Stokes's  Deep-  Water  Wa ves .  311 
These  give  approximately 
a=    L-054, 
£=-■068, 
7-— 015;  (5) 
and  these  values  lead  to 
A=-351,u--023yu3--005/a5 (6) 
This  value  of  h  increases  with  increasing  fractional  values 
of  //,,  and  reaches  the  value  *323  approximately  when  /Lt=l. 
Since  the  hypothesis  of  convergence  holds  good  for  any 
value  of  fi  finitely  less  than  unity,  we  may  take  this  to  give 
approximately  the  greatest  admissible  value  of  h. 
Using  the  numerical  values  (5)  again,  we  find  when  fi<l, 
F(^)}-^(w/?)*xU  .     (7) 
where 
iJe<p  2ik<t> 
U=l-(-018^-'023^--005ya5;e~  -(-01V--00V)  e  ~ 
2>ik<j)  iik<p  5iktp 
-(•0lV--003^5)^   T  -'006^  e~ir -'001^5  e~^  -•'  ' 
The  expression  for  the  velocity  (qel(°)  will  possess  a  series 
of  poles,  generally  above  the  water,  indicated  by 
fjue       c       =1. 
Using  the  numbers  in  (2)  and  (3) ,  I  find 
kc*l</  =  l  +  -12'5  ft2  +  '038/i4  +  -019  m6 
and 
*a  =  -351  yu  +  -041  fju3  +  '019  {M5. 
If  we  write  &  =  27r/\,  we  get 
^  ='112  fi  +  '01^fi3 +  '006  fi\      ...     (8) 
which  gives  for  the  height  of  the  greatest  rounded  wave 
2a/\='l?>l  approximately.  This  value  is  not  greatly  in 
defect  of  the  ratio,  viz.  '142,  found  by  Mr.  Michell  for  the 
pointed  wave.  There  appears  therefore  to  be  no  stage,  while 
f.L  is  finitely  less  than  unity,  at  which  the- expressions  cease  to 
give  rounded  waves  of  the  same  general  character. 
§  2.  In  order  to  demonstrate  that  in  the  critical  stage  when 
Phil.  Mag.  S.  6.  Vol.  11.  No.  63.  March  l^OU.  2  C 
