﻿Vibrations 
  of 
  Thin 
  Rods. 
  723 
  

  

  As 
  pointed 
  out 
  by 
  Lord 
  Kayleigh*, 
  whose 
  notation 
  we 
  have 
  

   adopted, 
  terms 
  depending 
  on 
  the 
  angular 
  motions 
  o£ 
  the 
  

   sections 
  o£ 
  the 
  bar 
  may 
  be 
  neglected 
  in 
  the 
  above 
  equations; 
  

   accordingly, 
  by 
  eliminating 
  N 
  from 
  (1) 
  and 
  (2) 
  we 
  obtain 
  

   finally 
  the 
  equation 
  

  

  g^-^^^S=«' 
  (^> 
  

  

  in 
  which 
  we 
  have 
  put 
  ^— 
  ^ 
  for 
  ^ 
  and 
  h^ 
  for 
  ^ 
  . 
  

   ^ 
  0^ 
  ^ 
  P 
  

  

  § 
  2. 
  Consider 
  now 
  frictionless 
  liquid 
  in 
  a 
  straight 
  canal 
  

   with 
  vertical 
  sides. 
  Take 
  the 
  origin 
  of 
  coordinates 
  at 
  point 
  

   0, 
  at 
  a 
  distance 
  h 
  above 
  the 
  undisturbed 
  level, 
  and 
  draw 
  OX 
  

   parallel 
  to 
  the 
  canal 
  and 
  OZ 
  vertically 
  downwards. 
  Let 
  the 
  

   motion 
  be 
  infinitesimal 
  and 
  let 
  f, 
  ^, 
  be 
  the 
  displacement 
  

   components 
  at 
  any 
  time 
  t 
  of 
  any 
  particle 
  of 
  water 
  whose 
  

   undisturbed 
  position 
  is 
  a', 
  r. 
  If 
  the 
  motion 
  of 
  the 
  water 
  be 
  

   started 
  primarily 
  from 
  rest 
  by 
  pressure 
  applied 
  to 
  the 
  free 
  

   surface, 
  the 
  hydrodynamical 
  equations 
  of 
  motion 
  take 
  the 
  

   well-known 
  form 
  

  

  j=^<^(.f,r,0; 
  i=^J{^,z,t); 
  . 
  . 
  (4) 
  

  

  from 
  which 
  we 
  obtain 
  by 
  integration 
  

  

  ?=|^.</)(^,c,0; 
  ^=^J(.v,z,t). 
  . 
  . 
  (5) 
  

  

  In 
  virtue 
  of 
  the 
  incompressibility 
  of 
  the 
  fluid 
  we 
  have 
  also 
  

   the 
  equation 
  

  

  which 
  shows 
  that 
  if 
  (/> 
  be 
  known 
  at 
  all 
  points 
  of 
  the 
  free 
  

   surface, 
  and 
  if 
  it 
  be 
  zero 
  at 
  points 
  infinitely 
  distant, 
  its 
  value 
  

   can 
  be 
  determined 
  at 
  all 
  points 
  throughout 
  the 
  fluid. 
  The 
  

   equation 
  to 
  be 
  fulfilled 
  at 
  the 
  free 
  surface, 
  according 
  to 
  a 
  

   result 
  given 
  by 
  Cauchy 
  and 
  Poisson, 
  is 
  

  

  (M) 
  _V^^ 
  . 
  . 
  . 
  (7) 
  

  

  where 
  a 
  denotes 
  gravity. 
  

  

  § 
  3. 
  "Remembering 
  now 
  that 
  every 
  function 
  derived 
  from 
  

   <j) 
  by 
  differentiations 
  or 
  integrations 
  with 
  respect 
  to 
  ^, 
  z, 
  t, 
  

  

  * 
  ' 
  Theory 
  of 
  Sound,' 
  vol. 
  i. 
  

  

  