576 Dr. A. C. Crehore on an Atomic Model 



The elements of charge in the rings are respectively 



dE 1 = a 1 p 1 d0 1 and dE 2 = a 2 p,d0 2 . . . (39) 



According to (38) the differential of cf> 2 with respect to the 

 phase angle 6 2 is more complex than the differential of $x 

 with respect to 6 X because c/> 2 involves R, which is a function 

 of 6 2 . For purposes of integration the elements of charge 

 are preferably expressed in terms of <f> instead of 0, as in 

 (39), since the sines and cosines in the force (30) are 

 in terms of </> and not 0. Differentiating (38) with respect 

 to gives 



dfa = d0 1 and dfa=(l-^^)d0 2 . . (40) 

 Hence the elements of charge become in terms of (p 



dE 1 = a i p 1 dfa and d'E 2 = a 2 p 2 \l -yzr) dfa- (41) 



\ C City 2 J 



Examining (30) with respect to fa it is found that it docs 

 not occur in the factor (RA 2 )~ 3 in (31), and that it only 

 occurs in the first power of Si and Ci within the brace of 

 (30). Since we have to integrate for U that is for fa, 

 between the limits of and 2ir to go once around the first 

 ring, all the odd powers of Sx and C\ give zero upon inte- 

 grating. Since there are no other terms in (30) except 

 constant terms independent of X , and each of these gives a 

 factor 2-rr upon integrating, the whole force upon the first 

 closed ring due to an element of the second ring is easily 

 found to be 



e^e 2 (.i-/3 2 2 ) r ; 



+ [_Y+A8 a ]j 



+ [-Z-AO f sin«l* { . (42) 



B. The force upon the whole of the first ring due to 

 the whole of the second ring. 



The process of integration around the second ring is to 

 substitute for dE 2 in (42) its value in (41), and for (RA 2 )~ a 

 its value in (31), and multiply the series of (31) by the 

 binomials in the brackets of (42), and integrate each term 

 separately. The terms thus found involve 2 only through 



