518 



Mr. J. W. L. Glaisher on the 



[June 15, 



duce the same error in 1 + 



+ 



200' 



1+: 



+ 



500' 



and 1+i 

 1 



calculation of the reciprocals beyond -— to 



+ Tooo' su PP osin s the 



have been accurately performed. 



It was therefore evident that the only means of ensuring freedom from 

 error in the harmonic series was to recalculate it. The lowest value of so 

 which would suffice for a verification was 1 00 ; the author therefore calcu- 

 lated the value of the series 1+i . . . to 100 places of decimals, 

 and the result was found to agree to that extent with that given by Mr. 

 Shanks in the paper previously referred to; the value of 1 + i . . . +-!- 



was also found to be correct. 



It may be mentioned that the calculation was abbreviated by a simple 

 artifice, suggested by Oettinger (Crelle's Journal, t. lx. p. 376), which will 

 be easily understood from an example. Suppose the sums of the recipro- 

 cals of the odd and even numbers up to 50 are known, and it is required 

 to find the sum of the reciprocals up to 100. 



Let 



1 



49' 



, , 1,1 



3 o 



then 



/ 3 = i+i+ 1 ...±; 



H 2 4 6 50' 



^^i + i+i +JL + JL 

 2 2 4 6* 93 100 



BO 



that 



1+i+i 

 2 3 



+1+-L 



99 100 



o+/3, ,1,1 ,1 

 = - — L+ a + — + — . , . + — , 



2 51 53 99 



and the calculation of the reciprocals of the even numbers between 50 and 

 1 00 is rendered unnecessary. 



The values of log 2 and log 5, which were required to form log 100, 

 log 200, log 500, and log 1000 were taken from Mr. Shanks's 'Rectifi- 

 cation of the Circle 5 (they are also given in the Proc. Hoy. Soc. vol. vi. 

 p. 397), and the summations of the harmonic series from Mr. Shanks's 

 paper in vol. xv. p. 431 of the Proc. Roy. Soc. The results were as 

 follows : — 



From x = 100:— 



y = -57721 56649 01532 86060 65120 



90082 40243 10421 59335 93992 



35988 05770 42799 90853 75858 



22362 13134 84454 84292 91195 . . . (A). 



