﻿484 
  Mr. 
  W. 
  J. 
  Harrison 
  on 
  the 
  

  

  period 
  and 
  modulus 
  o£ 
  decav 
  will 
  be 
  given 
  below 
  §3^ 
  on 
  

   account 
  of 
  their 
  intrinsic 
  interest, 
  and, 
  also, 
  in 
  order 
  that 
  

   they 
  may 
  be 
  recorded 
  on 
  account 
  o£ 
  the 
  possibility 
  of 
  their 
  

   usefulness 
  in 
  some 
  other 
  problem. 
  

  

  It 
  is 
  believed 
  that 
  the 
  real 
  cause 
  of 
  the 
  discrepancy 
  must 
  

   be 
  looked 
  for 
  alono- 
  the 
  foUowino; 
  lines. 
  Dr. 
  Houstoun 
  in 
  

   his 
  paper 
  treats 
  of 
  waves 
  of 
  the 
  type 
  sin 
  q.v 
  sin 
  ri/, 
  the 
  axes 
  of 
  

   A' 
  and 
  ?/ 
  being 
  horizontal, 
  but 
  his 
  analysis 
  is 
  only 
  suitable 
  for 
  

   waves 
  of 
  the 
  type 
  sin 
  qa\ 
  This 
  is 
  clearly 
  brought 
  into 
  

   evidence 
  by 
  the 
  fact 
  that 
  he 
  obtains 
  for 
  the 
  period 
  of 
  his 
  

   waves 
  to 
  a 
  first 
  approximation 
  the 
  expression 
  27r/{gr/h}^, 
  

   whereas 
  the 
  correct 
  expression 
  is 
  27r/ 
  ^g 
  {q- 
  -{• 
  r^)h}'^, 
  as 
  will 
  

   be 
  shown 
  later. 
  The 
  same 
  applies 
  to 
  the 
  modulus 
  of 
  decay; 
  he 
  

   obtains 
  it 
  as 
  a 
  function 
  of 
  ^^, 
  whereas 
  it 
  is 
  the 
  same 
  function 
  

   of 
  {q^-hr^). 
  But 
  as 
  will 
  be 
  seen 
  later, 
  this 
  in 
  itself 
  does 
  not 
  

   wholly 
  explain 
  the 
  discrepancy. 
  It 
  will 
  be 
  shown 
  that 
  we 
  

   cannot 
  adequately 
  take 
  account 
  of 
  the 
  influence 
  of 
  the 
  sides 
  

   of 
  the 
  trough 
  in 
  damping 
  the 
  motion. 
  Further 
  discussion 
  

   w^ill 
  be 
  deferred 
  till 
  the 
  question 
  has 
  been 
  analytically 
  con- 
  

   sidered. 
  

  

  2'he 
  damjnng 
  of 
  long 
  leaves 
  at 
  the 
  surface 
  of 
  a 
  single 
  liquid. 
  

  

  §2. 
  This 
  problem 
  was 
  considered 
  by 
  Hough 
  *, 
  and 
  he 
  

   obtained 
  an 
  expression 
  for 
  the 
  modulus 
  of 
  decay 
  r, 
  which 
  is 
  

   equivalent 
  to 
  

  

  a* 
  k^ 
  V' 
  

  

  T-^=2Pv 
  + 
  ^ 
  ,'^- 
  3- 
  , 
  

  

  2 
  a/2 
  cosh 
  'kh 
  sinh 
  ^kh 
  

  

  where 
  v 
  is 
  the 
  kinematical 
  coefficient 
  of 
  viscosity, 
  k 
  = 
  2TT/\ 
  

   \ 
  being 
  the 
  wave-length, 
  and 
  h 
  is 
  the 
  depth 
  of 
  the 
  liquid. 
  

   The 
  motion 
  under 
  consideration 
  is 
  two-dimensional. 
  The 
  

   term 
  depending 
  on 
  v 
  in 
  the 
  above 
  expression 
  is 
  not 
  correct 
  ; 
  

   it 
  should 
  be 
  

  

  cosh2 
  /.j^ 
  ^ 
  gij^i^2 
  7,/^- 
  

  

  [' 
  

  

  + 
  

  

  This 
  makes 
  an 
  appreciable 
  difference 
  in 
  the 
  case 
  of 
  lon^ 
  

   waves^ 
  since 
  we 
  now 
  have 
  

  

  1 
  ^ 
  I 
  f 
  .'^■^'^ 
  

  

  jg^kvy 
  

  

  This 
  additional 
  term 
  v/4/r 
  makes 
  a 
  difference 
  of 
  from 
  6 
  to 
  

   10 
  per 
  cent, 
  in 
  the 
  theoretical 
  damping 
  of 
  waves 
  such 
  as 
  those 
  

   * 
  Proc, 
  Lond. 
  Math. 
  Soc. 
  vol. 
  xxviii. 
  p. 
  276 
  (1897). 
  

  

  