10  Lord  Kelvin  on 
groups,  given  by  Stokes  in  his  Smith's  Prize  examination 
paper,  published  in  the  Cambridge  University  Calendar  for 
1876  :  and  for  rejecting  it  for  the  case  of  any  single  group  of 
waves.  In  reality  the  front  of  a  group,  left  to  itself,  travels 
with  accelerated  velocity  exceeding  the  velocity  of  periodic 
waves  of  the  given  wave-length,  instead  of  with  half  that 
velocity. 
§77.  Fig.  29  shows  the  steady  motion,  symmetrical  in 
front  and  rear  of  a  single  travelling  forcive,  which  is  a 
solution  of  our  problem  ;  but  it  is  an  unstable  solution  (as 
probably  are  the  solutions  of  the  problem  of  §  45  above, 
shown  in  figs.  13,  14,  15).  If  any  large  finite  portion  of  the 
water  is  given  in  motion  according  to  fig.  29,  say,  for 
example,  50  waves  preceding  O  (the  forcive)  and  50  waves 
following  0,  the  front  of  the  whole  procession,  to  the  right 
of  0,  will  become  dissipated  into  non-periodic  waves 
travelling  rightwards  and  leftwards  with  increasing  wave- 
lengths and  increasing  velocities ;  and  the  approximately 
steady  periodic  portion  of  it  will  shrink  backwards  relatively 
to  the  forcive.  Thus  before  the  forcive  has  travelled  fifty 
wave-lengths,  the  periodic  waves  in  front  of  it  are  all  gone  : 
but  there  is  still  irregular  disturbance  both  before  and  behind 
it.  After  the  forcive  has  travelled  a  hundred  wave-lengths, 
the  whole  motion  in  advance  of  it,  and  the  motion  for  perhaps 
30  wave-lengths  or  more  in  its  rear,  will  have  settled  to 
nearly  the  condition  represented  by  fig.  26,  in  which  there  is 
a  small  regular  elevation  in  advance  of  the  forcive,  and  a 
regular  train  of  approximately  sinusoidal  waves  in  its  rear  ; 
these  waves  being  of  double  the  wave-height  given  originally. 
This  motion,  as  said  above  in  §  68,  will  go  on,  leaving  behind 
the  forcive  a  train  of  steady  periodic  waves,  increasing  in 
number  ;  and  behind  these  an  irregular  train  of  waves, 
shorter  and  shorter,  and  less  and  less  high  the  farther  rear- 
ward we  look  for  them  (see  R  in  fig.  10  of  §§  26,  21  above). 
It  is  an  interesting,  but  not  at  all  an  easy  problem,  to  in- 
vestigate the  extreme  rear  (with  practically  motionless  water 
behind  it)  of  the  train  of  waves  in  the  wake  of  a  forcive 
travelling  uniformly  for  ever.  I  hope  to  return  to  this 
subject  when  wre  come  to  consider  the  work  done  by  the 
travelling  forcive. 
§  78.  Pass  now  to  the  investigation  of  the  formulas  by  the 
calculation  of  which  figs.  26,  27,  28,  29,  30  have  been  drawn, 
and  the  theorem  of  (97)  proved.  Go  back  to  the  problem  of 
§41   above:  but   instead   of  taking  e  =  '9,   as  in   §§46-61, 
