
  

   of 
  the 
  broken 
  ring 
  — 
  W 
  2 
  , 
  and 
  that 
  of 
  the 
  electron 
  in 
  the 
  

   secondary 
  circuit 
  — 
  W 
  2 
  ', 
  then 
  Debye' 
  s 
  application 
  of 
  Bohr's 
  

   frequency 
  law 
  gives 
  

  

  hv 
  = 
  W 
  1 
  -(W,+ 
  W 
  a 
  ') 
  (8) 
  

  

  Putting^? 
  = 
  3 
  and 
  t 
  — 
  2 
  he 
  finds 
  a 
  frequency 
  formula 
  which 
  

   gives 
  very 
  good 
  agreement 
  with 
  observations 
  for 
  values 
  of 
  N 
  

   smaller 
  than 
  30. 
  Above 
  this 
  value 
  there 
  is 
  a 
  considerable 
  

   deviation, 
  which 
  he 
  shows 
  to 
  be 
  due 
  to 
  the 
  fact 
  that 
  for 
  

   higher 
  atomic 
  numbers 
  the 
  mass 
  of 
  the 
  electron 
  will 
  increase 
  

   on 
  account 
  of 
  increase 
  of 
  velocity. 
  

  

  Adopting 
  a 
  similar 
  way 
  of 
  procedure 
  to 
  that 
  followed 
  by 
  

   Sommerfeld 
  in 
  his 
  theory 
  of 
  doublets, 
  Debye 
  calculates 
  the 
  

   energy 
  on 
  the 
  supposition 
  that 
  the 
  motion 
  takes 
  place 
  subject 
  

   to 
  the 
  principle 
  of 
  relativity. 
  The 
  equations 
  of 
  motion 
  under 
  

  

  