On  the  Mathematical  Theory  of  Atmospheric  Tides.         25 
being  premised,  the  following  known  differential  equations  to  the 
first  order  of  small  quantities,  expressed  in  the  usual  notation, 
will  be  employed  for  the  determination  of  the  motion  and  pres- 
sure : — 
d*<j>  _d*(f> 
a*dP  ~dx*  + 
dy* 
+  dz*' 
\dp) 
=  Xdsc  +  Ydy  +  Zdz—d . 
d$ 
dt' 
(1) 
(2) 
The  earth's  centre  being  the  origin  of  rectangular  coordinates, 
let  X  be  the  north  latitude,  and  6  the  longitude  west  from  Green- 
wich, at  the  time  /,  of  any  particle  of  the  fluid  distant  by  r  from 
the  origin.     Then 
#  =  rcos\cos#,     y  =  r  cosXsin  6,     £  =  rsin\. 
The  impressed  forces  X,  Y,  Z  are  the  resolved  parts  of  the  earth's 
attraction,  and  of  the  moon's  attraction  relative  to  her  attraction 
on  a  particle  at  the  earth's  centre.  Hence,  if  G  and  m  be  respec- 
tively the  attractions  of  earth  and  moon  at  the  unit  of  distance, 
R  the  moon's  distance  from  the  earth's  centre,  and  fit  her  an- 
gular distance  westward  from  the  meridian  of  Greenwich,  t  being 
the  time  reckoned  from  the  Greenwich  transit,  the  following 
equations  may  be  obtained  by  the  usual  process,  powers  of  the 
ratio  of  r  to  R  above  the  first  being  neglected : — 
X=--^  +  ^(#(3cosV-l)+  ^sin/**j, 
sm 
fit-\)  + 
Sx 
sin  fit), 
7_      Gz      mz 
~r*~W 
Consequently  Xdx  +  Ydy -\-Zdz  is  an  exact  differential,  and  the 
result  of  integrating  the  equation  (2),  regard  being  had  to  the 
expressions  for  x,  y,  and  z,  will  be  found  to  be 
#Nap.  logp=  i  +  gJ(8cos«Xcos«  (0_^)-l)-  J  +  f  (*).(3) 
It  will  now  be  convenient  to  employ  the  equation  (1)  under 
the  form  it  takes  when  its  coordinates  are  transformed  into  the 
polar  coordinates  r,  6,  and  X.  This  form  of  the  equation  is,  as 
is  known, 
d\r$      d*.r<j>         _1 d\r$        1   d^r$  __  tanX  d.r<j> 
aM*  dr*         r2cos*\    a&*    +  r*     dX*  f       dX  ' 
The  next  step  is  to  obtain,  by  a  particular  solution  of  this  equa- 
