442 Prof. Challis on the Principles of Hydrodynamics. 



«, being tlie constant rate of propagation. For the integral of 

 this equation is p=.Y{z—a^t). Now, since 



dp d.pu d.pv d.pw _„ 

 dt dx dy dz ' 



by substituting from the equation above, we have 

 dp _ d.pu d.pv d.pw 

 ^dz dx dy dz 



Since m=0 and v=0 for the motion along the axis of z, this 

 equation becomes 



. dp _ du dv dw 

 ^ ^ pdz ~ dx dy dz ' 



Also, because M=^-^, v=<f)-j-, w—f-j-,f=\, and 

 we obtain by substituting, 



/ d4>\dp _ h^<}> d^<p 

 V'~lk)~^z-~lF^d^ ^'^ 



The known general equation which gives the value of the density, 

 becomes for the motion along the axis of z, 



»».Nap.logp+/f +f f=FW. 



Differentiating with respect to z, and putting /=1, 

 a^.dp d^ d^d^_Q 

 pdz dzdt dz dz' 

 Eliminating p from (e) by this last equation, the result is 



by comparison of which equation with the equation (B), we obtain 



~dF^"'d^ + dz V'dz^ + dzdt)-^- • ^^^ 

 Now this equation vanishes identically if 



that is, if 6, -^, and -^ be functions of z—a.t, and by conse- 

 ^ dz dt 



quence p be a function of the same quantity, a result in accord- 

 ance with the original hypothesis respecting p. 



Let us, therefore, trace the consequence of introducing the 



