﻿Energy 
  of 
  Negative 
  Electrons 
  emitted 
  by 
  Hot 
  Bodies. 
  357 
  

  

  the 
  molecules 
  of 
  a 
  gas 
  which 
  start 
  from 
  any 
  surface 
  bounding 
  

   it 
  or 
  within 
  it, 
  then 
  the 
  above 
  functions 
  may 
  be 
  calculated 
  

   by 
  the 
  methods 
  of 
  the 
  kinetic 
  theory 
  of 
  gases. 
  They 
  are 
  

  

  F 
  {u 
  )du 
  = 
  2km 
  u 
  Q 
  e- 
  kmu 
  °*du 
  , 
  f(v 
  Q 
  ) 
  dv 
  = 
  /\J 
  — 
  e 
  - 
  kmv 
  * 
  2 
  dv 
  

  

  .f(w 
  )dw 
  Q 
  =\/—e- 
  kmw 
  <?dw 
  and 
  F'(W)dW 
  = 
  2kmWe-^ 
  r2 
  dy> 
  

  

  where 
  |-& 
  is 
  the 
  average 
  energy 
  of 
  translation 
  of 
  a 
  molecule 
  

   at 
  the 
  temperature 
  of 
  the 
  hot 
  body. 
  It 
  is 
  to 
  be 
  borne 
  in 
  

   mind 
  that 
  the 
  above 
  functions 
  are 
  expressed 
  as 
  fractions 
  of 
  

   the 
  total 
  number 
  of 
  ions 
  leaving 
  an 
  element 
  of 
  area 
  perpen- 
  

   dicular 
  to 
  the 
  axis 
  x 
  in 
  unit 
  time, 
  and 
  not 
  in 
  terms 
  of 
  the 
  

   number 
  in 
  unit 
  volume 
  as 
  is 
  usually 
  done. 
  

  

  If 
  w 
  T 
  e 
  substitute 
  these 
  values 
  of 
  the 
  functions 
  in 
  the 
  pre- 
  

   ceding 
  formulae 
  and 
  carry 
  out 
  the 
  integrations, 
  we 
  shall 
  

   obtain 
  the 
  current 
  to 
  the 
  upper 
  plate 
  as 
  a 
  function 
  of 
  the 
  

   potential-difference, 
  provided 
  the 
  law 
  of 
  distribution 
  of 
  velocity 
  

   among 
  the 
  emitted 
  electrons 
  is 
  Maxwell's 
  law. 
  Under 
  these 
  

   1 
  circumstances, 
  in 
  the 
  case 
  where 
  the 
  planes 
  are 
  of 
  indefinite 
  

   extent, 
  the 
  current 
  to 
  the 
  upper 
  plate 
  becomes 
  

  

  i 
  

  

  2(hn) 
  

   i 
  = 
  ii 
  e~ 
  — 
  - 
  

  

  ■ 
  1 
  duu€- 
  kmn2 
  \ 
  dv 
  e 
  - 
  kmv2 
  1 
  

  

  %) 
  / 
  e 
  %J 
  — 
  oo 
  %) 
  - 
  

  

  V2-V 
  

  

  =„«€-»▼? 
  = 
  Le- 
  2kY 
  ' 
  

  

  (10) 
  

  

  if 
  i 
  Q 
  is 
  the 
  value 
  of 
  the 
  current 
  at 
  the 
  initial 
  instant 
  when 
  

  

  V 
  = 
  0. 
  Since 
  k= 
  ^75-/) 
  where 
  R 
  : 
  is 
  the 
  constant 
  in 
  the 
  gas 
  

  

  equation 
  pv 
  = 
  ¥ii6, 
  calculated 
  for 
  a 
  single 
  molecule, 
  and 
  is 
  

   the 
  absolute 
  temperature, 
  we 
  have, 
  taking 
  logarithms 
  

  

  where 
  v 
  is 
  the 
  number 
  of 
  molecules 
  in 
  1 
  c.c. 
  of 
  gas 
  at 
  0° 
  C. 
  

   and 
  760 
  mms. 
  pressure, 
  and 
  R 
  is 
  the 
  constant 
  in 
  the 
  equation 
  

   pv 
  — 
  R# 
  taken 
  to 
  refer 
  to 
  the 
  quantity 
  of 
  gas 
  occupying 
  unit 
  

   volume 
  under 
  these 
  standard 
  conditions. 
  Assuming 
  what 
  is 
  

   now 
  fairly 
  well 
  established, 
  that 
  the 
  charge 
  e 
  on 
  an 
  electron 
  

   is 
  equal 
  to 
  that 
  carried 
  by 
  a 
  monovalent 
  ion 
  during 
  electro- 
  

   lysis, 
  ve 
  is 
  equal 
  to 
  the 
  quantity 
  of 
  electricity 
  required 
  to 
  

   liberate 
  half 
  a 
  cubic 
  centimetre 
  of 
  hydrogen 
  in 
  a 
  water 
  

   voltameter 
  under 
  standard 
  conditions 
  of 
  temperature 
  and 
  

  

  (9) 
  

  

  