56. 



NOTE ON THE VALUE OF EULER'S CONSTANT; LIKEWISE ON THE 

 VALUES OF THE NAPIERIAN LOGARITHMS OF 2, 3, 5, 7 AND 10, AND 

 OF THE MODULUS OF COMMON LOGARITHMS, ALL CARRIED TO 260 

 PLACES OF DECIMALS. 



[From the Proceedings of the Royal Society, Vol. XXVII. (1878).] 



IN the Proceedings of the Royal Society, Vol. xix., pp. 521, 522, 

 Mr Glaisher has given the values of the logarithms of 2, 3, 5, and 10, 

 and of Euler's constant to 100 places of decimals, in correction of some 

 previous results given by Mr Shanks. 



In Vol. xx., pp. 28 and 31, Mr Shanks gives the results of his re- 

 calculation of the above-mentioned logarithms and of the modulus of common 

 logarithms to 205 places, and of Euler's constant to 110 places of decimals. 



Having calculated the value of 31 Bernoulli's numbers, in addition to 

 the 31 previously known, I was induced to carry the approximation to 

 Euler's constant to a much greater extent than had been before practicable. 

 For this purpose I likewise re-calculated the values of the above-mentioned 

 logarithms, and found the sum of .the reciprocals of the first 500 and of 

 the first 1000 integers, all to upwards of 260 places of decimals. I also 

 found two independent relations between the logarithms just mentioned and 

 the logarithm of 7, which furnished a test of the accuracy of the work. 



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 determination 



81 



of log . With this exception, the logarithms given by Mr Shanks were 

 80 



found to be correct to 202 places of decimals. 



582 



