22 
The  Rev.  T.  K.  Abbott  on  the  Theory  of  the  Tides. 
the  remaining  quadrants — being  greatest  when  the  moon  is  in  the 
meridian,  and  zero  when  it  is  in  the  horizon.  Hence,  by  5  and 
4,  the  elevation  at  the  equator  (and  up  to  lat.  45°)  will  be  great- 
est (i.  e.  it  will  be  high  water)  90°  from  the  moon.  Beyond  lat. 
45°  the  depression  will  be  greatest  under  the  same  circumstances. 
In  these  latitudes,  therefore,  the  effect  of  the  former  component 
would  be  partially  counteracted.  It  is  easy,  however,  to  see  that 
the  variation  in  the  meridional  force  (which  alone  affects  the 
tide)  is  in  any  latitude  less  than  that  in  the  force  parallel  to  the 
equator  (in  the  proportion  of  sin  lat.  to  1) ;  so  that  while  the 
height  of  the  tide  would  be  lessened,  the  place  of  high  water 
would  be  as  before.  It  would  be  easy  to  calculate  the  height 
resulting  from  both  these  components  combined.  The  form  of 
the  surface  would  be  nearly  but  not  exactly  ellipsoidal,  with  the 
greatest  axis  equatorial  and  perpendicular  to  the  line  joining  the 
centres  of  the  earth  and  moon*. 
Let  us  now  consider  the  case  of  the  moon  having  a  declina- 
tion, which  for  simplicity  I  shall  suppose  less  than  22°  30f. 
This  limitation  will  not  affect  our  results.  We  shall,  as  before, 
take  the  two  components  separately. 
With  respect,  then,  to  the  component  which  acts  parallel  to 
the  equator.  Near  the  equator  itself  the  considerations  applied  in 
the  former  paper  still  hold  good.  Next  consider  a  place,  a, 
whose  polar  distance  is  less  than  the  moon's  declination,  to 
which  therefore  the  moon  is  circumpolar,  and  (with  the  as- 
sumed declination)  alternately 
north  and  south  of  the  zenith. 
If  a  b  c  d  be  the  circle  of  rotation 
of  such  a  place,  it  is  obvious  that 
the  water  will  be  accelerated 
through  the  whole  semicircle 
a  be,  a*nd  retarded  through  cda.x 
The  same  reasoning  as  already 
employed  will  show  that  it  will 
be  low  water  at  c  and  high  water 
at  a.  Now  take  an  intermediate 
place  whose  circle  of  rotation  is 
Imno.  Here  the  water  is  re- 
tarded and  rising  from  I  to  m  and  from  n  to  o ;  and  accelerated 
and  falling  from  m  to  n  and  from  o  to  Z,  and  the  interval  o  Im  is 
less  than  imno.  Hence  the  tide  is  lowest  at  n  and  not  so  low 
at  /,  and  it  is  high  water  at  m  and  of. 
The  meridional  component  at  the  equator  acts  during  half  a 
*  It  is  evident  that,  apart  from  the  meridional  force,  the  equatorial 
wave  would  itself  be  accompanied  by  a  slight  north  and  south  oscillation. 
t  In  the  figure  M  is  the  point  under  the  moon,  N  its  antipodes. 
