460 NOTE ON THE VALUE OF EULER'S CONSTANT, ETC. [56 



The error in the determination of log, 10, of course entirely vitiated 

 Mr Shanks' value of the modulus from the 103rd place onwards. As he 

 gives the complete remainder, however, after the division by his value of 

 log, 10, I was enabled readily to find the correction to be applied to the 

 erroneous value of the modulus. Afterwards I tested the accuracy of the 

 entire work by multiplying the corrected modulus by my value of log, 10. 



Mr Shanks' values of the sum of the reciprocals of the first 500 and 

 of the first 1000 integers, as well as his value of Euler's constant, were 

 found to be incorrect from the 102nd place onwards. 



Let S n , or S simply, when we are concerned with a given value of n, 

 denote the sum of the harmonic series, 



1 1 1 



1+ l + o + ...... +- 



23 n 



Also let R n , or R simply, denote the value of the semi-convergent 

 series, 



*' A, ^_ 



2ri> 4n< 6n" 

 where B v B 2 , B 3 , &c., are the successive Bernoulli's numbers. 



Then if Euler's constant be denoted by E, we shall have 



and the error committed by stopping at any term in the convergent part 

 of R n will be less than the value of the next term of the series. 



I have calculated accurately the values of the Bernoulli's numbers as 

 far as B^, and approximately as far as B IM , retaining a number of significant 

 figures varying from 35 to 20. 



When 'rt = 1000, the employment of the numbers up to B 6l suffices to 

 give the value of R im to 265 places of decimals. When n = 500, it is 

 necessary to employ the approximate values up to B 7i , in order to determine 

 R m with an equal degree of exactness. 



In order to reduce as much as possible the number of quantities which 

 must be added together to find S m and $ 1000 , I have resolved the reciprocal 

 of every integer up to 1000 into fractions whose denominators are primes 

 or powers of primes. 



Thus SM, and S^ may be expressed by means of such fractions, and 

 by adding or subtracting one or more integers, each of these fractions may 

 be reduced to a positive proper fraction, the value of which in decimals 



