ON CALCULATION OF MATHEMATICAL TABLES. 261 



To obtain the series of the first inverse powers of successive numbers as a 

 function of a; (a; — 1) it is necessary to have recourse to logarithms. 



T-T 1 , x+1 1,1,1, 



Now log — -L^ = -+ — + + . . . 



1, x+2 1,1, 



2 ° X x+l S{x+iy' 



1 , a;+3 1,1, 



, log ' = + + 



2 ^ x+l x+2 3(x+2f 



1 , x+r+l 1,1, 



_ log — ! 1 — = + + . . . 



2 ^ x+r-l x+r 3(a;+rf 



and the sum = 1 log i^+^)(^+''+^) = 1 log (x+r) (x+r+ 1) - _^ log a; (x- J ). 



A X \X — X ) ^ ~ 



Now the limiting value ofl + -+ . . . +- = log r+y 



2 r 



where y = -57721 56649 01532 86060 . . . 



= - log a;(a;— 1) + y' + ^-^ + ''tc. as above, 

 where y' becomes y in the limit when r -^ oo . 



Now the limit of ~ log (x+r) (a;+»'+ 1)— log '' is log i, i.e. zero, when r — ^ x . 



and, changing x into x+ 1, 



{'-^iHl-^)+ ■ ■ ■ =-^+^«g*(^+i)+3'2(^+ • • • 



If X is an integer, the left-hand side becomes 1 + ~ + . . . -f - • 



2 X 



We have therefore established this as a fimction of a;(a;+l), 



audl+-+ . . . + is the same function of X (a;— 1). 



2 ar— 1 



This completes the theorem regarding the series of all the odd powers. 



