902 Dr. A. C. Crehore on 



of coaxial circles the angle of latitude, X, which occurs in the 

 formula becomes tt/2, and cos A = A = 0. With this simplifi- 

 cation most of the terms in this equation drop out, leaving 

 in the notation there employed the r-component of the force 

 of one element of one ring upon one element of the other: 



F, = -^J^{A 0)0+ A 2 , 3r^ + A 4 , ^rw 



+ A 6 , S 3 r 3 A 6 + A 8 , ST 4 A 8 ...}, (11) 

 where the numerical coefficients follow the scheme 



A ,o=l ; A 2) o = 3A ,o^3 ; A 4 ,o=|A 2 ,o = 15/2 ; 

 i 6 ,o = |A 4> o = 35/2 ; A 8)0 = |A 6 , =315/8 ; etc. ; 



and the symbols are abbreviations for the following. The 

 distance between the centres of the circles is r when measured 

 in centimetres, and is v when measured in a small unit a^ 

 so that r — a^v. The radii of the orbits are a Y and a 2 , 

 or am and aji, so that ai = a # m and a 2 — an. 8 = mn, and 

 A = (v 2 + ra 2 + n 2 ) _1 /' 2 . r = cos7, where 7 is the phase 

 difference between the two elements of charge, k is the 

 dielectric constant. 



The first process is to integrate for the phase angle, 7, 

 around one circle between the limits of and 27r, which 

 ma} r be accomplished by replacing all odd powers of the 

 cosine by zero and the even powers by cos 2 7= 1/2, 

 cos 4 7 — 3/8, &c, as in averaging. These substitutions 

 enable us to derive from (11) the total force between coaxial 

 circles, 



F '= -^{l4 5 «WA* + ^«VA. + ...}. (12) 



The expansions of A 3 r, &c. in series give the following, 

 where M = m 2 + n 2 : 



A*v = v- 2 -*Mv-± + ^-M 2 v-*-p±Wv- s 

 2 o lb 



15 15 105 



—i~7n 2 tt 2 A 7 v = H- ~rin 2 n 2 v~ 6 ?r- M.m 2 n 2 v~ 8 



4 4 8 



+ ^M 2 m 2 n 2 z;- 10 ... 



^ m*n*A n v = +.~? m Vr 10 . . . ; 



64 64 



so that, when ?;, a u and a 2 are restored, the whole electro- 

 static force upon one ring due to a second coaxial with it 



