﻿Constants 
  of 
  Crystal 
  Faces. 
  145- 
  

  

  sein 
  wird. 
  Das 
  Volumen 
  dieses 
  Polyeders 
  kann 
  in 
  folgender 
  

   Form 
  geschrieben 
  werden 
  : 
  

  

  V=g(n 
  1 
  * 
  + 
  n.? 
  + 
  nJ 
  i 
  + 
  ...), 
  

  

  wobei 
  q 
  eine 
  Constante 
  ist. 
  

  

  " 
  Da 
  das 
  Polyeder 
  ein 
  Minimum 
  der 
  Oberflachenenergie 
  

   bei 
  constanten 
  Volumen 
  besitzen 
  muss, 
  so 
  mlissen 
  folgende 
  

   Bedingungen 
  erfullt 
  werden 
  : 
  

  

  dE 
  = 
  2p{k 
  l 
  n 
  1 
  dn 
  l 
  -+- 
  k 
  2 
  n 
  2 
  dn 
  2 
  + 
  k 
  d 
  ?i$dn 
  s 
  . 
  . 
  .) 
  = 
  0; 
  

   dY 
  — 
  3</ 
  (n{ 
  2 
  dn 
  x 
  + 
  n 
  2 
  dn 
  2 
  + 
  n§ 
  2 
  dn 
  3 
  -f- 
  . 
  . 
  .) 
  = 
  0, 
  

  

  dieses 
  ist 
  aber 
  nur 
  dann 
  moglich, 
  wenn 
  

  

  k 
  x 
  : 
  he, 
  : 
  k 
  s 
  . 
  . 
  . 
  — 
  n 
  x 
  '.n 
  2 
  :n 
  z 
  : 
  , 
  . 
  ." 
  

  

  This 
  proof, 
  however, 
  is 
  faulty; 
  for 
  in 
  writing 
  down 
  the 
  ex- 
  

   pressions 
  for 
  E 
  and 
  V 
  — 
  which 
  should 
  be 
  (J^p^i^ 
  + 
  k 
  2 
  p 
  2 
  n. 
  2 
  2 
  + 
  . 
  . 
  .) 
  

   and 
  }>(pin 
  1 
  i 
  -\-p 
  2 
  n 
  2 
  6 
  + 
  . 
  ..) 
  respectively, 
  where 
  ,p 
  l9 
  p 
  2 
  , 
  .. 
  . 
  are 
  

   constants 
  : 
  this, 
  however, 
  does 
  not 
  affect 
  his 
  argument, 
  — 
  

   he 
  has 
  assumed 
  the 
  ratios 
  n 
  x 
  : 
  n 
  2 
  : 
  n 
  3 
  . 
  . 
  . 
  constant, 
  and 
  there- 
  

   fore 
  in 
  the 
  expressions 
  for 
  dJZ 
  and 
  dY, 
  dn 
  x 
  : 
  dn^ 
  : 
  dn 
  3 
  . 
  . 
  . 
  will 
  

   be 
  constant 
  ; 
  but 
  in 
  deducing 
  k 
  l 
  : 
  k 
  2 
  : 
  k 
  3 
  . 
  . 
  — 
  n 
  i 
  \n^: 
  n 
  :l 
  . 
  . 
  . 
  

   from 
  dJOj=dV=0 
  he 
  tacitly 
  assumes 
  dn 
  lf 
  dn 
  2 
  , 
  dn 
  z 
  . 
  . 
  . 
  quite 
  

   independent. 
  His 
  result, 
  however, 
  is 
  correct, 
  as 
  the 
  following 
  

   proof 
  shows 
  : 
  — 
  

  

  Suppose 
  s 
  u 
  s 
  2 
  , 
  s 
  s 
  . 
  . 
  . 
  the 
  areas 
  of 
  the 
  faces 
  cr 
  1? 
  <r 
  2 
  , 
  a> 
  . 
  . 
  . 
  per- 
  

   pendicular 
  to 
  the 
  normals 
  through 
  any 
  point 
  whose 
  lengths 
  

   are 
  /*,, 
  n 
  2 
  , 
  ?? 
  3 
  . 
  . 
  . 
  respectively. 
  We 
  have 
  then 
  

  

  V=.\ 
  %ns 
  and 
  .*. 
  d\ 
  = 
  ^%(nds 
  + 
  sdn). 
  

  

  Now 
  if 
  a 
  polyhedron 
  undergoes 
  a 
  small 
  deformation, 
  the 
  

   normals 
  to 
  its 
  faces 
  remaining 
  fixed 
  in 
  direction, 
  we 
  have 
  

   dY=Xsdh; 
  for 
  suppose 
  the 
  polyhedron 
  to 
  be 
  immersed 
  in 
  a 
  

   weightless 
  incompressible 
  fluid 
  which 
  is 
  contained 
  in 
  a 
  cylinder 
  

   of 
  cross-section 
  A, 
  closed 
  at 
  the 
  top 
  with 
  a 
  movable 
  piston 
  of 
  

   weight 
  W. 
  Then 
  the 
  pressure 
  at 
  any 
  point 
  of 
  the 
  liquid 
  

   . 
  W 
  

  

  "a- 
  

  

  If 
  now 
  the 
  polyhedron 
  undergoes 
  a 
  small 
  deformation 
  of 
  

   the 
  kind 
  stated 
  above, 
  the 
  work 
  done 
  by 
  any 
  surface 
  lies 
  

   between 
  

  

  w 
  w 
  w 
  

  

  ~r 
  -(s 
  + 
  ds)dn 
  and 
  sdn, 
  and 
  .'. 
  = 
  r-sdn, 
  

  

  neglecting 
  small 
  quantities 
  of 
  the 
  second 
  order. 
  If 
  now 
  the 
  

   polyhedron's 
  volume 
  increases 
  by 
  a 
  quantity 
  dY, 
  the 
  piston 
  is 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  3. 
  No. 
  1>>. 
  Jan. 
  1902. 
  L 
  

  

  