ELEMENTARY PRINCIPLES OP KINEMATICS OF CONTINUOUS MEDIA. 25 



Cutting a vector- tube by any surface a let da denote the area of the section. A 

 being the vector and A its component normal to the section, let us consider the 

 product A n da. This product does not depend upon the angle contained between 

 the normal to the section and the axis of the tube; for as this angle varies, the 

 area da of the section and the vector-component A normal to it will vary in inverse 

 proportion to each other, always giving a product equal to that of the area of 

 the normal section into the tensor of the vector. We will call this product the 

 transport through the section. The name is derived from the case to be examined 

 more fully below, when the field represents motion. The bundle of tubes cutting 

 through a finite surface a divides this surface into elements da and determines a 

 certain transport A da through each of them. Forming their sum, we get 



(a) Transport through surface a = J Ada 



The excess of transport leading out of a closed surface over that leading into it may 

 be called the outflow, and the same quantity with the sign changed the inflow. The 

 outflow is obtained by taking the integral (a) over the closed surface, counting 

 the normal directed outward as positive. The inflow is obtained in the same way, 

 counting the normal directed inward as positive. 



Returning to an elementary vector-tube, let the section be moved from place to 

 place along it. The transport will then, as a rule, be found to vary. Measuring 

 its value from section to section in all tubes, we get numbers representing the field 

 of transport. This field can be represented in the common way by drawing surfaces 

 of equal transport. 



The tubes of flow in connection with the surfaces of equal transport will give 

 a representation of the vector field as complete as that given by the lines of flow 

 in connection with the surfaces of equal intensity; for, being sufficiently narrow, 

 the tubes will represent the direction of the vector equally well as the lines; and 

 from the value of the transport we can come back to the numerical value of the 

 vector dividing by the area of the cross-section of the tube. 



Though the field of transport thus performs a similar service as the field of 

 intensity for representing the numerical value of the vector, one important difference 

 should be observed. The intensity-field is uniquely determined, while the field 

 of transport has a definite sense only in connection with a given system of tubes. 

 Choosing new surfaces for defining the tubes, we shall as a rule get tubes which have 

 other cross-sections, and therefore lead to a new field of transport. 



112. Solenoidal Vector. A field may have the property that the outflow is 

 zero out of every closed surface. The transport will then be the same through every 

 section of one and the same tube. The surfaces of equal transport may then be left 

 out as superfluous. It will be sufficient to know the constant of transport for each 

 tube. It will in this case be found convenient to undertake the division of the field 

 into tubes in such a way that each tube gets the same transport, in the simplest 

 case unit transport. Choosing a unit of suitable magnitude, we can still get tubes 

 sufficiently narrow for the purpose of representation. These narrow, in the limiting 



