38 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



They are a kind of asymmetric hyperbolae having the axes of r and 2 for asymptotes, 

 but converging more rapidly toward the first of these axes than toward the second 

 (fig. 42). The axes are themselves vector-lines cutting each other at the neutral 

 point of the field. 



The vector is seen to have the constant numerical value A on the curve 



A 2 = R*+Al = cfr'+^tfz 1 



A A 



which is an ellipse of half-axes and \ These ellipses are drawn in fig. 42 for 



the values 1 , 2, 3, of A . 



We can now get a complete picture of the field. The meridian planes passing 

 through the axis of 2 form one set of surfaces of flow. The other set is generated by 

 the lines of flow of fig. 42, when this figure rotates around the axis of 2. Simultane- 

 ously the other curves of this figure will generate the surfaces of equal intensity. 

 We get thus the complete representation of the field by three sets of surfaces : two 

 sets of surfaces of flow cutting each other along the lines of flow in space, and one 

 set of surfaces representing equal scalar values of the vector. 



As the field is solenoidal, a representation can also be obtained where the last 

 set of surfaces is left out. A z is constant in a plane 2 = const. Thus there goes equal 

 transport through equal areas of this plane. A division of this plane into equal areas 

 is obtained if the radial lines defining the meridian planes are drawn with equal 

 angular intervals and the circles defining the other surfaces of flow are drawn with 

 radii proportional to the numbers VI, V2 , V3, V4, .... These intervals have been 

 chosen already for the meridian curves of fig. 42, which represent these surfaces of 

 flow. Thus the intersection of these surfaces with meridian planes which have 

 constant angular distance from each other will produce tubes of equal transport rep- 

 resenting the field completely. The surfaces of equal intensity may then be left out. 



As atmosphere and hydrosphere have a limited extent in vertical direction but 

 an enormous extent in horizontal direction, the best representations of fields of 

 motion in these media will be obtained by charts in horizontal projection. It will 

 be useful to consider different types of charts representing a simple field of motion, 

 as that which we have just examined. Fig. 43 gives four different types of such 

 charts. 



(A) In fig. 43 A, the full-drawn concentric circles are contour-lines representing 

 the topography of one of the surfaces of flow, namely, that of which a profile-curve 

 is drawn at the top of the figure. The radial lines represent the lines of flow on this 

 surface. Their vertical course is given directly by the topography of the surface. 

 Finally the stippled circles are curves for the equal intensity of the vector. Evi- 

 dently a set of charts of this kind each containing three sets of curves, contour-lines, 

 lines of flow, and curves of equal intensity, will give a complete representation of 

 the field. 



(B) A varied method of representation, derived from the solenoidal property, 

 is given in fig. 43 B. The contour-lines giving the topography of a surface of flow 

 are retained and the lines of flow on it are drawn as before. But these lines are 



