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



In a similar manner the following definite integrals that are 

 required have been obtained : — 



$& 2 d6iZ =2tt(-4/3 2 Y) j*C 2 W 2 =27r(|) 



p 2 C 2 d0 2 =O J\S 2 4 d0 2 =2tt(|) 



JO a W a =2w(i) JC 2 8 2 2 J# 2 =27r(-lftf) 



JS 2 W 2 =27r(i) JS 2 C 2 W 2 =.2ir(-iftY) 



JCW# 2 =*Mtftf) JS 2 2 C 2 2 d0 2 =27T(i) 



jS 8 Wj =2w(-iAY) JS 2 2 W 2 =JS 2 3 C 2 rf0 2 =O. 



■ • • (54) 



To illustrate the process of integration of (42) between 

 the limits 02 = and 27r, consider the {-component. Multi- 

 plying — (X + ftC 2 cos a) by the first two terms of (31) 

 gives only terms to the order of ft 2 as follows : — 



\ - (X + ftC 2 cos «) [1 + 3ft(YS 2 - f C 2 )] rfft 

 Jo 



= 2tt^ -X + ft 2 [fcosa + fX(Y 2 +f)]f. (55) 



Multiplying by the next three terms of (31) gives terms 

 both in ft 2 and in ft 4 as follows : — 



I ^-(X + ftCacosa) [ 6ft 2 (YS 2 - fC 2 ) 2 + 10yS 2 3 (YS 2 -fC 3 ) 3 



Jo 



+ 15A 4 (YS a -fCa) 4 ]^a 

 = 27r]-3ft 2 X(Y 2 + P)+ft 4 [ffcos«(Y 2 + f) 



_ ¥ X(Y 2 + f 2 ) 2 ]^ . (56) 



The next term of (31) gives terms of an order not lower 

 than ft 5 , and it will appear later that we need not even 

 retain terms in ft 4 , but they are now retained and rejected 

 later. Treating the/- and the ^-components in a similar way, 

 the complete force upon the first ring due to the second 

 becomes 



