﻿228 
  Mr. 
  S. 
  H. 
  Burbury 
  on 
  Irrever 
  

  

  have 
  the 
  logarithmic 
  form. 
  For 
  instance, 
  let 
  u, 
  v 
  be 
  two 
  

   functions 
  of 
  t 
  which 
  satisfy 
  at 
  every 
  point 
  in 
  a 
  given 
  space 
  

  

  the 
  conditions 
  

  

  du 
  , 
  dv 
  

  

  ^=kv, 
  and 
  rfl 
  =-*' 
  

  

  where 
  k, 
  k' 
  are 
  positive 
  constants. 
  These 
  correspond 
  to 
  

   circular 
  functions. 
  Then 
  if 
  S 
  =jTfw» 
  dx 
  dy 
  dz 
  throughout 
  the 
  

   space 
  in 
  question, 
  

  

  and 
  has 
  always 
  the 
  same 
  sign 
  until 
  on 
  average 
  of 
  the 
  whole 
  

   space 
  kv^ 
  — 
  k^. 
  Further, 
  when 
  this 
  state 
  is 
  reached, 
  ~T, 
  =°> 
  

  

  720 
  ' 
  etc 
  

  

  and 
  -^2- 
  = 
  — 
  4Lkk'ffiuv 
  dx 
  dy 
  dz, 
  and 
  S 
  retains 
  its 
  minimum 
  

  

  value 
  if 
  uv 
  = 
  0. 
  

  

  Planck's 
  Theory. 
  

   5. 
  A 
  vacuum 
  space 
  is 
  traversed 
  by 
  an 
  arbitrary 
  system 
  of 
  

   electromagnetic 
  waves. 
  In 
  this 
  space 
  is 
  a 
  linear 
  electric 
  

   resonator, 
  or 
  Dipol, 
  whose 
  proper 
  period 
  of 
  vibration 
  corre- 
  

   sponds 
  to 
  a 
  wave-length 
  very 
  great 
  in 
  comparison 
  with 
  its 
  

   own 
  linear 
  dimensions. 
  And 
  it 
  is 
  assumed 
  that 
  its 
  oscilla- 
  

   tions 
  are 
  damped 
  only 
  by 
  radiation 
  of 
  energy 
  into 
  the 
  

   surrounding 
  space, 
  and 
  not 
  in 
  any 
  degree 
  by 
  ohmic 
  resistance 
  

   or 
  other 
  internal 
  dissipative 
  process. 
  Let 
  fit) 
  denote 
  the 
  

   moment 
  at 
  time 
  t 
  of 
  the 
  resonator, 
  Z 
  the 
  component 
  at 
  time 
  t 
  

   in 
  the 
  direction 
  of 
  the 
  resonator 
  of 
  the 
  intensity 
  of 
  the 
  electric 
  

   field 
  at 
  the 
  point 
  where 
  the 
  resonator 
  is, 
  both 
  / 
  and 
  Z 
  being 
  

   expressed 
  in 
  absolute 
  electrostatic 
  measure. 
  Then 
  the 
  oscilla- 
  

   tion 
  of 
  the 
  resonator 
  is 
  given 
  by 
  the 
  equation 
  

  

  d 
  V 
  . 
  O 
  d 
  f 
  , 
  A 
  9 
  ZS 
  3C 
  3 
  CT 
  ry 
  

  

  in 
  which 
  c 
  is 
  the 
  velocity 
  of 
  light 
  in 
  vacuo, 
  v 
  the 
  number 
  of 
  

   oscillations 
  of 
  the 
  resonator 
  per 
  unit 
  of 
  time, 
  supposing 
  it 
  

   uninfluenced 
  by 
  any 
  other 
  bodies, 
  ;md 
  a 
  is 
  the 
  damping 
  or 
  

   logarithmic 
  decrement 
  of 
  the 
  amplitude 
  of 
  its 
  oscillations. 
  

   It 
  is 
  essential 
  to 
  the 
  theory 
  that 
  a, 
  and 
  also 
  av 
  , 
  be 
  very 
  

   small. 
  

  

  6. 
  The 
  vibrations 
  Z 
  in 
  the 
  surrounding 
  medium 
  may 
  con- 
  

   sist 
  of 
  waves 
  of 
  all 
  periods. 
  But 
  expressing 
  Z 
  in 
  a 
  series 
  of 
  

   the 
  form 
  

  

  -: 
  

  

  dvG 
  cos(2ttvc 
  — 
  6 
  v 
  ), 
  

  

  