Stokes's  Deep-  Water  Waves.  375 
Then  the  velocity  is  given  by 
1  c 
d(<l>-i<f)         \      c     ) 
where 
7Vc/        +      n6        °        ' 
At  the  free  surface  we  have  ^  =  0,  and  it  will  be  convenient 
in  this  case  to  write,  in  place  of  fl  — —  J, 
f(^)  =  1+27^.     .....    (1) 
The   condition  for   uniform  pressure  along  the  free  surface 
requires  that,  when  ^  =  0, 
9U~  isf2  =  constant. 
The  determination  of  the  successive  values  of  hn  resulting 
from  this  condition  is  only  a  matter  of  labour.    I  have  carried 
the  determination  to  the  sixth  order,  as  an  extended  number 
of  terms  is  important  as  the  basis  of  the  future  argument. 
Stating  results  only,  I  find 
7  22°, 
7  973    ,      1975 
32  _4     313 
hA=^h*+^h<, 
7  62575 
A.=,^*6,      .........     (2) 
where  I  have  written  It  in  place  of  liv 
Also  if  we  write  2a  to  denote  the  height  of  the  wave  from 
trough  to  crest,  I  find 
*«  =  /*+ ?7*3+  —hi6 (3) 
