﻿446 
  Lord 
  Kelvin 
  on 
  a 
  New 
  Specifying 
  Method 
  for 
  

  

  six 
  edges 
  of 
  the 
  tetrahedron, 
  denoted 
  as 
  follows 
  : 
  — 
  

  

  (P,F) 
  (Q,Q') 
  (R, 
  R') 
  .... 
  (3); 
  

  

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

   four 
  faces 
  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 
  p 
  , 
  p 
  \ 
  q 
  , 
  q 
  \ 
  r 
  , 
  r 
  Q 
  ' 
  be 
  the 
  values 
  of 
  

   the 
  specifying 
  elements 
  when 
  no 
  forces 
  are 
  applied 
  to 
  the 
  

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

   lengths 
  shown 
  in 
  formula 
  (2), 
  represent 
  the 
  strain 
  of 
  the 
  

   substance 
  when 
  under 
  the 
  stress 
  represented 
  by 
  (3). 
  

  

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

   commencing 
  at 
  zero, 
  are 
  gradually 
  increased 
  to 
  the 
  values 
  

   shown 
  in 
  (3). 
  In 
  the 
  course 
  of 
  this 
  process 
  we 
  have 
  

  

  dw^Ydp 
  + 
  Y'Jpf 
  + 
  ^ 
  + 
  Q'dtf 
  + 
  Bdr+Bldr' 
  . 
  . 
  (4). 
  

  

  § 
  3. 
  Hence 
  if 
  we 
  suppose 
  w 
  expressed 
  as 
  a 
  function 
  of 
  

   p, 
  p', 
  q, 
  q\ 
  r, 
  ?•', 
  we 
  have 
  

  

  ■T=P, 
  £? 
  = 
  F, 
  ^=Q, 
  ^ 
  = 
  Q', 
  '^=R, 
  ^=R' 
  (4). 
  

   dp 
  dp' 
  ' 
  dg 
  *' 
  dq' 
  ^' 
  dr 
  ' 
  dr' 
  J 
  

  

  This 
  completes 
  the 
  foundation 
  of 
  the 
  molar 
  dynamics 
  of 
  

   an 
  elastic 
  solid 
  of 
  the 
  most 
  general 
  possible 
  kind 
  according 
  

   to 
  Green's 
  theory, 
  expressed 
  in 
  terms 
  of 
  the 
  new 
  mode 
  of 
  

   specifying 
  stresses 
  and 
  strains, 
  without 
  restriction 
  to 
  in- 
  

   finitely 
  small 
  strains. 
  

  

  § 
  4:. 
  To 
  understand 
  thoroughly 
  the 
  state 
  of 
  strain 
  specified 
  

   by 
  (1) 
  or 
  (2), 
  let 
  the 
  tetrahedron 
  of 
  reference, 
  A 
  B 
  C 
  D 
  , 
  

   for 
  the 
  condition 
  of 
  zero 
  strain 
  and 
  stress, 
  be 
  equilateral 
  

   (that 
  is 
  to 
  say, 
  according 
  to 
  the 
  notation 
  of 
  § 
  2 
  (1) 
  let 
  i 
  of 
  

   each 
  edge=p 
  =q 
  — 
  r 
  =p 
  / 
  — 
  q 
  l 
  = 
  r 
  l 
  ). 
  In 
  A 
  B 
  C 
  Do 
  inscribe 
  

   a 
  spherical 
  surface 
  touching 
  each 
  of 
  the 
  six 
  edges. 
  Its 
  

   centre 
  must 
  be 
  at 
  K 
  , 
  the 
  centre 
  of 
  the 
  tetrahedron 
  ; 
  and 
  the 
  

   points 
  of 
  contact 
  must 
  be 
  the 
  middle 
  points 
  of 
  the 
  edges. 
  

   Alter 
  the 
  solid 
  by 
  homogeneous 
  strain 
  *, 
  to 
  the 
  condition 
  

   (ji, 
  q, 
  r, 
  p 
  f 
  , 
  q', 
  r') 
  in 
  which 
  A 
  B 
  O 
  D 
  o 
  becomes 
  ABCD. 
  

   The 
  inscribed 
  spherical 
  surface 
  becomes 
  an 
  ellipsoid 
  having 
  

   its 
  centre 
  at 
  K, 
  the 
  centre 
  of 
  ABCD, 
  and 
  touching- 
  

   its 
  six 
  edges 
  at 
  their 
  middle 
  points 
  t. 
  This 
  ellipsoid 
  shows 
  

  

  * 
  Thomson 
  and 
  Tait's 
  < 
  Natural 
  Philosophy,' 
  § 
  155 
  ; 
  ' 
  Elements,' 
  

   § 
  136. 
  

  

  t 
  Thus 
  we 
  have 
  an 
  interesting 
  theorem 
  in 
  the 
  geometry 
  of 
  the 
  tetra- 
  

   hedron 
  : 
  — 
  If 
  an 
  ellipsoid 
  touching 
  the 
  edges 
  of 
  a 
  tetrahedron 
  has 
  its 
  

   centre 
  at 
  the 
  centre 
  of 
  the 
  tetrahedion, 
  the 
  points 
  of 
  contact 
  are 
  at 
  the 
  

   middles 
  of 
  the 
  edges. 
  

  

  