﻿106 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  Action 
  of 
  Resonators. 
  

  

  We 
  now 
  abandon 
  the 
  idea 
  of 
  drawing 
  work 
  from 
  the 
  

   system 
  by 
  means 
  of 
  ^'^ 
  and 
  on 
  the 
  contrary 
  impose 
  the 
  

   condition 
  that 
  W 
  1 
  shall 
  do 
  no 
  work, 
  positive 
  or 
  negative. 
  The 
  

   effect 
  of 
  M*! 
  is 
  then 
  equivalent 
  to 
  a 
  change 
  in 
  the 
  inertia 
  

   a 
  n 
  , 
  or 
  spring 
  c 
  n 
  , 
  associated 
  with 
  this 
  coordinate 
  and 
  the 
  

   operation 
  may 
  be 
  regarded 
  as 
  a 
  tuning, 
  or 
  mistiming, 
  of 
  the 
  

   system. 
  If, 
  as 
  before, 
  fa, 
  due 
  to 
  the 
  given 
  force 
  ^ 
  2 
  , 
  be 
  

   unity, 
  and 
  > 
  l 
  r 
  1 
  = 
  K 
  1 
  * 
  M 
  i, 
  the 
  complete 
  value 
  of 
  fa 
  is 
  that 
  given 
  

   in 
  (34), 
  and 
  (35) 
  represents 
  the 
  work 
  done 
  altogether 
  by 
  S 
  & 
  1 
  

   in 
  unit 
  time. 
  Equating 
  this 
  to 
  zero, 
  we 
  get 
  as 
  the 
  relation 
  

   between 
  Ei 
  and 
  X 
  , 
  

  

  AjjRj 
  sina 
  n 
  = 
  sin 
  V 
  (46) 
  

  

  and 
  the 
  part 
  of 
  fa 
  due 
  to 
  M^ 
  is 
  

  

  r 
  (47; 
  

  

  sin 
  «u 
  

  

  The 
  modulus 
  of 
  this 
  is 
  a 
  maximum 
  when 
  sin 
  6 
  1 
  = 
  + 
  1, 
  and 
  

   the 
  value 
  of 
  the 
  maximum 
  is 
  cosec 
  a 
  n 
  . 
  In 
  this 
  case 
  (47) 
  

   becomes 
  

  

  — 
  l 
  + 
  icota 
  n 
  , 
  (48) 
  

  

  and 
  the 
  complete 
  value 
  of 
  fa 
  is 
  

  

  fa 
  = 
  icota 
  n 
  , 
  (49) 
  

  

  in 
  quadrature 
  with 
  the 
  former 
  value, 
  viz., 
  1, 
  

  

  We 
  may 
  regard 
  the 
  state 
  of 
  things 
  now 
  defined 
  as 
  being 
  

   in 
  a 
  sense 
  the 
  greatest 
  possible 
  disturbance 
  of 
  the 
  original 
  

   state 
  of 
  things. 
  If 
  the 
  system 
  be 
  quite 
  out 
  of 
  resonance, 
  

   forces 
  and 
  displacements 
  are 
  nearly 
  in 
  the 
  same 
  phase, 
  and 
  

   a 
  n 
  is 
  small. 
  The 
  altered 
  fa 
  is 
  then 
  a 
  large 
  multiple 
  of 
  the 
  

   original 
  value. 
  

  

  The 
  work 
  done 
  by 
  ^ 
  on 
  the 
  complete 
  value 
  of 
  fa 
  is 
  zero 
  

   by 
  supposition 
  ; 
  but 
  the 
  work 
  done 
  upon 
  the 
  part 
  of 
  fa 
  due 
  to 
  

   itself 
  is 
  by 
  (14) 
  in 
  unit 
  time 
  

  

  -ipR^An 
  sin 
  a 
  n 
  = 
  - 
  j— 
  

  

  J\ 
  u 
  Sin 
  a 
  u 
  

  

  This 
  corresponds 
  to 
  the 
  original 
  ^ 
  = 
  1, 
  or 
  s 
  $? 
  2 
  = 
  A 
  12 
  -~ 
  [ 
  e~ 
  ia 
  ^.. 
  

   If 
  the 
  prescribed 
  value 
  of 
  M^ 
  2 
  be 
  now 
  left 
  open, 
  we 
  have 
  as 
  

   the 
  work 
  in 
  question 
  

  

  j 
  P 
  A 
  12 
  2 
  MoW 
  2 
  

   ~~ 
  a 
  d^* 
  ' 
  • 
  • 
  • 
  • 
  • 
  v 
  ou 
  ; 
  

  

  f±H 
  ®m 
  Ctn 
  

  

  