58.] EULERIAX EQUATIONS. 



(2). 



7. We have thus three equations connecting the^t unknown 

 quantities u, v, w, p, p. We require therefore two additional 

 equations. One of these is furnished by a relation between p 

 and p, the form of which depends on the physical constitution 

 of the particular fluid which is the subject of investigation. For 

 the case of a gas kept at a uniform temperature we have Boyle s 

 Law 



P~ty ................................. (3). 



If we have a gas in motion of such a nature that we may neglect 

 the loss or gain of heat by an element due to conduction and radi 

 ation, the relation is 



where 7 = 1*41 for air. In the case of an incompressible fluid, 

 or liquid, we have 



p = constant ............................. (5). 



8. The remaining equation is a kinematical relation between 

 u, v, w, p obtained as follows. If V denote the volume of a 

 moving element of fluid, we have, on account of the constancy 

 of mass, 



!-+&amp;gt;- ........................ &amp;lt;&amp;gt; 



Now the rate of increase of volume of a moving region is evid 

 ently expressed by the surface-integral of the normal velocity 

 outwards, taken all over the boundary. If the region in question 

 be that occupied by the matter which at time t fills the rectangular 



