﻿110 
  Lord 
  Rayleigh 
  on 
  some 
  General 
  Theorems 
  

  

  f 
  +1 
  

   by 
  R 
  2 
  I 
  dfi, 
  that 
  emitted, 
  or 
  rather 
  diverted, 
  by 
  the 
  reso- 
  

  

  nator 
  will 
  be 
  

  

  Mod 
  2 
  a* 
  f^PnV)^. 
  

  

  P„ 
  2 
  (/*)^=-. 
  1 
  , 
  and 
  <fyt 
  = 
  2. 
  Also 
  

  

  Mod 
  2 
  a„= 
  (2n 
  &t 
  1)2 
  > 
  (63) 
  

  

  so 
  that 
  the 
  ratio 
  of 
  works 
  is 
  

  

  2n+l 
  

  

  (64) 
  

  

  PR 
  2 
  

  

  This 
  agrees 
  with 
  the 
  result 
  given 
  in 
  ' 
  Theory 
  of 
  Sound 
  ' 
  

   § 
  319 
  for 
  a 
  symmetrical 
  resonator 
  (ra 
  = 
  0). 
  

  

  Ln 
  order 
  to 
  express 
  (64) 
  in 
  terms 
  of 
  the 
  energy-flux 
  (per 
  

   unit 
  area) 
  of 
  the 
  primary 
  waves 
  at 
  the 
  place 
  of 
  the 
  resonator, 
  

   we 
  have 
  only 
  to 
  multiply 
  (64) 
  by 
  the 
  area 
  (47rR 
  2 
  ) 
  of 
  the 
  

   sphere 
  of 
  radius 
  R. 
  If 
  we 
  restore 
  2ir/\ 
  for 
  k, 
  we 
  get 
  as 
  

   the 
  equivalent 
  of 
  (64) 
  

  

  (2n 
  + 
  lU 
  2 
  /7r 
  (65*) 
  

  

  If 
  we 
  limit 
  the 
  resonator 
  to 
  one 
  definite 
  harmonic 
  vibration 
  

   of 
  order 
  n 
  and 
  suppose 
  that 
  the 
  primary 
  waves 
  may 
  be 
  

   incident 
  indifferently 
  in 
  all 
  directions, 
  the 
  mean 
  of 
  the 
  values 
  

   of 
  (65) 
  is 
  X 
  2 
  /7T 
  simply, 
  as 
  follows 
  from 
  known 
  properties 
  of 
  

   the 
  spherical 
  functions. 
  

  

  Before 
  we 
  can 
  apply 
  the 
  general 
  theorem 
  (52) 
  to 
  an 
  inde- 
  

   pendent 
  investigation 
  of 
  these 
  results, 
  it 
  is 
  necessary 
  to 
  consider 
  

   the 
  connexion 
  between 
  the 
  formulae 
  for 
  plane 
  and 
  spherical 
  

   waves 
  ; 
  and 
  for 
  this 
  purpose 
  it 
  is 
  desirable 
  to 
  use 
  a 
  method 
  

   which, 
  if 
  not 
  itself 
  quite 
  general, 
  is 
  of 
  a 
  character 
  susceptible 
  

   of 
  generalization. 
  If 
  <f> 
  denote 
  the 
  velocity-potential 
  due 
  to 
  a 
  

   " 
  force 
  " 
  <E>dV 
  acting 
  at 
  the 
  element 
  of 
  volume 
  dY 
  and 
  pro- 
  

   portional 
  to 
  the 
  periodic 
  introduction 
  and 
  abstraction 
  of 
  fluid 
  

   at 
  that 
  place, 
  we 
  may 
  write 
  

  

  *=BVdV'—, 
  (66) 
  

  

  * 
  It 
  will 
  be 
  observed 
  that 
  (65) 
  is 
  the 
  double 
  of 
  the 
  value 
  (53) 
  above 
  

   quoted. 
  

  

  Dec. 
  17. 
  — 
  I 
  have 
  since 
  learned 
  that 
  Prof. 
  Lamb's 
  calculations 
  for 
  the 
  

   acoustical 
  problem 
  have 
  already 
  been 
  published. 
  See 
  Math. 
  Soc. 
  Proc. 
  

   vol. 
  xxxii. 
  p. 
  11, 
  1900, 
  where 
  equation 
  (44) 
  is 
  identical 
  with 
  (65) 
  above. 
  

   Reference 
  may 
  also 
  be 
  made 
  to 
  Lamb, 
  Math. 
  Soc. 
  Proc. 
  vol. 
  xxxii. 
  p. 
  120, 
  

   1900. 
  

  

  