﻿146 
  Mr. 
  H. 
  Hilton 
  on 
  Capillarity 
  

  

  forced 
  through 
  a 
  height-r- 
  ; 
  and 
  therefore 
  the 
  work 
  done 
  by 
  

  

  . 
  dV 
  

   the 
  surfaces 
  of 
  the 
  polyhedron 
  is 
  W 
  —r- 
  ; 
  

  

  dV 
  ^ 
  W 
  

   and 
  .*. 
  W 
  — 
  i 
  =S 
  -r 
  sdn. 
  and 
  .*. 
  dV 
  — 
  Xsdn. 
  

   A 
  A 
  

  

  Hence 
  

  

  dY=l{2(nds)+dV} 
  and 
  .'. 
  Znds.=*2dV. 
  

  

  Now 
  we 
  have 
  relations 
  between 
  s 
  Y 
  , 
  s 
  2 
  , 
  s&. 
  . 
  . 
  due 
  to 
  the 
  fact 
  

   that 
  they 
  can 
  form 
  polyhedron 
  faces 
  whose 
  normals 
  have 
  

   fixed 
  directions. 
  For 
  take 
  any 
  point 
  of 
  reference 
  0, 
  and 
  

   take 
  planes 
  P 
  l5 
  P 
  2 
  , 
  P 
  3 
  . 
  . 
  . 
  through 
  0, 
  parallel 
  to 
  <r,, 
  <r 
  2 
  , 
  <r 
  3 
  . 
  . 
  . 
  . 
  

  

  Consider 
  any 
  point 
  A 
  ; 
  let 
  the 
  perpendiculars 
  from 
  on 
  

   the 
  surfaces 
  of 
  the 
  polyhedron 
  be 
  n 
  u 
  n 
  2l 
  n- 
  g 
  . 
  . 
  ., 
  and 
  from 
  A 
  

   be 
  7*!°, 
  n 
  2 
  a 
  , 
  n 
  3 
  a 
  . 
  . 
  . 
  ; 
  and 
  let 
  the 
  perpendiculars 
  from 
  A 
  on 
  

   P 
  1? 
  P 
  2 
  , 
  P 
  3 
  . 
  . 
  . 
  be 
  ai, 
  a 
  2 
  , 
  o 
  3 
  , 
  a 
  4 
  , 
  a 
  5 
  , 
  . 
  . 
  . 
  (considered 
  positive 
  or 
  

   negative, 
  as 
  A 
  and 
  the 
  corresponding 
  polyhedron 
  surface 
  are 
  

   on 
  the 
  same 
  or 
  different 
  sides 
  of 
  the 
  corresponding 
  plane 
  P). 
  

   Then 
  %ns 
  = 
  %n 
  a 
  s 
  ; 
  for 
  each 
  =3V; 
  and 
  

  

  .'. 
  2(>i— 
  n 
  a 
  )s 
  = 
  0, 
  and 
  .*. 
  2as 
  = 
  0, 
  (for 
  ;i 
  x 
  — 
  7ii 
  tt 
  = 
  a,, 
  &c). 
  

  

  Now 
  a 
  l9 
  a 
  2 
  , 
  a 
  z 
  . 
  . 
  . 
  are 
  sufficient 
  to 
  completely 
  fix 
  the 
  position 
  

   of 
  the 
  point 
  A 
  with 
  reference 
  to 
  0, 
  and 
  there 
  are 
  linear 
  

   relations 
  of 
  the 
  type 
  

  

  a 
  4 
  —p^cii 
  + 
  q±a 
  2 
  -f- 
  r±a 
  3 
  ; 
  a 
  5 
  —p 
  h 
  a 
  x 
  + 
  q 
  6 
  a 
  2 
  + 
  r 
  5 
  a 
  s 
  ; 
  &c. 
  

  

  Take 
  other 
  points 
  B, 
  C, 
  D, 
  . 
  .., 
  and 
  let 
  b, 
  c, 
  d, 
  . 
  . 
  . 
  cor- 
  

   respond 
  to 
  a. 
  Then 
  we 
  have 
  2as 
  = 
  0, 
  2&s 
  = 
  0, 
  2cs~0, 
  

   2^ 
  = 
  0,...; 
  an 
  indefinite 
  number 
  of 
  relations, 
  but 
  only 
  

   equivalent 
  to 
  3 
  independent 
  relations 
  ; 
  for 
  choosing 
  \, 
  fi, 
  v 
  so 
  

   that 
  

  

  \a 
  l 
  +iJ,b 
  1 
  + 
  vc 
  1 
  =d 
  1 
  ; 
  7<a 
  2 
  + 
  /jLb 
  2 
  + 
  vc 
  2 
  = 
  d 
  2 
  ; 
  A<2 
  3 
  + 
  fib 
  3 
  + 
  j/c 
  3 
  = 
  c? 
  3; 
  

  

  we 
  have 
  

  

  </ 
  4 
  =/>4^i 
  + 
  ^4^2 
  + 
  ^3 
  =lhO^^\ 
  + 
  A^i 
  + 
  yCi) 
  -f 
  y 
  4 
  (Xa 
  3 
  -f 
  fib 
  2 
  + 
  ^ 
  3 
  ) 
  

  

  4- 
  r 
  4 
  (Xa 
  3 
  4- 
  /^> 
  3 
  + 
  vc 
  3 
  ) 
  = 
  X 
  (p 
  4 
  a, 
  + 
  g 
  4 
  a 
  2 
  + 
  r 
  4 
  a 
  3 
  ) 
  + 
  /a(/? 
  4 
  &i 
  + 
  ^ 
  4 
  6 
  3 
  + 
  r 
  4 
  6 
  3 
  ' 
  

  

  + 
  K 
  Wi 
  + 
  ( 
  U 
  C 
  2 
  + 
  r 
  4 
  c 
  s) 
  = 
  *^4 
  + 
  f^h 
  + 
  ^4 
  ; 
  

  

  and 
  similarly 
  

  

  d 
  5 
  = 
  X<% 
  + 
  ytt/> 
  5 
  + 
  vc 
  5 
  , 
  &c 
  . 
  ; 
  

   and 
  hence 
  

  

  ^ds 
  ~ 
  XXas 
  +■ 
  /u,2^s 
  -f 
  vScs. 
  

   We 
  have 
  then 
  

  

  Inds 
  = 
  2dV; 
  2 
  Arrf* 
  = 
  dE 
  ; 
  and 
  2a<fc 
  = 
  Zbds 
  = 
  $cds 
  = 
  ; 
  

  

  