55. 



ON SOME PROPERTIES OF BERNOULLI'S NUMBERS. 



[IN 1872 a paper on this subject was communicated to the Cambridge 

 Philosophical Society. The paper contained a comparatively simple proof of 

 the theorem given above as Staudt's theorem, which was there attributed 

 to Clausen : another property of Bernoulli's numbers was also established, 

 viz.: "That if n be a prime number other than 2 or 3, then the numerator 

 of the nth number of Bernoulli will be divisible by ."] 



ON THE CALCULATION OF BERNOULLI'S NUMBERS. 



[A table of the values of the first sixty-two numbers of Bernoulli, as 

 given above, was printed in Vol. 85 of Civile' s Journal. A paper on 

 this subject was also published in the Report of the British Association 

 in 1877, of which the greater part is contained in the above paper, and 

 the remainder is given below.] 



Thirty-one of the numbers of Bernoulli are at present known to Mathe- 

 maticians, and are to be found in a communication by Ohm in Crelles 

 Journal, Vol. xx. p. 11. Of these numbers the first fifteen are given in 

 Euler's Institutions Calculi Differentially, Part 2, Chap. 5, and Ohm states 

 that the sixteen following numbers were calculated and communicated to him 

 by Professor Rothe of Erlangen. I find, however, that the first two of 

 these had been already given by Euler in a memoir contained in the 

 Acta Petropolitana for 1781. 



It may be sometimes useful to have the values of Bernoulli's numbers 

 expressed in integers and repeating decimals. 



