﻿Constants 
  of 
  Crystal 
  Faces. 
  147 
  

  

  and 
  for 
  V 
  constant 
  and 
  E 
  a 
  minimum 
  we 
  have 
  

  

  = 
  1nds 
  — 
  Skds 
  — 
  Xads 
  = 
  %bds 
  = 
  iLcds 
  ; 
  

  

  where 
  the 
  quantities 
  ds 
  are 
  connected 
  only 
  by 
  these 
  five 
  

   equations. 
  

  

  Hence 
  we 
  have 
  

  

  ay 
  — 
  xa 
  1 
  — 
  ftb 
  x 
  — 
  yci 
  — 
  pkx 
  — 
  0, 
  

   n 
  2 
  —aa 
  2 
  —ftb 
  2 
  —yc 
  2 
  —pk 
  2 
  = 
  0, 
  

   n 
  s 
  — 
  *a 
  d 
  — 
  ftb 
  3 
  —yc 
  3 
  -~ 
  'pk$~0, 
  

   &c. 
  &c. 
  

  

  where 
  a, 
  ft, 
  7, 
  p 
  are 
  (unknown) 
  constants. 
  

  

  Now 
  there 
  can 
  always 
  be 
  found 
  a 
  point 
  Oj, 
  whose 
  distances 
  

   from 
  P 
  1? 
  P 
  2 
  , 
  P 
  3 
  are 
  

  

  oai 
  + 
  ^i 
  + 
  yc], 
  <xa 
  2 
  + 
  ftb 
  2 
  + 
  yc 
  2 
  , 
  aa 
  3 
  + 
  ftb 
  3 
  -^yc 
  3 
  

   respectively 
  ; 
  the 
  distance 
  of 
  O 
  x 
  from 
  P 
  4 
  is 
  

   Pi 
  (<za 
  x 
  + 
  ftb 
  x 
  + 
  7c,) 
  + 
  (ji 
  {*a 
  2 
  + 
  fth 
  2 
  + 
  yc 
  2 
  ) 
  + 
  r±(aa 
  3 
  + 
  ftb 
  3 
  f 
  7C3) 
  

   = 
  a(p 
  4 
  a 
  t 
  4- 
  q±a, 
  2 
  + 
  *V*s) 
  + 
  ft(p 
  4 
  bi 
  + 
  </ 
  4 
  ^ 
  2 
  + 
  ? 
  * 
  4 
  W 
  

  

  + 
  7 
  (/^4 
  c 
  i 
  + 
  q 
  4 
  c 
  2 
  -f- 
  r 
  4 
  c 
  3 
  ) 
  = 
  aa 
  A 
  + 
  #6 
  4 
  + 
  7C4. 
  

   Similarly 
  its 
  distance 
  from 
  P 
  5 
  is 
  aa 
  5 
  + 
  @b 
  6 
  + 
  yc 
  5 
  , 
  &c. 
  

   Hence 
  if 
  the 
  perpendiculars 
  from 
  2 
  on 
  <x„ 
  <t 
  2 
  , 
  0-3, 
  . 
  . 
  . 
  are 
  

  

  '1 
  5 
  % 
  , 
  

  

  / 
  M 
  / 
  

  

  %' 
  ..., 
  we 
  have 
  

  

  a/ 
  = 
  h 
  1— 
  aa 
  x 
  — 
  /3fr, 
  — 
  7c, 
  = 
  /3.V1, 
  w 
  2 
  ' 
  = 
  rc 
  2 
  —aa 
  2 
  — 
  /3/> 
  2 
  — 
  yc 
  2 
  —pk 
  2 
  , 
  

   n 
  3 
  f 
  = 
  n 
  z 
  —aa 
  3 
  ~ 
  ftb 
  s 
  — 
  yc 
  3 
  =pk 
  3 
  , 
  &c. 
  <fec. 
  

  

  Hence 
  there 
  is 
  a 
  point 
  O 
  x 
  for 
  which 
  

  

  w/ 
  : 
  w 
  2 
  ' 
  : 
  nj 
  . 
  . 
  . 
  = 
  h\ 
  : 
  k 
  2 
  : 
  k 
  3 
  . 
  .., 
  

   which 
  is 
  Wulff's 
  theorem. 
  

  

  It 
  is 
  not 
  always 
  easy 
  to 
  apply 
  the 
  above 
  theorem 
  to 
  the 
  case 
  

   ,of 
  an 
  actual 
  crystal. 
  It 
  seems 
  therefore 
  worth 
  while 
  to 
  give 
  

   an 
  example 
  which 
  is 
  of 
  use 
  in 
  many 
  cases. 
  

  

  Let 
  three 
  planes 
  OBC, 
  OCA, 
  OAB 
  meet 
  along 
  the 
  lines 
  

   OA, 
  OB, 
  00 
  ; 
  and 
  let 
  them 
  be 
  met 
  by 
  two 
  planes 
  ABO, 
  

   A'B'C, 
  so 
  that 
  0A 
  = 
  ^, 
  OB 
  = 
  <*>/,, 
  00 
  = 
  arc 
  ; 
  0A!-ya\ 
  

   QB'=yb', 
  OG'^yv'j 
  where 
  a, 
  b, 
  c, 
  a!, 
  b', 
  c' 
  are 
  supposed 
  known. 
  

   Let 
  the 
  angles 
  BOO, 
  COA, 
  AOB 
  be 
  a, 
  ft, 
  7, 
  respectively. 
  

  

  Then 
  the 
  volume 
  of 
  the 
  figure 
  OABO 
  is 
  

  

  7' 
  /V/)/* 
  ___ 
  ____ 
  __________ 
  ___ 
  __ 
  __________^__ 
  

  

  — 
  7T- 
  ^1 
  + 
  2 
  cos 
  a. 
  cos/3 
  . 
  C0S7— 
  cos 
  2 
  a— 
  cos 
  2 
  /3— 
  cos 
  2 
  7, 
  

  

  and 
  of 
  OA'B'O 
  is 
  

  

  A'b'c' 
  f 
  

  

  - 
  — 
  : 
  — 
  - 
  vl 
  + 
  2 
  cos 
  a. 
  . 
  cos 
  ft 
  . 
  cos 
  7 
  — 
  cos" 
  2 
  a— 
  cos 
  2 
  ft 
  — 
  cos* 
  y 
  ; 
  

  

  L 
  2 
  

  

  