26 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



case infinitely narrow, tubes are called solenoids, and every vector which can be 

 represented completely by such tubes is called a solenoidal vector. 



The solenoidal vector is simpler than the general vector inasmuch as it can be 

 represented completely by two sets of surfaces, i. e., by two scalar fields, while the 

 general vector requires three. In other words, there is a dependency between the 

 three components of the solenoidal vector. Using the solenoidal condition, i. e., 

 the condition expressing the fact that the outflow from every closed surface is zero, 



(a) (Ada = o 



we can determine the third component of the vector, if we know the value of the two 

 others at all points of the field. 



113. Volume-Transport and Mass-Transport. Passing to concrete fields of 

 motion, we shall consider a tube of flow and a section of it having the area da. The 

 particles situated at a certain time / on this section and having the velocity v, will 

 an element of time dt later be situated on another section which is displaced the 

 distance vdt along the tube. The normal distance between the sections will be v n dt. 

 The two sections and the walls of the tube determine an elementary parallelepipedon 

 of volume v dt da, giving the elementary volume of the medium which during the 

 time dt has passed the section da. Multiplying by the density [> of the medium, we 

 get the mass contained in this volume, i. e., the mass which during the time dt has 

 passed the section da. When we remember that the product of density into 

 velocity gives the specific momentum V of the medium, we get as expression of this 

 elementary mass V dt da. Dividing by dt we get the expressions v da and V da 

 representing, according to our definition, the transport respectively in the field of 

 velocity and in the field of specific momentum. We thus arrive at this result : 



(A) In the field of velocity the transport through a surface 

 () jv n da 



gives the volume of the medium passing the surface per unit time. 



(B) In the field of specific momentum the transport through a surface 

 (b) j'v n da 



gives the mass of the medium passing the surface per unit time. 



Taken over a closed surface the integral (a) will represent the volume and (b) 

 the mass of the medium conveyed per unit time out through the closed surface. 



Considering the transport as given in our m. t. s. units, and returning to the 

 vectors, we arrive at these methods of measuring velocity and specific momentum, 

 which may be useful to bear in mind in the subsequent practical work : 



(C) Velocity is measured by the number of cubic meters and specific momentum 

 by the Jiumber of tons passing per second a square meter normal to the 

 direction of the motion. 



