giveriseto  Sonorous  Undulationsin  the  surrounding  Atmosphere.  441 
It  may  be  thought  that  the  interval  which  separates  a  state  of 
disturbance  of  this  kind  from  a  spherical  wave  consisting  of  a 
shell  of  condensation  immediately  succeeded  by  a  shell  of  rare- 
faction is  little  short  of  infinite. 
If,  however,  it  can  be  shown  that  in  the  case  of  motion  par- 
allel to  the  axis  of  a  cylindrical  tube  filled  with  air,  the  pressure 
in  part  depends  on  the'density  and  in  part  on  the  velocity;  and 
if  it  can  be  shown  further  that  under  the  same  circumstances 
the  portion  of  the  pressure  due  to  the  velocity  may  be  very  con- 
siderable, while  that  due  to  the  condensation  is  very  small;  I 
think  it  will  be  admitted  that  we  shall  have  made  a  great  ad- 
vance, I  do  not  say  towards  the  solution  of  the  problem,  but 
in  indicating  the  direction  in  which  the  solution  must  be  sought. 
Now  in  the  case  of  motion  last  spoken  of,  putting  p,  v9  p  for 
the  pressure,  velocity,  and  density  at  the  time  t  at  a  point  the 
coordinate  of  w?hose  point  of  rest  is  x}  we  shall  have 
p=  funct.  (x,  t)j 
v=  funct.  (x,  t)3 
/a=  funct.  (Xj  I); 
and,  eliminating  x  and  t  between  these  three  equations,  we  shall 
get 
p=  funct.  (p,  v), 
thus  proving  that  the  pressure  is  dependent  both  on  the  velocity 
and  density,  and  not,  as  is  erroneously  stated  in  the  ordinary 
theory  upon  the  subject,  upon  the  density  alone. 
Moreover,  putting?/  for  the  ordinate  of  the  particle  at  the  time 
/,  and  D  for  the  mean  density,  the  equation  of  motion  of  the 
particle  will  be 
dt2  ^  D  dx 
a  conclusion  which,  I  apprehend,  no  one  will  dispute. 
I  have  elsewhere  shown*  that  this  equation  is  satisfied  by  the 
three  following  relations,  viz. 
*   See  Philosophical  Magazine,  vol.  xxxvi.  p.  27. 
