﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  

  

  93 
  

  

  On 
  the 
  face 
  z 
  1 
  

  

  — 
  q 
  2 
  dxdy 
  

  

  — 
  q 
  1 
  dx 
  dy 
  

  

  -Pidxdy 
  

  

  *2 
  

  

  fe+ 
  -^7 
  d 
  is) 
  dxdy 
  

  

  (q 
  1 
  + 
  -J— 
  dz) 
  dxdy 
  

  

  (P 
  3 
  + 
  -Jjdz)dxdy 
  

  

  Attractions, 
  

  

  p 
  X 
  d 
  x 
  dy 
  d 
  z 
  

  

  pYdx 
  dy 
  dz 
  

  

  p 
  Z 
  d 
  x 
  dy 
  d 
  z 
  

  

  Taking 
  the 
  moments 
  of 
  these 
  forces 
  round 
  the 
  axes 
  of 
  the 
  particle, 
  we 
  find 
  

  

  ?,' 
  = 
  ?1 
  ?i 
  2 
  = 
  ?2 
  ?3 
  1 
  = 
  ?3> 
  

  

  and 
  then 
  equating 
  the 
  forces 
  in 
  the 
  directions 
  of 
  the 
  three 
  axes, 
  and 
  dividing 
  by 
  

   dx, 
  dy, 
  dz, 
  we 
  find 
  the 
  equations 
  of 
  pressures. 
  

  

  d 
  Pi 
  , 
  d 
  V 
  3 
  dq 
  2 
  

  

  + 
  

  

  dy 
  

  

  d 
  P-2 
  , 
  dq 
  ± 
  

  

  dy 
  

  

  dp 
  

   dz 
  

  

  dz 
  

  

  + 
  

  

  + 
  

  

  dq 
  2 
  

   dx 
  

  

  - 
  + 
  

  

  dz 
  

  

  dq 
  z 
  

   dx 
  

  

  dji 
  

  

  dy 
  

  

  + 
  p 
  X 
  = 
  

  

  Equations 
  of 
  Pressures. 
  

  

  . 
  . 
  . 
  (3.) 
  

  

  + 
  pZ 
  = 
  

  

  The 
  resistance 
  which 
  the 
  solid 
  opposes 
  to 
  these 
  pressures 
  is 
  called 
  Elasticity, 
  

   and 
  is 
  of 
  two 
  kinds, 
  for 
  it 
  opposes 
  either 
  change 
  of 
  volume 
  or 
  change 
  of 
  figure. 
  

   These 
  two 
  kinds 
  of 
  elasticity 
  have 
  no 
  necessary 
  connection, 
  for 
  they 
  are 
  possessed 
  

   in 
  very 
  different 
  ratios 
  by 
  different 
  substances. 
  Thus 
  jelly 
  has 
  a 
  cubical 
  elasticity 
  

   little 
  different 
  from 
  that 
  of 
  water, 
  and 
  a 
  linear 
  elasticity 
  as 
  small 
  as 
  we 
  please 
  ; 
  

   while 
  cork, 
  whose 
  cubical 
  elasticity 
  is 
  very 
  small, 
  has 
  a 
  much 
  greater 
  linear 
  

   elasticity 
  than 
  jelly. 
  

  

  Hooke 
  discovered 
  that 
  the 
  elastic 
  forces 
  are 
  proportional 
  to 
  the 
  changes 
  that 
  

   excite 
  them, 
  or, 
  as 
  he 
  expressed 
  it, 
  " 
  Ut 
  tensio 
  sic 
  vis." 
  

  

  To 
  fix 
  our 
  ideas, 
  let 
  us 
  suppose 
  the 
  compressed 
  body 
  to 
  be 
  a 
  parallelepiped, 
  

   and 
  let 
  pressures 
  P 
  15 
  P 
  2 
  , 
  P 
  3 
  * 
  act 
  on 
  its 
  faces 
  in 
  the 
  direction 
  of 
  the 
  axes 
  a, 
  /3, 
  y, 
  

   which 
  will 
  become 
  the 
  principal 
  axes 
  of 
  compression, 
  and 
  the 
  compressions 
  will 
  

   hp 
  da 
  8(3 
  dy 
  

  

  The 
  fundamental 
  assumption 
  from 
  which 
  the 
  following 
  equations 
  are 
  deduced 
  

   is 
  an 
  extension 
  of 
  Hooke's 
  law, 
  and 
  consists 
  of 
  two 
  parts. 
  

  

  I. 
  The 
  sum 
  of 
  the 
  compressions 
  is 
  proportional 
  to 
  the 
  sum 
  of 
  the 
  pressures. 
  

  

  II. 
  The 
  difference 
  of 
  the 
  compressions 
  is 
  proportional 
  to 
  the 
  difference 
  of 
  the 
  

   pressures. 
  

  

  These 
  laws 
  are 
  expressed 
  by 
  the 
  following 
  equations 
  : 
  — 
  

   I.(P, 
  + 
  P 
  ! 
  + 
  P 
  s 
  ) 
  = 
  3 
  M 
  (-^,^ 
  + 
  iX) 
  

  

  ^>=>»(¥-^) 
  

  

  II. 
  < 
  

  

  • 
  (4.) 
  

  

  Equations 
  of 
  Elasticity. 
  

  

  . 
  (5.) 
  

  

  