﻿148 
  (hi 
  Capillarity 
  Constants 
  of 
  Crystal 
  Faces. 
  

  

  the 
  area 
  of 
  ABC' 
  is 
  

  

  2~ 
  ^2 
  (l>' 
  2 
  c 
  2 
  sin 
  2 
  a.— 
  be 
  [fr 
  + 
  c 
  2 
  ) 
  cos 
  a. 
  -f 
  2a 
  2 
  6c 
  cos 
  (3 
  . 
  cos 
  7 
  ) 
  ; 
  

  

  A' 
  2 
  . 
  - 
  ytyc' 
  

  

  the 
  area 
  oi! 
  OBC 
  is 
  ~- 
  be 
  sin 
  a, 
  and 
  of 
  OB'C 
  is 
  — 
  ^ 
  -sina 
  ; 
  

  

  and 
  we 
  get 
  the 
  areas 
  of 
  A'B'C, 
  OCA, 
  OCA', 
  OAB, 
  OA'B' 
  

   similarly. 
  

  

  Suppose 
  now 
  that 
  the 
  figure 
  ABC 
  A'B'C 
  represents 
  a 
  crystal 
  

   which 
  is 
  such 
  that 
  its 
  surface 
  energy 
  is 
  a 
  minimum 
  for 
  a 
  

   given 
  volume. 
  Let 
  the 
  capillarity 
  constants 
  of 
  the 
  faces 
  

   BCB'C, 
  CAC'A', 
  ABA'B', 
  ABC, 
  A'B'C 
  be 
  *„ 
  k 
  2 
  , 
  * 
  s 
  , 
  k, 
  and 
  k\ 
  

   respectively 
  ; 
  then 
  the 
  volume 
  of 
  the 
  crystal 
  is 
  

  

  £ 
  ^ 
  ( 
  1 
  + 
  2 
  cos 
  ex. 
  . 
  cos 
  /3 
  . 
  cos 
  7 
  — 
  cos 
  2 
  a— 
  cos 
  2 
  £ 
  — 
  cos 
  2 
  7) 
  {x*abc—y 
  s 
  a'h'( 
  J 
  ), 
  

  

  and 
  the 
  surface 
  energy 
  is 
  Pi^ 
  2 
  — 
  P^ 
  2 
  , 
  where 
  

  

  Pi 
  = 
  I 
  (be 
  k\ 
  sin 
  a 
  -f 
  ca 
  k 
  2 
  sin 
  $-\-ab 
  k 
  3 
  sin 
  7 
  

  

  + 
  k 
  ^I^VsTn^: 
  - 
  be 
  (6 
  2 
  + 
  o 
  2 
  ) 
  cos 
  a. 
  + 
  2a 
  2 
  />r 
  cos 
  # 
  Tcos 
  7) 
  ), 
  

   P 
  2 
  = 
  J 
  (&V#, 
  sin 
  a 
  + 
  c'a^ 
  sin 
  £ 
  + 
  a'M 
  3 
  sin 
  7 
  

  

  - 
  A-' 
  V' 
  2(#V* 
  sin 
  2 
  a—b'c'(b 
  h2 
  + 
  f' 
  2 
  ) 
  cos 
  "a 
  + 
  2a'%'c' 
  cos 
  # 
  . 
  coTyj) 
  . 
  

  

  We 
  have 
  then 
  at 
  once 
  for 
  constant 
  volume 
  and 
  surface 
  

   energy 
  a 
  minimum, 
  on 
  differentiating 
  

  

  > 
  a 
  '/,y 
  t 
  m 
  P,V 
  2 
  //V 
  2 
  

  

  ' 
  T 
  - 
  . 
  with 
  the 
  condition 
  p 
  ... 
  ... 
  .j- 
  g 
  P.>0. 
  

  

  P 
  2 
  «/»c 
  ' 
  P 
  2 
  '«-AV 
  

  

  Magdalen 
  College, 
  Oxford. 
  

  

  