116 Prof. G. N. Watson on the 



On collecting our results, we now see that 



7rB n =2nlog e (27i) + 2n{7- log ir} . 



, a $ (-)™B m (l-2^)* 2m (-)^iB r+1 (l-2- 1 -^)<7 2y+2 

 + ™=i m.^" 1 ~ + ~^ (r + l).n*"+i 



This formula obviously has the property (which is sufficient 

 to give an asymptotic expansion) that the remainder after 

 r terms of the series on the right has the same sign and is 

 numerically less than the (r-f-l)thterm, and this (r+l)th 

 term tends to zero as n tends to infinity, r remaining fixed. 



By using the known formula o- 2m = 2 2m - 1 7 r 2m B m /(2m) ! 

 we may, if we wish, write 



(-)-B 2 ,7r 2 "(2^-2) for (-y-2B m (l-2'-^)a 2m 

 m . (2 m) I m\ 



in the series on the right. 



4. For purposes of tabulation we make use of the values 

 given for cr 2m in ChrystaFs ' Algebra/ ii. p. 367. 



The first few terms of the expansion then give the 

 approximate result 



S n =2tt[0-7329355992 x log 10 (2n) -0-1806453871] -n" 1 x 0*087266 

 + rc- 3 x0-01035-n- 5 x0-004 + ?i- 7 x 0*005. 



In the construction of the following table the term in 

 n~ 5 could be neglected throughout, and the term in n~ z 

 could be neglected when n>20. 



It is worth observing that, in the special case in which 

 the charge of the nucleus is numerically equal to the total 

 charge of the electrons composing the ring, the expression 

 for (o 2 given in § 1 yields a real value for co only when 

 n-JS n :>0, i. e. wheu' / n<472. 



5. The analogous series 



Tn= 2 



1 /mir 



cosec 



" 



n 2 J 



may be investigated in a manner similar to that in which 

 S„ was considered. This series occurs in the radial com- 

 ponent of force on an electron rotating round a nucleus in 

 company with n — 1 other electrons, when that particular 

 electron is displaced along the circle an angular distance j3 

 from its position in steady motion; it is supposed that 

 -2ir/n<l3<2>Tr/n. 



