an Expression for the Numbers of Bernoulli. 361 



The same equation in differences (2.) shows easily that 

 u . , =0, when n = or > m + 1, 

 if y = 0, when ra — 1 > w* ; 



«* m, n — 1 



but, by a well-known theorem, which in the present notation 

 becomes 



Ki$T (4,,) 



it is easy to prove, not only that 

 but also that 



^i,« = ' if » 2 > J ; ( 6 



we have therefore, generally, for all whole values of?«> 1, 

 and for all real values of n, 



w =0, unless n <-m . . . . (7.) 

 2. If then we make 



T m = ^^jw,m-2/t("" » .... (8.) 

 the sign 2 indicating a summation which may be extended 

 from as large a negative to as large a positive whole value of 

 Jc as we think fit, but which extends at least from Jc = to 

 Jc = m t m being here a positive whole number ; this sum will 

 in general (namely when m > 1) include only m — 1 terms 

 different from 0, namely those which correspond to Jc = 1, 2, 

 .... m — 1 ; but in the particular case m = 1, the sum will 

 have two such terms, instead of none, namely those answer- 

 ing to Jc = and Jc = 1, so that we shall have 



Multiplying the first member of the equation in differences 



(3.) by (— t) , and summing with respect to Jc, we obtain 

 m T , ,, m being here any whole number > 0. Multiplying 



and summing in like manner the second member of the same 

 equation (3.), the term my m m j t -2-2k°^ tnat memDer gives 

 — m t T , because we may change Jc to Jc + 1 before effect- 

 ing the indefinite summation ; Jc y m m _ 2 k gives t -j- T^ ; 



and(l-Jc)y mm + 2 _ 2li givest 2 TF T m ; but 



-^T ffl + H^T M= (l + f^(l + /rT w ; 



