24 
On  JDeep  Sea  Sid})- Waves. 
above  mid-level,  due  to  one  constituent  of  the  system  of 
crossing  hills  and  valleys  described  in  §  92.  Column  1  is 
—yjr.  Columns  2,  3  are  xja  and  y/a,  calculated  from  (127). 
Column   4    is    u,    calculated    by  (126)  from  columns   2,  3. 
Column  5  is   - .  d2 u j 'd-yjr2 ,  calculated  from   (124)  and  columns 
2,4. 
Column  6 
is  a  /  -— — —-,  calculated  from  columns 
1,  6.  u,  being,  as  we  have  seen,  a  maximum  for  values  of 
—yfr  from  0  to  3o°  16',  and  a  minimum  for  values  from  this 
to  90°,  we  see  that  the  proper  suffix  in  columns  4,  6,  for  the 
first  four  lines  of  each  column  is  1,  and  for  the  last  six 
lines  is  2. 
Col.  1. 
Col.  2. 
Col.  3. 
Col.  4. 
Col.  5. 
Col.  6. 
-* 
X 
a 
y 
a 
u 
r  dru 
~adj2 
/a    sec2  \b 
Vx'^r 
0° 
1-0000 
o-oooo 
1-00000 
1  -ooooo 
1-0000 
10° 
10145 
•1685 
•97239 
•93782 , 
10647 
20° 
T0497 
•3201 
•91587 
•73497 
1-3210 
30° 
1-0825 
•3750 
•87290 
•33333 
2  3094 
35°  16' 
1-0887 
•3849 
•86602 
o-ooooo 
oo 
40° 
1-0826 
•3773 
•87225 
-     -40830 
2-6660 
50° 
1-0201 
•3166 
•93624 
-  1-84070 
1-7839 
C0° 
•8750 
•2165 
1-10941 
-  5  00003 
1-7888 
70° 
■6441 
•1100 
1-53041 
-140987   : 
2-2793 
80° 
•3421 
•0297 
291222 
-63-3341 
4-1672 
90° 
o-oooo 
o-oooo 
00 
-oo 
OO 
§  94.  In  (130),  k  is  generally  a  function  of  ty;  but  if  the 
forcive  is  circular,  (§  81  above)  k  is  a  constant,  and  for 
points  on  one  of  the  isophasal  curves  (a  =  constant)  the  only 
variable  coefficients  of  the  sine  are  sec2  yjr,  and  /3~l.  But  for 
different  isophasal  curves  the  coefficient  in  (130)  expressing 
the  magnitude  of  the  range  above  and  below  mean  level, 
varies  inversely  as  \/a.  For  mid-wake  (^  =  0)  a  is  simply 
the  distance  from  the  forcive :  and  we  conclude,  not  merely 
for  our  point-forcive,  but  for  a  great  ship,  that  the  waves  at 
a  very  large  number  of  wave-lengths  right  astern,  are  smaller 
in  height  inversely  as  the  square  root  of  the  distance  from 
the  forcive  or  from  the  middle  of  the  ship. 
§  95.  The  infinity  for  yjr=  +35°  16f  represents  a  feature 
analogous  to  a  caustic  in  optics.  There  is  in  nature  no 
infinity  for  either  case,  if  the  source  is  finite  and  distributed, 
not  infinitely  intense  and  confined  to  an  infinitely  small  space. 
According  to  the  methods  followed  in   §§  1-72  above,    we 
