﻿Stress 
  and 
  Strain 
  in 
  an 
  Elastic 
  Solid. 
  447 
  

  

  full 
  j 
  and 
  clearly 
  the 
  state 
  of 
  strain 
  specified 
  by 
  p, 
  g, 
  r, 
  p 
  1 
  , 
  q', 
  r' 
  . 
  

   It 
  is 
  what 
  is 
  called 
  the 
  " 
  strain 
  ellipsoid 
  " 
  *. 
  

  

  § 
  5. 
  Two 
  ways 
  of 
  finding 
  the 
  ellipsoid 
  touching 
  the 
  six 
  

   edges 
  of 
  a 
  tetrahedron 
  are 
  obvious. 
  (1) 
  Through 
  AB 
  and 
  

   CD 
  draw 
  planes 
  respectively 
  parallel 
  to 
  CD 
  and 
  AB 
  ; 
  and 
  

   deal 
  similarly 
  with 
  the 
  two 
  other 
  pairs 
  of 
  non 
  -intersecting 
  

   edges. 
  The 
  three 
  pairs 
  of 
  parallel 
  planes 
  thus 
  found, 
  

   constitute 
  a 
  hexahedron 
  which 
  contains 
  the 
  required 
  ellipsoid 
  

   touching 
  the 
  six 
  faces 
  at 
  their 
  centres 
  ; 
  or 
  (2) 
  draw 
  AK, 
  

   BK, 
  OK, 
  DK, 
  and 
  produce 
  to 
  equal 
  distances 
  KA', 
  KB', 
  

   KC, 
  KD' 
  beyond 
  K. 
  We 
  thus 
  find 
  four 
  points, 
  A', 
  B', 
  0', 
  

   D', 
  which, 
  with 
  A, 
  B, 
  C, 
  D, 
  are 
  the 
  eight 
  corners 
  of 
  the 
  

   hexahedron 
  which 
  we 
  found 
  by 
  construction 
  (1). 
  A 
  cir- 
  

   cumscribed 
  hexahedron 
  being 
  thus 
  given, 
  the 
  principal 
  axes 
  

   of 
  the 
  ellipsoid, 
  and 
  their 
  orientation, 
  are 
  found 
  by 
  the 
  

   solution 
  of 
  a 
  cubic 
  equation. 
  

  

  § 
  6. 
  Another 
  way 
  of 
  finding 
  the 
  strain-ellipsoid, 
  which 
  is 
  

   in 
  some 
  respects 
  simpler, 
  and 
  which 
  has 
  the 
  advantage 
  that 
  

   in 
  its 
  construction 
  it 
  does 
  not 
  take 
  us 
  outside 
  the 
  boundary 
  of 
  

   our 
  fundamental 
  tetrahedron, 
  is 
  as 
  follows 
  : 
  — 
  In 
  the 
  equi- 
  

   lateral 
  tetrahedron 
  A 
  B 
  C 
  D 
  describe, 
  from 
  its 
  centre 
  K 
  , 
  a 
  

   spherical 
  surface 
  touching 
  any 
  three 
  of 
  its 
  faces. 
  It 
  touches 
  

   these 
  faces 
  at 
  their 
  centres 
  ; 
  and 
  it 
  also 
  touches 
  the 
  fourth 
  

   face, 
  and 
  at 
  its 
  centre. 
  Hence, 
  if 
  we 
  solve 
  the 
  determinate, 
  

   one-solutional, 
  problem 
  to 
  draw 
  an 
  ellipsoid 
  touching 
  at 
  their 
  

   centres 
  any 
  three 
  of 
  the 
  four 
  faces 
  of 
  any 
  tetrahedron 
  ABCD, 
  

   and 
  having 
  its 
  centre 
  at 
  K, 
  this 
  ellipsoid 
  touches 
  at 
  its 
  centre 
  

   the 
  fourth 
  face 
  of 
  the 
  tetrahedron 
  ; 
  and 
  it 
  is 
  the 
  strain- 
  

   ellipsoid 
  for 
  the 
  homogeneous 
  strain 
  by 
  which 
  an 
  equilateral 
  

   tetrahedron 
  of 
  solid 
  is 
  altered 
  to 
  the 
  figure 
  ABCD. 
  

  

  § 
  7. 
  To 
  bring 
  our 
  new 
  method 
  of 
  specifying 
  strain 
  and 
  

   stress 
  into 
  relation 
  with 
  the 
  ordinary 
  method 
  for 
  infinitesimal 
  

   strains 
  and 
  the 
  corresponding 
  stresses 
  : 
  — 
  Let 
  \ 
  denote 
  the 
  

   length 
  of 
  each 
  edge 
  of 
  the 
  equilateral 
  tetrahedron 
  of 
  reference, 
  

   A 
  B 
  C 
  D 
  ; 
  and 
  let 
  h 
  be 
  the 
  edge 
  of 
  the 
  cube 
  of 
  which 
  

   A 
  , 
  B 
  , 
  (; 
  , 
  D 
  are 
  four 
  corners 
  (this 
  cube 
  being 
  the 
  hexa- 
  

   hedron 
  found 
  by 
  applying 
  either 
  of 
  the 
  constructions 
  of 
  § 
  5 
  

   to 
  the 
  tetrahedron 
  A 
  B 
  C 
  D 
  ). 
  The 
  twelve 
  face-diagonals 
  of 
  

   this 
  cube 
  are 
  each 
  equal 
  to 
  A,, 
  and 
  therefore 
  \ 
  = 
  h 
  \/2. 
  Let 
  

   now 
  the 
  cube 
  be 
  infinitesimally 
  strained 
  so 
  that 
  its 
  edges 
  

   become 
  h(l 
  + 
  e), 
  h(l+f), 
  h(l+g) 
  ; 
  and 
  so 
  that 
  the 
  angles 
  in 
  

   its 
  three 
  pairs 
  of 
  faces 
  are 
  altered 
  from 
  right 
  angles 
  to 
  acute 
  

   and 
  obtuse 
  angles 
  differing 
  respectively 
  by 
  a, 
  b, 
  c 
  from 
  right 
  

   angles. 
  This 
  is 
  the 
  strain 
  (e, 
  f 
  g, 
  a, 
  b, 
  c) 
  in 
  the 
  notation 
  of 
  

  

  * 
  Thomson 
  and 
  Tait's 
  -Natural 
  Philosophy,' 
  §1G0; 
  < 
  Elements, 
  

   § 
  141. 
  

  

  