﻿1080 Mr. R. Hargreaves on Atomic System, 



and the process applied to (16) gives 



n-l 



2 

 zn s= i 



2T- 2 cosec sir In 



Vj- 



, , . 2.2.4.4... 1/1 1 1 . \ 



= logn + T-log 1- 3.3 -5 _ i -6- nl (p- |i + 3- 1 ...) 



60n 4 \l 4 2" 3 4 "/ 



(IT' 



= , ogn + 7 _l og __ 7 _ + __ oGs4 



and so 



7tt 3 



n fi i ""J ^ 



43200** 



^-2. n =-log e 2+^-^^.. 



!»-«»= ^{log.n + 7 + log-} +j| 



7T 49tt 3 



(18) 



144* 345600* 3 



§ 13. The method is also applicable to find the sum of 

 cubes, but with a vast difference in the labour entailed. 

 The cube of (15 a) has terms of types a 3 , 3a 2 b, and 6abc. 

 In the two latter it is necessary to separate each term into 

 partial fractions, and apply the approximate summations 



i+i+ 1 -S 1 



1 2-t- 2 2 " 1 ""-(^-l?~ 2 *""' 



A + i + X -S X - 



l3" + "2 3 " 1 ~--\72-l) 3 ~° 3 2?2 2 --' 



From these with (17 a) we have 



"z 1 1 , r + 1 V 1 



2 = Jos , 2, 



(17 6) 



"5 1 i_ _L / i i i 



,=i (rn + sf 2n z I r 2 7+T\ 2 J ' 



(17 c) 



which are sufficient to give a second term of order n~ 2 com- 

 pared with the main term. The separate summations are 

 very numerous, and it may be sufficient to add to this 

 account of the method the separate totals for the three 



