DEPENDING ON THE HEIGHT ABOVE THE EARTH'S SURFACE. 447 



the same in any medium which propagates a given state of density 

 with the uniform velocity b, as in the medium defined by the equa- 

 tion ji = h'p, provided the motion in both be subject to the condition 

 that V = where p = \. 



By help of what precedes the following Propositions may be solved. 



Prop. II. To find the impressed accelerative force which will alter 

 the rate of miiform propagation in a medium whose density varies as 

 the pressure. 



Let the medium be such that p = a^p, and let b be the altered rate 

 of propagation. It has been shewn that the motion is the same as in 

 a medium for which p = b-p, no impressed force acting. Hence the 

 effective accelerative force is the same. Hence 



d^x ¥dp 



df pdx ' 



But by the general equation, 



dp^ _ V- _ ^E 



pdx ~ df ' 



Hence, ^ = X + ^, 

 pdx pdx 



and X=(a'- b')-^. 



^ ^ pdx 



Prop. III. To find the relation between the pressure and the density 

 in a medium which propagates a given state of density uniformly. 



It is here supposed that there is no impressed force. The two 

 equations following are therefore to be applied : 



dp (dv\ dv dv , . 



dp _ d.pv _ ^ pdv vdp ,, 



dt ~ dx ~ dx " dx ' ^ '' 



