﻿154 
  ANNUAL, 
  REPORT 
  SMITHSONIAN 
  INSTITUTION, 
  1922. 
  

  

  successfully 
  to 
  account 
  for 
  these 
  phenomena 
  in 
  more 
  complicated 
  sys- 
  

   tems, 
  nor 
  for 
  the 
  magnetic 
  properties 
  of 
  atoms. 
  

  

  For 
  the 
  simplest 
  case, 
  a 
  relatively 
  heavy 
  nucleus 
  of 
  positive 
  charge 
  

   E 
  and 
  a 
  negative 
  electron 
  of 
  charge 
  e 
  and 
  mass 
  m 
  rotating 
  n' 
  times 
  

   per 
  second 
  about 
  the 
  nucleus 
  in 
  a 
  circular 
  orbit 
  of 
  radius 
  a, 
  we 
  have 
  

   equilibrium 
  if 
  the 
  electric 
  attraction 
  is 
  just 
  balanced 
  by 
  the 
  centrif- 
  

   ugal 
  force, 
  or 
  

  

  2= 
  = 
  (2x71 
  ') 
  a 
  

   a 
  2 
  

  

  The 
  total 
  energy, 
  kinetic 
  plus 
  potential, 
  is 
  easily 
  shown 
  to 
  be 
  

  

  2 
  a 
  

   By 
  the 
  quantum 
  law 
  fp 
  dq 
  = 
  h# 
  it 
  is 
  found 
  that 
  the 
  only 
  possible 
  

   values 
  of 
  W 
  are 
  those 
  for 
  which 
  W 
  = 
  — 
  y 
  2 
  s 
  h 
  n% 
  where 
  s 
  is 
  any 
  

   integer. 
  By 
  solving 
  these 
  three 
  equations 
  simultaneously 
  we 
  find 
  the 
  

   various 
  possible 
  energies 
  and 
  radii 
  of 
  the 
  atom 
  to 
  be 
  given 
  by 
  sub- 
  

   stituting 
  the 
  various 
  integral 
  values 
  of 
  s 
  in 
  the 
  equations 
  

  

  __ 
  2^me 
  2 
  E 
  2 
  s 
  2 
  h 
  2 
  

  

  W= 
  in 
  — 
  i 
  a 
  = 
  

  

  2 
  h 
  2 
  ' 
  u 
  '~47r 
  2 
  me£ 
  

  

  Since 
  the 
  difference 
  of 
  energy 
  in 
  any 
  two 
  states 
  equals 
  hn, 
  the 
  vari- 
  

   ous 
  possible 
  radiation 
  frequencies 
  are 
  given 
  by 
  n 
  = 
  (W 
  m 
  — 
  W 
  r 
  )/h, 
  or 
  

  

  _ 
  2TT 
  2 
  me 
  2 
  E 
  2 
  / 
  1 
  1 
  \ 
  

   n 
  ~ 
  h 
  3 
  K^'m 
  2 
  )' 
  

  

  where 
  r 
  and 
  m 
  are 
  any 
  two 
  integral 
  values 
  of 
  s. 
  E 
  is 
  simply 
  the 
  

   atomic 
  number 
  of 
  the 
  element 
  times 
  the 
  electronic 
  charge 
  e. 
  

  

  These 
  three 
  equations 
  are 
  in 
  exact 
  accord 
  with 
  all 
  experimental 
  

   evidence 
  available. 
  The 
  spectral 
  tests 
  of 
  the 
  latter 
  equation 
  are 
  par- 
  

   ticularly 
  severe 
  and 
  convincing, 
  since 
  substitution 
  of 
  the 
  known 
  

   values 
  of 
  the 
  constants 
  makes 
  the 
  term 
  outside 
  the 
  parenthesis 
  exactly 
  

   equal 
  to 
  the 
  observed 
  Rydberg 
  constant 
  A 
  7 
  ", 
  so 
  that 
  the 
  equation 
  is 
  

   identical 
  with 
  the 
  spectral 
  series 
  formula. 
  

  

  This 
  theory 
  has 
  been 
  extended 
  to 
  take 
  account 
  of 
  the 
  small 
  motion 
  

   of 
  the 
  nucleus 
  as 
  the 
  electron 
  revolves, 
  of 
  possible 
  elliptic 
  as 
  well 
  as 
  

   circular 
  orbits, 
  of 
  the 
  variation 
  of 
  the 
  mass 
  of 
  an 
  electron 
  with 
  its 
  

   speed 
  and 
  of 
  the 
  effect 
  of 
  placing 
  the 
  atom 
  in 
  a 
  strong 
  electric 
  field. 
  

   In 
  every 
  case 
  the 
  theory 
  leads 
  to 
  results 
  in 
  exact 
  accord 
  with 
  the 
  

   facts. 
  When 
  dealing 
  with 
  systems 
  with 
  several 
  negative 
  electrons 
  

   outside 
  the 
  nucleus, 
  the 
  problem 
  of 
  the 
  way 
  in 
  which 
  they 
  and 
  their 
  

   orbits 
  are 
  distributed 
  in 
  space 
  must 
  be 
  considered. 
  Models 
  with 
  

   coplanar, 
  parallel 
  and 
  crossed 
  orbits 
  have 
  been 
  considered, 
  with 
  the 
  

   latter 
  giving, 
  on 
  the 
  whole, 
  the 
  best 
  results. 
  But 
  the 
  computations 
  

   are 
  very 
  complicated, 
  and 
  but 
  little 
  progress 
  has 
  been 
  made 
  with 
  

   such 
  systems 
  or 
  with 
  molecules. 
  

  

  