46 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



The property of never intersecting each other very much limits the course of 

 the curves, and makes it possible to draw them as soon as the values of the scalar are 

 known in a relatively small number of points. But the course is never completely 

 determined by a limited number of observations. There will always be a certain 

 limited freedom in the way of drawing each curve. But as the number of observa- 

 tions is increased this freedom will be reduced, and finally the course of the curve 

 will be perfectly determined from a practical point of view, i. e., with a certain 

 finite degree of precision. 



The curves will obtain their characteristic features by the situation of the points 

 where the scalar has its extreme values. At the maximum points and the minimum 

 points the equiscalar curve will be reduced to a point. These points are surrounded 

 by closed equiscalar curves. Between the maximum and minimum points there 

 will be maximum-minimum points. In each a certain singular equiscalar curve 

 cuts itself. The two branches of this singular curve divide the field in the neighbor- 



Fig. 46. Maximum points, minimum points, and a 

 maximum-minimum point of a scalar field. 



Fig. 47. Maximum-minimum point of higher 

 order. 



hood of the maximum-minimum point into four angular areas. In two of them the 

 scalar has greater and in two of them smaller values than in the point of intersection 

 (fig. 46). More complex maximum-minimum points may also be mentioned, though 

 they will rarely be met with in practice. Thus three branches of the singular equi- 

 scalar curves may cut each other in this point, dividing the surrounding field into six 

 angular areas of alternately higher and lower values of the scalar (fig. 47), and so on. 

 The said features of the field give the practical rules for drawing the curves. 

 Examining the numbers we first look for the points where the scalar has its extreme 

 values. Around these points we then draw closed curves, proceeding subsequently 

 to the curves representing intermediate values of the scalar and having the more 

 complicated course between the areas of greater and those of smaller values. Among 

 these curves the ones intersecting themselves should not be avoided. They give 

 more information regarding the field than any other single curve. 



