﻿104 
  MR 
  JAMES 
  CLERK 
  MAXWELL 
  ON 
  THE 
  

  

  by 
  external 
  and 
  internal 
  normal 
  pressures 
  h 
  x 
  and 
  h 
  2 
  , 
  it 
  is 
  required 
  to 
  determine 
  

   the 
  equilibrium 
  of 
  the 
  elastic 
  solid. 
  

  

  The 
  pressures 
  at 
  any 
  point 
  in 
  the 
  solid 
  are 
  : 
  — 
  

  

  1. 
  A 
  pressure 
  p 
  in 
  the 
  direction 
  of 
  the 
  radius 
  ; 
  

  

  2. 
  A 
  pressure 
  q 
  in 
  the 
  perpendicular 
  plane. 
  

  

  These 
  pressures 
  depend 
  on 
  the 
  distance 
  from 
  the 
  centre, 
  which 
  is 
  denoted 
  by 
  r. 
  

  

  d 
  O 
  T 
  O 
  T 
  

  

  The 
  compressions 
  at 
  any 
  point 
  are 
  —r— 
  in 
  the 
  radial 
  direction, 
  and 
  — 
  in 
  the 
  

   tangent 
  plane, 
  the 
  values 
  of 
  these 
  compressions 
  are 
  : 
  — 
  

  

  —} 
  — 
  = 
  I 
  n 
  o 
  — 
  )(P 
  + 
  2?)+— 
  p- 
  ■ 
  ■ 
  (34.) 
  

  

  dr 
  \9 
  fx 
  3 
  m) 
  vr 
  *' 
  m 
  r 
  v 
  ' 
  

  

  — 
  = 
  (^--^-)(p 
  + 
  ^9) 
  + 
  — 
  9 
  • 
  • 
  • 
  (35-) 
  

  

  r 
  \9 
  fx 
  3 
  m) 
  ^ 
  * 
  y 
  in* 
  v 
  

  

  Multiplying 
  the 
  last 
  equation 
  by 
  r, 
  differentiating 
  with 
  respect 
  to 
  r, 
  and 
  

   equating 
  the 
  result 
  with 
  that 
  of 
  the 
  first 
  equation, 
  we 
  find 
  

  

  , 
  ( 
  1 
  _ 
  1 
  ) 
  (*Z 
  + 
  2p) 
  + 
  ±(4» 
  + 
  ? 
  -„) 
  =0 
  

   \9 
  Li 
  3 
  m) 
  \dr 
  dr) 
  m 
  \ 
  dr 
  ) 
  

  

  Since 
  the 
  forces 
  which 
  act 
  on 
  the 
  particle 
  in 
  the 
  direction 
  of 
  the 
  radius 
  must 
  

   balance 
  one 
  another, 
  or 
  2 
  qdrd 
  6 
  +p(rd 
  6f 
  = 
  (p 
  + 
  -j- 
  dr)(r 
  + 
  dr) 
  2 
  6 
  

  

  *~* 
  = 
  2 
  Tr 
  • 
  • 
  * 
  W 
  

  

  Substituting 
  this 
  value 
  of 
  q-p 
  in 
  the 
  preceding 
  equation, 
  and 
  reducing, 
  

  

  SP 
  + 
  a^o 
  

  

  dr 
  dr 
  

  

  Integrating, 
  p 
  + 
  2g=c 
  1 
  

  

  7* 
  d 
  t) 
  

  

  But 
  1 
  = 
  2d 
  +p 
  > 
  an( 
  ^ 
  ^ 
  e 
  e( 
  l 
  uat 
  i 
  on 
  becomes 
  

  

  d 
  / 
  + 
  3 
  p 
  - 
  + 
  c 
  -±-=0 
  

   dr 
  r 
  r 
  

  

  Since 
  p=h 
  x 
  when 
  r=a 
  iy 
  and 
  p=h 
  2 
  when 
  r=a 
  2 
  , 
  the 
  value 
  of 
  p 
  at 
  any 
  time 
  is 
  

   found 
  to 
  be 
  

  

  _ 
  o 
  1 
  3 
  A 
  1 
  — 
  o 
  2 
  3 
  h 
  2 
  a* 
  o 
  2 
  3 
  ^ 
  — 
  h 
  2 
  

   a,"°—aJ 
  

  

  _ 
  a 
  1 
  " 
  fl 
  1 
  — 
  a 
  2 
  n 
  2 
  _ 
  « 
  t 
  « 
  2 
  n 
  x 
  — 
  /t 
  2 
  ,„„ 
  . 
  

  

  (X-, 
  /1-, 
  & 
  2 
  flcf 
  d-, 
  Q> 
  2 
  fl-i 
  "2 
  /'QQ 
  \ 
  

  

  $ 
  = 
  7Tz 
  771 
  ^3 
  „ 
  3 
  „ 
  3 
  • 
  ' 
  V" 
  -/ 
  

  

  o^ 
  — 
  a 
  

  

  3"V 
  _ 
  o 
  $r 
  _ 
  «! 
  3 
  h 
  1 
  ~ 
  a 
  2 
  Z 
  K 
  1 
  3 
  « 
  x 
  3 
  a 
  2 
  3 
  h 
  x 
  — 
  h 
  2 
  1 
  

   V 
  " 
  r 
  a 
  i 
  3 
  — 
  a 
  2 
  3 
  /* 
  2 
  r 
  3 
  « 
  x 
  3 
  — 
  « 
  2 
  3 
  m 
  

  

  