148 Mr. E. A. Biedermann on the 



where 



11 = J l) n^? dx dy dz Jj) nW dx dy dZ: 



and 



i 2 



=Iw"'" 



These integrals are easily evaluated by transforming to 

 polar co-ordinates with origin at e^ and integrating first 

 with respect to <£, then with respect to r 2 treated as an 

 independent variable by means of the relation 



r 2 2 = r x 2 + r 2 — 2r l r cos 



(r 2 being of course regarded as positive for all points), and 

 finally with respect to r x . The results are 





From (1), (2), (4), and (5) we obtain for the total kinetic 

 energy of the system 



Cl r 



+ itt — | {u r u s + v r v s + 2w r ivj 



+ K*W + v, -K to .) ( 6i!: ^ 2 )} > ( 6 ) 



the double summation sign denoting that the charges are to 



be taken in every possible way in pairs. If the charges 



are spherical uniform volume charges instead of surface 



e 2 

 charges, it is easily shown that the coefficient of — becomes §, 



r 



and that of (— — ^- J becomes J instead of I. 



Now let these results be applied to the case of two circuits 

 carrying currents. 



