﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  99 
  

  

  the 
  cylinder 
  ; 
  n 
  is 
  the 
  difference 
  of 
  the 
  angle 
  of 
  rotation 
  of 
  the 
  two 
  indices 
  in 
  de- 
  

   grees. 
  

  

  This 
  is 
  the 
  most 
  accurate 
  method 
  for 
  the 
  determination 
  of 
  m 
  independently 
  

   of 
  /jl, 
  and 
  it 
  seems 
  to 
  answer 
  best 
  with 
  thick 
  cylinders 
  which 
  cannot 
  be 
  used 
  with 
  

   the 
  balance 
  of 
  torsion, 
  as 
  the 
  oscillations 
  are 
  too 
  short, 
  and 
  produce 
  a 
  vibration 
  

   of 
  the 
  whole 
  apparatus. 
  

  

  Case 
  III. 
  

  

  A 
  hollow 
  cylinder 
  exposed 
  to 
  normal 
  pressures 
  only. 
  When 
  the 
  pressures 
  

   parallel 
  to 
  the 
  axis, 
  radius, 
  and 
  tangent 
  are 
  substituted 
  for 
  p 
  v 
  p 
  2 
  , 
  and 
  p 
  3 
  , 
  Equa- 
  

   tions 
  (10) 
  become 
  

  

  -dx~ 
  = 
  (97I-3W 
  (°+P 
  + 
  9) 
  + 
  -o 
  . 
  . 
  (18.) 
  

  

  ~dF={g- 
  ] 
  J[-^)( 
  () 
  + 
  P 
  + 
  9) 
  + 
  -p 
  ■ 
  ■ 
  (19.) 
  

   d8(r6) 
  8r 
  ( 
  1 
  1 
  \ 
  , 
  , 
  1 
  

  

  By 
  multiplying 
  Equation 
  (20) 
  by 
  r, 
  differentiating 
  with 
  respect 
  to 
  r, 
  and 
  

  

  d 
  u 
  t 
  

  

  comparing 
  this 
  value 
  of 
  —7— 
  with 
  that 
  of 
  Equation 
  (19.) 
  

  

  p 
  — 
  g_ 
  / 
  1 
  1 
  \ 
  (do 
  dp 
  d 
  q\ 
  1 
  dq 
  

  

  rm 
  \9u 
  3 
  m) 
  \dr 
  dr 
  drj 
  m 
  dr 
  

  

  The 
  equation 
  of 
  the 
  equilibrium 
  of 
  an 
  element 
  of 
  the 
  solid 
  is 
  obtained 
  by 
  

   considering 
  the 
  forces 
  which 
  act 
  on 
  it 
  in 
  the 
  direction 
  of 
  the 
  radius. 
  By 
  equating 
  

   the 
  forces 
  which 
  press 
  it 
  outwards 
  with 
  those 
  pressing 
  it 
  inwards, 
  we 
  find 
  the 
  

   equation 
  of 
  the 
  equilibrium 
  of 
  the 
  element, 
  

  

  9^£ 
  = 
  ^ 
  .... 
  ( 
  2i.) 
  

   r 
  d 
  r 
  

  

  By 
  comparing 
  this 
  equation 
  with 
  the 
  last, 
  we 
  find 
  

  

  \9// 
  3 
  m/ 
  dr 
  \9 
  fx 
  2>m) 
  \dr 
  dr) 
  

  

  Integrating, 
  

  

  (97-3^) 
  ° 
  + 
  (wji 
  + 
  Wm) 
  <* 
  + 
  *)= 
  Pi 
  

  

  Since 
  0, 
  the 
  longitudinal 
  pressure, 
  is 
  supposed 
  constant, 
  we 
  may 
  assume 
  

  

  9 
  fx 
  dm 
  

   VOL. 
  XX. 
  PART 
  I. 
  2D 
  

  

  