﻿Motion 
  of 
  a 
  Viscous 
  Fluid. 
  783 
  

  

  so 
  that 
  

  

  ^-^ 
  = 
  (e^- 
  S 
  ^){i(f) 
  2 
  -(f) 
  2 
  } 
  

  

  4 
  sin 
  6 
  cos 
  ^ 
  (hjr 
  

   + 
  r 
  ~o¥ 
  dO 
  ' 
  

  

  -.v..„,«»,{(«)'-i@)-} 
  

  

  cos 
  2 
  # 
  — 
  sin 
  2 
  # 
  dfi/r 
  <^a/t 
  

   r 
  dr 
  dO 
  

  

  Thus 
  

  

  cos 
  d 
  sin 
  (V 
  2 
  - 
  v' 
  2 
  ) 
  - 
  (cos 
  2 
  6> 
  - 
  sin 
  2 
  0)wV 
  

  

  _ld^d^ 
  , 
  

  

  -~rTrde> 
  ' 
  ' 
  ' 
  [Zb) 
  

   and 
  (25) 
  becomes 
  

  

  — 
  = 
  -F'-2pff(^^drd0dz 
  . 
  (27) 
  

  

  dt 
  P 
  JJJi*dr 
  dd 
  ' 
  K 
  } 
  

  

  T', 
  F', 
  as 
  well 
  as 
  the 
  last 
  integral, 
  being 
  proportional 
  to 
  z. 
  

  

  We 
  suppose 
  the 
  motion 
  to 
  take 
  place 
  in 
  the 
  space 
  between 
  

   two 
  coaxal 
  cylinders 
  which 
  revolve 
  with 
  appropriate 
  velocities. 
  

   If 
  the 
  additional 
  motion 
  be 
  also 
  symmetrical 
  about 
  the 
  axis, 
  

   the 
  stream-lines 
  are 
  circles, 
  and 
  yjr 
  is 
  a 
  function 
  of 
  r 
  only. 
  

   The 
  integral 
  in 
  (27) 
  then 
  disappears 
  and 
  dT'/dt 
  reduces 
  to 
  

   — 
  F', 
  so 
  that 
  under 
  this 
  restriction 
  * 
  the 
  original 
  motion 
  is 
  

   stable. 
  The 
  experiments 
  of 
  Couette 
  t 
  and 
  of 
  Mallock 
  t, 
  made 
  

   with 
  revolving 
  cylinders, 
  appear 
  to 
  show 
  that 
  when 
  u', 
  v' 
  , 
  w' 
  

   are 
  not 
  specially 
  restricted 
  the 
  motion 
  is 
  unstable. 
  It 
  may 
  

   be 
  of 
  interest 
  to 
  follow 
  a 
  little 
  further 
  the 
  indications 
  

   of 
  (27). 
  

  

  The 
  general 
  value 
  of 
  ty 
  is 
  

  

  ^ 
  = 
  C 
  o 
  +C 
  1 
  cos0 
  + 
  S 
  1 
  sin0 
  + 
  . 
  . 
  . 
  + 
  C 
  n 
  cosn0 
  + 
  S 
  n 
  sin?20, 
  . 
  (28). 
  

   n 
  , 
  S 
  n 
  being 
  functions 
  of 
  r, 
  whence 
  

  

  jf 
  3* 
  =.*(*.% 
  -c„f), 
  . 
  (29) 
  

  

  n 
  being 
  1, 
  2, 
  3, 
  &c. 
  If 
  S», 
  C 
  n 
  differ 
  only 
  by 
  a 
  constant 
  

  

  * 
  We 
  may 
  imagine 
  a 
  number 
  of 
  thin, 
  coaxal, 
  freely 
  rotating 
  cylinders 
  

   to 
  be 
  interposed 
  between 
  the 
  extreme 
  ones 
  whose 
  motion 
  is 
  prescribed. 
  

   t 
  Ann. 
  d. 
  Chimie, 
  t. 
  xxi. 
  p. 
  433 
  (1890). 
  

   t 
  Proc. 
  Roy. 
  Soc. 
  vol. 
  lix. 
  p. 
  38 
  (1895). 
  

  

  