30  Prof.  Challis  on  the  Mathematical  Theory 
tion  of  the  moon  about  the  earth,  in  consequence  of  which  the 
moon's  attraction  might  produce  an  accumulation  of  waves  to  an 
unlimited  extent.  That  the  rate  of  this  kind  of  propagation  is 
independent  of  the  density  of  the  medium  is  proved  experimen- 
tally by  the  fact  that  mercury  and  water  are  propagated  at  the 
same  rate  in  a  rectangular  trough  by  the  action  of  gravity,  if 
only  the  depths  of  the  fluids  be  the  same.  With  respect  to  an 
unlimited  ocean,  if  the  uniform  depth  be  less  than  the  critical 
value  8*5  miles,  as  is  the  case  for  the  mean  depth  of  the  actual 
ocean,  C  is  negative. 
If  after  putting  8  for  p  in  the  equation  (5)  the  value  obtained 
above  be  substituted  for  a2  Nap.  log-r-,  and  if  V  be  put  for  r  in 
the  small  terms,  the  equation  of  the  upper  surface  of  the  atmo- 
sphere will  be  found  to  be 
,,  ,   Smb'4      2_    ,  GV /.       b5\       a.        „,,.       A 
rss  +4GP cos  x+7TV""Fy cos  Xco s2( 0-vO- 
Hence  as  C  is  positive,  high  tide  occurs  when  6—fit=0 — that 
IT 
is,  at  syzygies;  and  low  tide  when  Q—\it=-  ~,  or  at  quadratures. 
(The  contrary  is  the  case  for  ocean  tides,  because  with  respect  to 
the  actual  ocean  C  is  negative.)  The  difference  between  the 
high  and  low  tides  at  the  equator  is 
Again,  supposing  p1  to  be  the  density  of  the  atmosphere  at  any 
point  of  the  earth's  surface,  the  same  equation  (5)  gives 
21VT      ,     pf      3mb*      g_    ,   (3mb*  ,   10C62\      2^       0//J       A 
02Nap.log^  =  ^3Cos2X+  {jfi3+  — g-  Jcos2\cos2(0-/^). 
On  the  equator  let  p/  =  A(l  +  61)  at  syzygies,  and  p'=A(l  +  e2) 
at  quadratures,  €Y  and  e2  being  extremely  small  fractions.  Then, 
if  we  suppose  that  a^—gKD,  h  being  the  mean  height  of  the  mer- 
cury column  and  D  its  density,  we  shall  have  for  calculating 
h{e1  —  e2),  which  is  the  excess  of  the  height  of  the  barometer  at 
syzygies  above  that  at  quadratures,  the  formula 
Ab/Snd       20fiCb\ 
Vg\2R3  +      3      A 
To  obtain  the  foregoing  results  in  an  arithmetical  form,  it  would 
be  necessary  to  ascertain  the  numerical  value  of  C.  As  this 
constant  depends  on  the  height  of  the  earth's  atmosphere,  which 
is  an  unknown  element,  I  propose  to  perform  the  calculations  on 
the  hypothesis  that  the  height  of  the  atmosphere  is  sixty  miles. 
