Deep  Sea  Ship-  Waves,  21 
§87.    Geometrical  digression  on  a  system  of  autolomic, 
monoparametric  co-ordinates  *.     §§  87-90. 
In  (119)  put 
ru  =  a (126) 
where  a  denotes  the  parameter  0  W  of  the  curve  0  0  C, 
fig.  32,  which  we  are  about  to  describe  ;  being  the  curve 
given  intrinsically  by  (119)  and  (122)  with  suffix  lm* 
omitted  from  t.  In  the  present  paper  these  curves  may  be 
called  isophasals,  because  the  argument  o£  the  sine  in  (130) 
below  is  the  same  for  all  points  on  any  one  of  them. 
Solving  (119)  and  (122)  for  x  and  y,  we  find 
i-~rM  —t  ^1*)7^ 
X  ~a   (1  + $8)3/2  >       ^""a  (1+^)3/2        '        '        ^'^ 
The  largest  of  the  eight  curves  shown  in  fig.  32  has  been 
described  according  to  values  of  ar,  y  calculated  from  these 
two  equations,  by  giving  to  —  t  values  tan  0°,  tan  10°,  tan  20°, 
....  tan  90°.  The  seven  other  isophasals  partially  shown  in 
fig.  32,  all  similar  to  the  largest,  have  been  drawn  to  corre- 
spond to  seven  equidifferent  smaller  values,  19X,  18X  ....  13\, 
of  the  parameter  a,  if  we  make  the  largest  equal  to  20X. 
§  88.  It  is  seen  in  the  diagram  that  every  two  of  these 
isophasals  cut  one  another  in  two  points,  at  equal  distances 
on  the  two  sides  of  0  W.  If  we  continue  the  system  down 
to  parameter  0,  every  point  within  the  angle  C  0  C  is  the 
intersection  of  two  and  only  two  of  the  curves  given  by 
(127),  with  two  different  values  of  the  parameter  a.  If  we 
are  to  complete  each  curve  algebraically,  we  must  duplicate 
our  diagram  by  an  equal  and  similar  pattern  on  the  left  of 
0  :  and  the  doubled  pattern,  thus  obtained,  would  show  a 
system  of  waves,  equal  and  similar  in  the  front  and  rear, 
which  (§77  above)  is  possible  but  instable.  We  are,  how- 
ever, at  present  only  concerned  with  the  stable  ship-waves 
contained  in  the  angle  +  19°  28;  on  the  two  sides  of  the 
mid-wake  ;  and  we  leave  the  algebraic  extension  with  only  the 
remark  that  all  points  in  the  angle  COC  of  the  diagram, 
and  the  opposite  angle  leftward  of  0,  can  be  specified  by  real 
values  of  the  parameter  a :  while  imaginary  values  of  it 
would  specify  real  points  in  the  two  obtuse  angles. 
*  Of  this  land  of  co-ordinates  in  a  plane,  we  have  a  well-known  case 
in  the  elliptic  co-ordinates  consisting  of  confocal  ellipses  and  hyperbolas. 
