﻿96 
  On 
  Stress 
  and 
  Strain 
  in 
  an 
  Elastic 
  Solid. 
  

  

  small. 
  As 
  a 
  notational 
  method 
  it 
  has 
  the 
  inconvenience 
  

   that 
  the 
  specifying 
  elements 
  are 
  of 
  two 
  essentially 
  different 
  

   kinds 
  (in 
  the 
  notation 
  of 
  Thomson 
  and 
  Tait 
  <?, 
  /, 
  g, 
  simple 
  

   elongations; 
  a, 
  h, 
  e, 
  shearings). 
  Both 
  these 
  faults 
  are 
  avoided 
  

   if 
  we 
  take 
  the 
  six 
  lengths 
  of 
  the 
  six 
  edges 
  of 
  a 
  tetrahedron 
  of 
  

   the 
  solid, 
  or 
  what 
  amounts 
  to 
  the 
  same, 
  though 
  less 
  simple, 
  

   the 
  three 
  pairs 
  of 
  face-diagonals 
  of 
  a 
  hexahedron 
  *, 
  as 
  the 
  

   specifying 
  elements. 
  This 
  I 
  have 
  thought 
  of 
  for 
  the 
  last 
  thirty 
  

   years, 
  but 
  not 
  till 
  to-day 
  (Dec. 
  16) 
  have 
  I 
  seen 
  how 
  to 
  make 
  

   it 
  conveniently 
  practicable, 
  especially 
  for 
  application 
  to 
  the 
  

   generalized 
  dynamics 
  of 
  a 
  crystal. 
  

  

  We 
  shall 
  suppose 
  the 
  solid 
  to 
  be 
  a 
  homogeneous 
  crystal 
  of 
  

   any 
  possible 
  character. 
  Cut 
  from 
  it 
  a 
  tetrahedron 
  ABCD 
  

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

   pairs 
  (AB, 
  CD), 
  (BO, 
  AD), 
  (CA, 
  BD) 
  of 
  its 
  six 
  edges 
  be 
  

   denoted 
  by 
  

  

  (/,/'), 
  (9,9% 
  . 
  (MO 
  (1). 
  

  

  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 
  six 
  edges 
  of 
  the 
  

   tetrahedron, 
  denoted 
  as 
  follows 
  : 
  — 
  

  

  (P,F), 
  (Q..Q0, 
  (R,R') 
  (2); 
  

  

  and 
  we 
  suppose 
  that 
  these 
  forces, 
  applied 
  as 
  they 
  are 
  to 
  the 
  

   surface 
  of 
  the 
  solid, 
  are 
  balanced 
  in 
  virtue 
  of 
  the 
  mutual 
  

   forces 
  between 
  its 
  particles, 
  when 
  its 
  edges 
  are 
  of 
  the 
  lengths 
  

   specified 
  as 
  in 
  (1). 
  Let/ 
  ,/(/, 
  g 
  0i 
  gj, 
  // 
  , 
  h 
  ', 
  be 
  the 
  values 
  of 
  

   the 
  specifying 
  elements 
  in 
  (1) 
  when 
  no 
  forces 
  are 
  applied 
  to 
  

   the 
  faces. 
  Thus 
  the 
  differences 
  from 
  these 
  values, 
  of 
  the 
  six 
  

   lengths 
  shown 
  in 
  formula 
  (1), 
  represent 
  the 
  strain 
  of 
  the 
  

   substance 
  when 
  under 
  the 
  stress 
  represented 
  by 
  (2). 
  

  

  Let 
  w 
  be 
  the 
  work 
  done 
  when 
  pulls 
  upon 
  the 
  faces, 
  each 
  

   commencing 
  at 
  zero, 
  are 
  gradually 
  increased 
  to 
  the 
  values 
  

  

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

   planes, 
  is 
  admitted 
  in 
  crystallography 
  : 
  hut 
  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 
  rect- 
  

   angular 
  equilateral 
  hexahedron 
  is 
  a 
  cube. 
  

  

  