Deep  Sea  Ship-  Waves.  8 
5  is  2  tan-1  8  .  &/c,  or  z-rr-.Trblc,  and   the  total 
loU 
area  of  the  diagram    extended  to  infinity   on   each  side  is 
1  A£ 
irbk.     Hence  the  area  o£  fig.  25  is  j-tttt,  or  *92,  of  the  total 
area.  This  total  area,  irbk,  I  call,  for  brevity,  the  forcive- 
area  ;  and  Trh,  I  call  the  mean  breadth  of  the  forcive-aroa. 
The  breadth  of  the  forcive  where  z='8k  (as  shown  by  the 
dotted  line  BE  in  the  diagram)  is  b. 
§  67.  Now  let  the  forcive  be  suddenly  set  in  motion,  and 
kept  moving  uniformly  with  any  velocity  v  in  the  rightward 
direction  of  our  diagrams.  This  will  produce  a  great  com- 
motion, settling  ultimately  into  more  and  more  nearly  steady 
motion  through  greater  and  greater  distances  from  0.  The 
investigation  of  §§  1-10  (Phil.  Mag.  June  1904),  and 
particularly  the  results  described  in  §§5,  6,  and  illustrated 
in  figs.  2,  3,  show  that  in  our  present  case  the  commotion, 
however  violent,  even  if  including  splashes*,  divides  itself 
into  two  parts  which  travel  away  in  the  two  directions  from  0, 
ultimately  at  wave-speed  increasing  in  proportion  to  square 
root  of  distance  (according  to  the  law  of  falling  bodies);  and 
leaving  in  their  rears,  through  ever  broadening  spaces,  what 
would  be  more  and  more  nearly  absolute  quiescence  if  the 
forcive  were  suddenly  to  cease  after  having  acted  for  any 
time,  long  or  short. 
§  68.  But  if  the  forcive  continues  acting,  and  travelling 
rightwards  with  constant  speed,  v,  according  to  §  67,  the 
travelling  away  of  the  two  parts  of  the  initial  commotion  in 
the  two  directions  from  0  (itself  merely  a  point  of  reference, 
moving  uniformly  rightwards),  leaves  the  water,  as  shown  by 
fig.  26,  in  a  state  of  more  and  more  nearly  quite  steady 
motion  through  an  ever  broadening  space  on  the  rear  side  of 
O,  and  through  a  small  space  in  advance  of  0;  provided 
certain  moderating  conditions  are  fulfilled  in  respect  to 
k,  b,  v. 
§  69.  To  illustrate  and  prove  §  C>8  ;  first  suppose  v  infinitely 
small.  The  water  will  be  infinitely  little  disturbed  from  the 
static  forcive-curve  shown  in  fig.  25,  and  described  in  ^  66. 
Small  enough  velocities  will  make  very  small  disturbance 
with  any  finite  value  of  kjb. 
§  70.  But  now  go  to  the  other  extreme  and  let  v  be  very 
great.  It  is  clear,  on  dynamical  principles  without  calcula- 
tion, that    v   may  be  great  enough,  to    make    but  very    little 
*  However  sudden  and  great  the  commotion  is,  the  motion  of  the 
liquid  is,  and  continues  to  be,  irrotational  throughout, 
B'2 
