﻿1056 
  Mr. 
  J. 
  K. 
  Wilton 
  on 
  the 
  

  

  This 
  result 
  is 
  in 
  agreement 
  with 
  the 
  work 
  o£ 
  Stokes 
  and 
  

   of 
  Michell 
  on 
  the 
  " 
  highest 
  wave 
  " 
  in 
  water. 
  The 
  ratio 
  of 
  

   wave-length 
  to 
  amplitude 
  is 
  also 
  in 
  fair 
  agreement 
  with 
  the 
  

   approximate 
  value 
  given 
  by 
  Michell*, 
  but 
  the 
  velocity 
  is 
  

   considerably 
  greater. 
  He 
  gives 
  

  

  a/\ 
  = 
  -l£2, 
  or 
  \=7'04a 
  

  

  and 
  c 
  2 
  =-191^X, 
  or 
  ^-=1'20. 
  

  

  ij 
  

  

  Michell's 
  series 
  appear 
  to 
  converge 
  fairly 
  rapidly, 
  but 
  the 
  

   addition 
  of 
  terms 
  of 
  higher 
  order 
  would 
  certainly 
  tend 
  to 
  

   increase 
  his 
  estimate 
  of 
  the 
  velocity. 
  

  

  The 
  wave-profile 
  evidently 
  consists 
  of 
  a 
  succession 
  of 
  arcs 
  

   of 
  cycloids 
  which 
  meet 
  one 
  another 
  at 
  an 
  angle 
  of 
  120°. 
  If 
  

  

  2c 
  2 
  6 
  

   we 
  put 
  y= 
  — 
  sin 
  2 
  ^-, 
  the 
  equation 
  of 
  the 
  free 
  surface 
  takes 
  

  

  the 
  form 
  9 
  

  

  *-£(*+ 
  Bin*)] 
  

  

  \ 
  \> 
  (3) 
  

  

  y=J:(l-cOB*)J 
  

  

  where 
  the 
  whole 
  wave-length 
  is 
  included 
  between 
  6= 
  — 
  7r/3 
  

   and 
  = 
  7r/3. 
  Certain 
  easily 
  determinable 
  constants 
  of 
  in- 
  

   tegration 
  must 
  be 
  added 
  to 
  the 
  expression 
  for 
  x 
  in 
  the 
  various 
  

   cases 
  when 
  6 
  has 
  any 
  other 
  of 
  its 
  possible 
  ranges 
  of 
  values 
  

  

  on 
  the 
  free 
  surface: 
  — 
  e, 
  a. 
  when 
  - 
  — 
  <#<^r 
  the 
  constant 
  is 
  

  

  'J 
  

  

  Cl-^). 
  

  

  "We 
  may 
  verify 
  this 
  result 
  by 
  determining 
  the 
  steady 
  

   motion 
  for 
  which 
  equations 
  (3) 
  represent 
  a 
  free 
  surface. 
  It 
  

   will 
  be 
  given 
  by 
  

  

  z=-[6+ 
  sin0 
  + 
  *(l- 
  cos*?)], 
  

  

  if 
  

  

  w=-\ 
  \/lc 
  2 
  -2c 
  2 
  (l- 
  cos 
  .0)"|2(1+ 
  cos 
  6)d0, 
  

   f\/2cos(9-lcos^0; 
  

  

  whence 
  

  

  2c 
  3 
  

   9 
  

  

  % 
  = 
  «> 
  f'x/l 
  -4 
  sin»| 
  + 
  ^sin" 
  1 
  (2 
  sin*). 
  

   * 
  Phil. 
  Mag. 
  Noyember 
  1893. 
  

  

  