﻿80 
  Dr. 
  A. 
  C. 
  Crehore 
  on 
  the 
  Formation 
  of 
  the 
  

  

  Limiting 
  Value 
  of 
  the 
  Angular 
  Velocity 
  of 
  the 
  Electrons 
  

   to 
  remain 
  within 
  the 
  Positive 
  Sphere. 
  

  

  It 
  is 
  evident 
  that 
  the 
  term 
  containing 
  a 
  L 
  2 
  in 
  (73) 
  may 
  

   be 
  omitted 
  as 
  being 
  small 
  compared 
  with 
  unity, 
  since 
  

  

  1 
  ^mKft) 
  2 
  

   W 
  2e* 
  

   we 
  have 
  

  

  =/3 
  is 
  probably 
  small 
  compared 
  with 
  unity. 
  Hence, 
  if 
  

  

  ^- 
  (76) 
  

  

  o 
  1 
  /S 
  u\ 
  ■ 
  . 
  

  

  ^(g+i) 
  • 
  W 
  

  

  To 
  find 
  the 
  limiting 
  value 
  of 
  the 
  angular 
  velocity 
  so 
  that 
  

   the 
  electrons 
  will 
  just 
  remain 
  within 
  the 
  sphere, 
  put 
  a 
  = 
  b, 
  

  

  and 
  a*=- 
  C) 
  o 
  ,z 
  , 
  whence, 
  when 
  S 
  = 
  l, 
  P 
  = 
  l 
  ? 
  u 
  = 
  0, 
  andp 
  = 
  2, 
  

  

  2 
  le 
  2 
  

   for 
  the 
  two-electron 
  atom, 
  co 
  2 
  = 
  5.. 
  Taking 
  the 
  mass 
  of 
  

  

  hna 
  i 
  

  

  10 
  

  

  the 
  electron 
  as 
  '878 
  X 
  10" 
  27 
  gram, 
  and 
  e 
  = 
  4-77x10" 
  

  

  n 
  1 
  21300 
  A 
  l 
  

  

  electrostatic 
  unit, 
  we 
  find 
  co 
  = 
  3 
  ~ 
  • 
  As 
  has 
  been 
  

  

  explained 
  in 
  another 
  place, 
  it 
  seems 
  probable 
  that 
  the 
  orbits 
  

   of 
  the 
  electrons 
  within 
  the 
  atoms, 
  and 
  the 
  atoms 
  themselves, 
  

   are 
  many 
  times 
  smaller 
  than 
  the 
  dimensions 
  of 
  the 
  molecules, 
  

   though 
  the 
  exact 
  amount 
  is 
  uncertain. 
  Using 
  the 
  smaller 
  

   value 
  of 
  "a" 
  calculated 
  above 
  (p. 
  56) 
  as 
  -72X10 
  -12 
  cm., 
  

   we 
  find 
  n 
  = 
  *556x 
  10 
  22 
  . 
  This 
  is 
  nearly 
  1000 
  times 
  greater 
  

   than 
  X-ray 
  frequencies, 
  and 
  is 
  an 
  indication 
  that 
  the 
  linear 
  

   velocity 
  of 
  the 
  electron 
  might 
  approach 
  close 
  to 
  the 
  velocity 
  

   of 
  light, 
  at 
  which 
  speed 
  the 
  law 
  of 
  the 
  forces 
  rapidly 
  changes, 
  

   before 
  the 
  electron 
  would 
  leave 
  the 
  positive 
  sphere 
  due 
  to 
  its 
  

   own 
  centrifugal 
  force. 
  

  

  •« 
  v 
  

  

  The 
  Correction 
  Term. 
  

   In 
  finding 
  the 
  expression 
  for 
  the 
  radius 
  of 
  the 
  outside 
  

   ring 
  (73) 
  or 
  (77), 
  the 
  assumption 
  was 
  made 
  that 
  the 
  u 
  electrons 
  

   which 
  are 
  not 
  in 
  the 
  outside 
  ring 
  are 
  all 
  concentrated 
  at 
  the 
  

   centre 
  of 
  the 
  sphere. 
  It 
  can 
  be 
  shown 
  that 
  if 
  the 
  same 
  

   number 
  of 
  electrons 
  is 
  distributed 
  in 
  a 
  second 
  ring 
  within 
  

   the 
  outside 
  ring, 
  instead 
  of 
  being 
  at 
  the 
  centre 
  of 
  the 
  sphere, 
  

   the 
  outside 
  ring 
  w 
  7 
  ould 
  have 
  a 
  greater 
  radius. 
  To 
  maintaiii 
  

   the 
  outside 
  ring 
  the 
  same 
  more 
  central 
  electrons 
  are 
  required 
  

   than 
  if 
  they 
  were 
  distributed 
  in 
  inside 
  rings. 
  That 
  is, 
  the 
  

   true 
  value 
  of 
  a 
  l 
  may 
  be 
  found 
  by 
  the 
  formulae 
  (74) 
  or 
  (75) 
  

   by 
  adding 
  to 
  ?/, 
  the 
  total 
  number 
  of 
  electrons 
  in 
  the 
  inside, 
  a 
  

  

  