Deep  Sea  Ship-  Waves.  23 
§  92.  Explaining  quite  similarly  the  determination  of 
d(/£,  y)  for  every  point  of  each  of  the  smaller  curves  which 
we  see  in  the  diagram  cutting  the  longer  arcs  of  the  largest 
curve,  we  arrive  at  the  following  conclusions  as  the  complete 
solution  of  our  problem. 
The  whole  system  of  standing  waves  in  the  wake  of  the 
travelling  forcive  is  given  by  the  superposition  of  constituents 
calculated  according  to  (127),  with  greater  and  smaller  values 
of  the  parameter  a  with  infinitely  small  successive  differences. 
Hence,  what  we  see  in  looking  at  the  waves  from  above  is 
exactly  a  system  of  crossing  hills  and  valleys,  with  ridges 
and  beds  of  hollows,  all  shaped  according  to  the  isophasal 
curves  shown  in  fig.  32.  Looking  at  any  one  of  the  short 
arc-ridges  and  following  it  through  the  cusps,  we  find  it 
becoming  the  middle  line  of  a  valley  in  each  of  the  long  arcs 
of  the  curve.  And  following  a  short  arc  mid-valley  through 
the  cusps,  we  find,  in  the  continuation  of  the  curve,  two  long 
ridges.  Every  ridge,  long  or  short,  is  furrowed  by  valleys. 
All  the  curved  ridges  and  valleys  are  parts  of  one  continuous 
system  of  curves,  illustrated  by  fig.  32  and  expressed  by  the 
algebraic  equation  (129). 
With  these  explanations  we  may  write  (118)'  as  follows  r 
,,       N      4tt2/>/c  sec2  ^    .    2tt/         X\  /<rt^ 
where  /w      d*u\  /ll)1N 
§  93.  An  important,  perhaps  the  most  important,  feature 
of  the  wave-system  which  we  actually  see  on  the  two  sides 
of  the  mid-wake  of  a  steamer  travelling  through  smooth 
water  at  sea,  or  of  a  duckling  *  swimming  as  fast  as  it  can  in 
a  pond,  is  the  steepness  of  the  waves  in  two  lines  which  we 
know  to  be  inclined  at  19°  28'  to  the  mid-wake.  The  theory  of 
this  feature  is  expressed  by  the  coefficient  of  the  sine  in  (130), 
and  is  well  illustrated    by  the  calculation   of   \/-r»  — tt 
for  eleven  points  of  any  one  of  the  curves  of  fig.  32,  the 
results  of  which  are  shown  in  column  6  of  the  following 
table.     They  express   the    depression    below,    and    elevation 
*  In  the  case  of  even  the  highest  speed  attained  by  a  duckling-,  this 
angle  is  perhaps  perceptibly  greater  than  L9°  28',  because  of  the  dynamic 
effect  of  the  capillary  surface  tension  of  water.  See  •  Baltimore  Lectures, 
p.  593  (letter  to  ProfessoT  'Tail,  of  date  23rd  Aug.  18711  and  pn.  600, 
('01  (letter  to  William  Froude,  reprinted  from  'Nature'  of  2«th  Oct 
.1871). 
