104 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



at a great number of points. Afterwards we can draw curves for integer values of <p, 

 and thus arrive at the common representation of the field of the scalar <p, which is the 

 derivative of the scalar a. 



In the way described we arrive at the field of tp by a discontinuous process. 

 But it can be changed at once into a continuous one. Instead of moving the differ- 

 entiating sheet alongthe curves s, we move it along the curves a = const., and measure 

 the line-elements which are contained between two curves a = a a and a = a, . When 

 we come to places where the element ds is seen to give one of the required integer 

 values of <p we make a mark through the slit of the sheet. In this manner we mark 

 points through which the required curves for integer values of <p are to go. After- 

 wards these curves can be drawn continuously through the points determined. 



Vice versa the problem of determining the field of a when that of <p is given, *. e., 

 the problem of integration, will be determinate when an initial value of a is given at 



Fig. S3. The curves 5 are represented by thick lines with arrow-heads; the curves = 12, 11, 10, . 



da 

 continuous lines; and the curves (f= = . . . 2, 1, o, 1, 2, . . . by stippled lines. 



as 



by fine 



one point of each curve s, for instance when an initial curve a = a is given. The meas- 

 urements which are to give the values of a at other points can be performed along 

 one after another of the curves 5 as described in the preceding section. Or they can 

 be performed first along the initial curve a = a in order to determine the points of 

 the next curve a = a, ; then along this second curve in order to determine the next 

 curve a = a 2 , and so on. Both methods are continuous. 



It will much facilitate the drawing of the field of the derivative to observe that 

 the curve <p = o can be drawn at once, without any use of the differentiating sheet; 

 for evidently this curve will pass through all the points of tangency of the curves 5 

 with the curves a = const., including the points of maximum, minimum, or maximum- 

 minimum, at which the curve a = const, is reduced to a point or cuts itself. Vice 

 versa we conclude that when the field of <p is given, and that of a shall be determined 

 by integration, the curves a = const, must have tangency with the curves s at the 

 points where these curves are cut by the curve <p = o. 



