8 REPORT 1877. 



to make it appear that he had put a wrong interpretation upon his own results. 

 I found, however, that the time at my disposal would not enable me to make 

 myself sufficiently master of the whole subject to treat it in the way that I wished, 

 and I have therefore been obliged to content myself with merely making this 

 allusion to it, as an illustration of the more general considerations to which I have 

 ventured to ask your attention. 



Mathematics. 



On the Calculation of Bernoulli's Numbers up to B r , t by means of Staudt's 

 Theorem. By Professor J. C. Adams, M.A., F.11.S. 



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

 and are to be found in a communication by Ohm in Crelle's Journal, vol. xx. p. 11. 

 Of these numbers the first fifteen are given in Eider's ' Institutiones Calculi Differ- 

 entialis,' 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, how- 

 ever, that the first two of these had been already given by Eider in a memoir con- 

 tained in the 'Acta Petropolitana ' for 1781. 



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 these numbers 

 by reducing all the requisite calculations to operations with integers only. 



The theorem may be thus stated : — 



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

 these divisors so as to form the series 2, 3, rt-f-1, «'-f-i .... 2w-|-l, and if from this 

 series only the prime numbers 2, 3, p, p' .... be selected, then the fractional part 

 of the «tli number of Bernoulli will be 



<-»-G+H + >-)- 



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 

 peiformed, so that I feel entire confidence in the accuracy of the results. 



I propose to publish some of the principal steps of these calculations in an Ap- 

 pendix to the twenty-second volume of the ' Cambridge Observations,' which is now 

 in the press. In making them I have received very efficient aid from my Assis- 

 tants, Mr. Graham and Mr. Todd. 



The following is an outline of the method employed : — 



Bernoidli's numbers B u B a , &c. are defined by the equation 



T -,=l-<5.t'+r^- ~i 9 3 4? +«c.-r-(-I) —X -KVc. 



If we multiply by e* — 1, and equate to zero the coefficient of ar" +I on the right- 

 hand side of the resulting equation, we shall find 



(-1)" o; p., +(-i)" _1 c;;_ 1 b b _ x +&c.+(-i) c 1 "b i +»-i=o, 



in which C denotes the coefficient of x in the expansion of (1+*) 



