[    24    ] 
III.   On  the  Mathematical  Theory  of  Atmospheric  Tides,     By  the 
Rev.  Professor  Challis,  M.A.,  LL.D.,  F.R.S.,  F.R.A.S* 
THE  object  of  this  communication  is  to  indicate  a  method 
of  deriving  the  solution  of  the  Problem  of  Atmospheric 
Tides  from  the  general  equations  of  Hydrodynamics.  I  have 
already  applied  the  same  method  to  Oceanic  Tides,  on  the  par- 
ticular suppositions  that  the  whole  of  the  earth's  surface  is 
covered  by  water  of  uniform  depth,  and  that  the  body  which  by 
its  attraction  produces  the  tides  revolves  in  the  plane  of  the 
earth's  equator.  (See  two  articles  in  the  Numbers  of  the 
Philosophical  Magazine  for  January  and  April  1870,  and  a 
Supplement  in  the  Number  for  June  1870.)  The  problem 
with  these  limitations  is  one  which  we  must  know  how  to 
treat  mathematically  before  we  can  hope  to  arrive  at  a  general 
theory  of  tidal  motion.  But  although  in  point  of  generality  this 
problem  is  a  step  in  advance  of  that  in  which  the  hypothesis  of 
an  "  equatorial  canal "  is  made,  its  solution  falls  far  short  of 
giving  results  in  accordance  with  the  facts  of  nature,  on  account 
both  of  the  irregularities  of  the  ocean-bed,  and  of  the  interrup- 
tion of  the  water-surface  by  islands  and  continents.  The  case, 
however,  is  not  the  same  with  respect  to  the  atmosphere,  which 
may  be  regarded  as  a  fluid  envelope  of  nearly  uniform  thickness, 
covering  the  whole  of  the  earth,  and  of  such  height  that  its  tides 
are  but  little  affected  by  the  irregularities  of  the  earth's  surface. 
Accordingly  the  following  mathematical  treatment  of  the  tides  of 
the  atmosphere  is  closely  analogous  to  that  which  was  applied  (I 
think,  not  unsuccessfully)  to  the  above-mentioned  hypothetical 
case  of  oceanic  tides. 
As  it  is  not  my  intention  to  discuss  the  problem  completely, 
but  rather  to  demonstrate  the  applicability  of  a  particular  pro- 
cess of  reasoning,  I  make,  for  the  sake  of  simplicity,  the  following 
suppositions  : — (1)  The  inferior  boundary  of  the  atmosphere  is  a 
spherical  surface  the  radius  of  which  is  equal  to  the  earth's  mean 
radius ;  (2)  the  attracting  body  is  the  moon  revolving  eastward 
about  the  earth  in  the  plane  of  the  equator  at  its  mean  distance 
with  its  mean  angular  velocity ;  (3)  the  earth  has  no  motion  of 
revolution,  the  moon  being  conceived  to  revolve  about  it  west- 
ward with  the  mean  relative  angular  velocity  (fi).  As  tidal  mo- 
tion is  oscillatory,  udx  +  vdy  -f  wdz  is  assumed  to  be  an  exact 
differential  (d<f>).  Centrifugal  force  will  be  left  out  of  account, 
as  having,  under  the  above  conditions,  no  appreciable  effect  on 
the  oscillations.  The  relation  between  the  pressure  (p)  and 
density  (/?)  is  assumed  to  be  always  p  =  a?p  at  all  points;  so  that 
the  effect  of  variations  of  temperature  is  not  considered.  This 
*  Communicated  by  the  Author. 
