﻿470 
  Mr. 
  J. 
  S. 
  Townsend 
  on 
  Applications 
  of 
  

  

  Section 
  I. 
  

  

  1. 
  The 
  conditions 
  to 
  be 
  satisfied 
  by 
  p, 
  the 
  pressure 
  of 
  the 
  

   gas 
  A, 
  are 
  : 
  — 
  

  

  "W 
  + 
  dy* 
  + 
  dzV 
  P 
  ~dt 
  ; 
  

  

  \dx 
  2 
  dy 
  2, 
  di. 
  

  

  p 
  = 
  at 
  the 
  boundary 
  <j)(x, 
  ?/, 
  z)=Q 
  } 
  for 
  all 
  values 
  

   of 
  I; 
  

  

  p=p 
  initially 
  throughout 
  the 
  space 
  bounded 
  bv 
  

   0=0. 
  

  

  Let 
  the 
  gases 
  be 
  contained 
  between 
  two 
  parallel 
  plates. 
  

   The 
  boundary 
  will 
  then 
  be 
  the 
  two 
  planes 
  x 
  = 
  and 
  a? 
  = 
  a. 
  

   In 
  this 
  case 
  the 
  differential 
  equation 
  reduces 
  to 
  

  

  d 
  2 
  p 
  _ 
  dp 
  

  

  K 
  d^*~Tt> 
  

  

  the 
  general 
  solution 
  of 
  which 
  is 
  

  

  p— 
  tAe-« 
  2Kt 
  sin 
  (aa 
  + 
  0) 
  9 
  . 
  . 
  . 
  . 
  (1) 
  

  

  where 
  A, 
  a, 
  and 
  /3 
  are 
  to 
  be 
  determined 
  by 
  the 
  conditions 
  

  

  p 
  = 
  when 
  x 
  = 
  and 
  x 
  = 
  a, 
  for 
  all 
  values 
  of 
  t 
  ; 
  

   p=Po 
  when 
  t 
  = 
  for 
  all 
  values 
  of 
  x 
  between 
  x 
  — 
  and 
  

   x 
  — 
  a. 
  

  

  The 
  first 
  condition 
  is 
  satisfied 
  by 
  making 
  /3 
  = 
  and 
  

  

  11TT 
  

  

  a 
  — 
  — 
  . 
  

   a 
  

  

  The 
  coefficients 
  A 
  are 
  determined 
  from 
  the 
  second 
  condition 
  

  

  by 
  multiplying 
  both 
  sides 
  of 
  the 
  equation 
  (1) 
  by 
  sin 
  dx 
  

  

  and 
  integrating 
  from 
  x 
  = 
  to 
  x 
  = 
  a. 
  a 
  

  

  Since 
  

  

  . 
  nirx 
  . 
  n*TTX/ 
  n 
  

  

  sin 
  sin 
  dx 
  — 
  0, 
  

  

  o 
  a 
  a 
  

  

  we 
  obtain 
  the 
  following 
  value 
  of 
  A», 
  the 
  coefficient 
  of 
  the 
  

   term 
  

  

  _(^) 
  2 
  ^ 
  . 
  nirx 
  

  

  6 
  V 
  a 
  / 
  gin 
  

  

  a 
  

   in 
  the 
  Fourier's 
  series 
  (1), 
  

  

  ,x=a 
  

  

  nir 
  L 
  a 
  J 
  x=0 
  

  

  Hence, 
  when 
  n 
  is 
  even 
  An 
  = 
  0, 
  and 
  when 
  n 
  is 
  odd 
  A 
  n 
  = 
  — 
  . 
  

  

  