﻿696 
  Prof. 
  0. 
  W. 
  Richardson 
  on 
  the 
  

  

  have 
  had 
  difficulties 
  with 
  it, 
  so 
  that 
  it 
  may 
  not 
  be 
  out 
  o£ 
  

   place 
  to 
  give 
  the 
  deduction 
  here. 
  

  

  It 
  is 
  important 
  to 
  remember 
  that 
  the 
  above 
  expression 
  

   does 
  not 
  represent 
  the 
  probability 
  that 
  the 
  velocity 
  o£ 
  a 
  

   molecule 
  selected 
  from 
  a 
  given 
  volume 
  at 
  any 
  instant 
  should 
  

   lie 
  within 
  given 
  limits. 
  The 
  selection 
  has 
  to 
  be 
  made 
  from 
  

   among 
  those 
  molecules 
  which 
  leave 
  a 
  given 
  surface 
  in 
  a 
  given 
  

   time. 
  It 
  will 
  probably 
  conduce 
  to 
  clearness 
  to 
  get 
  rid 
  of 
  the 
  

   notion 
  of 
  probability 
  altogether 
  and 
  to 
  use 
  instead 
  the 
  

   number 
  ^(u)du 
  which 
  leave 
  unit 
  of 
  surface 
  with 
  a 
  velocity 
  

   component 
  perpendicular 
  to 
  the 
  surface 
  between 
  7i 
  and 
  

   u 
  + 
  du. 
  If 
  formula 
  (1) 
  is 
  true 
  it 
  is 
  clear 
  that 
  

  

  ^(u)du 
  = 
  2kmuNe-^''''''du 
  .... 
  (2) 
  

  

  where 
  N 
  is 
  the 
  number 
  of 
  all 
  sorts 
  which 
  leave 
  unit 
  area 
  in 
  

   unit 
  time. 
  It 
  is 
  necessary 
  therefore 
  to 
  prove 
  formula 
  (2j. 
  

   It 
  is 
  proved 
  in 
  practically 
  every 
  text-book 
  of 
  the 
  Kinetic 
  

   Theory 
  of 
  Gases 
  that 
  if 
  n 
  is 
  the 
  total 
  number 
  of 
  molecules 
  of 
  

   any 
  gas 
  per 
  cubic 
  centimetre 
  at 
  any 
  instant, 
  the 
  number 
  

   which 
  leave 
  unit 
  area 
  of 
  surface 
  in 
  unit 
  time 
  with 
  components 
  

   of 
  velocity 
  perpendicular 
  to 
  that 
  surface 
  between 
  u 
  and 
  

   u-\-du 
  is 
  

  

  ^(u)du 
  = 
  n{—yue-^"''''du 
  . 
  . 
  . 
  . 
  fS) 
  

  

  Substituting 
  for 
  n 
  in 
  (3) 
  in 
  terms 
  of 
  N 
  we 
  get 
  

  

  whence 
  (1) 
  follows. 
  

  

  It 
  is 
  interesting 
  to 
  observe 
  that 
  this 
  expression 
  is 
  un- 
  

   changed 
  when 
  the 
  geometrical 
  surface 
  to 
  which 
  we 
  have 
  so 
  

   far 
  confined 
  our 
  attention 
  is 
  replaced 
  by 
  a 
  physical 
  boundary 
  

   in 
  escaping 
  through 
  which 
  the 
  molecules 
  have 
  to 
  do 
  work. 
  

   The 
  author 
  has 
  shown 
  * 
  that 
  the 
  velocity 
  component 
  uq 
  

   normal 
  to 
  the 
  surface 
  after 
  escape 
  is 
  related 
  to 
  the 
  correspond- 
  

   ing 
  quantity 
  ui 
  impinging 
  on 
  the 
  surface 
  by 
  the 
  equation 
  

  

  2 
  

  

  m 
  

  

  where 
  m 
  is 
  the 
  mass 
  of 
  the 
  molecule 
  and 
  ^ 
  is 
  the 
  change 
  in 
  

   the 
  work 
  function 
  in 
  passing 
  through 
  the 
  surface. 
  The 
  

  

  * 
  Phil. 
  Trans. 
  A. 
  vol. 
  201, 
  p. 
  501 
  (1903). 
  

  

  Whence 
  

  

  