﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  113 
  

  

  ddr 
  ( 
  1 
  1\, 
  ' 
  ,,j) 
  

  

  — 
  — 
  = 
  ( 
  s 
  5 
  — 
  )(o+p 
  + 
  q)+*- 
  -c 
  v 
  

  

  dr 
  \9 
  (A 
  3 
  m) 
  v 
  *- 
  m 
  3 
  

  

  Br 
  fl 
  1 
  \, 
  . 
  ? 
  

  

  r 
  \9/i 
  3 
  w/ 
  v 
  z/ 
  m 
  3 
  

  

  The 
  equation 
  of 
  equilibrium 
  is 
  

  

  f=*+rg 
  • 
  • 
  (2D 
  

  

  and 
  since 
  the 
  tube 
  is 
  supposed 
  to 
  be 
  of 
  a 
  considerable 
  length 
  

  

  —r^- 
  =c 
  i 
  a 
  constant 
  quantity. 
  

   From 
  these 
  equations 
  we 
  find 
  that 
  

  

  ^ 
  + 
  e 
  » 
  9 
  "(Tii'W^ 
  { 
  2p 
  + 
  r 
  Jr) 
  

  

  

  _L 
  JL 
  

  

  9 
  fi 
  3 
  »a 
  

  

  and 
  hence 
  v=c 
  x 
  log 
  r 
  + 
  c 
  2 
  , 
  p 
  may 
  be 
  found 
  in 
  terms 
  of 
  r. 
  

  

  / 
  2 
  1 
  \ 
  -i 
  . 
  1 
  

  

  Hence 
  g= 
  (A- 
  + 
  ^) 
  " 
  \ 
  < 
  3 
  log 
  r- 
  ^ 
  + 
  , 
  6 
  + 
  (A 
  + 
  _L) 
  ft 
  , 
  s 
  

  

  (2 
  i 
  \ 
  -i 
  1 
  

  

  9^7 
  + 
  3--j 
  C 
  l 
  C 
  3- 
  26WC 
  5^2 
  

  

  the 
  rings 
  seen 
  in 
  this 
  case 
  will 
  differ 
  from 
  those 
  described 
  in 
  Case 
  III. 
  only 
  by 
  

   the 
  addition 
  of 
  a 
  constant 
  quantity. 
  

  

  When 
  no 
  pressures 
  act 
  on 
  the 
  exterior 
  and 
  interior 
  surfaces 
  of 
  the 
  tube 
  

   A,=A 
  2 
  =0, 
  and 
  

  

  U- 
  (1. 
  + 
  J-\ 
  ~\ 
  c 
  Jw 
  r 
  I 
  «i 
  8 
  « 
  2 
  2 
  loggi-log« 
  8 
  , 
  gi 
  g 
  log«i-a 
  8 
  8 
  loga 
  a 
  l 
  

   62 
  L- 
  f 
  — 
  4-—^ 
  "V 
  c 
  (logr 
  < 
  a 
  2 
  2 
  lQ 
  g 
  a 
  i- 
  lo 
  g 
  q 
  2 
  Vlog^-a/loga, 
  1 
  

   1-6 
  fiL 
  + 
  M 
  - 
  1 
  . 
  c 
  eofi 
  o 
  ^^log^i-log^ 
  l 
  

  

  There 
  will, 
  therefore, 
  be 
  no 
  action 
  on 
  polarized 
  light 
  for 
  the 
  ring 
  whose 
  radius 
  

   is 
  r 
  when 
  

  

  r 
  2_Q 
  a 
  i 
  a 
  2 
  \ 
  ff^L 
  

   «l 
  2 
  -°2 
  2 
  «2 
  

  

  CASE 
  X. 
  

  

  Sir 
  David 
  Brewster 
  has 
  observed 
  (Edinburgh 
  Transactions, 
  vol. 
  viii.), 
  that 
  

   when 
  a 
  solid 
  cylinder 
  of 
  glass 
  is 
  suddenly 
  heated 
  at 
  the 
  cylindric 
  surface 
  a 
  polar- 
  

   izing 
  force 
  is 
  developed, 
  which 
  is 
  at 
  any 
  point 
  proportional 
  to 
  the 
  square 
  of 
  the 
  

   distance 
  from 
  the 
  axis 
  of 
  the 
  cylinder 
  ; 
  that 
  is 
  to 
  say, 
  that 
  the 
  difference 
  of 
  retarda- 
  

  

  