﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  107 
  

   and 
  this 
  b}- 
  (20) 
  is 
  the 
  same 
  as 
  

  

  jA 
  12 
  2 
  Mod^ 
  2 
  ^ 
  ^ 
  _ 
  ^ 
  (51) 
  

  

  1^22^12 
  "+" 
  ^33 
  -"-13 
  4" 
  ••• 
  

  

  6 
  U 
  , 
  6 
  12 
  , 
  &c. 
  being 
  supposed 
  to 
  be 
  zero. 
  This 
  expression 
  

   differs 
  from 
  (43) 
  only 
  as 
  regards 
  the 
  numerical 
  factor. 
  If 
  

   b 
  33 
  , 
  &c. 
  also 
  vanish, 
  (51) 
  becomes 
  

  

  -2b— 
  "*&• 
  

  

  If 
  in 
  (51) 
  we 
  introduce 
  the 
  suppositions 
  of 
  (44), 
  we 
  get 
  as 
  

   in 
  (45) 
  for 
  the 
  sum 
  of 
  all 
  the 
  values 
  

  

  Mod 
  2V 
  P 
  

  

  t 
  ( 
  52 
  > 
  

  

  In 
  an 
  interesting 
  paper 
  entitled 
  " 
  An 
  Electromagnetic 
  

   Illustration 
  of 
  the 
  Theory 
  of 
  Selective 
  Absorption 
  of 
  Light 
  in 
  

   a 
  Gas"*, 
  Prof. 
  Lamb 
  has 
  developed 
  a 
  general 
  law 
  for 
  the 
  

   maximum 
  energy 
  emitted 
  by 
  a 
  resonator 
  situated 
  in 
  a 
  uniform 
  

   medium 
  when 
  submitted 
  to 
  incident 
  plane 
  waves. 
  " 
  The 
  rate 
  

   at 
  which 
  energy 
  is 
  carried 
  outwards 
  by 
  the 
  scattered 
  waves 
  

   is, 
  in 
  terms 
  of 
  the 
  energy-flux 
  in 
  the 
  primary 
  waves, 
  

  

  -^V 
  (53) 
  

  

  where 
  \ 
  is 
  the 
  wave-length, 
  and 
  n 
  is 
  the 
  order 
  of 
  the 
  spherical 
  

   harmonic 
  component 
  of 
  the 
  incident 
  waves 
  which 
  is 
  effective/' 
  

   Prof. 
  Lamb 
  remarks 
  that 
  the 
  law 
  expressed 
  by 
  (53) 
  " 
  is 
  of 
  a 
  

   very 
  general 
  character, 
  and 
  is 
  independent 
  of 
  the 
  special 
  

   nature 
  of 
  the 
  conditions 
  to 
  be 
  satisfied 
  at 
  the 
  surface 
  of 
  the 
  

   sphere. 
  It 
  presents 
  itself 
  in 
  the 
  elastic 
  solid 
  theory, 
  and 
  

   again 
  in 
  the 
  much 
  simpler 
  acoustical 
  problem 
  where 
  there 
  is 
  

   synchronism 
  between 
  plane 
  waves 
  of 
  sound 
  and 
  a 
  vibrating 
  

   sphere 
  on 
  which 
  they 
  impinge.'* 
  

  

  The 
  generality 
  claimed 
  by 
  Lamb 
  for 
  (53) 
  seemed 
  to 
  me 
  to 
  

   indicate 
  a 
  still 
  more 
  general 
  theorem 
  in 
  the 
  background 
  ; 
  

   and 
  it 
  was 
  upon 
  this 
  suggestion 
  that 
  the 
  investigations 
  of 
  the 
  

   preceding 
  pages 
  were 
  developed. 
  An 
  initial 
  difficulty, 
  how- 
  

   ever, 
  stood 
  in 
  the 
  way. 
  The 
  occurrence 
  of 
  n, 
  a 
  quantity 
  

   special 
  to 
  the 
  spherical 
  problem, 
  seemed 
  to 
  constitute 
  a 
  limita- 
  

   tion 
  ; 
  and 
  the 
  further 
  question 
  suggested 
  itself 
  as 
  to 
  why 
  the 
  

   efficiency 
  of 
  the 
  resonator 
  should 
  rise 
  with 
  increasing 
  n. 
  For 
  

   example, 
  why 
  in 
  the 
  acoustical 
  problem 
  should 
  a 
  resonator 
  

   formed 
  by 
  a 
  rigid 
  sphere, 
  moored 
  to 
  a 
  fixed 
  point 
  by 
  elastic 
  

  

  * 
  Cambr. 
  Trans, 
  vol. 
  xviii. 
  p. 
  348 
  (1899). 
  

  

  