﻿Motion 
  of 
  Electrons 
  in 
  Solids. 
  221 
  

  

  right, 
  vre 
  ot* 
  course 
  obtain 
  the 
  expectation 
  of 
  the 
  value 
  of 
  

   the 
  term 
  on 
  the 
  left. 
  In 
  taking 
  these 
  expectations 
  of 
  value,. 
  

   we 
  put 
  

  

  tu 
  = 
  ij/e, 
  S 
  r 
  = 
  5) 
  ly 
  = 
  . 
  

   X?i2 
  = 
  Sy2 
  = 
  Szr 
  = 
  IlT/;«, 
  Xuv= 
  ... 
  =0. 
  

  

  Also 
  there 
  is 
  no 
  correlation 
  between 
  velocities 
  and 
  position,, 
  

   and 
  we 
  may 
  put 
  

  

  and 
  the 
  same 
  for 
  all 
  differential 
  coefficients 
  of 
  odd 
  order. 
  

   Equation 
  (78) 
  now 
  assumes 
  the 
  form 
  

  

  (^ 
  \" 
  ■ 
  

   -7. 
  j 
  ^x—P>^^xt 
  ' 
  ' 
  (79) 
  

  

  where 
  pn 
  depends 
  only 
  on 
  RT/??i, 
  e/m, 
  and 
  on 
  averages 
  of 
  

   differential 
  coefficients, 
  and 
  of 
  products 
  of 
  differential 
  

   coefficients 
  of 
  Y 
  *. 
  

  

  Corresponding 
  to 
  a 
  definite 
  value 
  ii 
  of 
  i. 
  at 
  time 
  ti, 
  the 
  

   expectation 
  of 
  ix 
  at 
  time 
  fg* 
  say 
  2*2, 
  is 
  given 
  by 
  

  

  in 
  which 
  the 
  expectation 
  of 
  every 
  term 
  has 
  to 
  be 
  taken. 
  

   Thus 
  from 
  equation 
  (79) 
  

  

  h 
  = 
  ii-^ 
  {t.-t,} 
  2^,ii 
  + 
  i 
  (h-hy 
  F2h 
  + 
  •- 
  , 
  • 
  (80) 
  

   also, 
  for 
  great 
  values 
  of 
  ^2—^1^ 
  the 
  relation 
  is 
  known 
  to 
  be 
  

  

  = 
  h-{t2-'h)ei,-{-}, 
  (t2-hye\ 
  + 
  , 
  . 
  . 
  . 
  (81> 
  

  

  where, 
  as 
  before, 
  6 
  = 
  l^e^/mK. 
  

  

  For 
  great 
  values 
  of 
  (^2~^i)^ 
  equations 
  (80) 
  and 
  (81)- 
  

  

  * 
  Relation 
  (79) 
  can 
  be 
  deriTed 
  more 
  simply, 
  and 
  by 
  a 
  method 
  which 
  

   some 
  may 
  think 
  more 
  rigorous, 
  by 
  an 
  application 
  of 
  statistical 
  mechanics. 
  

  

  Let 
  all 
  possible 
  states 
  of 
  the 
  electrons 
  in 
  a 
  unit 
  of 
  volume 
  be 
  repre- 
  

   sented 
  in 
  the 
  appropriate 
  generalized 
  space. 
  Let 
  S 
  denote 
  the 
  region 
  or 
  

   regions 
  of 
  this 
  space 
  for 
  which 
  '%ii 
  = 
  ix/e, 
  where 
  ix 
  has 
  a 
  definite 
  given 
  

   value. 
  Equation 
  (78) 
  is 
  true 
  for 
  the 
  configuration 
  represented 
  by 
  every 
  

   point 
  in 
  the 
  generalized 
  space 
  : 
  it 
  is 
  therefore 
  true 
  throughout 
  S. 
  

   Multiply 
  it 
  by 
  the 
  element 
  of 
  volume 
  in 
  the 
  generalized 
  space, 
  integrate 
  

   throughout 
  S, 
  divide 
  by 
  the 
  volume 
  of 
  S, 
  and 
  we 
  obtain 
  equation 
  (79). 
  

   at 
  once. 
  

  

  