﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  103 
  

  

  P 
  being 
  the 
  extending 
  force, 
  b 
  the 
  length 
  of 
  the 
  rod, 
  s 
  the 
  sectional 
  area, 
  and 
  

   8 
  x 
  the 
  elongation, 
  which 
  may 
  be 
  determined 
  by 
  the 
  deflection 
  of 
  a 
  wire, 
  as 
  in 
  the 
  

   apparatus 
  of 
  S' 
  Gravesande, 
  or 
  by 
  direct 
  measurement. 
  

  

  Case 
  IV. 
  

  

  The 
  only 
  known 
  direct 
  method 
  of 
  finding 
  the 
  compressibility 
  of 
  liquids 
  is 
  

   that 
  employed 
  by 
  Canton, 
  (Ersted, 
  Perkins, 
  Aime, 
  &c. 
  

  

  The 
  liquid 
  is 
  confined 
  in 
  a 
  vessel 
  with 
  a 
  narrow 
  neck, 
  then 
  pressure 
  is 
  applied, 
  

   and 
  the 
  descent 
  of 
  the 
  liquid 
  in 
  the 
  tube 
  is 
  observed, 
  so 
  that 
  the 
  difference 
  between 
  

   the 
  change 
  of 
  volume 
  of 
  the 
  liquid 
  and 
  the 
  change 
  of 
  internal 
  capacity 
  of 
  the 
  vessel 
  

   may 
  be 
  determined. 
  

  

  Now, 
  since 
  the 
  substance 
  of 
  which 
  the 
  vessel 
  is 
  formed 
  is 
  compressible, 
  a 
  

   change 
  of 
  the 
  internal 
  capacity 
  is 
  possible. 
  If 
  the 
  pressure 
  be 
  applied 
  only 
  to 
  the 
  

   contained 
  liquid, 
  it 
  is 
  evident 
  that 
  the 
  vessel 
  will 
  be 
  distended, 
  and 
  the 
  compressi- 
  

   bility 
  of 
  the 
  liquid 
  will 
  appear 
  too 
  great. 
  The 
  pressure, 
  therefore, 
  is 
  commonly 
  

   applied 
  externally 
  and 
  internally 
  at 
  the 
  same 
  time, 
  by 
  means 
  of 
  a 
  hydrostatic 
  

   pressure 
  produced 
  by 
  water 
  compressed 
  either 
  in 
  a 
  strong 
  vessel 
  or 
  in 
  the 
  depths 
  

   of 
  the 
  sea. 
  

  

  As 
  it 
  does 
  not 
  necessarily 
  follow, 
  from 
  the 
  equality 
  of 
  the 
  external 
  and 
  inter- 
  

   nal 
  pressures, 
  that 
  the 
  capacity 
  does 
  not 
  change, 
  the 
  equilibrium 
  of 
  the 
  vessel 
  must 
  

   be 
  determined 
  theoretically. 
  CErsted, 
  therefore, 
  obtained 
  from 
  Poisson 
  his 
  

   solution 
  of 
  the 
  problem 
  and 
  applied 
  it 
  to 
  the 
  case 
  of 
  a 
  vessel 
  of 
  lead. 
  To 
  find 
  the 
  

   cubical 
  elasticity 
  of 
  lead, 
  he 
  applied 
  the 
  theory 
  of 
  Poisson 
  to 
  the 
  numerical 
  

   results 
  of 
  Tredgold. 
  As 
  the 
  compressibility 
  of 
  lead 
  thus 
  found 
  was 
  greater 
  than 
  

   that 
  of 
  water, 
  (Ersted 
  expected 
  that 
  the 
  apparent 
  compressibility 
  of 
  water 
  in 
  a 
  

   lead 
  vessel 
  would 
  be 
  negative. 
  On 
  making 
  the 
  experiment 
  the 
  apparent 
  compres- 
  

   sibility 
  was 
  greater 
  in 
  lead 
  than 
  in 
  glass. 
  The 
  quantity 
  found 
  by 
  Tredgold 
  from 
  

   the 
  extension 
  of 
  rods 
  was 
  that 
  denoted 
  by 
  E, 
  and 
  the 
  value 
  of 
  jjl 
  deduced 
  from 
  E 
  

  

  alone 
  by 
  the 
  formulae 
  of 
  Poisson 
  cannot 
  be 
  true, 
  unless 
  £-= 
  ^ 
  ; 
  and 
  as 
  Bl 
  for 
  lead 
  

  

  mom 
  

  

  is 
  probably 
  more 
  than 
  3, 
  the 
  calculated 
  compressibility 
  is 
  much 
  too 
  great. 
  

  

  A 
  similar 
  experiment 
  was 
  made 
  by 
  Professor 
  Forbes, 
  who 
  used 
  a 
  vessel 
  of 
  

   caoutchouc. 
  As 
  in 
  this 
  case 
  the 
  apparent 
  compressibility 
  vanishes, 
  it 
  appears 
  that 
  

   the 
  cubical 
  compressibility 
  of 
  caoutchouc 
  is 
  equal 
  to 
  that 
  of 
  water. 
  

  

  Some 
  who 
  reject 
  the 
  mathematical 
  theories 
  as 
  unsatisfactory, 
  have 
  conjec- 
  

   tured 
  that 
  if 
  the 
  sides 
  of 
  the 
  vessel 
  be 
  sufficiently 
  thin, 
  the 
  pressure 
  on 
  both 
  sides 
  

   being 
  equal, 
  the 
  compressibility 
  of 
  the 
  vessel 
  will 
  not 
  affect 
  the 
  result. 
  The 
  fol- 
  

   lowing 
  calculations 
  shew 
  that 
  the 
  apparent 
  compressibility 
  of 
  the 
  liquid 
  depends 
  

   on 
  the 
  compressibility 
  of 
  the 
  vessel, 
  and 
  is 
  independent 
  of 
  the 
  thickness 
  when 
  the 
  

   pressures 
  are 
  equal. 
  

  

  A 
  hollow 
  sphere, 
  whose 
  external 
  and 
  internal 
  radii 
  are 
  a 
  1 
  and 
  a 
  2 
  , 
  is 
  acted 
  on 
  

  

  VOL. 
  XX. 
  PART 
  I. 
  2 
  E 
  

  

  