The  Rev.  T.  K.  Abbott  on  the  Theory  of  the  Tides.    .     21 
versa.  If,  then,  when  this  form  is  spherical  the  difference 
mentioned  in  3  were  always  in  the  same  direction,  it  would  con- 
tinue to  act  until  a  certain  permanent  alteration  was  produced. 
If  the  difference  were  constant  in  amount,  a  state  of  equilibrium 
would  be  attained ;  but  if  it  alternately  increases  and  diminishes, 
then  the  mean  form  of  the  surface  will  be  the  same  that  would 
be  produced  by  a  constant  force  equal  to  the  mean  amount  of 
the  actual  force.  The  alternate  excess  and  defect  of  the  latter 
will  cause  a  periodical  motion,  just  as  if  it  were  an  independent 
force  *. 
First,  then,  the  moon  being  still  supposed  to  be  in  the  equator, 
let  the  earth  be  uniformly  covered  with  water.  The  tangential 
force  may  be  resolved  into  two  components,  one  touching  the 
parallel  of  latitude  (i.  e.  east  and  west),  the  other  meridional f. 
These  may  be  regarded  as  giving  rise  to  two  distinct  waves — 
one  north  and  south,  the  other  east  and  west.  Now,  with  respect 
to  the  latter,  the  reasoning  in  the  first  paper  (in  the  case  of  an 
equatorial  canal  with  the  moon  in  its  plane)  still  holds  good; 
and  if  this  force  were  alone  (that  is,  if  the  water  moved  in  canals 
parallel  to  the  equator),  the  ocean  in  every  circle  of  latitude 
would  take  the  form  of  an  ellipse,  with  its  short  axis  towards 
the  moon.  All  the  axes  in  this  direction  being  shortened,  and 
those  at  right  angles  being  elongated  similarly  (for  this  compo- 
nent varies  according  to  the  same  law  in  every  latitude),  the 
entire  ocean  would  assume  the  form  of  an  ellipsoid  with  its 
greatest  and  least  axes  in  the  plane  of  the  equator,  the  least  being 
directed  towards  the  moon.  The  polar  diameter  being  unaffected, 
would  be  the  mean  axis  of  the  ellipsoid. 
The  effect  of  the  meridional  component  is  of  a  different  kind. 
This  constantly  acts  in  the  same  direction,  viz.  towards  the 
equator,  and  therefore  causes  an  accumulation  there  correspond- 
ing to  its  mean  amount,  and  a  proportionate  depression  at  the 
poles.  From  the  equator  to  lat.  45°  it  is  an  elevating  force, 
being  greater  as  the  particles  are  further  from  the  equator ;  from 
that  to  the  poles  it  is  depressing.  In  every  case,  however,  the 
force  is  in  excess  of  the  mean  for  half  a  rotation,  viz.  from  45° 
on  each  side  of  the  meridian  under  the  moon,  and  in  defect  in 
*  If  the  reader  wishes  to  apply  these  considerations  to  the  case  of  an 
equatorial  canal  treated  in  the  first  paper,  it  must  be  observed  that  there 
the  elevating  force  is  the  excess  of  easterly  force  acting  on  any  particles  of 
water  above  that  which  affects  those  in  advance,  i.  e.  to  the  east  of  them. 
This  excess  is  positive  from  45°  west  of  the  moon  to  45°  east  (i.  e.  while  the 
moon  passes  from  45°  east  zenith  distance  to  45°  west),  then  negative  for 
90',  and  so  on. 
t  The  equatorial  component  is  proportional  to  cos  lat.  sin  2  (hour-angle 
of  moon  from  meridian) ;  the  meridional  to  sin  2  lat.  cos'2  (hour-angle  of 
moon  from  meridian). 
