Deep  Sea  Ship-  Waves.  I  9 
From  the  second  of  (115)  we  find  d\jr  =  dql/  (@lx/7r\  where 
*-\Z\kFwK aifi)- 
Dealing  similarly  in  respect  to  yfr2  and  values  of  -v/r  differing 
but  little  from  it,  we  take  -\-q2  instead  of  the  —  q2  of  (114), 
and  (d'2u/dyjr2)2  instead  of  the  —  (d2ujd'^r2)l  of  (115) ;  because 
U\  is  the  maximum  and  u2  the  minimum.  Calling  &l5  k2  the 
values  of  k  corresponding  to  i/rb  yjr2,  and  using  these  ex- 
pressions properly  in  (113),  we  find,  for  the  depression  of  the 
water  at  (a?,  ?/), 
+ 
(S2COS 
^—-  I      <fy2  sin  (a2  +  </22)]  .  (117). 
TsJ-oo  -J 
The  limits  go,  —go  are  assigned  to  the  integrations 
relatively  to  <7X  and  ^2  because  the  greatness  of  r/\  in  (115) 
and  corresponding  formula  relative  to  ^2  makes  q^  and  q2 
each  very  greats  (positive  or  negative,)  for  moderate  properly 
small  positive  or  negative  values  of  -^r— ^  and  ^r—  i/r2. 
Now  as  discovered  by  Buler  or  Laplace  (see  Gregory's 
Examples,  p.  179),  we  have 
„  CO  /^  CO 
I       dq  sin  q2  =  I       dq  cos  q2  —  \Zirj2, 
J    —CO  J    —  CO 
and  using  these  in  (117)  we  find 
At*  ,A-  2\/2tt25  r^  (sin  ^-cosaQ      £2(sin«2  +  cos«2)-|    m^ 
a(''y)"  X  L  AOOBF.^        ~+-      /32COS2*2         J'  (  }' 
Substituting  for  ab  a2  values  by  (115)  we  find 
,  .        N        4tt2/>  r         *!  .     27T  /  \\ 
a(#,  ?/)  =  — » —    -5 K-y  sm  — -    rw., ; 
K    JJ        X     LA  cos-  i^i         \  \     :      si 
j32  COS2  yjr2  X  \     "       8/J     v 
§85.  To  determine  the  quantities  denoted  by  p\.  A  in 
(11G)  ....  (118)',  we  write  (112)  as  follows  :— 
ru  =  (x  +  yt)\/l  +  l'2,  where  /  =  tan  i/r.     .     .(119). 
C2 
