28  Prof.  Challis  on  the  Mathematical  Theory 
atoms  below  the  surface  will  exceed  the  downward  accelerative 
force  due  to  that  of  the  atoms  above  by  just  the  accelerative 
force  of  gravity.  Consequently,  if  by  reason  of  the  diminution 
of  the  density  the  former  force  eventually  becomes  only  equal  to 
the  force  of  gravity,  it  is  evident  that  there  can  be  no  more 
downward  molecular  action,  and  that  thus  a  superior  limit  of  the 
fluid  will  be  reached.  Also,  since  at  the  very  boundary  the 
density  cannot  be  finite,  there  will  be  an  abnormal  increase  of 
density  downwards  from  the  boundary,  where  it  is  zero,  to  a  cer- 
tain small  distance  at  which  the  variation  of  density  becomes 
regular — that  is,  unaffected  by  the  abrupt  termination.  Within 
this  stratum,  which,  although  extremely  thin,  must  be  supposed 
to  exceed  in  extent  the  sphere  of  molecular  action  on  a  given 
atom,  the  variation  of  density  satisfies  the  condition  of  making 
the  upward  molecular  action  on  each  atom  exceed  the  downward 
by  just  the  force  of  gravity.  If  8  be  the  density  at  the  distance 
Z>',  where  the  variation  ceases  to  be  abnormal,  the  result  there  of 
the  combined  action  of  the  molecular  forces  and  the  force  of  gra- 
vity is  equivalent  to  a  pressure  a?8  applied  at  all  points  of  the 
surface  of  radius  b',  and  the  terminal  density  may,  without  sen- 
sible error,  be  supposed  to  have  the  finite  value  8.  Although 
this  reasoning  applies  strictly  to  a  state  of  equilibrium  of  the 
atmosphere,  it  will  clearly  not  be  perceptibly  affected  by  the 
slow  oscillatory  motions  we  are  considering. 
It  may  here  be  stated  that  the  idea  of  a  particular  molecular 
condition  at  the  superior  boundary  of  the  atmosphere  was  enter- 
tained by  Poisson,  and  that  it  was  regarded  by  him  as  analogous 
to  a  gradation  of  superficial  density  assumed  to  exist  at  the  sur- 
faces of  liquids  and  solids.  In  these  substances,  however,  the 
gradation  of  density  would  be  maintained  by  combined  molecular 
attraction  and  repulsion,  independently  of  the  force  of  gravity. 
Supposing,  therefore,  b'  to  be  the  value  of  r  for  the  top  of  the 
aerial  column  which  has  its  base  at  the  earth's  pole,  according 
to  the  foregoing  argument  the  equation  (5)  gives 
which  determines  the  relation  between  the  terminal  density  8 
and  V—b  the  height  of  the  atmosphere. 
This  being  understood,  since  a  particle  at  the  superior  surface 
may  be  assumed  to  remain  at  the  surface  in  successive  instants, 
we  shall  have,  with  respect  to  such  a  particle,  (-£)  =  0,  because 
the  superficial  density  will  at  all  points  be  the  same  in  successive 
instants.     Hence,  differentiating  the  equation  (5)  with  respect 
