﻿1078 Mr. R. Hargreaves on Atomic Systems 



is written as 



x 2 (l-cos0 r + Z){2x(l-cos6 r ) + % 2 }-y 2 , 

 we find 



yg) = ^U2 1 + ^.,v^ 3 — 3 ||^- 1 '%- 1| 3 2 6 ... . (12 6) 



For a vacant centre 



N =/g) -K=/C*)+/(i)-«w-{^)-W. 



in which the bracket vanishes as the condition determining *r. 

 Hence 



•1/2 _u -y-3/2 



32 v 

 or 



¥ ( ' ^ 



N = 6' 2n -2^-f- g2 ^ 6 g) (S3 + Si) (13 a) 



on using Si = 2(c2»— c n ). 



The equation of condition /(#) = ic tt by use of £ in (12 a) 

 becomes 



= i ( C2 ^2cg-|(l-?/)(2 3 + 2 1 ) + -^.... . (14a) 

 If this is multiplied by f/2 and added to (13 a), we have 



N= (, 2 „-2 c „)(l + |)-^ + ^| 5 .... (18.JJ 

 Also, if (4 a) is multiplied by 1 + 3f/4, and % d % 3 neglected, 



ff (Ss + ^n-C'an) = 4:{c 2n -2c n ) + 3? %/8 . . . , (14 5) 



where on the right hand only the main term in the value of f 

 is to be applied. 



The values of f or x— 1 and N are thus dependent on the 

 sums of three series, which now require consideration *. 



* A slightly different treatment may be based on Taylor's theorem 

 with the values 



f(l)=Uc2n-Cn), 



-/'(l)=l(S3+2xj = i [2 w 3 T,+3nT 2 (logn+y+log| - ^)] , 

 r(l) = -i-/'(l), and /"'(l)-18T 5 « 5 /#. 



