ISOGONAL CURVES. 



65 



The examination of the figures leads to the following results : 



(1) The positive singular point of isogons corresponds to a point of divergence or 



convergence, the negative singular point to a neutral point of the vector-field. 



(2) The rotation of the system of isogons of a positive point has as a conse- 



quence that the vector-lines take the form of spiral curves of all types, 

 including the limiting cases of straight radial lines and of circles. 



(3) The rotation of the system of isogons of a negative point has as a conse- 



quence a rotation of the system of hyperbolic vector-lines without 

 any change in their form; the angle of rotation of the vector-lines 

 is half as great as that of the isogonal curves. 



When the isogonal curves are no longer straight radii with constant angular 

 intervals, but curves with more irregular intervals, the vector-lines of the corre- 

 sponding vector-field will no longer be true logarithmic spirals or true hyperbolae; 

 but otherwise the character of the field will remain unchanged. If the numbers 

 1 to 64 are repeated twice or a greater number of times on a contour surrounding 

 the singular point, always increasing in the same direction, the singular point will 

 be of higher order. Only the negative singular points will be physically possible ; 

 but even they will occur rarely and be of small practical interest. (Cf. fig. 38.) 



142. Further Remarks on the Field of Isogonal Curves and their Relation 

 to the Vector-Field. When the isogonal curves are to be drawn, the first thing 

 will be to discover the situation of the singular points. For this we have to examine 

 whether closed contours can be found on which the numbers representing the 

 observations always increase in the same direction. If this be the case we are 

 sure that there must be a singular point within the contour. As these singular points 

 will always coincide with the singular points of the vector-field, we can also find these 

 points by the use of rules which we have developed in the preceding chapters. 



Fig. 58. Closed isogonal curves. 

 Inflexions of vector-lines. 



Fig. 59. Parallel isogonal 



Fig. 60. Concentric circles 

 as isogonal curves. 



The situation of the singular points being found, in which the curves intersect 

 each other, the drawing of the curves will involve no other difficulties than those 

 connected with the drawing of the equiscalar curves of the single-valued scalars; 

 for isogonal curves representing different angles can never intersect each other in 

 other points. Besides curves issuing from or entering into the singular points there 

 will be found closed curves surrounding places of what may be called maxima or 



