378  Range  of  Stokes's  Deep-Water  Waves. 
ft=l,  this  method  leads  to  expressions  identical  with  those 
of  Mr.  Mich  ell,  I  return  to  the  general  free-surface  con- 
dition, and,  for  convenience,  write  £  in  place  of  <j)/c. 
On  differentiating  the  free-surface  condition  in  respect  to 
f,  we  get  an  equation  of  the  form 
t{F(^-F(-tt0}=<^{F(ttf)F(-i*?)}-i; 
and  in  this  I  write 
F(^)=(l-^^)-*u-i. 
F{-ikQ=(l-fjLe-^K)-iY-\ 
where  U  and  V  are  rapidly  converging  series  of  the  type 
1  +  0^  +  0^2**  +  ..., 
so  that 
fJ  f  1  1  \ 
i   l(l-^«*;)*U       {l-fie-m)W) 
=  ^{C1-a*^*(1-a*^-««)*UV}.    ...     (9) 
This  may  be  put  in  the  form 
2&m  sin  frg tt*v>  1 
+    3    (l-A***«)4(l-#*-«QiL  V   J      *      ^10) 
which  becomes,  when  /a=1,  identical,  except  for  the  chanoe 
in  units,  with  the  equation  which  Mr.  Michell  has  employed 
to  obtain  the  form  of  the  Highest  Wave  in  Water. 
If,  however,  the  left-hand  side  of  (9)  is  rationalized  the 
right-hand  side  is  also  found  to  be  rational,  and  this  form 
may  be  found  more   convenient  whether  jju  is  less  than  or 
equal  to  unity.     The  values  of  the  constants  in  IT  and  V  are 
of  course,  different  in  these  two  cases. 
The  simplification  on  which  I  have  relied  would  appear  to 
be  applicable  to  all  cases  of  irrotational  waves  whether  in 
shallow  or  in  deep  water,  and  it  will  probably  be  found  that 
such  waves  in  either  case  will  be  related  to  poles,  as  I  have 
shown  to  be  the  case  with  deep-water  waves. 
