34 Prof. Challis on the Principles of Hydrodynamics. 

 Consequently, by integration, 



dt dx dy dz ^^ ' ^ ' 



and substituting the foregoing values of u, v, w, 



This is the equation it was requu-ed to find. The equations 

 (1.), (2.), (3.), with the equations 



^ dilr ^ d-Jr . d-Jr 



and a given relation between the pressure {p) and density [p), 

 suffice for determining the seven unknown quantities -x/r, X, u, v, 

 tv, p, p. 



Corollary. By substituting -j:, -j-, -r- respectively for u, v, w 



in the equation (A.), it will appear that 



and by integration, 



This equation shows that the value of i^ for a given element may 

 vary in an arbitraiy manner with the time ; and, on account of 

 the arbitrary constant C, that it may be of arbitraiy value at a 

 given time. Hence this function has not necessarily the same 

 value at the same time for difi'erent elements. 



The fundamental hydi-odynamical equations having now been 

 investigated, I proceed to apply them to particular cases. 



Example I. Let the relation between the pressure and the 

 density be expressed by the equation j9 = o^p, and let the velocity 

 and density be functions of the distance from a fixed jilane, and 

 the fluid be supposed to be acted upon by no accelerative force : 

 it is required to determine the motion. 



Assuming that the fixed plane is parallel to the plane yz, we 



shall have 



„ d^y d'^z 



r = 0, to = 0, — 7 =0, -Tw =0. 



' dt^ ' dt- 



Hence the equations (1.) and (2.) become for this case, 

 a^dp du du _ 

 pdx dt dx ' 



dp dp du _^ .^ 



pdt pdx dx~ 



