EXAMPLES OF SOLENOIDAL FIELDS. 



41 



(c) 



dx _ dy _ dz 

 ax b az 



Integrating each of the three equations contained in this system, we get 



v ' x = e b xz = c 2 z = e b 



The surfaces for equal scalar values A of the vector are given by the equation 



(e) A 3 = Al+Al+Al = 6 2 +a 2 (.r'+3') 



representing for every constant value of A a circular cylinder around the axis of y. 

 The second equation (d) shows that the lines of flow in space project them- 

 selves as equilateral hyperbolae on the plane of xz. As the cylindrical surfaces of 

 equal intensity cut this same plane along concentric circles, we get in this plane a 

 figure precisely similar to that of fig. 37. The third equation (d) shows that the 

 lines of flow in space project themselves on the yz-plane as exponential curves 

 converging asymptotically toward positive y. The first equation (d) shows in the 

 same manner that the lines of flow in space project themselves on the xy-plane as 

 exponential curves diverging out asymptotically from negative y (fig. 44). 



Fig. 44. Lines of flow in the *}i-plane diverging from, and in the yz-plane 

 converging to the axis of y, which is a singular line of flow. 



As the planes of xy and yz are themselves surfaces of flow, fig. 44 represents 

 directly the lines of flow contained in these planes. The axis of y is itself a singular 

 line of flow, and toward this line an infinity of lines of flow converge in asymp- 

 totically in the vertical plane and diverge out asymptotically in the horizontal 

 plane. 



In order to get a more complete view of the field, we can use the different repre- 

 sentations by charts in horizontal projection. 



(A) Fig. 45 a gives the topographical representation of two surfaces of flow 

 which cut the xz-plane along two equilateral hyperbolae. The course in space of 



