5.20 Mr. J. W. L. Glaisher on the [June 15, 



By subtraction we find : 



(B) - ( A) = ... (GO ciphers) . 48590 86852 

 94425 30244 89703 42646 

 54039 73172 (E) 



(D) - (C) = . . . (60 ciphers) . . . 48590 86852 

 94425 30244 89703 42646 

 54039 73171 . . . (E) 



which only differ by a unit in the 100th decimal place. 



Leaving out of consideration the terms involving Bernoulli's number?, 

 the difference between the series when # = 200 and when #=100 is 



(i + i ■ ■ ■ +4- log 200 )- + t * * * +m - log 10 °) 

 i i i 



~ 101 + 102 ' * * + 200 ~ log 2 * 



Similarly, the difference between the series when 1000 and when 

 x = 500 is 



1_ J_ 1 



~501 + 502 ' ' ' + 1000 ~ log 2 * 

 Now as it is extremely improbable that two errors, exactly equal in 

 amount, should have been made in the calculations of 



_L _L _L J_ _L 1 



101 + 102 " " ' +200 501 + 502 * ' ' + 1000' 



w r e have very strong evidence that the value of log 2 is inaccurate, and 

 that (E) is the correction to be applied to it. 

 By subtracting (C) from (A), we obtain 



(C)-(A)= ... (59 ciphers) ... 1 11064 84235 

 30114 97702 62179 26049 23519 

 38678 (F) 



The difference of the corresponding series (omitting as before the terms 

 involving the Bernoulli's numbers) 



~101 + 102 ' ' * + 500 ~ l0§ °' 



Having only this one* difference-result involving log 5, it is impossible 

 to decide from (A), (B), (C), and (D) whether the harmonic series or log 

 5 or both are in error ; but the following reasoning places it beyond all doubt 

 that (F) is a correction to log 5, and that the sum of the harmonic series is 

 correct. 



* By subtracting (B) from (D) wo might get another, but the portion 9^+9^ 

 + ^77) of the harmonic series, as well as log 5, would be common to both.— June 16. 



