﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  

  

  107 
  

  

  The 
  effect 
  of 
  pressure 
  on 
  the 
  surface 
  of 
  a 
  spherical 
  cavity 
  on 
  any 
  other 
  part 
  

   of 
  an 
  elastic 
  solid 
  is 
  therefore 
  inversely 
  proportional 
  to 
  the 
  cube 
  of 
  its 
  distance 
  

   from 
  the 
  centre 
  of 
  the 
  cavity. 
  

  

  When 
  one 
  of 
  the 
  surfaces 
  of 
  an 
  elastic 
  hollow 
  sphere 
  has 
  its 
  radius 
  rendered 
  

   invariable 
  by 
  the 
  support 
  of 
  an 
  incompressible 
  sphere, 
  whose 
  radius 
  is 
  a 
  v 
  then 
  

  

  dr 
  

  

  =0, 
  when 
  r=a 
  1 
  

  

  p 
  = 
  h 
  2 
  

  

  3a 
  2 
  3 
  /J. 
  

   2a 
  l 
  z 
  m 
  + 
  3a 
  2 
  z 
  }x 
  

  

  + 
  0— 
  2 
  - 
  

  

  2m 
  

  

  r 
  % 
  2 
  a/ 
  m 
  + 
  3a 
  2 
  3 
  fx 
  

  

  " 
  a 
  2 
  M 
  I 
  

  

  a, 
  a„ 
  

  

  m 
  

  

  ** 
  2 
  2a 
  1 
  3 
  rrc 
  + 
  3a 
  2 
  3 
  ju 
  " 
  2 
  r 
  3 
  2a 
  x 
  z 
  m 
  + 
  3a 
  2 
  z 
  fx 
  

  

  - 
  = 
  h 
  

  

  o^ 
  3 
  

  

  (45.) 
  

  

  When,=« 
  2 
  ir 
  =A 
  2 
  , 
  2, 
  r 
  

  

  ^a^m 
  + 
  Sa^/i 
  " 
  8 
  r* 
  2a 
  1 
  3 
  m 
  + 
  3a 
  2 
  3 
  )U 
  

   3« 
  2 
  3 
  — 
  3a 
  x 
  s 
  

  

  Case 
  V. 
  

  

  On 
  the 
  equilibrium 
  of 
  an 
  elastic 
  beam 
  of 
  rectangular 
  section 
  uniformly 
  bent. 
  

  

  By 
  supposing 
  the 
  bent 
  beam 
  to 
  be 
  produced 
  till 
  it 
  returns 
  into 
  itself, 
  we 
  may 
  

   treat 
  it 
  as 
  a 
  hollow 
  cylinder. 
  

  

  Let 
  a 
  rectangular 
  elastic 
  beam, 
  whose 
  length 
  is 
  2 
  tt 
  c, 
  be 
  bent 
  into 
  a 
  circular 
  

   form, 
  so 
  as 
  to 
  be 
  a 
  section 
  of 
  a 
  hollow 
  cylinder, 
  those 
  parts 
  of 
  the 
  beam 
  which 
  lie 
  

   towards 
  the 
  centre 
  of 
  the 
  circle 
  will 
  be 
  longitudinally 
  compressed, 
  while 
  the 
  op- 
  

   posite 
  parts 
  will 
  be 
  extended. 
  

  

  The 
  expression 
  for 
  the 
  tangential 
  compression 
  is 
  therefore 
  

  

  dr 
  

   r 
  

  

  c 
  — 
  r 
  

  

  8 
  

  

  Comparing 
  this 
  value 
  of 
  — 
  with 
  that 
  of 
  Equation 
  (20.) 
  

  

  c 
  — 
  r 
  

   r 
  

  

  = 
  (97r-3^) 
  («+*+*) 
  + 
  

  

  q=p 
  + 
  r 
  

  

  dp 
  

   dr' 
  

  

  becomes 
  

  

  and 
  by 
  (21.) 
  

  

  By 
  substituting 
  for 
  q 
  its 
  value, 
  and 
  dividing 
  by 
  r 
  (^ 
  — 
  + 
  37") 
  > 
  * 
  ne 
  equation 
  

  

  9m 
  ll 
  

  

  dp 
  2m 
  + 
  3/x 
  p 
  _9mfx 
  — 
  (m 
  — 
  S 
  /x) 
  

  

  dr 
  m 
  + 
  6 
  /jl 
  r 
  ~ 
  (ni 
  + 
  6 
  /x)r 
  (m 
  + 
  6/x)c 
  

  

  a 
  linear 
  differential 
  equation, 
  which 
  gives 
  

  

  2m+ 
  3 
  fi 
  

  

  P 
  = 
  G,r~ 
  

   VOL. 
  XX. 
  PAET 
  I. 
  

  

  m 
  + 
  6|* 
  3m 
  ll 
  r 
  9f 
  Xm-(m-3fj.)o 
  

  

  m 
  + 
  3 
  [X 
  c 
  

  

  2m 
  + 
  3 
  fx 
  

  

  (46. 
  

  

  2f 
  

  

  