40 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



tour-lines, giving the topography of a second surface of flow relatively to the first. 

 The stippled circles of fig. 43 b are these contour-lines. A set of charts of this kind, 

 each containing three sets of curves, lines of flow, curves of absolute and curves of 

 relative topography, will also give a complete representation of the field. 



Instead of using surfaces of flow, as in the cases (A) and (B), we can use arbi- 

 trary surfaces cutting through the field. We can then simplify by choosing sur- 

 faces of simple configuration, instead of the surfaces of flow, which as a rule will not 

 be simple. But in return we must give special representations of the component 

 fields tangential to and normal to the surface. In the case before us it will be easiest 

 to cut the field by horizontal planes 2 = const. As above, we shall then get two 

 different representations according as we make explicit use or not of the solenoidal 

 property of the field. We shall then arrive at the following two representations, 

 (C) and (D) : 



(C) Let us imagine the field in space to be given by tubes of equal transport 

 i. e., by the meridian planes and the surfaces of revolution which form the walls of 

 these tubes. The two sets of surfaces will cut the horizontal plane in two sets of 

 curves, the radii and the circles of fig. 43 c. These curves divide the plane into 

 areas which are sections of the unit tubes, and thus areas of equal transport normal 

 to the plane. While these areas represent the normal component-field, the radial 

 lines of flow represent the tangential field. Evidently the field in space can be 

 represented completely by a set of charts of this description. 



(D) Instead of using the solenoidal property of the field, we can draw the vector- 

 lines and the curves of equal intensity which represent the tangential field contained 

 in the plane z = const, and the curves of equal intensity which represent the normal 

 field, as developed in section 118. In the case before us the vector tangential to any 

 of the planes z = const, is R = ar. It has radial lines of flow and curves of equal 

 intensity which are concentric circles with radii increasing in arithmetical series 

 (fig. 43 d). As in the case before us the normal component A z = 2az is independent 

 of the coordinates x and y, no curves for representing the normal field are required. 

 Only the constant value of the component will have to be noted for each plane. 



123. Solenoidal Field in Space with Asymptotic Line. As another example of a 

 solenoidal field in space, we shall consider that defined by the rectangular components 

 (a) A x = ax A y = b A z =az 



It consists of two partial fields which we have examined already (section 120), the 

 field of the constant vector A, and the field of the linear vectors A x and A z defining 

 a two-dimensional deformation-field in planes parallel to the xz-plane. Each of 

 these partial fields being solenoidal, that produced by their co-existence will also be 

 solenoidal. 



The vector-lines of the field thus produced will be represented by the differ- 

 ential equations 

 ... dx _dy _ dz 



A X Ay A z 



or, substituting the values of the components, 



