﻿Solid 
  Sphere 
  in 
  contact 
  with 
  a 
  Liquid 
  Surface. 
  929 
  

  

  large 
  compared 
  with 
  a, 
  and 
  that 
  the 
  angle 
  "o£ 
  contact 
  of 
  the 
  

   liquid 
  with 
  the 
  material 
  of 
  the 
  sphere 
  is 
  zero. 
  The 
  extension 
  

   of 
  the 
  formulae 
  to 
  liquids 
  of 
  finite 
  contact-angles 
  presents 
  no 
  

   difficulty. 
  

  

  Considering 
  the 
  vertical 
  components 
  of 
  the 
  forces 
  acting 
  

   on 
  the 
  sphere, 
  we 
  have 
  (1), 
  a 
  downward 
  pull 
  equal 
  to 
  

  

  27r/T 
  sin 
  cf)i 
  ; 
  

  

  (2), 
  and 
  (3), 
  the 
  vertical 
  components 
  of 
  the 
  hydrostatic 
  

   forces 
  on 
  the 
  concave 
  and 
  convex 
  sides 
  respectively 
  of 
  the 
  

   sphere. 
  

  

  Let 
  the 
  atmospheric 
  pressure 
  be 
  II, 
  and 
  let 
  rr 
  x 
  be 
  the 
  

   pressure 
  at 
  any 
  point 
  (#, 
  y) 
  in 
  the 
  liquid 
  and 
  on 
  the 
  surface 
  

   of 
  the 
  sphere. 
  

  

  Then 
  the 
  vertical 
  component 
  of 
  the 
  force 
  due 
  to 
  (2) 
  and 
  

   (3) 
  on 
  the 
  element 
  of 
  surface 
  swept 
  out 
  by 
  the 
  revolution 
  of 
  

   an 
  element 
  ds 
  of 
  the 
  circle 
  at 
  (x, 
  y) 
  will 
  be 
  

  

  = 
  2irxds 
  cos 
  </>(fI 
  — 
  Tl 
  x 
  ) 
  

  

  = 
  '27rxdx{ll-U 
  l 
  ) 
  

  

  = 
  2irgpxy 
  dx 
  

  

  = 
  2irgpz(B,- 
  SR*-a*)dx, 
  

  

  since 
  

  

  U^U-gpy 
  

  

  and 
  

  

  The 
  total 
  hydrostatic 
  pull 
  is 
  therefore 
  

  

  = 
  27rg 
  P 
  \ 
  'x(R-\/R 
  2 
  -x 
  2 
  )dx 
  

   9 
  [%,*■'« 
  , 
  (R 
  2 
  -r' 
  2 
  ) 
  f 
  R3-i 
  

  

  = 
  2 
  wL 
  R 
  x 
  + 
  — 
  3 
  d- 
  

  

  And 
  therefore, 
  if 
  we 
  suppose 
  that 
  these 
  surface-forces 
  are 
  

   balanced 
  by 
  an 
  upward 
  pull 
  of 
  mg 
  dynes, 
  the 
  equation 
  of 
  

   equilibrium 
  of 
  the 
  sphere 
  will 
  be 
  

  

  mg 
  = 
  2irrTsm<l> 
  1 
  + 
  27rgp\-f- 
  + 
  ^ 
  . 
  - 
  J 
  — 
  -J. 
  (xu.) 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  20. 
  No. 
  155. 
  Nov. 
  1913. 
  3 
  ft 
  

  

  