54. 



ON THE CALCULATION OF THE BERXOULLIAN NUMBERS FROM 5 82 TO B K 



[From Appendix I. to the Cambridge Observations, Vol. xxii.] 



IN the year 1877 I communicated to the meeting of the British 

 Association at Plymouth the values of 31 of Bernoulli's numbers which I 

 had obtained in addition to the 31 of those numbers already known, and 

 I stated that it was my intention to publish some of the steps of the 

 calculation in an Appendix to the Cambridge Observations. 



The following Tables accordingly contain some of the principal steps of 



the calculations, together with more detailed specimens of the work in the 



cases of the 32nd and the 62nd Bernoulli's numbers, the first and last of 

 those which I have calculated. 



In order to render the Tables intelligible, the substance of my com- 

 munication to the British Association is here reproduced. 



A remarkable theorem, due to Staudt, gives at once the fractional part 

 of any one of Bernoulli's numbers, and thus greatly facilitates the finding 

 of those numbers by reducing all the requisite calculations to operations 

 with integers only. 



The theorem may be thus stated : 



If 1, 2, a, a'...2n be all the divisors of 2n, and if unity be added to 

 each of these divisors so as to form the series 2, 3, a+l, a'+l...2n+l, 



