of  Atmospheric  Tides.  31 
For  this  purpose  we  have  6  =  3956  miles,  #=4016  miles, 
/t=^x  the  earth's  rate  of  rotation,  and !-7r-  =  c-—  ==  ^-i—r : 
o.^  tr  g        oil '4 
and  it  will  be  assumed  that  ^  =  t^t-t  and  ~  =  ^-      With 
B,       603  G      70 
these   data    the    calculation    from    the    formula   for   C    gives 
n 
=  0*0000008296.      Hence  from  the  foregoing  formulse  the 
following  results  may  be  obtained : — 
For  the  equation  of  the  surface, 
r-b'=  1-0841  ft.  cos2 X+ 1-2753  ft.  cos2 X  cos  2(0-/**); 
The  difference  between  high  and  low  tide  =2*5507  ft. ; 
The  barometer  is  higher  at  syzygies  than  at  quadratures  by 
0-00278  in. 
Also,  from  the  equation  which  gave  the  relation  between  the 
terminal  density  8  and  the  height  6'  —  b  of  the  atmosphere,  it  is 
found,  by  calculating  on  the  supposition  of  a  height  of  sixty  miles, 
that  8  is  equal  to  one  six-millionth  part  of  the  density  A  at  the 
surface. 
These  results  do  not  admit  of  comparison  with  observation, 
excepting  in  the  case  of  the  difference  of  barometric  heights. 
By  observations  made  at  St.  Helena,  first  by  Captain  Lefroy  and 
afterwards  by  Captain  Smythe,  the  details  of  which  are  given  by 
Sir  Edward  Sabine  in  the  Philosophical  Transactions  for  1847 
(p.  45),  it  appears  that  the  height  was  greater  at  syzygy  than 
at  quadrature  by  0*00365  in.  Nearly  contemporaneous  obser- 
vations (1841-45)  made  by  Captain  Elliott  at  Singapore  gave  a 
difference  equal  to  000570  in.  (see  the  Philosophical  Transac- 
tions for  1852,  p.  125).  Both  these  values  exceed  that  given 
by  the  theory  on  the  hypothesis  of  an  atmosphere  sixty  miles 
high.  By  assuming  a  less  height  a  nearer  agreement  with  ob- 
servation would  be  obtained.  So  far,  therefore,  as  the  theory 
is  trustworthy,  we  may  infer  from  it  that  the  height  of  the 
atmosphere  is  less  than  sixty  miles. 
Thus,  although  the  theory  cannot  be  put  to  a  strict  numerical 
test,  inasmuch  as  it  makes  the  high  tide  of  the  atmosphere 
occur  under  the  moon  it  is  so  far  confirmed  by  observation.  I 
think  also  that  it  may  be  regarded  as  no  little  confirmation  of 
the  theory  that  it  explains  why  in  this  particular  the  tide  of  the 
atmosphere  differs  from  that  of  an  unlimited  ocean  of  uniform 
depth  less  than  8-5  miles,  for  which  there  would  be  low  tide 
under  the  moon. 
The  foregoing  paper  was  read  in  the  Mathematical  Section,  at 
the  Meeting  of  the  British  Association  at  Edinburgh ;  and  an 
