﻿Long 
  Air-Bubbles 
  in 
  a 
  Vertical 
  Tube. 
  

  

  955 
  

  

  Table 
  II. 
  shows 
  values 
  of 
  V 
  calculated 
  by 
  this 
  formula 
  

   against 
  those 
  obtained 
  by 
  experiment. 
  

  

  Table 
  II. 
  

  

  Diam. 
  cm. 
  

  

  •536 
  

  

  •560 
  

  

  •788 
  

  

  1-321 
  

  

  1-680 
  

  

  1-796 
  

  

  2-695 
  

  

  5-09 
  

  

  7-78 
  

  

  Velocity 
  [measured 
  

  

  •028 
  

  

  •336 
  

  

  3-23 
  

  

  103 
  

  

  15-0 
  

  

  16-3 
  

  

  23-4 
  

  

  29-9 
  

  

  32 
  9 
  

  

  32-8 
  

  

  cm. 
  per 
  sec. 
  ) 
  , 
  , 
  , 
  -, 
  

   r 
  I 
  calculated 
  

  

  •031 
  

  

  •320 
  

  

  3-20 
  

  

  10-6 
  

  

  15-0 
  

  

  16-3 
  

  

  23-5 
  

  

  29-6 
  

  

  When 
  <i=*535 
  cm. 
  the 
  formula 
  makes 
  V=0. 
  As 
  indi- 
  

   cated 
  in 
  Table 
  I., 
  no 
  motion 
  is 
  possible 
  in 
  a 
  tube 
  *42 
  cm. 
  in 
  

   diameter, 
  and 
  further 
  investigation 
  showed 
  that 
  this 
  is 
  also 
  

   the 
  case 
  with 
  a 
  diameter 
  of 
  '51 
  cm. 
  Under 
  these 
  circum- 
  

   stances 
  there 
  appears 
  to 
  be 
  no 
  connexion 
  between 
  the 
  films 
  

   forming 
  the 
  upper 
  and 
  lower 
  surfaces 
  of 
  the 
  bubble. 
  Some 
  

   idea 
  of 
  the 
  limiting 
  diameter 
  at 
  which 
  motion 
  becomes 
  

   possible 
  may 
  be 
  obtained, 
  independently, 
  from 
  a 
  consideration 
  

   of 
  the 
  conditions 
  of 
  equilibrium 
  of 
  the 
  fluid 
  surrounding 
  the 
  

   head 
  of 
  the 
  bubble 
  when 
  motion 
  is 
  on 
  the 
  point 
  of 
  beginning. 
  

   At 
  this 
  instant 
  the 
  upper 
  surface 
  of 
  the 
  bubble 
  becomes 
  very 
  

   nearly 
  spherical 
  and 
  of 
  approximately 
  the 
  same 
  radius 
  as 
  the 
  

   tube. 
  If 
  R 
  be 
  the 
  radius 
  of 
  the 
  tube, 
  the 
  difference 
  between 
  

   the 
  internal 
  and 
  external 
  pressures 
  at 
  its 
  apex, 
  on 
  these 
  as- 
  

   sumptions, 
  is 
  2T/R, 
  while 
  at 
  its 
  point 
  of 
  tangency 
  the 
  

   difference 
  is 
  T/R. 
  Expressing 
  the 
  fact 
  that 
  the 
  difference 
  

   of 
  external 
  pressure 
  at 
  these 
  points, 
  due 
  to 
  the 
  difference 
  of 
  

   level, 
  is 
  R/ot, 
  where 
  sr, 
  the 
  weight 
  of 
  water 
  per 
  cub.cm., 
  is 
  

   unity, 
  we 
  get 
  

  

  2T 
  T 
  

  

  R- 
  

  

  R 
  

  

  R' 
  

  

  R^yTrry-0738 
  

  

  = 
  •2718 
  cm. 
  

  

  a 
  value 
  in 
  fair 
  agreement 
  with 
  that 
  (*2675 
  cm.) 
  deduced 
  

   from 
  experiment. 
  

  

  4. 
  Ripple 
  Formation 
  on 
  the 
  Surface 
  of 
  a 
  Bubble. 
  — 
  The 
  

   system 
  of 
  ripples 
  on 
  the 
  surface 
  of 
  a 
  long 
  bubble 
  appears 
  

   to 
  be 
  analogous 
  to 
  that 
  found 
  at 
  the 
  upstream 
  side 
  of 
  an 
  

   obstacle 
  piercing 
  the 
  surface 
  of 
  a 
  running 
  stream, 
  the 
  dis- 
  

   turbance 
  produced 
  by 
  the 
  sudden 
  change 
  of 
  section 
  and 
  of 
  

   velocity 
  in 
  the 
  rear 
  of 
  the 
  bubble 
  being 
  analogous 
  to 
  that 
  

   due 
  to 
  the 
  presence 
  of 
  the 
  obstacle 
  in 
  the 
  latter 
  case. 
  

  

  In 
  the 
  case 
  of 
  wave-formation 
  on 
  the 
  walls 
  of 
  a 
  vertical 
  

  

  3T2 
  

  

  