﻿492 
  A 
  Doctrine 
  on 
  Material 
  Stresses. 
  

  

  in 
  terms 
  of 
  the 
  elements 
  o£ 
  stress, 
  we 
  may 
  eliminate 
  the 
  former 
  

   by 
  differentiation, 
  and 
  obtain 
  a 
  set 
  of 
  relations 
  involving 
  only 
  

   the 
  elements 
  of 
  stress 
  (stress-relations). 
  

  

  (6) 
  I 
  shall 
  now 
  suppose 
  the 
  set 
  of 
  three 
  mechanical 
  stress- 
  

   equations 
  to 
  be 
  introduced 
  which 
  correspond 
  to 
  the 
  state 
  of 
  

   rest 
  or 
  motion 
  of 
  the 
  body 
  as 
  a 
  rigid 
  body. 
  With 
  these 
  

   equations 
  I 
  shall 
  suppose 
  the 
  stress-relations 
  of 
  the 
  preceding 
  

   paragraph 
  to 
  be 
  combined. 
  From 
  this 
  combination 
  we 
  

   may 
  obtain 
  Stress-Equations, 
  by 
  which 
  we 
  may 
  regard 
  the 
  

   elements 
  of 
  stress 
  to 
  be 
  defined 
  : 
  as 
  far 
  as 
  they 
  are 
  capable 
  of: 
  

   being 
  defined 
  so 
  long 
  as 
  the 
  surface-traction 
  conditions 
  have 
  

   not 
  been 
  considered. 
  

  

  (7) 
  Up 
  to 
  this 
  point 
  I 
  have 
  been 
  contemplating 
  such 
  

   cases 
  as 
  a 
  cube 
  of 
  metal 
  on 
  a 
  rough 
  inclined 
  plane, 
  or 
  a 
  

   connecting-rod 
  moving 
  in 
  a 
  prescribed 
  manner, 
  the 
  geo- 
  

   metrical 
  conditions 
  for 
  rigidity 
  being 
  preserved. 
  The 
  stresses 
  

   are 
  not 
  to 
  be 
  regarded 
  as 
  undefined, 
  but 
  as 
  being 
  determinate. 
  

  

  (8) 
  The 
  Theory 
  of 
  Elasticity 
  is 
  now 
  introduced 
  by 
  the 
  

   definition 
  of 
  an 
  Elastic 
  Body 
  as 
  a 
  body 
  such 
  that 
  the 
  strain, 
  

   hitherto 
  considered 
  as 
  a 
  virtual 
  or 
  potential 
  strain, 
  is 
  the 
  

   actual 
  strain 
  experienced 
  by 
  the 
  parts 
  of 
  the 
  body. 
  The 
  

   stresses 
  being 
  definite, 
  the 
  strains 
  become 
  definite, 
  and 
  the 
  

   displacement 
  may 
  be 
  deduced. 
  

  

  The 
  acceptance 
  of 
  the 
  principles 
  involved 
  in 
  this 
  doctrine 
  

   would 
  have 
  the 
  effect 
  of 
  removing 
  the 
  subject 
  of 
  stresses 
  

   from 
  its 
  accepted 
  place 
  in 
  the 
  Theory 
  of 
  Elasticity, 
  and 
  of 
  

   making 
  it 
  an 
  integral 
  part 
  of 
  the 
  Statics 
  and 
  Dynamics 
  of 
  a 
  

   Rigid 
  Body. 
  Apart 
  from 
  the 
  introduction 
  of 
  the 
  cases 
  of 
  

   motion 
  of 
  the 
  body, 
  this 
  would 
  appear, 
  at 
  first 
  sight, 
  to 
  be 
  

   merely 
  a 
  matter 
  of 
  exposition 
  of 
  the 
  Elastic 
  Theory. 
  But 
  a 
  

   further 
  consequence 
  would 
  be 
  that 
  the 
  determination 
  of 
  the 
  

   elastic 
  strains 
  from 
  the 
  Rigid 
  Body 
  Stresses 
  would 
  be 
  only 
  

   the 
  first 
  stage 
  in 
  the 
  Theory 
  of 
  Elasticity. 
  A 
  closer 
  approxi- 
  

   mation 
  to 
  the 
  values 
  of 
  the 
  stresses 
  would 
  follow 
  from 
  the 
  

   estimated 
  alteration 
  of 
  the 
  surface-traction 
  conditions 
  con- 
  

   sequent 
  on 
  the 
  displaced 
  condition 
  of 
  the 
  surface 
  of 
  the 
  body, 
  

   and 
  the 
  subject 
  would 
  become 
  one 
  of 
  continued 
  approxi- 
  

   mations 
  on 
  the 
  lines 
  of 
  certain 
  other 
  subjects 
  of 
  Mathematical 
  

   Physics, 
  and 
  an 
  opportunity 
  would 
  be 
  provided 
  for 
  a 
  theory 
  

   of 
  permanent 
  set 
  and 
  of 
  rupture. 
  

  

  Lymm, 
  Cheshire. 
  

  

  