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  XLVII. 
  On 
  Wood's 
  Criticism 
  of 
  Wien's 
  Distribution 
  Law. 
  

   By 
  Harold 
  Jeffreys, 
  M.A., 
  D.Sc* 
  

  

  IN 
  the 
  February 
  number 
  of 
  the 
  Philosophical 
  Magazine,, 
  

   pp. 
  190-203, 
  Mr. 
  F. 
  E. 
  Wood 
  offers 
  a 
  criticism 
  of 
  

   Wien's 
  law 
  of 
  the 
  distribution 
  of 
  energy 
  in 
  the 
  spectrum 
  

   of 
  a 
  radiating 
  gas. 
  If 
  X 
  denote 
  the 
  wave-length 
  and 
  6 
  the 
  

   absolute 
  temperature, 
  Wien's 
  law 
  is 
  that 
  the 
  energy 
  of 
  the 
  

   part 
  of 
  the 
  radiation 
  with 
  wave-lengths 
  between 
  X 
  and 
  

   X 
  + 
  dX 
  is 
  

  

  0(X, 
  0)dX=^e~ 
  k/K9 
  d\, 
  

  

  where 
  C 
  and 
  k 
  are 
  constants. 
  Wood 
  shows 
  from 
  the 
  same 
  

   assumptions 
  as 
  Wien 
  that 
  the 
  omission 
  of 
  a 
  factor 
  by 
  Wien 
  

   in 
  his 
  first 
  equation 
  led 
  to 
  an 
  error 
  in 
  the 
  final 
  result 
  and 
  

   that 
  the 
  correct 
  formula 
  based 
  on 
  these 
  hypotheses 
  is 
  

  

  <j>(X, 
  0)dX= 
  -%e~ 
  k,K6 
  d\. 
  

  

  X 
  2 
  

  

  With 
  reference 
  to 
  this 
  law 
  it 
  must 
  be 
  noted 
  that 
  the 
  total 
  

   radiation 
  is 
  obtained 
  by 
  integrating 
  cj>dX 
  from 
  zero 
  to 
  infinity,, 
  

   and 
  as 
  this 
  must 
  by 
  Stefan's 
  law 
  be 
  proportional 
  to 
  4 
  , 
  it 
  

   can 
  easily 
  be 
  shown 
  that 
  Ck"" 
  in 
  Wood's 
  equation 
  is 
  not 
  a 
  

   constant, 
  but 
  is 
  proportional 
  to 
  0~$. 
  As 
  Wood's 
  argument 
  

   is 
  involved 
  and 
  in 
  places 
  somewhat 
  obscure 
  f, 
  I 
  offer 
  an 
  

   alternative 
  proof 
  which 
  is 
  shorter 
  and 
  appears 
  to 
  be 
  equally 
  

   satisfactory 
  subject 
  to 
  substantially 
  the 
  same 
  assumptions. 
  

  

  If 
  N 
  be 
  the 
  total 
  number 
  of 
  molecules 
  in 
  a 
  mass 
  of 
  gas, 
  

   the 
  number 
  whose 
  absolute 
  velocities 
  lie 
  between 
  v 
  aud 
  

   v 
  + 
  dv 
  is 
  

  

  dN=4Nw-*a-*i*e-* 
  , 
  *dv 
  9 
  . 
  . 
  . 
  . 
  (1) 
  

  

  where 
  a 
  is 
  a 
  velocity 
  whose 
  square 
  is 
  proportional 
  to 
  the 
  

   temperature. 
  Then 
  Wien's 
  first 
  assumption 
  is 
  that 
  the 
  

   wave-length 
  and 
  intensity 
  of 
  the 
  radiation 
  emitted 
  by 
  a 
  

   molecule 
  depends 
  on 
  v 
  alone. 
  If 
  e 
  be 
  the 
  rate 
  of 
  emission 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  t 
  The 
  argument 
  at 
  the 
  foot 
  of 
  p. 
  197 
  and 
  the 
  top 
  of 
  the 
  next 
  page 
  

   implies 
  that 
  the 
  number 
  of 
  molecules 
  with 
  velocities 
  between 
  v 
  and 
  

  

  v+dv 
  is 
  kv>6-*e- 
  lv2/9 
  dt? 
  instead 
  of 
  kv^e'^'W 
  

  

  The 
  introduction 
  of 
  an 
  ideal 
  type 
  of 
  gas 
  on 
  p. 
  198, 
  with 
  the 
  notion 
  of 
  

   corresponding- 
  velocities, 
  is 
  unnecessary 
  ; 
  so 
  is 
  the 
  use 
  of 
  the 
  meaningless 
  

   law 
  XS= 
  constant, 
  which 
  requires 
  such 
  careful 
  interpretation 
  that 
  it 
  is 
  

   more 
  difficult 
  to 
  apply 
  than 
  the 
  second 
  and 
  third 
  hypotheses 
  of 
  the 
  

   present 
  paper, 
  expressed 
  in 
  equations 
  (5) 
  and 
  (14), 
  which 
  are 
  equivalent 
  

   to 
  it. 
  

  

  