KINEMATIC PROGNOSIS. 1 59 



correspond to two different mathematical forms of the equation. The first theorem 

 deals with the velocity of expansion of a given mass and states its identity with the 

 integral of the normal component of velocity taken over the limiting surface of the 

 mass. K being the volume of the mass, the velocity of expansion will be expressed 

 by the individual time-derivative of K. Thus we can write the equation 



dK 



(a) - = jv n d<T 



dt 



The theorem 114 (B) deals with the varying mass M which is contained within a 

 stationary volume, and states that the diminution of this mass per unit time is equal 

 to the mass-outflow through the limiting surface of the volume. Evidently the 

 diminution of the mass M per unit time in a stationary volume is expressed by the 

 negative local time-derivative of M. When we identify this derivative with the 

 well-known expression of the mass-outflow, we get this other form of the equation 

 of continuity 



In order to bring the equations to forms more easily used we can apply them 

 to infinitely small volumes K. The integrals appearing in the second member of (a) 

 or (b) will then be expressed by the product of this volume K into the divergence of 

 the vector. When at the same time we express the volume K of the moving mass 

 by the product of its mass M into its specific volume, K = aM, and remember that 

 this mass M is constant, we get equation (a) in the form 



M^ = Kdwv 

 dt 



M 

 Dividing by K, and remembering that the ratio is the reciprocal specific volume, 



K 



we get 



, \ 1 da j. 



(c) - = div v 



a at 



In the same manner, when in (b) we express the mass M as the product of its density 

 p into its volume K, and remember that here the volume is stationary in space and 

 therefore constant, we get 



(d) -% = div V 



When we use the relation existing between local and individual derivative 

 (section 177) as well as the relations existing between density and specific volume 

 and between velocity and specific momentum, we can verify at once the fact that (c) 

 and (d) are merely different forms of the same equation. 



193. Equation of Continuity as a Prognostic Equation. Equation (d) of the 

 previous section directly tells us that if we know the field of specific momentum at 



any moment, we can find a field representing the rate of decrease of density 



