16  Lord  Kelvin  on 
§  82.  Let  now  the  forcive,  whether  circular  or  not,  be 
kept  travelling  in  the  direction  of  x  negative  *  with  velocity 
v  :  and  let  X  denote  the  corresponding  free  wave-length 
given  by  the  formula  2irv^lg.  This  is  the  wave-length  of  the 
constituent  train  of  waves  corresponding  to  -\jr  =  0.  For 
the  ^-constituent,  the  component  velocity  perpendicular 
to  the  front  is  v  cos  -^,  and  the  wave-length,  is  X  cos2  ^fr. 
Looking  now  to  fig.  26,  with  Xcos2-^  instead  of  X  ;  and  to 
fig.  31  ;  and  to  equations  (97),  (98)  ;  we  see  that  the  portion 
of  the  depression  at  (x,  y)  due  to  the  constituent  of  forcive 
shown  under  the  integral  in  (108)  is 
4:7r2bJc  dyjr    .     2-7T  (x  cos  i|r  + 1]  sin  ^  ) 
•  X  COS2  -y\r  X  cos2  i/r 
provided  x  cos  ty+y  sin  yjr  is  considerably  greater  than 
■JX  cos2  i/r.  Hence  for  the  depression  at  (x,  y)  due  to  the 
whole  travelling  forcive,  we  have 
„      ^  *nL*l  '  '  '      (U0)> 
kdyjr       .     27T  [x  cos  ty  -j-  y  sin  yjr)       . 
X  cos2  yjr  Xcos2  yjr  '    ^       '' 
§83.  The  reason  for  choosing   the   limits   —(— —0\   to 
~-  is  that  each  constituent  forcive  gives  a  train  of  sinusoidal 
waves  in  its  rear,  and  no  perceptible  disturbance  in  its  front 
at  distances  from  it  exceeding  half  a  wave-length.  Look 
now  to  fig.  31,  and  consider  the  infinite  number  of  medial 
lines  of  the  forcives  included  in  the  integrals  (108),  (111)  ; 
all  as  lines  passing  through  0.  Four  examples,  Q  P,  Y'  Y, 
LK,  XX'  of  these  lines  are  shown  in  the  diagram  :  corre- 
sponding respectively  to  -\jr=  —  (  —  —  0\  -*}r  =  0,  ty=  any 
positive   acute  angle,    ty=—.     On  each    of  the    first   three 
of  these  lines  EE  indicates  the  rear.  The  fourth,  XX', 
is  in  the  direction  of  the  motion,  and  has  neither  front  nor 
rear.  The  integral  (111)  must  include  all,  and  only  all,  the 
medial  lines  which  have  rears  towards  P.  Hence  Q  P  is  one 
limit  of  ylr  in  (111)  because  it  passes  through  P  ;  XX7  is 
the  other  limit  because  it  has  neither  front  or  rear.  Thus 
all  the  lines  included  in  the  integral,  lie  in  the  obtuse  angle 
*  This  is  opposite  to  the  direction  of  the  motion  of  the  forcive  in 
fit?.  26. 
