DEPENDING ON THE HEIGHT ABOVE THE EARTH'S SURFACE. 449 



any density p at a distance a- from the origin, the air being in vibration. 

 Then 



dp 



and h" =— = d' 

 dp 



dx 



d.pc[>' 

 dx 



The same result may be obtained by means of Proposition II. For 

 we may consider the effect of the heat developed or absorbed by the 

 sudden condensation or rarefaction of the air in vibration to be the 

 same as that of an impressed force, which alters the rate of uniform 

 propagation. The velocity of propagation, supposing the temperature 

 constant and equal to 6,, is ay/l + aG^. Hence, by what has been 

 proved, 



dp 



X= \a-{\ +a0,) -¥ 



^dx' 



But the effective accelerative force which urges the element pdx 



in the direction of x, is 4-; and 



pax 



dp 



pdx 



a^d. 



^(l+a^J-a=a 



d . p<p 

 pdx 



The first term of the right-hand side of this equation is the accelerative 

 force which would act supposing no change of temperature ; the other 

 is due to variation of temperature. Consequently, 



d .p(p 



= — a'a 



{a'il+aOD-b'] 

 This leads to the value of b' obtained above 



dp 

 pdx 



odx 



The vibrations which take place in the propagation of sound are 

 so rapid as not to allow sufficient time for any sensible alteration of the 

 difference of temperature of two contiguous portions of the air by 

 communication of heat from one to the other. This difference may 



