10 



REPORT 1877. 



is that with the above assumption the quantities l v I 2 , I 3 , I 4 , I 5 , I s all vanish, so 

 that we have fewer quantities to calculate. 



The numbers C I ,, which are required in order to find the value of (2«+l) I", 



»— i 



can be readily derived from the numbers C r I , which have been already em- 

 ployed in finding the value of the similar quantity (2m — 1) I w _ 1 which immediately 

 precedes it. For since 



p"_ (2m+1)2w it-i _ m(2m+1) c «-i 



r (2n-2r+l)(2n-2r) ' (n— r) (2»-2r+l) * ' 



we have 



i* t _ n(2n+Y) 



C" I — "^■a« I *-) r\ n 1 J . 



r r (n-r)(2n-2r+\) r r> 



and a test of the correctness of the work is supplied by the divisions by (?i — r) and 

 2re— 2r+l being performed without leaving any remainder. 



I have proved that if n be a prime number, other than 2 or 3, then the numerator 

 of the wth number of Bernoulli will be divisible by n. 



This forms another excellent test of the correctness of the work. 



I have also observed that if q be a prime factor of n, which is not likewise a factor 

 of the denominator of B B , then the numerator of B„ will be divisible by q. I have 

 not succeeded, however, in obtaining a general proof of this proposition, though I 

 have no doubt of its truth. 



For the sake of convenience of reference, I include in the following Table the 

 numbers of Bernoulli which have been previously calculated as well as those which 

 have been added by myself. 



Table of Bernoulli's Numbers expressed in Vulgar Fractions. 



