22  Lord  Kelvin  on 
§  89.  By  differentiation  of  (127),  we  find 
^  =  -*=-tan^ (128); 
dy  r 
which  proves  that  tan-1£  is  the  angle  measured  an ti- clockwise 
from  0  Y  to  the  tangent  to  the  curve  at  any  point  (#,  y),  in 
the  lower  half  of  the  diagram.  Elimination  of  t  between  the 
two  equations  of  (127)  gives,  as  the  cartesian  equation  of  our 
curve, 
(^+.y2)3  +  «2(8^/4-20^y-^4)  +  16«y  =  0      •  (129). 
But  the  implicit  equations  (127)  are  much  more  convenient 
for  all  our  uses.  It  is  interesting  to  verify  (129)  for  the 
case  — 1=  ±\/i  in  (127),  corresponding  to  either  of  the  two 
cusps  shown  in  the  diagram. 
§  90.  Going  back  now  to  §  86  and  the  continuous  varia- 
tions considered  in  it,  we  see  that  —  ^  and  — 12  are  re- 
spectively the  tangents  of  the  inclinations,  reckoned  from 
0  Y  clockwise,  of  portions  of  the  long  arc  0  C  and  of  the 
short  arc  W  C,  in  the  upper  half  of  the  diagram.  Thus,  if 
we  carry  a  point  from  0  to  C  in  the  long  arc,  and  from  0  to 
W  in  the  short  arc,  we  have  the  change  of  inclinations  to 
0  Y  represented  continuously  by  the  decrease  of  tan-1(  —  ^) 
from  90°  to  35°  16',  while  y  increases  from  0  to  x\/S;  and 
the  farther  decrease  of  tan_1(  — 12)  from  35°  16/  to  0°,  while 
y  diminishes  from  x  */8  to  0  again.  The  inclination  to  O  Y 
of  the  two  branches  meeting  in  the  cusp,  C,  is  35°  1 6'  (or 
tan_1x/^).  For  any  point  in  the  short  arc  CWC  of  the 
curve  u  or  cos  (yfr  —  0)/cos2  i/r,  is  a  minimum.  In  each  of  the 
long  arcs  u  is  a  maximum.  At  every  point  of  the  curve  the 
value  of  u,  whether  minimum  or  maximum,  is  a/r.  Hence 
for  different  points  of  the  curve,  u  is  inversely  proportional 
to  the  radius  vector  from  0. 
§  91.  Going  back  to  (118)'  we  now  see  that  for  all  points  on 
any  one  of  our  curves,  rux  and  ru2  have  both  the  same  value, 
being  the  parameter  0  W  of  the  curve.  The  first  part  of 
(118/  is  one  constituent  of  the  depression  at  any  point  on 
either  of  the  long  arcs  ;  and  the  second  part  of  (118/  is  one 
constituent  of  the  depression  at  any  point  on  the  short  arc. 
Taking  for  example  the  largest  of  the  curves  shown  in  fig.  32, 
we  now  see  that  for  any  point  of  either  of  its  long  arcs,  the 
second  constituent  of  the  depression  of  the  water  is  to  be 
calculated  from  the  second  part  of  (118/  ;  while  for  any 
point  of  its  short  arc,  the  second  constituent  of  the  depression 
is  to  be  calculated  from  the  first  part  of  (118)'. 
