﻿in 
  which 
  

  

  Processes 
  and 
  Planch's 
  Tlieory. 
  229 
  

  

  6 
  V 
  are 
  functions 
  of 
  v, 
  and 
  0„ 
  is 
  always 
  pi 
  

   s 
  (p. 
  443) 
  for 
  the 
  solution 
  of 
  (1) 
  

  

  3c 
  3 
  f 
  00 
  C 
  

  

  where 
  C„ 
  and 
  6 
  V 
  are 
  functions 
  of 
  v, 
  and 
  0„ 
  is 
  always 
  positive, 
  

   Planck 
  obtains 
  (p. 
  443) 
  for 
  the 
  solution 
  of 
  (1) 
  

  

  v 
  2 
  -v 
  2 
  

  

  COt 
  y 
  v 
  = 
  -2 
  7T. 
  

  

  (TV 
  V 
  

  

  Now 
  cr 
  is 
  assumed 
  very 
  small. 
  Therefore 
  sin 
  y 
  v 
  is 
  negligible 
  

   for 
  all 
  values 
  of 
  v 
  except 
  values 
  nearly 
  equal 
  to 
  v 
  . 
  Whence 
  

  

  1 
  , 
  y 
  - 
  — 
  y 
  

  

  also 
  we 
  may 
  write 
  coty,,— 
  — 
  2ir. 
  Planck 
  concludes 
  that 
  

  

  <TV 
  

  

  only 
  those 
  waves 
  which 
  have 
  a 
  period 
  very 
  nearly 
  equal 
  to 
  

   that 
  of 
  the 
  resonator 
  affect 
  it 
  or 
  are 
  affected 
  by 
  it. 
  

  

  Henceforward 
  we 
  will 
  use 
  v 
  without 
  the 
  suffix 
  to 
  denote 
  the 
  

   frequency 
  of 
  the 
  resonator. 
  

  

  7. 
  Planck 
  now 
  distinguishes 
  between 
  rapidly 
  and 
  slowly 
  

   varying 
  quantities. 
  Z 
  is 
  a 
  rapidly 
  varying 
  quantity, 
  and 
  on 
  the 
  

   average 
  of 
  a 
  time 
  r, 
  which 
  though 
  very 
  short 
  contains 
  many 
  

  

  complete 
  periods 
  -, 
  the 
  mean 
  value 
  of 
  Z 
  is 
  zero. 
  But 
  the 
  

  

  important 
  thing 
  is 
  not 
  its 
  mean 
  value 
  but 
  the 
  mean 
  value 
  of 
  

   its 
  square. 
  And 
  Z 
  2 
  , 
  if 
  taken 
  on 
  average 
  for 
  two 
  different 
  

   periods 
  of 
  time, 
  each 
  equal 
  to 
  t, 
  will 
  generally 
  vary. 
  It 
  

   belongs 
  to 
  the 
  class 
  of 
  sloidy 
  varying 
  quantities. 
  And 
  Planck 
  

   now 
  defines 
  Z 
  2 
  to 
  be 
  the 
  intensity 
  of 
  the 
  exciting 
  oscillation. 
  In 
  

   like 
  manner 
  the 
  mean 
  energy 
  of 
  the 
  resonator 
  is 
  understood 
  to 
  

   be 
  the 
  mean 
  taken 
  over 
  an 
  interval 
  of 
  time 
  r 
  many 
  times 
  greater 
  

   than 
  the 
  period 
  of 
  the 
  resonator, 
  and 
  other 
  quantities 
  are 
  

   treated 
  in 
  like 
  manner. 
  The 
  assumption 
  in 
  Planck's 
  theory 
  

   is 
  fundamental, 
  that 
  we 
  may 
  use 
  these 
  mean 
  values, 
  p. 
  445 
  & 
  

   p. 
  457. 
  

  

  8. 
  Let 
  U 
  be 
  the 
  energy 
  of 
  the 
  resonator. 
  Then 
  by 
  known 
  

   formulas 
  

  

  u=4K/2+ 
  i 
  L 
  gy 
  (2) 
  

  

  in 
  which 
  

  

  16*-V 
  t 
  _4ttV 
  

  

  and, 
  a 
  being 
  very 
  small, 
  K/ 
  2 
  = 
  L(-^ 
  \ 
  on 
  average. 
  

  

  Further, 
  Planck 
  obtains, 
  analysing 
  Z, 
  

  

  ^+^vU=^I„(22)p.455 
  ; 
  

  

  