Deep  Sea  Ship- Waves.  17 
P  0  X;.  Thus  the  integral  (111)  expresses  the  depression 
at  P(#,  y)  due  to  the  joint  action  of  all  the  constituent 
forcives,  because  none  except  those  whose  medial  lines  lie  in 
the  angle  P  0  X',  contribute  anything  to  the  disturbance  of 
the  water  at  P. 
§  84.  For  interpreting  and  approximately  evaluating  the 
definite  integral,  we  may  conveniently  put 
r=^?+7,     and     u=C0S^~e)   ■     ■     -(112), 
and  write  (111)  as  follows: 
d(«,y)  =  4**6f*        JWsin2™    .     .  (H3), 
1  \cosJ\^  A- 
-G-0 
Now  if  we  suppose  r/\  very  great,  there  will  be  exceedingly 
rapid  transitions  between  equal  positive  and  negative  values 
of  sin  (27rru/X),  which  will  cause  cancelling  of  all  portions  of 
the  integral  except  those,  if  any  there  are,  for  which  du/di/r 
vanishes.  We  shall  see  presently  that  there  are  two  such 
values,  -*/rl5  fa,  both  real  if  tan  6  <  ^/\  ;  u  being  a  maximum 
(mx)  for  one  of  them,  and  a  minimum  (u2)  for  the  other  ; 
and  that,  when  6  has  any  value  between  tan-^J  and 
2tt— tan-\/l  the  values  of  fa,  fa  are  both  imaginary. 
Consideration  of  this  last-mentioned  case  shows  that,  in  tne 
whole  area  of  sea  in  advance  of  two  lines  through  the  centre 
of  the  travelling  forcive  inclined  at  equal  angles  of  tan"1  s/-}, 
(or  19°  280  on  eacn  S1(ie  of  the  mid-wake,  there  is  no 
perceptible  disturbance  at  distances  of  much  more  than  a 
half  wave-length  from  the  centre  of  the  forcive.  The  main 
disturbance  by  ship-waves,  therefore,  lies  in  the  rearward 
angular  space  between  these  two  lines.  It  is  illustrated  by 
fig.  32,  as  we  now  proceed  to  prove  by  the  proper  interpre- 
tation of  (113).  Expanding  the  argument  of  the  sin  in  (113) 
by  Taylor's  theorem  for  values  of  f  differing  from  ^  by 
small  fractions  of  a  radian,  we  find 
7rru  .    2wr  f         .  fd2u\   ,  ,        .  X2"|  a      Min 
XT=  IT  b^-h(W)W-W-]  =  *i-^  ■  (114), 
27rru 
where 
«1=-  -  l,    and     ?1=(^-Vr1)yT(-^3j|.  (115) 
Phil,  Mag.  S.  6.  Vol.  1  I.  No.  61.  Jan.  lyOfi.  0 
