﻿EQUILIBRIUM 
  OF 
  ELASTIC 
  SOLIDS. 
  91 
  

  

  d 
  8 
  x 
  d 
  8 
  x 
  d 
  8 
  x 
  

  

  dx 
  dy 
  dz 
  

  

  d8y 
  d8y 
  d8y 
  

  

  dx 
  dy 
  dz 
  

  

  d 
  8 
  z 
  d 
  8 
  z 
  d 
  8 
  z 
  

  

  dx 
  dy 
  dz 
  

  

  Since 
  the 
  number 
  of 
  these 
  quantities 
  is 
  nine, 
  if 
  nine 
  other 
  independent 
  quan- 
  

   tities 
  of 
  the 
  same 
  kind 
  can 
  be 
  found, 
  the 
  one 
  set 
  may 
  be 
  found 
  in 
  terms 
  of 
  the 
  

   other. 
  The 
  quantities 
  which 
  we 
  shall 
  assume 
  for 
  this 
  purpose 
  are 
  — 
  

  

  1. 
  Three 
  compressions, 
  — 
  , 
  -£-, 
  — 
  -, 
  in 
  the 
  directions 
  of 
  three 
  principal 
  axes 
  

   a, 
  @, 
  y. 
  

  

  2. 
  The 
  nine 
  direction-cosines 
  of 
  these 
  axes, 
  with 
  the 
  six 
  connecting 
  equations, 
  

   leaving 
  three 
  independent 
  quantities. 
  (See 
  Gregory's 
  Solid 
  Geometry). 
  

  

  3. 
  The 
  small 
  angles 
  of 
  rotation 
  of 
  this 
  system 
  of 
  axes 
  about 
  the 
  axes 
  of 
  x, 
  y, 
  z, 
  

   The 
  cosines 
  of 
  the 
  angles 
  which 
  the 
  axes 
  of 
  x, 
  y, 
  z 
  make 
  with 
  those 
  of 
  a, 
  /3, 
  7 
  

  

  are 
  — 
  

  

  cos 
  (aO 
  x) 
  = 
  a 
  v 
  cos 
  (j3 
  x) 
  = 
  b 
  v 
  cos 
  (70*) 
  = 
  ^, 
  

   cos 
  (aOy) 
  = 
  a 
  2 
  , 
  cos 
  ((3 
  0y) 
  = 
  b 
  2 
  , 
  cos 
  (7 
  y) 
  — 
  c 
  2 
  , 
  

   cos 
  (a 
  z) 
  =a 
  3 
  , 
  cos 
  (/3 
  z) 
  = 
  b 
  3 
  , 
  cos 
  (y 
  z) 
  = 
  c 
  3 
  , 
  

  

  These 
  direction-cosines 
  are 
  connected 
  by 
  the 
  six 
  equations, 
  

  

  a 
  \ 
  + 
  V 
  + 
  c 
  i 
  = 
  I 
  a 
  1 
  a 
  2 
  + 
  b 
  L 
  b 
  2 
  + 
  c 
  1 
  c 
  2 
  = 
  

  

  « 
  2 
  2 
  + 
  b 
  2 
  2 
  + 
  c 
  2 
  2 
  = 
  1 
  a 
  2 
  a 
  3 
  + 
  b 
  2 
  b 
  3 
  + 
  c, 
  2 
  c 
  3 
  = 
  

  

  a 
  3 
  2 
  + 
  6 
  3 
  2 
  + 
  c 
  3 
  2 
  = 
  1 
  a 
  3 
  a 
  l 
  + 
  b 
  3 
  b 
  1 
  +c 
  3 
  c 
  l 
  = 
  

  

  The 
  rotation 
  of 
  the 
  system 
  of 
  axes 
  a, 
  /3, 
  7, 
  round 
  the 
  axis 
  of 
  

  

  x, 
  from 
  y 
  to 
  z, 
  =8 
  6 
  V 
  

   y, 
  from 
  z 
  to 
  x, 
  = 
  8 
  6 
  2 
  , 
  

   z, 
  from 
  x 
  to 
  y, 
  =8 
  6 
  3 
  ; 
  

  

  By 
  resolving 
  the 
  displacements 
  8 
  a, 
  8(3, 
  8 
  y, 
  V 
  6 
  V 
  <9 
  3 
  , 
  in 
  the 
  directions 
  of 
  the 
  

   axes 
  x, 
  y, 
  z, 
  the 
  displacements 
  in 
  these 
  axes 
  are 
  found 
  to 
  be 
  

  

  8x 
  = 
  a 
  1 
  8a 
  + 
  b 
  1 
  8(3 
  + 
  c 
  l 
  8y 
  — 
  6 
  2 
  z 
  + 
  6 
  3 
  y 
  

  

  8 
  y 
  = 
  a 
  2 
  8 
  a 
  + 
  b 
  2 
  8 
  (3 
  + 
  c 
  2 
  8 
  7 
  — 
  6 
  3 
  x 
  + 
  6 
  X 
  z 
  

  

  8 
  z 
  = 
  a 
  3 
  8 
  a 
  + 
  b 
  3 
  8 
  {3 
  + 
  c 
  2 
  8y 
  — 
  6 
  1 
  y 
  + 
  6 
  2 
  x 
  

  

  But 
  8a 
  = 
  a<L? 
  8(3 
  = 
  (3 
  ^, 
  and 
  8y 
  = 
  7 
  ** 
  

  

  and 
  a 
  = 
  a 
  1 
  x 
  + 
  a 
  2 
  y 
  + 
  a 
  3 
  z, 
  (3=b 
  1 
  x 
  + 
  b 
  2 
  y 
  + 
  b 
  3 
  z 
  ) 
  and 
  y 
  = 
  c 
  x 
  x 
  + 
  c 
  2 
  y 
  + 
  c 
  3 
  z. 
  

  

  Substituting 
  these 
  values 
  of 
  8 
  a, 
  8 
  (3, 
  and 
  £7 
  in 
  the 
  expressions 
  for 
  8x, 
  8y, 
  

   vol, 
  xx. 
  part 
  1. 
  2 
  b 
  

  

  