﻿concerning 
  Forced 
  Vibrations 
  and 
  Resonance. 
  113 
  

  

  These 
  conditions 
  are 
  satisfied 
  in 
  the 
  present 
  case 
  if 
  we 
  identify 
  

   each 
  ^ 
  with 
  the 
  force 
  <$>dY 
  acting 
  over 
  the 
  various 
  equal 
  

   elements 
  dY 
  into 
  which 
  infinite 
  space 
  may 
  be 
  divided, 
  the 
  

   value 
  of 
  <£ 
  being 
  everywhere 
  the 
  same. 
  Each 
  point-source 
  

   is 
  regarded 
  as 
  the 
  origin 
  of 
  plane 
  waves 
  which 
  fall 
  upon 
  the 
  

   resonator. 
  The 
  efficiency 
  of 
  the 
  sources 
  which 
  lie 
  in 
  a 
  given 
  

   direction 
  still 
  depends 
  upon 
  the 
  distance, 
  the 
  waves 
  as 
  they 
  

   reach 
  the 
  resonator 
  being 
  attenuated 
  by 
  the 
  resistance 
  and 
  

   also 
  in 
  the 
  usual 
  manner 
  according 
  to 
  the 
  law 
  of 
  inverse 
  

   squares. 
  

  

  Let 
  us 
  compare 
  the 
  efficiency 
  of 
  the 
  element 
  <&dY 
  at 
  

   distance 
  r 
  with 
  the 
  efficiency 
  of 
  an 
  equal 
  element 
  at 
  distance 
  

   unity, 
  the 
  value 
  of 
  a 
  being 
  so 
  small 
  that 
  no 
  perceptible 
  

   attenuation 
  due 
  to 
  it 
  occurs 
  in 
  distance 
  unity. 
  The 
  element 
  

   of 
  volume 
  

  

  dY 
  = 
  do 
  .d(\r% 
  (77) 
  

  

  in 
  which 
  for 
  the 
  present 
  do- 
  is 
  kept 
  unchanged. 
  The 
  efficiency 
  

   of 
  the 
  element 
  at 
  distance 
  r 
  varies 
  as 
  

  

  (®dY) 
  2 
  .e~ 
  2ar 
  .r- 
  2 
  ; 
  

   and 
  

  

  t{<&dSfYe- 
  2 
  **r-* 
  = 
  <&da 
  dY 
  f 
  " 
  e~^ 
  dr 
  = 
  J^(<$>dY)\ 
  

  

  Hence 
  for 
  the 
  sum 
  of 
  all 
  the 
  elements 
  lying 
  within 
  do 
  we 
  

   have 
  

  

  1 
  r- 
  -= 
  x 
  efficiency 
  of 
  (<£> 
  dY) 
  at 
  distance 
  unity. 
  

  

  This 
  has 
  now 
  to 
  be 
  again 
  integrated 
  with 
  respect 
  to 
  do. 
  

   The 
  result 
  may 
  be 
  expressed 
  by 
  the 
  statement 
  that 
  the 
  sum 
  

   of 
  all 
  the 
  works 
  emitted 
  by 
  the 
  resonator 
  is 
  

  

  v 
  X 
  mean 
  work 
  emitted 
  by 
  resonator 
  corresponding 
  to 
  

  

  the 
  various 
  positions 
  of 
  the 
  point-source 
  on 
  the 
  sphere 
  r 
  = 
  l. 
  

   By 
  the 
  theorem 
  (52) 
  this 
  sum 
  of 
  all 
  the 
  works 
  is 
  also 
  expressed 
  

  

  by 
  

  

  (<$>dY) 
  2 
  

  

  2b 
  dY 
  ' 
  

   or 
  in 
  accordance 
  with 
  (76) 
  is 
  equal 
  to 
  

  

  y^ 
  — 
  yr 
  T 
  x 
  work 
  emitted 
  by 
  (<E> 
  dY) 
  itself. 
  

  

  We 
  see 
  therefore 
  that 
  the 
  mean 
  work 
  emitted 
  by 
  the 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  3. 
  No. 
  13. 
  Jan. 
  1902. 
  I 
  

  

  