1871.] Calculation of Eider's Constant. 515 



Mr. Shanks obtains the value of y from this formula by making #=10, 

 20, 50, 100, 200, 500, and 1000, and remarks, as a curious coincidence, 

 that the number of decimal places obtained from x being made equal to 

 10, 20, 50, and 100 is nearly proportional to 3/10, |/20, |/50, and */ 100. 

 On p. 432 he gives a Table of the number of decimal places obtainable 

 from the formula when x has the values 2, 5, 10, 20, 50, 100, 200, 500, 

 and 1000, and from it draws the inference that "we may fairly infer that 

 when n is increased in a geometrical ratio, the corresponding number of 

 decimals obtained in the value of E increases only in something like an 

 arithmetical one, and that probably from 50,000 to 100,000 terms in the 

 Harmonic Progression would require to be summed in order to obtain 100 

 places of decimals in the value of E, Euler's constant." 



Algebraically, of course, y is independent of x in the formula (i), but 

 arithmetically, since the series ultimately becomes divergent, the value of y 

 is so far dependent on x that for a given value of x the series will only 

 afford a certain number of decimal places. The number of decimal places 

 directly obtainable is equal to the number of ciphers which precede the first 

 significant figure in the value of the numerically least term of the series. 



The wth term (considering only the terms after ^ in (i), so that the 



Mth term is the term involving B„) 



_ B w 



2nx 2 "' 



which, since 



B _ 2(1.2.3...2n) / 11 \ 



(2tt) 2m V 3 / 



is very nearly equal to 



2{1.2.3. . . (2n-l)} 

 (2-nx) 2 * 5 



so that the ratio of the nth to the (n — l)th term 



_ (2n—2)(2n—l) _ wV, 3_\ 

 4ttV n 2 x\ 2n) 



very nearly. 



Let 77i be the greatest integer contained in xtt so that #7r=m4-/, / 

 being a proper fraction, then the ratio of the mth to the (m— l)th term 



0+/A W 



which is always less than unity. 



The ratio of the (w-f-l)th to the with term is found in a similar manner 

 to be 



roU / 



2 s 2 



