CHAPTER VII. 



ISOGONAL CURVES. 



140. Isogonal Curves. The drawing of vector-lines from the observed direc- 

 tions of a vector is an operation of the nature of an integration (section 127). On 

 account of the incompleteness of the observations this integration is combined with 

 interpolations. But it will be possible to separate from each other these two hetero- 

 geneous operations of interpolation and of integration. This is obtained by the 

 method of isogonal curves devised by Mr. Sandstrom.* 



We have agreed to represent observed directions by numbers (section 98). In- 

 stead of inscribing the arrows we can inscribe these numbers on a chart. Then we 

 can draw curves joining the points where these numbers are equal. In all points 

 of such a curve the vector will have the same direction, i. e., form the same angle 

 with the north-south line. These curves may therefore be called isogonal curves 

 or isogons. 



A chart containing these curves may be considered a completely interpolated 

 representation of the differential equation determining the vector-curves. This 

 representation being obtained, the integration will cause no difficulty. Across each 

 isogonal curve we can draw short lines of the direction represented by the curve. 

 These will be line-elements of the vector-lines. In this manner we can get the whole 

 plane filled with such line-elements, and joining them to continuous curves we get 

 the vector-lines. 



141. Singular Points in the Field of a Multiple-Valued Scalar. The isogonal 

 curves represent the field of a multiple- valued scalar, the angle. The angle has no 

 true greatest and no true smallest value. From the highest number, 64, used in our 

 representation, we interpolate to the lowest, 1 ; f or 1 represents the same angle as 

 65 would do. 



In order to see the consequences which this peculiarity of the scalar has on 

 the appearance of the field, let us suppose observations to have been taken at the 

 points of a closed curve and to have given in succession the numbers from 1 to 64 ; 

 in this case isogonal curves representing all angles must run in through the closed 

 curve, in order to cut each other somewhere in the area contained within it. The 

 point of intersection will be a singular point. 



In the diagrams of figs. 56 and 57 the isogonal curves passing through the singu- 

 lar point are for the sake of simplicity drawn as straight radii. The numbers belong- 

 ing to these radii may be arranged in two different ways : they can increase in the 



*J. W. Sandstrom: Ueber die Bewegung der Fliissigkeiten. Annalen der Hydrographie und der maritimen 

 Meteorologie. Berlin, 1909. 



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