Deep  Sea  Skip-  Waves.  5 
disturbance  of  the  water-surface,  however  steep  be  the  static 
forcive  curve.  A  "  skipping  stone "  and  a  ricoehetting 
cannon  shot,  illustrate  the  application  of  the  same  dynamical 
principle  in  three-dimensional  hydrokinetics.  By  mathe- 
matical calculation  (§79  below)  we  shall  see  that,  when  v  is 
great  enough,  we  have 
'«¥4 (y7)> 
where  h  denotes  the  height  of  crests  above  mean  water-level 
in  the  train  of  sinusoidal  free  waves  left  in  the  rear  of  the 
travelling  forcive;  A  denotes  the  area  of  the  forcive-curve 
(fig.  25)  ;  being  given  in  §  66  by  the  equation 
A  =  M (98): 
and  X,  given  [§  39,  (71)]  by 
X=M\g (99), 
denotes  the  wave-length  of  free  waves  travelling  with 
velocity  v. 
§  71.  A  very  important  theorem  in  respect  to  ship-waves 
is  expressed  by  (97).  Without  calculation  we  see  that,  i/X 
is  very  great  in  comparison  with  irb,  (the  "  mean  breadth " 
of  the  forcive-curve  according  to  §66),  h  must  be  simply 
proportional  to  A,  for  different  forcives  travelling  at  the 
same  speed.  This  we  see  because,  for  the  same  value  of  b, 
hjk  is  the  same,  and  because  superposition  of  different 
forcives  within  any  breadth  small  in  comparison  with  X, 
gives  for  h  the  sum  of  the  values  which  they  would  give 
separately.  Farther  without  calculation,  we  can  see,  by 
imagining  altered  the  scale  of  our  din  grams,  that  likjA.  must 
be  constant.  But  without  calculation  I  do  not  see  how  we 
could  find  the  factor  4cir  of  (97),  as  in  §  79  below. 
§  72.  The  effect  of  the  condition  prescribed  in  §  71  is 
illustrated  and  explained  by  considering  cases  in  which  it  is 
not  fulfilled.  For  example,  let  two  forcives  be  superposed 
with  their  middles  at  distance  ^X  ;  they  will  give  /t  =  0,  that 
is  to  say  no  train  of  waves.  The  displaced  water  surface  for 
this  case  is  represented  in  fig.  27.  Or  let  their  distance  be 
J.X  or  |X  ;  the  two  will  give  the  same  value  of  //  as  that 
given  by  one  only.  Or  let  the  two  be  at  distance  X  ;  they 
will  make  h  twice  as  great  as  one  Eorcive  makes  ir. 
§73.  In  figs.  26,  '27,  29,  30,  representing  result-  of  the 
calculations  of  §§78,  79  below,  the  abscissas  are  all  marked 
according  to   wave-length.      The  scale  of   ordinate-    eorro- 
