ELEMENTARY PRINCIPLES OP KINEMATICS OF CONTINUOUS MEDIA. 27 



114. Equation of Continuity. The physical significance of the integral express- 

 ing transport in a field of motion being thus known, it will be easy to give in 

 quantitative form the dependency of the future fields of mass upon the present 

 field of motion. 



Measuring the elementary volume conveyed out of a closed surface in an 

 element of time dt, we evidently get the elementary increase of volume during the 

 time dt of that mass which is momentarily contained in the closed surface. Reducing 

 to unit time we get the velocity of expansion of this mass. Thus : 



(A) The integral of the normal component of velocity taken over a closed 

 surface 



(a) Jvja 



is equal to the increase of volume per unit time of the mass momentarily 



contained in the surface. 

 Measuring on the other hand the elementary mass conveyed out of a closed 

 surface in the element of time dt, we get the elementary decrease during this time 

 of the mass stored within the surface. Reducing to unit time, we get : 



(B) The integral of the normal component of the specific momentum taken 

 over a closed surface 



(b) fV n da 



is equal to the diminution per unit time of the mass contained in the 



surface. 

 The dependency of the future field of mass upon the present field of motion is 

 expressed by these two theorems in two different ways, in the first case by the 

 change of volume of moving masses, in the second by the change of mass within 

 stationary volumes. We shall later write in explicit form the "equation of con- 

 tinuity, " expressing in mathematical symbols the contents of any of these theorems. 

 Provisionally it will be found more convenient to work directly with the physical 

 facts as contained in the theorems (A) and (B). 



115. Conditions Leading to Solenoidal Fields of Motion. Every reference to 

 variations in time of the field of mass will disappear from the theorem 114 (A) if 

 the mass momentarily contained in the closed surface does not change its volume. 

 In this case the field of velocity will fulfil the solenoidal condition 



(a) J v n da = o 



In the same manner, the reference to future fields of mass will drop out of the 

 theorem 114 (B) when the content of mass of every stationary volume is constant. 

 Specific momentum will then fulfil the solenoidal condition 



(jb) fV n da = o 



We thus get the important results : 



(A) Velocity is a solenoidal vector if the moving medium is incompressible. 



(B) Specific momentum is a solenoidal vector if the field of mass is stationary 

 in space. 



