﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  89 
  

  

  tional 
  to 
  the 
  linear 
  compression 
  in 
  the 
  direction 
  of 
  the 
  pressure, 
  while 
  the 
  other 
  

   is 
  proportional 
  to 
  the 
  diminution 
  of 
  volume. 
  As 
  this 
  hypothesis 
  admits 
  two 
  co- 
  

   efficients, 
  it 
  differs 
  from 
  that 
  of 
  this 
  paper 
  only 
  in 
  the 
  values 
  of 
  the 
  coefficients 
  

   selected. 
  They 
  are 
  denoted 
  by 
  K 
  and 
  k, 
  and 
  TL=\i—\m,k=m. 
  

  

  The 
  theory 
  of 
  Professor 
  Stokes 
  is 
  contained 
  in 
  Vol. 
  viii., 
  Part 
  3, 
  of 
  the 
  Cam- 
  

   bridge 
  Philosophical 
  Transactions, 
  and 
  was 
  read 
  April 
  14, 
  1845. 
  

  

  He 
  states 
  his 
  general 
  principles 
  thus 
  : 
  — 
  " 
  The 
  capability 
  which 
  solids 
  possess 
  

   of 
  being 
  put 
  into 
  a 
  state 
  of 
  isochronous 
  vibration, 
  shews 
  that 
  the 
  pressures 
  called 
  

   into 
  action 
  by 
  small 
  displacements 
  depend 
  on 
  homogeneous 
  functions 
  of 
  those 
  

   displacements 
  of 
  one 
  dimension. 
  I 
  shall 
  suppose, 
  moreover, 
  according 
  to 
  the 
  

   general 
  principle 
  of 
  the 
  superposition 
  of 
  small 
  quantities, 
  that 
  the 
  pressures 
  due 
  

   to 
  different 
  displacements 
  are 
  superimposed, 
  and, 
  consequently, 
  that 
  the 
  pressures 
  

   are 
  linear 
  functions 
  of 
  the 
  displacements." 
  

  

  Having 
  assumed 
  the 
  proportionality 
  of 
  pressure 
  to 
  compression, 
  he 
  proceeds 
  

   to 
  define 
  his 
  coefficients. 
  — 
  "Let 
  — 
  A£be 
  the 
  pressures 
  corresponding 
  to 
  a 
  uni- 
  

   form 
  linear 
  dilatation 
  8 
  when 
  the 
  solid 
  is 
  in 
  equilibrium, 
  and 
  suppose 
  that 
  it 
  

   becomes 
  m 
  A 
  8, 
  in 
  consequence 
  of 
  the 
  heat 
  developed 
  when 
  the 
  solid 
  is 
  in 
  a 
  state 
  

   of 
  rapid 
  vibration. 
  Suppose, 
  also, 
  that 
  a 
  displacement 
  of 
  shifting 
  parallel 
  to 
  the 
  

   plane 
  xy, 
  for 
  which 
  8x=kx,8y 
  =—ky, 
  and 
  8z=0, 
  calls 
  into 
  action 
  a 
  pressure 
  

   — 
  B 
  k 
  on 
  a 
  plane 
  perpendicular 
  to 
  the 
  axis 
  of 
  x, 
  and 
  a 
  pressure 
  B 
  k 
  on 
  a 
  plane 
  

   perpendicular 
  to 
  the 
  axis 
  of 
  y\ 
  the 
  pressure 
  on 
  these 
  planes 
  being 
  equal 
  and 
  of 
  

   contrary 
  signs 
  ; 
  that 
  on 
  a 
  plane 
  perpendicular 
  to 
  z 
  being 
  zero, 
  and 
  the 
  tangential 
  

   forces 
  on 
  those 
  planes 
  being 
  zero." 
  The 
  coefficients 
  A 
  and 
  B, 
  thus 
  defined, 
  when 
  

  

  expressed 
  as 
  in 
  this 
  paper, 
  are 
  A 
  = 
  3 
  jjl, 
  B= 
  ^-. 
  

  

  Professor 
  Stokes 
  does 
  not 
  enter 
  into 
  the 
  solution 
  of 
  his 
  equations, 
  but 
  gives 
  

   their 
  results 
  in 
  some 
  particular 
  cases. 
  

  

  1. 
  A 
  body 
  exposed 
  to 
  a 
  uniform 
  pressure 
  on 
  its 
  whole 
  surface. 
  

  

  2. 
  A 
  rod 
  extended 
  in 
  the 
  direction 
  of 
  its 
  length. 
  

  

  3. 
  A 
  cylinder 
  twisted 
  by 
  a 
  statical 
  couple. 
  

  

  He 
  then 
  points 
  out 
  the 
  method 
  of 
  finding 
  A 
  and 
  B 
  from 
  the 
  two 
  last 
  cases. 
  

  

  While 
  explaining 
  why 
  the 
  equations 
  of 
  motion 
  of 
  the 
  luminiferous 
  ether 
  are 
  

   the 
  same 
  as 
  those 
  of 
  incompressible 
  elastic 
  solids, 
  he 
  has 
  mentioned 
  the 
  property 
  

   of 
  plasticity 
  or 
  the 
  tendency 
  which 
  a 
  constrained 
  body 
  has 
  to 
  relieve 
  itself 
  from 
  a 
  

   state 
  of 
  constraint, 
  by 
  its 
  molecules 
  assuming 
  new 
  positions 
  of 
  equilibrium. 
  This 
  

   property 
  is 
  opposed 
  to 
  linear 
  elasticity 
  ; 
  and 
  these 
  two 
  properties 
  exist 
  in 
  all 
  

   bodies, 
  but 
  in 
  variable 
  ratio. 
  

  

  M. 
  Wertheim, 
  in 
  Annales 
  de 
  Chimie, 
  S 
  e 
  Serie, 
  xxiii., 
  has 
  given 
  the 
  results 
  of 
  

   some 
  experiments 
  on 
  caoutchouc, 
  from 
  which 
  he 
  finds 
  that 
  K=k, 
  or 
  /x=|m 
  ; 
  and 
  

   concludes 
  that 
  k=K 
  in 
  all 
  substances. 
  In 
  his 
  equations, 
  jjl 
  is 
  therefore 
  made 
  

   equal 
  to 
  I 
  m,. 
  

  

  