﻿112 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  in 
  'Theory 
  of 
  Sound/ 
  § 
  277. 
  If 
  we 
  compare 
  our 
  present 
  

   (71) 
  with 
  (10), 
  § 
  245, 
  we 
  6nd 
  that 
  

  

  B 
  =4^ 
  ( 
  73 
  > 
  

  

  so 
  that 
  according 
  to 
  the 
  present 
  interpretation 
  of 
  <£, 
  (66) 
  

   gives 
  

  

  1 
  p—ikr 
  

  

  *=4Vp* 
  rfV 
  V 
  < 
  74 
  > 
  

  

  whereas 
  in 
  the 
  notation 
  of 
  § 
  277 
  

  

  We 
  are 
  now 
  to 
  some 
  extent 
  prepared 
  for 
  the 
  application 
  of 
  

   (52), 
  but 
  the 
  difficulty 
  remains 
  that 
  (52) 
  deals 
  in 
  the 
  first 
  

   instance 
  with 
  a 
  finite 
  system 
  subject 
  to 
  dissipative 
  forces 
  ; 
  

   whereas 
  the 
  uniform 
  medium 
  is 
  infinite, 
  and 
  need 
  not 
  be 
  

   supposed 
  subject 
  to 
  any 
  forces 
  truly 
  dissipative. 
  There 
  is, 
  

   however, 
  no 
  objection 
  to 
  the 
  introduction 
  of 
  a 
  small 
  dissipative 
  

   force 
  of 
  the 
  character 
  supposed 
  in 
  the 
  general 
  theorem, 
  that 
  is, 
  

   proportional 
  everywhere 
  to 
  clfyjdt. 
  Under 
  this 
  influence 
  plane 
  

   waves 
  are 
  attenuated 
  as 
  they 
  advance 
  ; 
  the 
  law 
  of 
  attenuation 
  

   being 
  represented 
  by 
  the 
  introduction 
  into 
  (67) 
  of 
  the 
  factor 
  

   e~ 
  ax 
  , 
  where 
  a 
  is 
  a 
  small 
  quantity, 
  real 
  and 
  positive. 
  

  

  The 
  connexion 
  between 
  a 
  and 
  b 
  may 
  be 
  investigated 
  by 
  

   considering 
  the 
  action 
  of 
  the 
  dissipative 
  force 
  — 
  /><£ 
  operative 
  

   over 
  a 
  slice 
  hx 
  at 
  x 
  = 
  in 
  causing 
  the 
  attenuation. 
  By 
  (67) 
  

   the 
  effect 
  at 
  x 
  of 
  this 
  force 
  is 
  represented 
  by 
  

  

  so 
  that 
  

  

  &/>/</>= 
  — 
  27ra/>B 
  Sx. 
  

  

  By 
  supposition 
  this 
  must 
  be 
  the 
  same 
  as 
  —*8x; 
  and 
  

   accordingly 
  

  

  a 
  = 
  27rabB 
  (75) 
  

  

  If 
  we 
  use 
  this 
  result 
  to 
  eliminate 
  B 
  from 
  (72), 
  we 
  get 
  as 
  

   the 
  work 
  emitted 
  from 
  a 
  point-source 
  

  

  P 
  * 
  f 
  Mo& 
  2 
  (<P 
  JY) 
  (76) 
  

  

  4tt/, 
  

  

  The 
  formula 
  (52) 
  expresses 
  the 
  sum 
  of 
  all 
  the 
  works 
  

   emitted 
  by 
  the 
  resonator 
  when 
  submitted 
  ' 
  successively 
  to 
  all 
  

   the 
  various 
  forces 
  ^, 
  subject 
  themselves 
  to 
  conditions 
  (44). 
  

  

  