﻿Energy 
  of 
  Negative 
  Electrons 
  emitted 
  by 
  Hot 
  Bodies. 
  355 
  

  

  and 
  V 
  is 
  the 
  difference 
  of 
  potential 
  between 
  the 
  two 
  plates. 
  

   There 
  is 
  no 
  reason 
  why 
  all 
  the 
  ions 
  should 
  be 
  emitted 
  from 
  

   the 
  hot 
  metal 
  with 
  the 
  same 
  velocity. 
  If 
  we 
  consider 
  a 
  

   sufficiently 
  great 
  number 
  the 
  Telocity 
  or 
  energy 
  will 
  be 
  

   distributed, 
  anions: 
  them 
  according 
  to 
  some 
  regular 
  law. 
  

   Out 
  of 
  any 
  large 
  number 
  of 
  ions 
  emitted 
  by 
  the 
  plate 
  let 
  us 
  

   denote 
  the 
  fraction 
  having 
  velocity 
  components 
  between 
  n 
  

   and 
  u 
  + 
  du 
  by 
  J?(n 
  )du 
  , 
  those 
  between 
  v 
  and 
  v 
  ±dv 
  , 
  and 
  

   ivq 
  and 
  ic 
  + 
  dw 
  by 
  f(v 
  )dv 
  and 
  f(iv 
  Q 
  )dw 
  respectively. 
  Here 
  

   r 
  and 
  ic 
  are 
  the 
  initial 
  components 
  of 
  velocity 
  parallel 
  to 
  

   y 
  and 
  z 
  respectively 
  ; 
  if 
  the 
  planes 
  are 
  of 
  sufficient 
  extent 
  

   the 
  condition 
  that 
  the 
  ions 
  should 
  reach 
  the 
  upper 
  plane 
  will 
  

   be 
  independent 
  of 
  these 
  components 
  of 
  velocity. 
  If 
  n 
  is 
  the 
  

   total 
  number 
  of 
  ions 
  emitted 
  by 
  the 
  lower 
  plane 
  per 
  second 
  

   the 
  current 
  i 
  to 
  the 
  upper 
  plane 
  is 
  clearly 
  

  

  i=c—r=ne\ 
  _T(u 
  )du 
  \ 
  /(i^dvA 
  f(w 
  )dw 
  , 
  (3) 
  

  

  "* 
  J 
  / 
  e 
  */— 
  oo 
  J— 
  qo 
  

  

  V 
  2-V 
  

   v 
  m 
  

  

  where 
  c 
  is 
  the 
  capacity 
  of 
  the 
  system. 
  

  

  Circular 
  Plate. 
  

  

  In 
  practice 
  it 
  is 
  impossible 
  to 
  use 
  planes 
  of 
  indefinite 
  

   extent, 
  so 
  that 
  it 
  is 
  important 
  to 
  determine 
  the 
  effect 
  of 
  the 
  

   finite 
  size 
  of 
  the 
  two 
  plates. 
  In 
  the 
  experiments 
  the 
  plates 
  

   were 
  circular, 
  the 
  lower 
  being 
  somewhat 
  larger 
  than 
  the 
  

   upper 
  one. 
  The 
  upper 
  plate 
  was 
  surrounded 
  by 
  a 
  guard- 
  

   ring. 
  The 
  hot 
  metal 
  could 
  be 
  treated 
  with 
  sufficient 
  appro- 
  

   ximation 
  as 
  a 
  point 
  at 
  the 
  centre 
  of 
  the 
  lower 
  plate. 
  We 
  

   shall 
  now 
  calculate 
  the 
  number 
  of 
  ions 
  which 
  reach 
  the 
  upper 
  

   plate, 
  everything 
  being 
  supposed 
  to 
  be 
  symmetrical 
  about 
  

   the 
  axis 
  of 
  the 
  two 
  circles. 
  Consider 
  a 
  charged 
  particle 
  

   whose 
  distance 
  from 
  this 
  axis 
  is 
  p, 
  and 
  whose 
  distance 
  from 
  

   the 
  lower 
  plate 
  is 
  x. 
  Its 
  equations 
  of 
  motion 
  are 
  

  

  dt 
  2 
  

  

  

  3V 
  

  

  and 
  

  

  d?p_ 
  

   df- 
  

  

  0. 
  

  

  (4) 
  

  

  Integrating 
  these 
  subject 
  to 
  the 
  conditions 
  that 
  when 
  

  

  dx 
  

  

  and 
  

  

  t=0, 
  x 
  = 
  0, 
  ^="o, 
  p=0, 
  

   dp 
  

  

  dt 
  

  

  =w. 
  

  

  2B2 
  

  

  