﻿and 
  the 
  Constitution 
  of 
  the 
  Atom. 
  295 
  

  

  which 
  proved 
  to 
  be 
  of 
  very 
  great 
  importance. 
  He 
  intro- 
  

   duces 
  generalized 
  coordinates 
  qi 
  (coordinates 
  of 
  position) 
  

   and 
  pi 
  (coordinates 
  of 
  momentum), 
  and 
  takes 
  as 
  the 
  general 
  

   quant-condition 
  for 
  stable 
  orbits 
  : 
  

  

  r 
  

  

  p 
  i 
  dq 
  i 
  = 
  nh, 
  . 
  ; 
  (3) 
  

  

  where 
  the 
  integral 
  is 
  to 
  be 
  extended 
  over 
  one 
  period. 
  

  

  This 
  general 
  quant-condition 
  involves 
  the 
  new 
  question 
  as 
  

   to 
  the 
  choice 
  of 
  coordinates. 
  

  

  Sommerfeld 
  carries 
  out 
  the 
  calculation 
  in 
  polar 
  coordinates 
  

   (?"\Jr3) 
  for 
  a 
  single 
  electron 
  moving 
  round 
  a 
  positive 
  charge. 
  

   The 
  three 
  quant-conditions 
  then 
  take 
  the 
  form 
  

  

  \ 
  P^ 
  y 
  l 
  r 
  = 
  n 
  ih 
  jjtv2 
  r 
  =>i'#, 
  \p^d$ 
  = 
  n 
  2 
  h. 
  . 
  

  

  (4) 
  

  

  If 
  we 
  merely 
  consider 
  orbits 
  in 
  a 
  plane, 
  only 
  the 
  first 
  two 
  

   conditions 
  will 
  be 
  wanted. 
  

  

  Carrying 
  out 
  the 
  calculation 
  in 
  the 
  case 
  of 
  hydrogen 
  

   he 
  gets 
  : 
  

  

  V 
  = 
  R 
  Xin^n'y" 
  {m^m'yj 
  ' 
  * 
  " 
  ' 
  (5) 
  

  

  mi, 
  m! 
  being 
  the 
  value 
  of 
  n^ 
  n' 
  for 
  the 
  orbit 
  from 
  which 
  the 
  

   electrons 
  recombine. 
  

  

  In 
  order 
  to 
  give 
  the 
  ordinary 
  Balmer 
  series 
  it 
  is 
  

   necessary 
  that 
  

  

  N 
  = 
  n 
  1 
  + 
  w' 
  = 
  2 
  

  

  and 
  that 
  M 
  = 
  m 
  1 
  -\-m' 
  is 
  an 
  integer 
  number 
  > 
  2. 
  Further, 
  

   the 
  number 
  of 
  combinations 
  is 
  limited 
  by 
  the 
  conditions 
  

  

  m' 
  5: 
  n 
  . 
  

  

  This 
  new 
  form 
  of 
  the 
  frequency 
  formula 
  is 
  so 
  far 
  identical 
  

   with 
  that 
  of 
  Bohr 
  that 
  it 
  gives 
  the 
  same 
  position 
  of 
  the 
  

   lines, 
  but 
  it 
  is 
  different 
  with 
  regard 
  to 
  the 
  way 
  in 
  which 
  we 
  

   may 
  imagine 
  the 
  lines 
  to 
  be 
  produced. 
  Thus 
  the 
  H-line 
  

   may 
  be 
  produced 
  in 
  four 
  different 
  ways, 
  as, 
  e.g., 
  by 
  re- 
  

   combination 
  from 
  an 
  elliptic 
  orbit 
  (ni 
  1 
  = 
  2, 
  m' 
  = 
  l). 
  The 
  

   case 
  of 
  circular 
  orbits 
  treated 
  by 
  Bohr 
  is 
  only 
  a 
  special 
  

   case 
  of 
  manv, 
  and 
  corresponds 
  to 
  a 
  recombination 
  from 
  

   (m 
  l 
  = 
  Z 
  f 
  m' 
  = 
  0) 
  to 
  (^ 
  = 
  2, 
  n' 
  = 
  0). 
  

  

  The 
  H-lines 
  which 
  are 
  produced 
  in 
  these 
  different 
  ways 
  

   are 
  only 
  with 
  certaint3 
  r 
  identical 
  provided 
  we 
  may 
  treat 
  the 
  

   system 
  as 
  consisting 
  of 
  two 
  constant 
  masses 
  attracting 
  each 
  

  

  Y2 
  

  

  