54] CALCULATION OF THE BERNOULLIAN NUMBERS FROM TO B K . 427 



and if from this series only the prime numbers 2, 3, p, p' ... be selected, 

 then the fractional part of the nth number of Bernoulli will be 



Having found, several years ago, a simple and elementary proof of this 

 theorem, I was induced to apply the theorem to the calculation of several 

 additional numbers of Bernoulli, and I ultimately obtained the values of 

 the thirty-one numbers which are given in the present paper. 



The method which has been employed affords numerous tests, throughout 

 the course of the work, of the correctness with which the requisite operations 

 have been performed, so that I feel entire confidence in the accuracy of 

 the results. 



In making these calculations I have received very efficient aid from 

 my Assistants, Mr Graham and Mr Todd. 



The following is an outline of the method employed : 

 Bernoulli's numbers B lt B. a &c. are defined by the equation 



x 



1.2.3.4 





where n takes all positive integer values from 1 to oo . 



If we multiply by e* 1, and equate to zero the coefficient of rr" +I on 

 the right-hand side of the resulting equation, we shall find 



in which C r n denotes the coefficient of of in the expansion of ( 1 + cc) 2 "" 1 " 1 . 

 This equation gives B n when B u B n ,...B n _ l are known. 



Now let B n = / + (_!) (/-!), 



where ( !)"/ is the fractional part of B n given by Staudt's Theorem, so 

 that / is an integer. 



542 



