of  Atmospheric  Tides.  29 
both  to  space  and  time,  putting  for  -7-  or  ~  the  foregoing  value, 
and  bf  for  r,  and  omitting  terms  of  the  second  order  with  respect 
to  m,  it  will  be  found  that  the  condition  (-?.)  =  0  gives  the  fol- 
lowing equation  for  determining  C  : — 
(•♦*»)- 
b*         G      V       W*)t  ~  4GR3  * 
Thus  the  three  arbitrary  constants  have  been  determined,  and 
at  the  same  time  the  general  hydrodynamical  equations  (1)  and 
(2)  are  satisfied.  It  might  hence  be  argued  that  the  solution 
we  have  obtained  is  by  this  means  proved  to  be  the  true  one, 
inasmuch  as  only  one  solution  can  satisfy  all  the  conditions  re- 
quired to  be  fulfilled.  It  will,  however,  be  worth  while  to  test 
this  inference  by  other  considerations. 
First,  it  may  be  remarked  that  C  becomes  infinite  if  the  factor 
which  multiplies  it  vanishes.  Putting  x  for  -7-,  this  will  be  the 
case  if 
79 
Since  #  =  32*191   feet,  //,=  —  x    the  earth's  rotation  in  one 
second,  and  6  =  3956  miles,  it  follows  that  -'—  =  yv^-,  and  the 
value  of  x  which  satisfies  the  above  equation  will  be  found  to  be 
1  +  -r^T  nearly.     Consequently 
U—b=     ft     =  8*58  miles  nearly. 
Hence,  if  the  height  of  the  atmosphere  had  this  particular  value, 
C  would  be  infinite,  and  there  would  be  unlimited  tide.  Ac- 
cording as  C  is  positive  or  negative,  b'  —  bi%  greater  or  less  than 
this  quantity.  Hence  for  the  atmosphere  C  is  positive,  its  height 
being  known  to  be  much  greater  than  8*6  miles. 
It  is  remarkable  that  the  theory  of  oceanic  tides  conducted  to 
a  like  critical  value  of  the  depth  of  the  ocean,  and  in  amount 
very  nearly  the  same  as  this  critical  height  of  the  atmosphere. 
The  reason  is,  that  for  this  particular  depth  of  the  ocean,  or 
height  of  the  atmosphere,  the  rate  of  propagation  of  waves,  as 
due  to  the  action  of  gravity  independently  of  the  elasticity  of  the 
medium,  is  very  nearly  the  same  as  the  rate  of  the  relative  rota- 
