﻿Stress 
  and 
  Strain 
  in 
  an 
  Elastic 
  Solid. 
  445 
  

  

  hexahedron*, 
  as 
  the 
  specifying 
  elements. 
  This 
  I 
  have 
  

   thought 
  of 
  for 
  the 
  last 
  thirty 
  years, 
  but 
  not 
  till 
  a 
  few 
  weeks 
  

   ago 
  have 
  I 
  seen 
  how 
  to 
  make 
  it 
  conveniently 
  practicable, 
  

   especially 
  for 
  application 
  to 
  the 
  generalized 
  dynamics 
  of 
  a 
  

   crystal. 
  

  

  § 
  1. 
  We 
  shall 
  suppose 
  the 
  solid 
  to 
  be 
  a 
  homogeneous 
  

   crystal 
  of 
  any 
  possible 
  character. 
  Cut 
  from 
  it 
  a 
  tetrahedron 
  

   ABUD 
  of 
  any 
  shape 
  and 
  orientation. 
  Let 
  the 
  three 
  non- 
  

   intersecting 
  pairs 
  (AB, 
  CD), 
  (BC, 
  AD), 
  (CA, 
  BD) 
  of 
  its 
  

   six 
  edges 
  be 
  denoted 
  by 
  

  

  (8p,V), 
  (&L,W)> 
  (3»-,3,-0 
  . 
  . 
  . 
  (1). 
  

   This 
  notation 
  gives 
  

  

  (P,P')> 
  (<J>9'), 
  (r,r 
  / 
  ) 
  .... 
  (2) 
  

  

  for 
  the 
  six 
  edges 
  of 
  a 
  tetrahedron, 
  similar 
  to 
  ABCD, 
  formed 
  

   by 
  taking 
  for 
  its 
  corners 
  (a, 
  /3, 
  <y, 
  8) 
  the 
  centres 
  of 
  gravity 
  f 
  

   of 
  the 
  four 
  triangular 
  faces 
  BCD, 
  CDA, 
  DAB, 
  ABC 
  

   respectively, 
  so 
  that 
  we 
  have 
  £> 
  = 
  a/3, 
  q 
  = 
  fSy, 
  r 
  = 
  y<z, 
  y 
  = 
  yS, 
  

   q 
  / 
  =aS 
  J 
  / 
  = 
  j88. 
  Consider 
  now, 
  in 
  advance, 
  the 
  amounts 
  of 
  

   work 
  done 
  by 
  the 
  six 
  pairs 
  of 
  balancing 
  forces 
  constituting 
  

   the 
  six 
  stress-components 
  described 
  in 
  § 
  2, 
  when 
  the 
  strain 
  - 
  

   components 
  vary 
  ; 
  for 
  example, 
  the 
  balancing 
  pulls 
  P, 
  

   parallel 
  to 
  AB, 
  when 
  a/3 
  increases 
  from 
  p 
  to 
  p 
  + 
  dp, 
  all 
  the 
  

   other 
  rive 
  lengths 
  g, 
  r, 
  p, 
  g', 
  r' 
  remaining 
  constant. 
  For 
  

   the 
  reckoning 
  of 
  work 
  we 
  may 
  suppose 
  the 
  opposite 
  forces, 
  

   P, 
  to 
  be 
  applied 
  at 
  a 
  and 
  /3, 
  instead 
  of 
  being 
  equably 
  

   distributed 
  over 
  the 
  faces 
  ADC, 
  BDC. 
  Hence 
  the 
  work 
  

   which 
  they 
  do 
  is 
  Ydp 
  ; 
  and 
  other 
  five 
  pairs 
  of 
  balancing- 
  

   pulls, 
  Q, 
  B, 
  P', 
  Q', 
  P', 
  do 
  no 
  work. 
  

  

  § 
  2. 
  Parallel 
  to 
  the 
  edge 
  AB 
  apply 
  to 
  the 
  faces 
  ADC, 
  

   BDC 
  equal 
  and 
  opposite 
  pulls, 
  P, 
  equally 
  distributed 
  over 
  

   them. 
  These 
  two 
  balancing 
  pulls 
  we 
  shall 
  call 
  a 
  stress 
  or 
  

   a 
  stress-component. 
  Similarly, 
  parallel 
  to 
  each 
  of 
  the 
  five 
  

   other 
  edges 
  apply 
  balancing 
  pulls 
  on 
  the 
  pair 
  of 
  faces 
  cutting- 
  

   it. 
  Thus 
  we 
  have 
  in 
  all 
  six 
  stress-components 
  parallel 
  to 
  the 
  

  

  * 
  This 
  name, 
  signifying 
  a 
  figure 
  bounded 
  by 
  three 
  pairs 
  of 
  parallel 
  

   planes, 
  is 
  admitted 
  in 
  crystallography 
  ; 
  but 
  the 
  longer 
  and 
  less 
  expressive 
  

   '■ 
  parallelepiped 
  " 
  is 
  too 
  frequently 
  used 
  instead 
  of 
  it 
  by 
  mathematical 
  

   writers 
  and 
  teachers. 
  A 
  hexahedron, 
  with 
  its 
  angles 
  acute 
  and 
  obtuse, 
  

   is 
  what 
  is 
  commonly 
  called, 
  both 
  in 
  pure 
  mathematics 
  and 
  crystallography, 
  

   a 
  rhombohedron. 
  A 
  right-angled 
  hexahedron 
  is 
  a 
  brick, 
  for 
  which 
  no 
  

   Greek 
  or 
  other 
  learned 
  name 
  is 
  hitherto 
  to 
  the 
  front 
  in 
  usage. 
  A 
  

   rectangular 
  equilateral 
  hexahedron 
  is 
  a 
  cube. 
  

  

  t 
  For 
  brevity 
  I 
  shall 
  henceforth 
  call 
  the 
  centre 
  of 
  gravity 
  of 
  a 
  triangle, 
  

   or 
  of 
  a 
  tetrahedron, 
  simply 
  its 
  centre. 
  

  

  