based upon Electromagnetic Theory. 575 



uhere only the first line of: (RA)~ 3 in (29) need be used, 

 namely, 



(RA 2 )~ 3 = [l + 3/3^ f 6A»gj" + 10/5 2 3 Q 3 . . . .] r" 3 , (31) 



in which i>/r = S 2 Y-C 2 £ (32) 



and £ = Xcos«-f Z sin a (33) 



These equations, (30) and (31), express the instantaneous 

 force upon the first point charge due to the second with 

 completeness so far as the inverse square terms at great 

 distances are concerned, and to all orders of [S\ and /3 2 . 

 The next operation is to integrate the force fur two closed 

 rings. 



II. 



The Force upon a First Closed Ring due to a Second Closed 

 Ring at great Distances, each Ring being in a Fixed 

 Position. 



A. The force upon the whole of the first ring due to an 

 element of length only of the second ring. 



Let dFt l and dFi 2 denote the charges per element of length 

 of the rings respectively, and pi and p 2 the charges per unit 

 of length of the circumference. The whole charge of each 

 ring is then 



E 1 = 2to 1/5i , (34) 



and E 2 = 27ra 2 p 2 - ' (35) 



The angles measured from the centres of the rings to the 

 elements of charge respectively are by (7) 



(p l = co 1 t + 6 1 , (36) 



2 = G) 2 (*-R,/c)+0 2 (37) 



In summing the forces around each ring the time is to be 

 regarded as constant and may as well be assumed to be zero. 

 The whole force of the second ring upon the first is the same 

 at all times, that is, does not vary with time. For £ = 0, 

 we have 



<j) l = d l and 00 = 0,-^1?. . . . (38) 



c 



