406  Prof.  Challis  on  the  Hijdrodynamical 
points  of  the  fluid  at  all  times ;  and  if  the  fluid  be  disturbed 
within  a  limited  space  and  be  of  unlimited  extent,  there  must 
be   distant  points  at  which  -^-,  or  I  l-^dx-\-  -r:dy+  ~n  dz\ 
vanishes  together  with  V,  and  the  density  has  a  constant  value 
o0.     Hence 
d$        V2 
p  =  p0e~a'*dt~2al* (C) 
This  equation  applies  generally  to  unsteady  motion.     Now  it  is 
to  be  observed  that,  whether  the  motion  be  steady  or  unsteady, 
the  investigation  of  the  equation  (a) 'is  the  same,  because  in  both 
cases  the  mean  quantity  of  fluid  which  passes  a  given  transverse 
area  in  a  given  time  is  not  sensibly  altered  by  the  reaction  of  one 
dV 
or  more  small  spheres.     Hence,  eliminating  -=-  from  (a)  by 
az 
means  of  (c),  the  result  is 
dHp        V*dD         d*6 
j    =  71 — ty^-t  +  -ri:  very  nearly,     .     .     (a) 
pdz       (l—J))dz       dzdt       *  *'  v  > 
which  equation  differs  from  that  for  steady  motion  by  having  an 
additional  term  on  the  right-hand  side.  If  this  equation  be  ap- 
plied in  a  case  in  which  V  represents  the  velocity  in  vibratory 
motion,  the  additional  term  will  have  as  much  positive  as  nega- 
tive value,  so  that  the  mean  effect  of  the  impulses  it  indicates 
will  be  zero.  The  other  term  is  indicative  of  impulses  towards 
the  denser  part  of  the  substance,  whether  V  be  positive  or  nega- 
tive, just  as  when  the  motion  is  steady. 
9.  Recurring  now  to  the  before-mentioned  theories  of  atomic 
repulsion  and  molecular  attraction,  according  to  which  the  repul- 
sion which  keeps  the  atoms  asunder  is  due  to  vibrations  emana- 
ting from  individual  atoms,  while  the  counteracting  attractions 
result  from  composite  vibrations  emanating  from  a  congeries  of 
atoms  constituting  a  molecule,  it  will  appear  that  the  maximum 
velocity  and  breadth  must  be  supposed  to  be  much  greater  in  the 
latter  vibrations  than  in  the  others.  Also  it  is  presumable  that 
the  maximum  velocity  may  very  much  exceed  the  velocity  of  the 
earth  in  its  orbit,  or  that  of  the  solar  system  in  space,  and  yet  be 
small  compared  with  the  value  of  a\  which  is  about  190,000 
miles  per  second.  In  the  supposed  case  of  the  atoms  being  con- 
strained to  take  positions  such  as  to  produce  a  regular  gradation 
of  atomic  density  from  one  end  to  the  other  of  a  steel  bar,  at- 
traction-vibrations, propagated  in  the  direction  from  the  denser 
to  the  rarer  end,  will  continually  counteract  the  atomic  repulsions 
urging  the  atoms  towards  the  rarer  end,  without  being  neutra- 
lized by  attraction-vibrations  of  the  same  order  propagated  in 
