DEPENDING ON THE HEIGHT ABOVE THE EARTH'S SURFACE. 445 



The solution of the problem requires, therefore, the knowledge of 



do fid 



expressions for the partial differential coefficients —- and -~ . There 



dz dp 



are at present no means of finding these by a method entirely a priori ; 



and recourse must consequently be had to experiment and observa- 



dd 

 tion. To obtain the value of -j- we shall refer to the experimental 



determination of the velocity of sound, beginning, first, with some 

 Propositions for finding the velocity theoretically. 



Prop. I. To find an expression for the velocity with which a given 

 state of density is propagated in any medium. 



The motion is supposed to be in parallel lines. Take an axis 

 parallel to the direction of motion, and let v, p, be the velocity and 

 density of a particle in motion at the distance x from a fixed origin, 

 and at the time t. Then we have the equation, 



dt dx ' 



(Poisson, Traife de Mecanique, Tom. II. p. 674.) 



The differential coefficients are partial with respect to time and 

 space. Let now p be the density at the same time at the distance 

 X + Sx. Then, 



p = p +^ Sx + &c. 



After the small time St let the density at the distance x + Sx be- 

 come p. Consequently, 



p = p — -ji Si + &c. 



