﻿114 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  resonator 
  for 
  positions 
  of 
  the 
  point-sources 
  distributed 
  uni- 
  

   formly 
  over 
  the 
  sphere 
  r 
  = 
  l 
  is 
  equal 
  to 
  the 
  work 
  emitted 
  by 
  

   each 
  of 
  the 
  point-sources 
  themselves 
  divided 
  by 
  k 
  2 
  . 
  If 
  the 
  

   point-sources 
  are 
  supposed 
  to 
  lie 
  at 
  a 
  distance 
  r 
  in 
  place 
  of 
  

   unity, 
  the 
  divisor 
  becomes 
  Jc?r 
  2 
  in 
  place 
  of 
  h 
  2 
  . 
  

  

  Although 
  the 
  above 
  deduction 
  may 
  stand 
  in 
  need 
  of 
  some 
  

   supplementing 
  before 
  it 
  could 
  be 
  regarded 
  as 
  rigorous 
  at 
  all 
  

   points, 
  it 
  suffices 
  at 
  any 
  rate 
  to 
  show 
  that 
  the 
  general 
  theorem 
  

   (52) 
  really 
  does 
  include 
  the 
  more 
  special 
  cases 
  which 
  sug- 
  

   gested 
  it. 
  In 
  some 
  applications, 
  e. 
  g. 
  to 
  an 
  elastic 
  solid, 
  we 
  

   should 
  have 
  at 
  first 
  to 
  suppose 
  the 
  forces 
  introduced 
  at 
  any 
  

   element 
  of 
  volume 
  dY 
  to 
  act 
  in 
  various 
  directions, 
  but 
  no 
  

   great 
  complication 
  thence 
  arises, 
  and 
  the 
  general 
  result 
  

   finally 
  takes 
  the 
  same 
  form. 
  

  

  Energy 
  stored 
  in 
  Resonators. 
  

  

  In 
  preceding 
  investigations 
  we 
  have 
  been 
  concerned 
  with 
  

   energy 
  emitted 
  from 
  a 
  resonator. 
  We 
  now 
  turn 
  to 
  the 
  con- 
  

   sideration 
  of 
  some 
  general 
  theorems 
  relating 
  to 
  the 
  energy 
  

   stored, 
  as 
  it 
  were, 
  in 
  the 
  resonator 
  when 
  the 
  applied 
  forces 
  

   have 
  frequencies 
  in 
  the 
  neighbourhood 
  of 
  the 
  natural 
  frequency 
  

   of 
  the 
  resonator. 
  And 
  we 
  will 
  treat 
  first 
  the 
  simple 
  case 
  of 
  

   one 
  degree 
  of 
  freedom. 
  

  

  As 
  in 
  (2) 
  we 
  have 
  

  

  T=i«t 
  2 
  , 
  F 
  = 
  p^, 
  V=ic^, 
  . 
  . 
  (78) 
  

   giving 
  as 
  the 
  equation 
  of 
  vibration 
  

  

  a^ 
  + 
  ^ 
  + 
  c^r=^ 
  = 
  E^ 
  (79) 
  

  

  The 
  time 
  factor 
  being 
  suppressed, 
  the 
  solution 
  of 
  (79) 
  is 
  

  

  + 
  = 
  ^— 
  r-y, 
  (80) 
  

  

  c 
  — 
  ap' 
  + 
  ipo 
  ' 
  

  

  whence 
  

  

  Mod 
  2 
  ^ 
  

  

  "♦■ 
  ^^ 
  • 
  • 
  • 
  • 
  < 
  81 
  ) 
  

  

  7i, 
  equal 
  to 
  s/ 
  {of 
  a), 
  being 
  the 
  value 
  of 
  p 
  corresponding 
  to 
  

   maximum 
  resonance. 
  If, 
  as 
  we 
  suppose, 
  b 
  is 
  very 
  small, 
  the 
  

   important 
  values 
  of 
  Mod 
  2 
  sfr 
  are 
  concentrated 
  in 
  the 
  neigh- 
  

   bourhood 
  of 
  p 
  = 
  n, 
  and 
  we 
  may 
  substitute 
  n 
  for 
  p 
  in 
  the 
  

   term 
  p 
  2 
  b 
  2 
  . 
  Also 
  n 
  2 
  —p 
  2 
  may 
  be 
  identified 
  with 
  2n(n— 
  p). 
  

   Accordingly 
  (81) 
  becomes 
  

  

  „ 
  , 
  2 
  , 
  1 
  Mod 
  2 
  ^ 
  /QO 
  , 
  

  

  