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



This mode of finding S m and S 1<m is attended with the advantage that 

 if an error were made in the calculation of the former of these quantities, 

 it would not affect the latter. 



The logarithms required have been found in the following manner : 



10 25 81 50 , , , 126 



Let log T = o, ^g- = b, log- = c, log- = d, and log =e. 



Then we have 



\la-3b + 5c, log 5 = 



Also Iog7 = |(39a-106+17o-d); 



LJ 



or again, log 7 = 19a-4& + 8c + e, 



and we have the equation of condition 



a-2b + c = d + 2e, 



which supplies a sufficient test of the accuracy of the calculations by which 

 a, b, c, d, and e have been found. 



Since log =-log(l- 



9 5 10 



25 



50 

 log -= - 



126 



If we have settled beforehand on the number of decimal places which we 

 wish to retain, and have already formed the decimal values of the reciprocals 

 of the successive integers to the extent required, then the formation of 

 the values of a, b, c, d, and e, will only involve operations which, though 

 numerous, are of extreme simplicity. 



In this way have been found the following results : 



