﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  109 
  1 
  

  

  Accordingly 
  

  

  + 
  (-l)»fe„P„( 
  T 
  4-)™7-'' 
  (55) 
  

  

  V 
  a 
  . 
  ikrj 
  kr 
  J 
  

  

  Now 
  the 
  only 
  condition 
  imposed 
  upon 
  the 
  appliances 
  

   introduced 
  at 
  the 
  surface 
  of 
  the 
  sphere 
  is 
  that 
  they 
  shall 
  do 
  

   no 
  work. 
  The 
  velocity 
  is 
  d<j>/dr 
  + 
  d>fr/dr, 
  and 
  the 
  pressure, 
  

   proportional 
  to 
  d<fi/dt 
  + 
  dsfr/dt, 
  is 
  in 
  quadrature 
  with 
  (55) 
  . 
  All 
  

   therefore 
  that 
  is 
  required 
  is 
  that 
  (55) 
  and 
  its 
  derivative 
  with 
  

   respect 
  to 
  r 
  be 
  in 
  the 
  same 
  phase, 
  or 
  that 
  the 
  ratio 
  of 
  these 
  

   symbolic 
  expressions 
  be 
  real. 
  Since 
  P„ 
  is 
  a 
  wholly 
  odd 
  or 
  

   wholly 
  even 
  function, 
  this 
  requirement 
  is 
  satisfied 
  if 
  

  

  2n 
  + 
  l-(-l) 
  n 
  ika 
  n 
  

  

  ; 
  ,\ 
  . 
  be 
  real. 
  

  

  (-L) 
  n 
  ka 
  n 
  

  

  If 
  A», 
  which 
  may 
  be 
  complex, 
  be 
  written 
  Ae 
  ia 
  , 
  we 
  get 
  

  

  -M?=-(-l) 
  n 
  (2w+t)sina. 
  . 
  . 
  . 
  (56) 
  

  

  Thus 
  A 
  is 
  a 
  maximum 
  when 
  

  

  sin 
  «=—( 
  — 
  1)", 
  (57) 
  

  

  and 
  the 
  maximum 
  value 
  is 
  

  

  A 
  = 
  {2n 
  + 
  l)/k 
  (58) 
  

  

  By 
  (57), 
  (58) 
  

  

  a.==-(-l)-*(9n 
  + 
  l)/A 
  (59) 
  

  

  so 
  that 
  in 
  (55) 
  

  

  2n 
  + 
  l-(-l)"/fe 
  = 
  0, 
  .... 
  (60) 
  

  

  but 
  (j> 
  n 
  + 
  i/r„ 
  does 
  not 
  itself 
  vanish. 
  

  

  If 
  the 
  incident 
  plane 
  waves 
  are 
  regarded 
  as 
  due 
  to 
  a 
  source 
  

   at 
  a 
  great 
  distance 
  R, 
  we 
  have 
  in 
  correspondence 
  with 
  (53) 
  

  

  * 
  =! 
  V' 
  ■•■•..■ 
  (61) 
  

   with 
  which 
  we 
  may 
  compare 
  

  

  ^.= 
  ^V^ 
  P 
  »0") 
  ( 
  62 
  ) 
  

  

  The 
  work 
  emitted 
  from 
  the 
  primary 
  source 
  being 
  represented 
  

  

  