based upon Electromagnetic Theory. 579 



+ /3/[|?cos«;Y 2 + f 2 ) 



-X(Y 2 +f) 2 ]]; 



8 



+ [-Y+/9 2 2 [Y-gY(Y 2 + f 2 )] 

 + /3 2 *[§Y(Y 2 + f) 



-JjfYCYs+f 2 )' 2 ] 



+ [- Z 4 /3 2 2 [f sin «-a Z (Y 2 + f 2 )] 

 + /3 2 4 [ffsina(Y 2 + fj 



• • • (67) 

 This force is constant independent of time, but depends 

 upon the relative positions of the orbits. The direction of 

 the force is not necessarily along the line joining the centres 

 of the orbits. To find the component which does act along 

 the line of centres, multiply the coefficient of i by X, of j by 

 Y, and of k by Z, and add the three products. The resuming 

 force is 



. . . (59) 



It may be shown that the quantity (Y 2 +f 2 ), when multi- 

 plied by r 2 } is equal to the square of the perpendicular 

 distance from the centre of the first ring* to the axis of 

 rotation of the second ring. In fact, this quantity is the 

 equivalent of 



1-C-Xsina-f Zcosa) 2 , .... (GO) 



which is the form given it in equation (54) of an earlier* 

 paper. 



III. 



The Average Force upon the First Ring due to the 

 Second Ring for all Orientations. 



The only quantity in (59) that varies as the axes are 

 turned relatively to each other, without changing the dis- 

 tance between centres, is (Y 2 + £ 2 ). When the orbits are 



* Phys. Rev. Juno L917, p. 456. 



