﻿102 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  This 
  corresponds 
  to 
  Mod 
  A 
  J 
  r 
  = 
  A 
  _i 
  ; 
  and 
  as 
  the 
  work 
  abstract- 
  

   able 
  is 
  proportional 
  to 
  Mod 
  2 
  M*, 
  we 
  have 
  in 
  general 
  for 
  the 
  

   maximum 
  

  

  - 
  pA 
  2 
  Mod 
  2 
  ¥ 
  

  

  8A 
  sin 
  a 
  

  

  Now, 
  as 
  we 
  see 
  from 
  the 
  values 
  of 
  A 
  and 
  a 
  in 
  (24), 
  or 
  other- 
  

   wise 
  by 
  (20), 
  

  

  — 
  p" 
  1 
  Asina 
  = 
  />A 
  2 
  ; 
  

  

  and 
  thus 
  the 
  maximum 
  work 
  that 
  can 
  be 
  abstracted 
  is 
  

  

  Mod"* 
  , 
  

  

  It 
  may 
  at 
  first 
  occasion 
  surprise 
  that 
  the 
  work 
  obtainable 
  

   should 
  be 
  independent 
  of 
  a 
  and 
  c, 
  upon 
  which 
  the 
  behaviour 
  

   of 
  the 
  system 
  as 
  a 
  resonator 
  depends. 
  But 
  the 
  truth 
  is 
  that 
  

   by 
  suitable 
  choice 
  of 
  W 
  we 
  have 
  in 
  effect 
  tuned 
  the 
  system, 
  

   and 
  so 
  reduced 
  it 
  to 
  the 
  condition 
  of 
  evanescent 
  a 
  and 
  c 
  — 
  in 
  

   the 
  electrical 
  illustration 
  to 
  a 
  merely 
  resisting 
  circuit. 
  Had 
  

   we 
  assumed 
  the 
  evanescence 
  of 
  a 
  and 
  c 
  from 
  the 
  beginning, 
  

   we 
  could 
  of 
  course 
  have 
  arrived 
  more 
  simply 
  at 
  the 
  expres- 
  

   sion 
  (31). 
  

  

  In 
  the 
  case 
  of 
  maximum 
  withdrawal 
  of 
  energy 
  the 
  com- 
  

   plete 
  symbolical 
  value 
  of 
  ty 
  in 
  (27) 
  becomes 
  

  

  ^=1+^— 
  = 
  l-£ 
  + 
  i/cot«, 
  . 
  . 
  . 
  (32) 
  

   2 
  sin 
  a 
  z 
  z 
  K 
  ' 
  

  

  the 
  part 
  of 
  the 
  complete 
  value 
  which 
  is 
  in 
  the 
  same 
  phase 
  as 
  

   before 
  being 
  halved. 
  

  

  It 
  is 
  not 
  difficult 
  to 
  recognise 
  that 
  the 
  result 
  as 
  to 
  the 
  

   maximum 
  work 
  abstract-able 
  admits 
  of 
  further 
  generalization. 
  

   So 
  far 
  we 
  have 
  considered 
  the 
  case 
  of 
  a 
  single 
  degree 
  of 
  

   freedom, 
  e.g., 
  a 
  single 
  electric 
  circuit. 
  Other 
  degrees 
  of 
  

   freedom, 
  e, 
  g., 
  neighbouring 
  electric 
  circuits, 
  do 
  not 
  affect 
  the 
  

   result, 
  provided 
  that 
  the 
  forces 
  in 
  them 
  all 
  vanish 
  and 
  that 
  

   the 
  only 
  dissipation 
  is 
  that 
  already 
  considered. 
  If 
  in 
  equa- 
  

   tions 
  (3) 
  all 
  the 
  quantities 
  b 
  except 
  b 
  u 
  vanish, 
  as 
  well 
  as 
  the 
  

   forces 
  ^ 
  2 
  , 
  ^3, 
  &c, 
  the 
  second, 
  third, 
  and 
  following 
  equations 
  

   determine 
  real 
  ratios 
  between 
  all 
  the 
  other 
  coordinates 
  and 
  

   yfr^ 
  and 
  virtually 
  reduce 
  the 
  system 
  to 
  a 
  single 
  degree 
  of 
  

   freedom. 
  The 
  reaction 
  of 
  the 
  other 
  parts 
  of 
  the 
  system 
  will 
  

   influence 
  the 
  force 
  M*/ 
  required 
  in 
  order 
  to 
  abstract 
  most 
  

   work, 
  but 
  not 
  the 
  maximum 
  value 
  itself. 
  

  

  