448 PROFESSOR CHALLIS, ON ATMOSPHERIC TEMPERATURE 



Now by supposition a given state of density is propagated with a 

 uniform velocity. Hence if b equal the velocity of propagation, by 



Proposition I, 



dv J dp 

 dx ~ ' pdx' 



Integrating, 



® = 6 Nap. log p + <f){t) ; 



and introducing the condition that v = wherever p = I, which can be 

 satisfied when, as we suppose, the propagation is in a single direction 

 only, it follows that (^ {t) — 0. Hence, differentiating with respect to 

 time only, 



dv , dp 



dt pdt 



civ civ 



Substituting these values of -t- and -ri in the equations (a) and (b), 



we obtain, 



dp^^vdp^bdp ^^ 

 pdt pdx pdx ' 



dp bdp J dp 

 — f- + .-± + i» — f- = 0. 

 pdx pat pdx 



Multiplying the first of these by b and subtracting, the result is, 



^ = b' -^ • 

 dx ' dx'' 



and integrating, 



p^b'p + ^{t). 



We can now find an expression for the velocity of the propagation 

 of sound in the atmosphere, assuming the velocity to be uniform. Let 

 0, be the temperature of the air when at rest. Experiments shew that 

 by sudden compression the temperature is increased, and by sudden 

 dilatation diminished. Let 0, + <p be the temperature corresponding to 



