﻿198 
  Mr. 
  F, 
  E. 
  Wood 
  : 
  A 
  Criticism 
  of 
  

  

  However, 
  the 
  areas 
  associated 
  with 
  corresponding 
  velocity 
  

   intervals 
  have 
  the 
  same 
  ratio, 
  by 
  (25), 
  as 
  the 
  ratio 
  of 
  the 
  

   total 
  areas, 
  and 
  so 
  we 
  have 
  

  

  Theorem 
  I. 
  The 
  number 
  of 
  molecules 
  in 
  a 
  gas 
  at 
  temper- 
  

   ature 
  1 
  which 
  have 
  velocities 
  in 
  a 
  given 
  interval 
  is 
  equal 
  to 
  

   the 
  number 
  of 
  molecules 
  of 
  this 
  gas 
  at 
  temperature 
  2 
  having 
  

   velocities 
  in 
  the 
  corresponding 
  velocity 
  interval. 
  

  

  From 
  this 
  theorem 
  we 
  have 
  the 
  

  

  Corollary. 
  Hie 
  number 
  of 
  molecules 
  in 
  a 
  gas 
  at 
  temper- 
  

   ature 
  1 
  which 
  have 
  velocities 
  less 
  than 
  or 
  equal 
  to 
  an 
  arbitrary 
  

   velocity 
  v 
  1 
  is 
  equal 
  to 
  the 
  nurnber 
  of 
  molecules 
  in 
  the 
  same 
  gaf 
  

   at 
  temperature 
  2 
  which 
  have 
  velocities 
  less 
  than 
  or 
  equal 
  to 
  

  

  the 
  corresponding 
  velocity 
  \J 
  —Vj 
  *. 
  

  

  The 
  velocity 
  of 
  any 
  particular 
  molecule 
  of 
  this 
  gas 
  at 
  

   temperature 
  1 
  is 
  constantly 
  changing, 
  owing 
  to 
  impacts, 
  etc. 
  ;. 
  

   so 
  it 
  is 
  difficult 
  to 
  consider 
  the 
  change 
  of 
  velocity 
  due 
  to 
  a 
  

   rise 
  of 
  temperature 
  alone. 
  Let 
  us 
  consider 
  an 
  ideal 
  gas 
  in 
  

   which 
  the 
  velocities 
  of 
  the 
  various 
  molecules 
  are 
  such 
  that 
  

   (1) 
  the 
  distribution 
  of 
  velocities 
  obeys 
  Maxwell's 
  law 
  for 
  

   any 
  temperature, 
  and 
  (2) 
  the 
  velocity 
  of 
  each 
  molecule 
  does 
  

   not 
  change 
  while 
  the 
  temperature 
  remains 
  constant. 
  We 
  

   assume 
  that 
  the 
  results 
  obtained 
  from 
  a 
  study 
  of 
  this 
  ideal 
  

   gas 
  will 
  hold 
  also 
  for 
  an 
  actual 
  gas. 
  By 
  this 
  device, 
  we 
  can 
  

   consider 
  a 
  single 
  molecule, 
  or 
  a 
  group 
  of 
  molecules, 
  at 
  a 
  

   given 
  velocity 
  instead 
  of 
  the 
  succession 
  of 
  molecules, 
  or 
  

   groups, 
  which 
  have 
  that 
  velocity 
  at 
  various 
  moments 
  of 
  time* 
  

   Moreover, 
  if 
  the 
  temperature 
  of 
  this 
  gas 
  be 
  raised 
  to 
  2 
  , 
  each 
  

   molecule, 
  in 
  general, 
  will 
  have 
  a 
  new 
  velocity, 
  since 
  the 
  

   mean-square 
  velocity 
  has 
  been 
  changed, 
  and 
  the 
  new 
  distri- 
  

   bution 
  of 
  velocities 
  will 
  also 
  obey 
  MaxwelPs 
  law. 
  The 
  new 
  

   velocity 
  of 
  a 
  molecule 
  for 
  given 
  values 
  of 
  0i 
  and 
  2 
  wll 
  l 
  De 
  a 
  

   continuous 
  function 
  of 
  the 
  former 
  velocity 
  ; 
  also 
  molecules 
  

   whose 
  velocities 
  for 
  @ 
  = 
  1 
  are 
  zero, 
  will 
  again 
  have 
  zero 
  (or 
  

   nearly 
  zero) 
  velocities 
  for 
  6 
  = 
  2 
  . 
  Then 
  the 
  molecules 
  

   having 
  velocities 
  in 
  the 
  interval 
  §Sv<v 
  x 
  for 
  = 
  1 
  must 
  have 
  

   their 
  velocities 
  in 
  some 
  interval 
  0<v<v 
  2 
  f° 
  r 
  = 
  02' 
  From 
  

  

  * 
  If 
  the 
  velocity 
  at 
  which 
  the 
  maximum 
  number 
  of 
  molecules 
  move 
  

   be 
  called 
  the 
  mode, 
  then 
  it 
  follows 
  from 
  this 
  theorem 
  and 
  corollary, 
  

   (1) 
  that 
  the 
  mode 
  varies 
  directly 
  as 
  the 
  square 
  root 
  of 
  the 
  absolute 
  tem- 
  

   perature, 
  (2) 
  that 
  when 
  the 
  temperature 
  is 
  increased, 
  the 
  number 
  of 
  

   molecules 
  whose 
  velocity 
  is 
  at 
  the 
  new 
  mode 
  is 
  less 
  than 
  the 
  number 
  

   whose 
  velocity 
  was 
  at 
  the 
  former 
  mode, 
  varying 
  inversely 
  as 
  the 
  tem- 
  

   perature, 
  and 
  (3) 
  that 
  the 
  fraction 
  of 
  the 
  total 
  number 
  of 
  molecules 
  

   which 
  have 
  velocities 
  less 
  than 
  or 
  equal 
  to 
  the 
  mode 
  is 
  independent 
  of 
  

   the 
  temperature. 
  

  

  