Deep  Sea  Ship-Wares.  13 
c=  l/(2;  +  l).     By  (86)  and  (87;  of 
ving  solution 
d  =  J+^ (100) 
take    ^  =  1-10"4;  and  c=  1/(2/4-1).     By  (86)  and  (87)  oi 
§  45  we  have  the  following  solution 
where 
«*=d+-»  (4  an  0  +  iW  tan-1  vx_g 
-  *  cos  O  +  iJ^og^ v,cos^  +  ,|    •     (101) 
and 
-v     i        1  <?cos$      <?2cos20  .  .    ,■••/!       /in.-n 
^=^7n+*FT  +  ^r  + +      7    -(0)" 
Fig.  29  has  been  calculated  by  putting  0  =  — .- — j,   and 
A-    7  +  2 
taking  j=20.     The  explanation  is  that,  as  we  shall  see  by 
(78)  of  §43  above,  (100),  (101),  (102),  express  the  water 
disturbance  due  to  an  infinite  row  of  forcives  at  consecutive 
distances  each  equal  to  (20 J)  X  ;  the  expression  for  each 
foreive  being 
r:W?ZL_ (103), 
where  n  is  zero  or  any  positive  or  negative  integer  ;  and  by 
(79)  we  have 
/>=2°-5-10"4^    ....     (104). 
LIT 
Thus  we  see  that  the  pressure  at  O  due  to  each  of  the  forcives 
next  to  0,  on  the  two  sides,  is  l/{l-f  (27T.  104)2}  of  the 
pressure  due  to  the  foreive  whose  centre  is  O.  Thus  we 
see  that  the  pressures  due  to  all  the  forcives,  except  the 
last  mentioned,  may  be  neglected  through  several  wave- 
lengths on  each  side  of  O  :  and  we  conclude  that  (100).  (10 1\ 
(10^)  express,  to  a  very  high  degree  of  approximation,  the 
disturbance  produced  in  the  water  by  the  single  travelling 
foreive  whose  centre  is  at  O. 
§79.  To  prove  (97)  take  0=180°  in  (100),  (101),  (102); 
we  thus  rind 
(-iyd(180°)  =e>  [\^/e  tan-1^4-1-^  4  C  •  •  •  • 
