Deep  Sea  Ship- Waves.  15 
greater  than  half  a  wave-length,  we  were  able  to  simplify  the 
calculations  by  taking  6  =  0. 
§  81.  For  the  three-dimensional  system  let,  in  fig.  31,  -v/r  be 
the  inclination  to  0  X  of  the  rearward  wave-normal  of  one 
of  the  constituent  systems  of  waves.  This  is  also  the  incli- 
nation to  0  Y  of  the  medial  line  of  the  travelling  forcive  to 
which  that  set  of  waves  is  due.  Take  now  for  the  forcive 
obtained    by   the    superposition    of    an   infinite    number    of 
Fig.  31. 
/ 
constituents,  as  described  in  §  80, 
9 
n  (*,  y) 
j>[(-- 
bvc 
COS  ty -I- y  sin  \jr)'2  +  b2] 
(108), 
where  k  may  be  a  function  of  \fr,  and  b  is  the  same  lor  all 
values  of  ijr. 
For  the  case  of  a  circular  forcive  system  we  must  take  k 
constant  ;  and  we  find 
Jn«=7^b)  where' 'w+'"v-  •  -(109)- 
