24 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



lead to a fundamental relation connecting these fields with that of mass. For as 

 motion consists in the displacement of invariable masses having to fill space con- 

 tinuously, the knowledge of the present field of motion involves a certain knowledge 

 regarding the future field of mass. Thus the two fundamental suppositions regard- 

 ing the medium lead to an intrinsic relation of prognostic nature, in its mathematical 

 form called the equation of continuity. In special cases time drops out, and the 

 equation is reduced to a diagnostic one, submitting the fields of motion to certain 

 restrictive conditions. In connection with the geometrical principles for representing 

 the fields of motion, we shall therefore develop this prognostic equation and pay 

 special attention to the cases when it is reduced to a diagnostic equation. 



no. Vector-Lines, Vector-Surfaces, and Tensor-Surfaces. In Statics we have 

 considered the methods of representing geometrically certain special vectors, the 

 ascendants or the gradients of a scalar quantity (sections 16, 17). The field of 

 the scalar quantity gave a complete representation also of the vector derived from 

 it. But in the general case a vector will, for its geometrical as well as for its ana- 

 lytical representation, require the use of three instead of only one scalar quantity. 



In order to represent first the direction of a vector at every point of the field, we 

 can draw a set of curves running tangentially to the direction of the vector. These 

 lines are called vector-lines, or for a field of motion lines of flow. A set of curves in 

 space is obtained by the intersection of two sets of surfaces. Each set of surfaces 

 being the equiscalar surfaces of a certain scalar field, we see that the representation 

 of the direction of a vector by vector-lines involves the use of two scalar fields. 



The surfaces used to represent the vector lines may be chosen in an infinite 

 number of ways; but they have the common property of being surfaces generated by 

 vector-lines. Any surface generated in this way will be called a vector-surface, or 

 for the field of motion a surface of flow. 



The direction of the vector being thus given by two scalar fields, we can use a 

 third for representing its numerical value or its tensor. An equiscalar surface of this 

 third field will pass through all points where the vector has a certain constant 

 numerical value. These surfaces may be called tensor-surfaces, or surfaces of equal 

 intensity. 



The vectors considered by us will have a uniquely determined direction at every 

 point where it is different from zero. As intersections of vector-lines under finite 

 angles would give two or more different directions for the vector in the point of 

 intersection we conclude : 



Vector-lines can intersect each other only at zero-points of the field. 



Nothing prevents vector-lines from touching each other ; for, having a common 

 tangent, both lines indicate the same direction at the point of tangency. 



in. Vector-Tubes and Surfaces of Equal Transport. The two sets of vector- 

 surfaces cutting each other along the vector lines will divide the field into a set 

 of elementary tubes which have parallelogrammatic cross-sections. These may be 

 called vector-tubes, or, for a field of motion, tubes of flow. 



