﻿724 
  Mr. 
  G. 
  Green 
  on 
  Flexuml 
  

  

  satisfies 
  all 
  the 
  equations 
  satisfied 
  by 
  the 
  function 
  <^, 
  we 
  see 
  

   that 
  we 
  have 
  

  

  at 
  a 
  free, 
  or 
  level, 
  liquid 
  surface. 
  Equations 
  (6) 
  and 
  (8) 
  

   combined 
  give 
  us 
  the 
  relation 
  

  

  B.r^ 
  ~ 
  g^'bt^ 
  ' 
  ^^ 
  

  

  and 
  when 
  we 
  put 
  y 
  in 
  place 
  of 
  (j>, 
  K.h 
  in 
  place 
  of 
  -, 
  and 
  

  

  interchange 
  x 
  and 
  t, 
  this 
  equation 
  may 
  be 
  written 
  in 
  the 
  

   form 
  

  

  P...^g.0, 
  (3) 
  

  

  which 
  is 
  the 
  equation 
  already 
  found 
  for 
  flexural 
  vibrations 
  

   of 
  an 
  elastic 
  bar, 
  when 
  terms 
  depending 
  upon 
  the 
  angular 
  

   motions 
  of 
  the 
  sections 
  of 
  the 
  bar 
  may 
  be 
  neglected. 
  Ac- 
  

   cordingly, 
  equation 
  (9) 
  proves 
  that 
  every 
  hydrodynamical 
  

   potential 
  function, 
  with 
  x 
  and 
  t 
  interchanged, 
  is 
  a 
  solution 
  

   of 
  equation 
  (3) 
  above. 
  (The 
  converse 
  is 
  not 
  always 
  true.) 
  

   In 
  relation 
  to 
  the 
  flexural-waves 
  problems 
  z 
  is 
  in 
  general 
  to 
  

   be 
  treated 
  as 
  a 
  constant. 
  

  

  § 
  4. 
  It 
  may 
  be 
  of 
  some 
  interest 
  to 
  note 
  that, 
  in 
  all 
  solutions 
  

   for 
  elastic 
  bars 
  obtained 
  by 
  applying 
  the 
  above 
  result, 
  the 
  

   displacement 
  y 
  at 
  each 
  point 
  of 
  the 
  bar 
  may 
  be 
  regarded 
  as 
  

   derived 
  from 
  a 
  single 
  function 
  V(i, 
  r, 
  x) 
  such 
  that 
  

  

  y=^^ 
  («'> 
  

  

  Further, 
  it 
  follows 
  from 
  the 
  relationship 
  established 
  between 
  

   such 
  a 
  function 
  V 
  and 
  the 
  hydrodynamic 
  potential 
  function, 
  

   that 
  every 
  function 
  derived 
  from 
  Y 
  by 
  differentiations 
  or 
  

   integrations 
  with 
  respect 
  to 
  t^ 
  r, 
  a-, 
  is 
  a 
  solution 
  of 
  the 
  dif- 
  

   ferential 
  equation 
  (3). 
  Thus 
  any 
  solution 
  of 
  (3) 
  may 
  be 
  a 
  

   displacement 
  potential, 
  or 
  a 
  displacement, 
  or 
  a 
  velocity. 
  If 
  

   the 
  function 
  Y 
  represents 
  displacement, 
  w^e 
  readily 
  find 
  from 
  

   equation 
  (7) 
  that 
  the 
  curvature 
  at 
  each 
  point 
  x 
  of 
  the 
  bar 
  

   can 
  be 
  obtained 
  by 
  a 
  single 
  differentiation 
  with 
  respect 
  to 
  ^, 
  

   being 
  given 
  by 
  the 
  equation 
  

  

  R^/S'Bl^ 
  ^^^) 
  

  

  from 
  which 
  the 
  potential 
  energy 
  can 
  easily 
  be 
  obtained. 
  

  

  