﻿472 
  Mr. 
  J. 
  S. 
  Townsend 
  on 
  Applications 
  of 
  

  

  The 
  coefficients 
  A 
  are 
  determined 
  by 
  using 
  the 
  second 
  

   condition 
  which 
  p 
  must 
  satisfy: 
  p=Po 
  when 
  £ 
  = 
  0. 
  

   Hence 
  

  

  p 
  = 
  A 
  1 
  J 
  (a 
  1 
  r) 
  + 
  A 
  2 
  J 
  (a 
  3 
  r) 
  + 
  &C. 
  

  

  for 
  all 
  values 
  of 
  r. 
  

  

  Multiplying 
  each 
  side 
  of 
  this 
  equation 
  by 
  rj 
  (et 
  n 
  r)dr 
  ) 
  and 
  

   integrating 
  from 
  r=0 
  to 
  r 
  — 
  a^ 
  we 
  obtain 
  

  

  A»= 
  ~ 
  2p0 
  = 
  2p{) 
  

  

  aotnJJ 
  [aa 
  n 
  ) 
  aoc 
  n 
  3 
  x 
  (aa 
  n 
  ) 
  9 
  

  

  since 
  

  

  I 
  rJ 
  (a 
  H 
  r) 
  J 
  (a 
  n 
  r)dr~Q, 
  \ 
  rJ 
  2 
  (otnr)dr 
  = 
  ^-Jo' 
  2 
  («««)> 
  

  

  I 
  rJ 
  (a 
  n 
  r)dr 
  = 
  J 
  '(a» 
  a) 
  * 
  

  

  Jo 
  «» 
  

  

  Hence 
  the 
  value 
  of 
  p 
  expressed 
  as 
  a 
  function 
  of 
  r 
  and 
  t 
  is 
  

  

  = 
  2j>_ 
  r 
  J,(«,r) 
  ^ 
  + 
  J,(«,r) 
  £ 
  _^ 
  + 
  &c 
  _T 
  

   a 
  L 
  a 
  idi(«i«) 
  a 
  2 
  d 
  1 
  (a 
  2 
  a) 
  J 
  

  

  The 
  ratio 
  of 
  q 
  t 
  , 
  the 
  mass 
  of 
  the 
  gas 
  A 
  which 
  remains 
  

   unabsorbed 
  at 
  the 
  time 
  t, 
  to 
  the 
  original 
  mass 
  in 
  the 
  cylinder 
  is 
  

  

  2 
  1 
  prdr 
  9 
  . 
  » 
  t 
  

  

  3. 
  When 
  the 
  gases 
  are 
  contained 
  inside 
  a 
  spherical 
  boundary 
  

   the 
  differential 
  equation 
  becomes 
  

  

  re 
  d 
  / 
  2 
  dp\_ 
  dp 
  

   r*dr\ 
  r 
  dsrfdb' 
  

  

  which 
  can 
  also 
  be 
  written 
  

  

  d 
  2 
  d{rp) 
  

  

  d^ 
  V1 
  ' 
  dt 
  

   The 
  solution 
  of 
  which 
  is 
  

  

  rp 
  = 
  X 
  A 
  sin 
  (ar 
  -f 
  /3) 
  e~ 
  a 
  ' 
  2Kt 
  . 
  

  

  The 
  condition 
  ^> 
  = 
  at 
  the 
  boundary 
  r 
  = 
  a 
  is 
  satisfied 
  if 
  /3 
  = 
  

  

  -. 
  nir 
  

  

  and 
  a= 
  — 
  ■. 
  

  

  * 
  Lord 
  Rayleigh 
  : 
  ' 
  Theory 
  of 
  Sound/ 
  sections 
  203, 
  204. 
  

  

  