66 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



minima. Within these regions the lines of flow will have points of inflexion (fig. 58) . 

 As in the fields of the single-valued scalar, there may appear complexes of such 

 maxima and minima , containing between them a maximum-minimum point where 

 a certain singular isogonal curve cuts itself (fig. 46). 



It is remarkable that no special singularity of the isogonal curves corresponds 

 to lines of convergence or of divergence in the field of motion. Fig. 59 shows a case 

 where such lines appear in the case of rectilinear and parallel isogonal curves, fig. 60 

 a case where they appear in the case of circular concentric isogonal curves. The 

 feature of the isogonal curves in the case of the wave-motions described in section 

 131 is remarkably simple. Let the numbers on the rectilinear and parallel isogonal 

 curves oscillate between two extreme values for instance, 52 and 12. If the isogonal 

 curves run parallel to the average wind-direction, we get the parallel and equidistant 

 lines of convergence and divergence of fig. 61 a. As the angle between the average 



eo i t \a 8 1 eosesi ss w i 



$6 60 i 8 12 t i 60S6S1S6600 4 I 



Fig. 61. Isogonal curves for combined wave-motion and motion of translation. 



A. Isogonal curves parallel to the main wind-direction. 



B. Isogonal curves oblique to the main wind-direction. 



C. Isogonal curves normal to the main wind-direction. 



wind-direction and the isogonal curves increases, the singular lines are displaced 

 relatively to each other, until finally two and two join into one, as in fig. 61 b. For 

 still smaller angles we get sinusoidal lines of flow, the case of symmetry (fig. 61 c) 

 arising when the isogonal curves are normal to the main wind-direction. 



143. Sandstrom's Integration-Machines. Mr. Sandstrom has based a method 

 for graphical integration of differential-equations upon the representation of these 

 equations by isogonal curves.* These curves being drawn, the tracing of the curves 

 representing the integral, i. e., the vector-curves, will cause no difficulty. Still, 

 the draftsman will find it time-wasting to measure out the precise angles which 

 these curves will have as they pass the different isogonal curves. But the work of 

 drawing the vector-lines is very much facilitated by special machines constructed by 

 Mr. Sandstrom, which trace automatically line-elements of the required direction 

 across the isogonal curves. The construction of these machines will depend upon 

 the system of coordinates to which the angles are referred. If the angles are referred 

 to the meridians of a chart drawn in conical projection, very simple devices may be 

 used. Fig. 62 shows a simple instrument serving the purpose in this case. A rule 



*See note, p. 63. 



