14  Lord  Kelvin  on 
Instead  now  of  taking  e  =  l  — 10-4,  as  we  took  in  our  calcu- 
lations for  d(0),  let  us  take  e=l.     This  reduces  (105)  to 
(-iyd(i80°)=|+i-|  +  i.... 
I— IV-1      If  —  IV 
+  Vr-+2¥+I-  •  -(106)- 
Lastly  take  7   an  infinitely  great  odd  or  even   integer,  and 
we  find 
d(180°)  =  (-iy.| (107). 
Now  fig.  26  is,  as  we  have  seen,  found  by  superimposing  on 
the  motion  represented  by  fig.  29  an  infinite  train  of  periodic 
2ttx 
waves   represented    by   —^h.  sin— — ■,   and  therefore   h  =  7r, 
which  proves  (97). 
§  80.  To  pass  now  from  the  two-dimensional  problem  of 
canal-ship-waves  to  the  three-dimensional  problem  of  sea- 
ship-waves,  we  shall  use  a  synthetic  method  given  by 
Rayleigh  at  the  end  of  his  paper  on  "The  form  of  standing 
waves  on  the  surface  of  running  water/'  communicated  to 
the  London  Mathematical  Society  in  December  1883  *.  In 
an  infinite  plane  expanse  of  water,  consider  two  or  more 
forcives,  such  as  that  represented  by  (95)  of  §  66,  with  their 
horizontal  medial  generating  lines  in  different  directions 
through  one  point  0,  travelling  with  uniform  velocity,  v,  in 
any  direction.  The  superposition  of  these  forcives,  and  of 
the  disturbances  of  the  water  which  they  produce,  each 
calculated  by  an  application  of  (100),  (101),  (102),  gives  us 
the  solution  of  a  three-dimensional  wave  problem  ;  which 
becomes  the  ship-wave-problem  if  we  make  the  constituents 
infinitely  small  and  infinitely  numerous.  Rayleigh  took 
each  constituent  forcive  as  confined  to  an  infinitely  narrow 
space,  and  combated  the  consequent  troublesome  infinity  by 
introducing  a  resistance  to  be  annulled  in  interpretation  of 
results  for  points  not  infinitely  near  to  O.  I  escape  from  the 
trouble  in  the  two-dimensional  system  of  waves,  by  taking 
(95)  to  express  the  distribution  of  pressure  in  the  forcive, 
and  making  b  as  small  as  we  please.  Thus,  as  indicated  in 
§§79,  73,  76,  by  taking  b=  10^X/(104 .  ir)  we  calculated  a 
finite   value  for    d(0).     But  for   values    of    x,  considerably 
*  Proc.  L.M.S.,   1883;    republished  in  Rayleigh's  Scientific  Papers, 
vol.  ii.  art.  109. 
