﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  105 
  

  

  . 
  (39.) 
  

  

  8 
  V 
  a* 
  h.-a^h, 
  1 
  ,3 
  , 
  h 
  x 
  

  

  2 
  „ 
  3. 
  

  

  ^ 
  /a^ 
  3a^\ 
  K 
  a 
  2 
  3 
  /l 
  3 
  \ 
  

   a 
  x 
  3 
  — 
  a 
  2 
  3 
  \ 
  )U 
  2m/ 
  a 
  x 
  z 
  — 
  a^\jl 
  2m) 
  

  

  When 
  the 
  external 
  and 
  internal 
  pressures 
  are 
  equal 
  

  

  a 
  =h„=p=g, 
  and 
  -==- 
  =-A=^ 
  . 
  . 
  (40.) 
  

  

  the 
  change 
  of 
  internal 
  capacity 
  depends 
  entirely 
  on 
  the 
  cubical 
  elasticity 
  of 
  the 
  

   vessel, 
  and 
  not 
  on 
  its 
  thickness 
  or 
  its 
  linear 
  elasticity. 
  

  

  When 
  the 
  external 
  and 
  internal 
  pressures 
  are 
  inversely 
  as 
  the 
  cubes 
  of 
  the 
  

   radii 
  of 
  the 
  surfaces 
  on 
  which 
  they 
  act, 
  

  

  a, 
  3 
  K 
  = 
  a 
  2 
  s 
  h»p=^h 
  v 
  q=-^-± 
  V 
  

  

  8V_ 
  _3a 
  1 
  3 
  A 
  1 
  

   V 
  ~ 
  2 
  r 
  s 
  m 
  

  

  , 
  8 
  V 
  3 
  h, 
  

  

  when 
  r—a,-^ 
  r 
  = 
  — 
  7i 
  — 
  

  

  1 
  V 
  Am 
  

  

  (41.) 
  

  

  In 
  this 
  case 
  the 
  change 
  of 
  capacity 
  depends 
  on 
  the 
  linear 
  elasticity 
  alone. 
  

  

  M. 
  Regnault, 
  in 
  his 
  researches 
  on 
  the 
  theory 
  of 
  the 
  steam 
  engine, 
  has 
  given 
  

   an 
  account 
  of 
  the 
  experiments 
  which 
  he 
  made 
  in 
  order 
  to 
  determine 
  with 
  accuracy 
  

   the 
  compressibility 
  of 
  mercury. 
  

  

  He 
  considers 
  the 
  mathematical 
  formulae 
  very 
  uncertain, 
  because 
  the 
  theories 
  

   of 
  molecular 
  forces 
  from 
  which 
  they 
  are 
  deduced 
  are 
  probably 
  far 
  from 
  the 
  truth 
  ; 
  

   and 
  even 
  were 
  the 
  equations 
  free 
  from 
  error, 
  there 
  would 
  be 
  much 
  uncertainty 
  in 
  

   the 
  ordinary 
  method 
  by 
  measuring 
  the 
  elongation 
  of 
  a 
  rod 
  of 
  the 
  substance, 
  for 
  it 
  

   is 
  difficult 
  to 
  ensure 
  that 
  the 
  material 
  of 
  the 
  rod 
  is 
  the 
  same 
  as 
  that 
  of 
  the 
  hollow 
  

   sphere. 
  

  

  He 
  has, 
  therefore, 
  availed 
  himself 
  of 
  the 
  results 
  of 
  M. 
  Lame 
  for 
  a 
  hollow 
  

   sphere 
  in 
  three 
  different 
  cases, 
  in 
  the 
  first 
  of 
  which 
  the 
  pressure 
  acts 
  on 
  the 
  inte- 
  

   rior 
  and 
  exterior 
  surface 
  at 
  the 
  same 
  time, 
  while 
  in 
  the 
  other 
  two 
  cases 
  the 
  pres- 
  

   sure 
  is 
  applied 
  to 
  the 
  exterior 
  or 
  interior 
  surface 
  alone. 
  Equation 
  (39.) 
  becomes 
  

   in 
  these 
  cases, 
  — 
  

  

  1. 
  When 
  h 
  x 
  =h 
  2 
  -^r 
  = 
  -77 
  and 
  the 
  compressibility 
  of 
  the 
  enclosed 
  liquid 
  being 
  

   jjl 
  2 
  , 
  and 
  the 
  apparent 
  diminution 
  of 
  volume 
  8' 
  V, 
  -^- 
  = 
  \ 
  ( 
  ) 
  . 
  . 
  (42.) 
  

  

  2. 
  When 
  h 
  , 
  = 
  0, 
  -==- 
  = 
  -^- 
  = 
  - 
  h 
  2 
  — 
  ^ 
  — 
  3I- 
  + 
  0— 
  ) 
  .... 
  (43.) 
  

  

  1 
  V 
  V 
  2 
  a 
  1 
  3 
  — 
  a 
  2 
  3 
  \/x 
  2 
  m) 
  K 
  ' 
  

  

  8Y 
  

  

  o 
  wu 
  1 
  8 
  V 
  h, 
  /a 
  3 
  3 
  a 
  3 
  \ 
  

  

  3. 
  When 
  K- 
  0, 
  -^- 
  r 
  - 
  = 
  — 
  =-i 
  — 
  5 
  ( 
  — 
  + 
  o 
  — 
  ) 
  

  

  2 
  V 
  a^ 
  — 
  aj 
  \ 
  /J. 
  2m) 
  

  

  8'Y 
  h, 
  /a, 
  3 
  3 
  a 
  

  

  V 
  

  

  _ 
  K 
  (< 
  1 
  3 
  < 
  . 
  (a 
  3 
  a 
  z) 
  1 
  \ 
  

  

  -«,»-«,» 
  U 
  + 
  2m 
  + 
  ^ 
  a 
  ^nJ 
  

  

  