﻿224 
  Prof. 
  J. 
  H. 
  Jeans 
  on 
  the 
  

  

  The 
  Law 
  of 
  Force. 
  

  

  43. 
  To 
  reconcile 
  experiment 
  with 
  electron-theory, 
  we 
  have- 
  

   found 
  that 
  at 
  low 
  tHmperatures 
  the 
  -value 
  of 
  /c 
  must 
  be 
  of 
  the 
  

   form 
  given 
  by 
  equation 
  (87). 
  But 
  actual 
  calculation 
  has. 
  

   shown 
  that 
  k 
  must 
  have 
  the 
  form 
  given 
  by 
  equation 
  (83;. 
  

   Comparing 
  these 
  two 
  values 
  for 
  /c, 
  we 
  find 
  

  

  E2 
  = 
  EPw/6VySy. 
  ..... 
  (89) 
  

  

  Thus 
  the 
  average 
  square 
  of 
  the 
  force 
  (^^E^) 
  acting 
  on 
  an 
  

   electron 
  must 
  be 
  of 
  the 
  form 
  gX}^ 
  where 
  g 
  is 
  the 
  absolute 
  

   constant 
  Ilm/a^/3^7^. 
  We 
  can 
  easily 
  see 
  why 
  ^^E^ 
  increases 
  

   with 
  T 
  ; 
  at 
  higher 
  temperatures 
  the 
  kinetic 
  energies 
  of 
  the 
  

   electrons 
  are 
  greater, 
  and 
  the 
  electrons 
  consequently 
  penetrate 
  

   to 
  regions 
  at 
  which 
  the 
  force 
  is 
  greater. 
  The 
  knowledge 
  of 
  

   the 
  exact 
  relation 
  between 
  E^ 
  and 
  T 
  enables 
  us 
  to 
  find 
  the^ 
  

   law 
  of 
  force 
  under 
  which 
  the 
  electrons 
  move. 
  

  

  44. 
  Consider 
  firsi 
  a 
  temperature 
  T 
  = 
  0. 
  The 
  electrons^ 
  

   have 
  no 
  kinetic 
  energy, 
  and 
  so 
  fall 
  into 
  positions 
  of 
  equilibrium.. 
  

   Thus 
  E^ 
  = 
  0, 
  in 
  accordance 
  with 
  equation 
  (89). 
  

  

  Considering 
  next 
  small 
  values 
  of 
  T, 
  we 
  can 
  find 
  the 
  nature 
  

   of 
  these 
  positions 
  of 
  equilibrium. 
  Suppose, 
  first, 
  that 
  these^ 
  

   positions 
  were 
  represented 
  by 
  ordinary 
  minima 
  of 
  potential 
  

   (if 
  such 
  could 
  exist), 
  then 
  at 
  a 
  small 
  temperature 
  T 
  the 
  

   electrons 
  would 
  oscillate 
  about 
  these 
  positions. 
  The 
  average 
  

   squared 
  force 
  (EV^) 
  would 
  be 
  proportional 
  to 
  the 
  average 
  

   square 
  of 
  the 
  displacement, 
  and 
  therefore 
  to 
  T, 
  and 
  equation 
  

   (89) 
  would 
  not 
  be 
  satisfied. 
  Thus 
  the 
  zero 
  value 
  of 
  E^" 
  

   which 
  indicates 
  rest 
  in 
  a 
  position 
  of 
  minimum 
  potential 
  is 
  

   not 
  a 
  limiting 
  solution 
  of 
  equation 
  (89). 
  

  

  There 
  is 
  only 
  one 
  alternative— 
  E^ 
  must 
  vanish 
  when 
  T 
  = 
  0, 
  

   on 
  account 
  of 
  the 
  electrons 
  being 
  at 
  infinity, 
  or 
  being 
  so 
  far 
  

   removed 
  from 
  the 
  centre 
  of 
  force 
  that 
  the 
  forces 
  acting 
  on 
  

   them 
  are 
  negligible. 
  

  

  Suppose, 
  first, 
  that 
  each 
  electron 
  is 
  acted 
  on 
  by 
  only 
  one 
  

   centre 
  of 
  force 
  of 
  law 
  yi\r^. 
  The 
  potential 
  is 
  {^i—\)yi.jr^~^^ 
  

   so 
  that 
  at 
  temperature 
  T 
  the 
  law 
  of 
  distribution 
  of 
  distances 
  

   is 
  ^-(^-l)/^/^^^^"~S'Vr, 
  and 
  the 
  value 
  of 
  e'W 
  (the 
  average 
  

   value 
  of 
  /i7^") 
  is 
  found 
  to 
  be 
  proportional 
  to 
  T^"/^-^ 
  This 
  

   gives 
  relation 
  (89) 
  if 
  72 
  = 
  3, 
  and 
  if 
  /x 
  is 
  the 
  same 
  for 
  all 
  kinds 
  

   of 
  matter. 
  

  

  If 
  the 
  electrons 
  are 
  acted 
  on 
  by 
  more 
  than 
  one 
  centre 
  of 
  

   force, 
  it 
  is 
  readily 
  found 
  that 
  E^ 
  will 
  not 
  vary 
  exactly 
  as 
  any 
  

   power 
  of 
  T, 
  and 
  must 
  moreover 
  involve 
  the 
  distances 
  between 
  

   the 
  different 
  centres 
  of 
  force 
  ; 
  these 
  would 
  be 
  different 
  for 
  

   different 
  kinds 
  of 
  matter. 
  

  

  45. 
  Thus 
  we 
  must 
  suppose 
  that, 
  in 
  all 
  kinds 
  of 
  matter 
  

  

  