﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  117 
  

  

  ■c 
  V2 
  are 
  small 
  (b 
  l2 
  has 
  been 
  already 
  made 
  zero), 
  so 
  that 
  the 
  

   resonating 
  coordinate 
  may 
  vibrate 
  with 
  a 
  considerable 
  degree 
  

   ■of 
  independence. 
  We 
  are 
  also 
  to 
  suppose 
  that 
  n 
  corre- 
  

   sponds 
  to 
  these 
  comparatively 
  independent 
  vibrations, 
  so 
  that 
  

   r 
  u 
  — 
  n 
  2 
  a 
  n 
  = 
  approximately, 
  while 
  c 
  22 
  — 
  n 
  2 
  a 
  22 
  is 
  relatively 
  

   large. 
  These 
  simplifications 
  reduce 
  the 
  bracket 
  in 
  the 
  

   denominator 
  of 
  (95) 
  to 
  

  

  whence 
  we 
  obtain 
  finally 
  

  

  Mod 
  2 
  ¥ 
  2 
  45 
  M 
  l 
  ; 
  

  

  In 
  this 
  expression, 
  which 
  is 
  of 
  the 
  same 
  form 
  as 
  (84) 
  , 
  the 
  

   numerator 
  on 
  the 
  left 
  may 
  be 
  considered 
  to 
  represent 
  the 
  

   integrated 
  energy 
  of 
  the 
  resonator. 
  It 
  must 
  not 
  be 
  overlooked 
  

   that 
  the 
  suppositions 
  involved 
  are 
  to 
  some 
  extent 
  antagonistic. 
  

   For 
  example, 
  the 
  coefficient 
  of 
  b 
  22 
  in 
  (90) 
  is 
  treated 
  as 
  constant 
  

   when 
  p 
  varies, 
  although 
  (c 
  lx 
  — 
  » 
  a 
  a 
  u 
  ) 
  is 
  small. 
  The 
  theorem 
  

   should 
  be 
  regarded 
  as 
  one 
  applicable 
  in 
  the 
  limit 
  when 
  b 
  22 
  is 
  

   exceedingly 
  small. 
  

  

  If 
  there 
  be 
  more 
  than 
  two 
  degrees 
  of 
  freedom, 
  the 
  result 
  is 
  

   unaffected, 
  provided 
  that 
  the 
  forces 
  N^,^, 
  &c, 
  of 
  the 
  new 
  types 
  

   vanish 
  and 
  that 
  the 
  only 
  dissipation 
  is 
  that 
  represented 
  by 
  b 
  22 
  . 
  

   By 
  the 
  3rd, 
  4th, 
  &c. 
  of 
  (3) 
  the 
  new 
  coordinates 
  may 
  be 
  

   eliminated. 
  In 
  this 
  process 
  b 
  22 
  is 
  undisturbed, 
  and 
  everything 
  

   remains 
  as 
  if 
  there 
  were 
  only 
  two 
  coordinates 
  as 
  above. 
  

  

  The 
  idea 
  of 
  the 
  integration 
  with 
  respect 
  to 
  p 
  is 
  borrowed 
  

   from 
  a 
  paper 
  by 
  Prof. 
  Planck 
  (Ann. 
  d. 
  Pliys. 
  i. 
  p. 
  99, 
  1900), 
  

   in 
  which 
  is 
  considered 
  the 
  behaviour 
  of 
  an 
  infinitely 
  small 
  

   electromagnetic 
  resonator 
  under 
  incident 
  plane 
  waves. 
  The 
  

   proof 
  of 
  the 
  general 
  theorem 
  covering 
  Prof. 
  Planck's 
  case 
  

   would 
  require 
  a 
  process 
  similar 
  to 
  that 
  by 
  which 
  (51) 
  was 
  

   established. 
  Subject 
  to 
  the 
  condition 
  

  

  Mod 
  s 
  ¥ 
  Mod 
  s 
  ¥ 
  _ 
  = 
  Mod 
  2 
  ^ 
  

  

  K 
  ~ 
  *- 
  ~~ 
  " 
  h 
  ' 
  

  

  ■we 
  might 
  expect 
  to 
  find, 
  as 
  in 
  (52), 
  

  

  (97) 
  

  

  ^ 
  r,,r 
  '■,- 
  . 
  7 
  77 
  Mod 
  2 
  "^ 
  /rt 
  _ 
  N 
  

  

  S4c 
  u 
  JMod 
  2 
  t,^= 
  Tb 
  . 
  . 
  . 
  . 
  (98) 
  

  

  Terling 
  Place, 
  Witham. 
  

   November 
  1901. 
  

  

  