1878.] 



on the Value of Eulers Constant. 



On comparing my results with those of Mr. Shanks, I found that 

 the latter were all affected by an error in the 103rd and 104th places 

 of decimals, in consequence of an error in the 104th place in the deter- 

 81 



mination of log g^. With this exception, the logarithms given by 



Mr. Shanks were found to be correct to 202 places of decimals. 



The error in the determination of log e 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 e 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 e 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 



2 S n 



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

 series, 



Bi B 2 B 3 



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

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



E=S tt + R n -^-log^, 



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

 part of R w 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 62 , and approximately as far as B 100 , retaining a number of 

 significant figures varying from 35 to 20. 



When ^=1000, the employment of the numbers up to B 61 suffices to 

 give the value of R 1000 to 265 places of decimals. When n=500, it is 

 necessary to employ the approximate values up to B 74 , in order to 

 determine R 500 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 Ssoo and S 1000 , I have resolved the 

 reciprocal of every integer up to 1000 into fractions whose denomina- 

 tors are primes or powers of primes. 



Thus Sgoo and S 1000 may be expressed by means of such fractions, 

 and by adding or subtracting one or more integers, each of these 



