374  Mr.  Ii.  F.  Gwyther  on  the  Range  of 
Hence  the  term  to  be  added  to  —  EL/  is 
and  for  a  first  approximation, 
Jh_ E£ 
Vi      -08194  prf  +-3.«iZ8+-6I/Z 
This  example  is  sufficient  to  indicate  the  procedure  to  be 
adopted  in  all  such  cases. 
University  College,  Bristol, 
October  1905. 
XXIX.   On  the  Range  of  Stokes's  Deep-Water  Waves. 
By  R.  F.  Gwyther*. 
THE  waves  with  which  I  propose  to  deal  are  the  waves  of 
finite  amplitude  of  which  the  investigation  was  initiated 
by  Sir  George  Stokes  f  in  his  paper  on  the  Theory  of 
Oscillatory  Waves,  and  continued  in  the  Supplement  to  that 
paper.  The  object  of  the  paper  is  to  establish  the  correctness 
of  the  opinion  expressed  in  the  paper  quoted  (p.  227) 
"After  careful  consideration  I  feel  satisfied  ...  that  we  may 
approach  as  near  as  we  please  to  the  form  in  which  the  cur- 
vature at  the  vertex  becomes  infinite,  and  the  vertex  becomes 
a  multiple  point  where  the  two  branches  ...  enclose  an  angle 
of  120°."  The  method  of  the  Supplement  is  adhered  to  as 
closely  as  is  convenient.  The  result  is  to  establish  that  the 
velocity  in  all  waves  of  the  series,  small  as  well  as  great,  is 
represented  by  a  function  which  possesses  poles  which  as  the 
amplitude  increases  approach  nearer  to  the  fluid  surface,  and 
in  the  limiting  form  is  identical  with  that  investigated  by 
Mr.  Michell  J  in  his  paper  on  the  "  Highest  Wave  in  Water," 
in  which  the  poles  lie  on  the  water-surface. 
§  1.  Making  a  change  in  the  notation  used  by  Stokes  in 
the  Supplement,  in  order  to  make  the  analytical  form  of  the 
velocity  more  obvious,  I  write 
<£  +  n/r  hn     -_ 
x  +  iy  =  — L  —  iz—- ,e        c         ' 
J  c  nk 
where  n  is  an  integer,  and  k  is  a  constant  defining  the  wave- 
length.    Also  let  -yjr  =  0  define  the  free  surface  of  the  water. 
*  Communicated  by  the  Author. 
t  Mathematical  and  Physical  Papers,  toI.  i.  p.  197,  and  Supplement, 
p.  314. 
X  Phil.  Mag.  November  1893. 
