12 Mr. A. Schuster's Electrical Notes. 



lines of force due to an indefinitely small magnet of finite 

 magnetic moment. 



Substituting the above values of u, v, w into (3), it will be 

 shown further on (13) that 



Sr r 



q, __ ^xy 



w 



r 



27771', 



...... (9) 



J 



But these values of F ; , G', W neither satisfy equations (1), 

 nor do they satisfy the condition J = for 



dF dW_ dH! = _ 8 wa m 

 dx dy dz 3 r 3 



It seems well at this stage to investigate the conditions 

 which have to be satisfied in order that J = 0. 



By a well-known transformation we obtain from equa- 

 tion (3), 



dW d& , dW CCCl/du , dv , dw\. . , 



dx dy dz JJJr\dx dy 



lu + mv + nw 



-K 



rfS. 



The triple integral vanishes, and the surface integral has to 

 be taken over the boundary of the region which is considered. 

 If this region includes all space which contains electric cur- 

 rents, the second integral will also vanish provided the space 

 includes no singular points. In the example which has been 



x 

 given, however, (f> — -3, which makes the components infinitely 



large at the origin. If this point is excluded by an infinitely 

 small sphere surrounding it, the surface integral in the above 

 equation has to be taken over this infinitely small sphere and 

 will be finite. It is owing to this fact that the condition 



dW dG' dW . 

 dx dy dz 



is not satisfied in the special case treated above. 



This may perhaps be more clearly seen in another way. 

 Maxwell's equations fall to the ground unless the currents 

 flow everywhere in closed lines. The example we have chosen 



