of  Atmospheric  Tides.  27 
Also 
**  =  -2/4(02  +  CV-3)  cos2  X  cos  2(0-^). 
To  obtain  the  motion  and  density  of  the  fluid  at  any  point,  it  is 
now  only  required  to  find  the  values  of  the  arbitrary  quantities 
C,  C,  and  tJt(^)  from  the  given  couditions  of  the  problem. 
One  condition  is  that  at  the  earth's  surface  u'  is  constantly 
zero.     Hence,  if  the  earth's  radius  —b}  we  have 
2C£-3C7r-4=0,  or  ~  =  %- 
C  o 
Hence,  by  eliminating  C, 
#  on/„      °5 
=     2C 
and 
(r  -  -4)  cos2  \  sin  2  (0 -//i), 
,2 
(Q  =  -2H,C(r*+  |^)  cos2  X  cos  2(0-^) . 
For  determining  yjr(t)  we  may  employ  a  condition  indicated  by 
the  foregoing  expressions  for  u\  v',  u/,  and  -^-,   namely   that 
IT 
these  quantities  are  all  constantly  zero  where  \=  ~ — that  is,  at 
the  pole  of  the  earth  and  in  the  fluid  column  incumbent  upon  it. 
Consequently  the  density  of  the  fluid  at  the  pole  will  be  con- 
stant; and  if  A  be  its  value,  we  have,  by  the  equation  (3), 
Nap.logA=^-^3+tM, 
which  proves  that  yjr{t)  is  independent  of  the  time.  By  elimi- 
nating this  quantity,  and  substituting  the  foregoing  value  of 
-p  the  equation  (3)  becomes 
^Nap.log^  =  -Gg-1;)+^^+J(l-3sin^)) 
+  (S^+2^c('"2+£))cos2xcos3(e-^)-  •  ® 
The  third  condition  necessarily  has  reference  to  the  circum- 
stances of  the  atmosphere  at  its  superior  limit.  On  the  hypo- 
thesis of  the  atomic  constitution  of  bodies,  it  may  be  shown  a3 
follows  that  at  a  certain  height  the  atmosphere  must  terminate 
abruptly.  Conceive  a  horizontal  surface  to  be  drawn  through 
the  position  of  a  given  atom.  Then,  on  that  hypothesis,  the 
upward  accelerative  force  due  to  the  molecular  action  of  the 
