﻿Highest 
  Wave 
  in 
  Deep 
  Water. 
  1057 
  

  

  We 
  may 
  write 
  the 
  equation 
  for 
  z 
  in 
  the 
  form 
  

  

  <•■ 
  

  

  Then 
  die 
  2c* 
  n 
  ^— 
  T 
  Odd 
  

  

  i— 
  = 
  — 
  v 
  2 
  cos 
  U 
  — 
  1 
  cos 
  ~ 
  -7- 
  

  

  ^2 
  (/ 
  2 
  flfe 
  

  

  This 
  gives 
  q 
  = 
  

  

  = 
  c\/l 
  — 
  r' 
  e 
  + 
  e' 
  

  

  = 
  c 
  2 
  (2 
  cos 
  6—1), 
  when 
  # 
  is 
  real, 
  

  

  i. 
  e. 
  q 
  2 
  z=c 
  2 
  — 
  zgy 
  

  

  Further, 
  the 
  bottom 
  of 
  the 
  liquid 
  corresponds 
  to 
  0= 
  —ceo 
  

   (z/= 
  — 
  co 
  , 
  ^ 
  = 
  — 
  go 
  ), 
  which 
  makes 
  

  

  as 
  it 
  should 
  be. 
  

  

  Hence 
  equations 
  (3), 
  where 
  it 
  is 
  understood 
  that 
  

  

  \ 
  < 
  cos 
  0< 
  1, 
  

  

  and 
  that, 
  when 
  # 
  does 
  not 
  lie 
  between 
  — 
  - 
  — 
  and 
  ^- 
  . 
  the 
  

  

  o 
  o 
  

  

  appropriate 
  constant 
  is 
  to 
  be 
  added 
  to 
  x 
  in 
  order 
  to 
  make 
  

  

  the 
  arcs 
  of 
  the 
  various 
  cycloids 
  " 
  fit 
  on," 
  represent 
  a 
  possible 
  

  

  form 
  of 
  the 
  " 
  highest 
  wave 
  " 
  in 
  deep 
  water. 
  

  

  The 
  necessity 
  of 
  " 
  fitting 
  on 
  " 
  the 
  different 
  arcs 
  of 
  the 
  

   cycloid 
  leads 
  to 
  a 
  difficulty 
  which 
  cannot 
  satisfactorily 
  be 
  

   overcome. 
  There 
  is 
  a 
  discontinuity 
  in 
  the 
  motion 
  across 
  the 
  

   vertical 
  lines 
  through 
  the 
  crests 
  of 
  the 
  wave, 
  for 
  the 
  vertical 
  

   velocity 
  does 
  not 
  vanish 
  on 
  these 
  lines, 
  except 
  at 
  the 
  crest 
  of 
  

   the 
  wave 
  and 
  at 
  the 
  bottom 
  of 
  the 
  fluid 
  ; 
  and, 
  moreover, 
  its 
  

   direction, 
  but 
  not 
  its 
  magnitude, 
  is 
  changed 
  in 
  passing 
  across 
  

   these 
  lines. 
  This 
  necessitates 
  a 
  constant 
  change 
  of 
  momentum 
  

   in 
  passing 
  across 
  these 
  lines, 
  which 
  change, 
  it 
  will 
  be 
  seen, 
  

   is 
  vertically 
  upiuards, 
  so 
  that 
  it 
  can 
  be 
  supplied, 
  and 
  the 
  above 
  

   equations 
  will 
  therefore 
  represent 
  an 
  exact 
  solution 
  of 
  the 
  

   problem, 
  if 
  we 
  suppose 
  that 
  the 
  bottom 
  of 
  the 
  fluid 
  is 
  a 
  rigid 
  

   horizontal 
  plane. 
  

  

  The 
  solution 
  is, 
  however, 
  more 
  interesting 
  when 
  regarded 
  

   as 
  an 
  approximation 
  to 
  the 
  actual 
  physical 
  solution 
  of 
  the 
  

   problem 
  of 
  determining 
  the 
  form 
  of 
  the 
  highest 
  wave 
  in 
  

   deep 
  water. 
  That 
  it 
  can 
  be 
  so 
  regarded 
  is 
  due 
  to 
  the 
  fact 
  

   that 
  along 
  the 
  vertical 
  lines 
  through 
  the 
  crests 
  r 
  is 
  small. 
  

  

  I 
  find, 
  for 
  x 
  = 
  — 
  I 
  ~- 
  -f 
  -J, 
  the 
  following 
  table 
  connecting 
  

   u 
  and 
  v 
  with 
  y 
  : 
  — 
  

  

  