﻿Solid 
  Sphere 
  in 
  contact 
  with 
  a 
  Liquid 
  Surface, 
  927 
  

   This 
  gives 
  at 
  once 
  from 
  (vi.) 
  

  

  (vii.) 
  

  

  y(H- 
  ? 
  2 
  )-^ 
  4a 
  2 
  

  

  and 
  the 
  positive 
  sign 
  must 
  be 
  chosen 
  as 
  y 
  2 
  >2a 
  2 
  . 
  

  

  Hence, 
  substituting 
  the 
  values 
  from 
  (v.) 
  in 
  the 
  small 
  terms 
  

   of 
  (iv.), 
  and 
  the 
  values 
  from 
  (vii.) 
  in 
  the 
  small 
  terms 
  of 
  the 
  

   resulting 
  equation, 
  we 
  obtain 
  on 
  integration 
  

  

  7^7) 
  = 
  J 
  ~ 
  i 
  (4a2 
  - 
  /)f 
  " 
  ¥ 
  + 
  i*"- 
  *■>• 
  • 
  ^ 
  

  

  which, 
  substituting 
  the 
  above 
  values 
  for 
  h 
  L 
  2 
  , 
  and 
  assuming 
  

   that 
  p 
  = 
  when 
  y 
  = 
  d, 
  gives 
  

  

  1 
  3 
  4a 
  3 
  2\/2 
  — 
  1 
  1 
  l\/2a 
  z 
  

  

  rf 
  ~ 
  + 
  3r 
  l 
  ; 
  3r 
  ^/2 
  3r' 
  2 
  " 
  ' 
  l 
  j 
  

  

  And, 
  putting 
  d 
  2 
  = 
  4a 
  2 
  in 
  the 
  small 
  terms 
  of 
  (ix.), 
  we 
  have 
  

   finally 
  

  

  d 
  2 
  = 
  4a 
  3 
  -^, 
  (ix.a) 
  

  

  w 
  T 
  hich 
  value, 
  substituted 
  in 
  equation 
  (ii.)> 
  gives 
  

  

  F 
  = 
  2A 
  ff 
  p(a-Q, 
  (x.) 
  

  

  a 
  more 
  exact 
  equation 
  from 
  which 
  to 
  calculate 
  a 
  2 
  . 
  

  

  It 
  is 
  to 
  be 
  noted 
  (fig. 
  1) 
  that 
  r 
  is 
  not 
  the 
  radius 
  of 
  the 
  

   disk. 
  The 
  terms 
  in 
  r 
  being 
  assumed 
  small, 
  we 
  may, 
  as 
  a 
  

   first 
  approximation 
  put 
  r 
  = 
  r 
  Y 
  . 
  More 
  closely, 
  if 
  we 
  assume 
  

   the 
  meniscus 
  to 
  be 
  circular 
  in 
  outline 
  *, 
  we 
  have 
  

  

  r 
  1 
  = 
  r+h 
  = 
  r 
  + 
  a<y2 
  ) 
  

  

  which, 
  substituted 
  in 
  (x.), 
  gives 
  finally 
  

  

  This 
  result 
  is 
  in 
  agreement 
  with 
  that 
  of 
  Laplace 
  if 
  the 
  

   term 
  in 
  r 
  x 
  2 
  be 
  suppressed. 
  Whether 
  such 
  a 
  suppression 
  is 
  

  

  * 
  This, 
  of 
  course, 
  is 
  only 
  very 
  approximately 
  true. 
  It 
  would 
  be 
  

   easy 
  to 
  find 
  a 
  more 
  accurate— 
  and 
  more 
  complex— 
  correction. 
  But 
  the 
  

   above 
  assumption 
  is 
  close 
  enough 
  for 
  the 
  estimation 
  of 
  a 
  second 
  order 
  

   term. 
  

  

  