TUANSACTIONS OF THE SECTION'S. J/ 



■\vlicuce it follows tlial 



auJ 



r 



""' e"'"'~°^'dv = \ . f r (l\l]4-n»rf->l\Y\ 



V e 



''■^rft>=i.|rg)u+a"r(-^)v|, .... (4) 



the foniuila iu question. 



When n=2i+l, it -will be found that this gives, after use of the formula 



rO»i) T(l—tn)= -t-^^ — and reduction, as the value of the inteor.al 

 smmn 



■(!){ 



(-)'2"a'' 



(1. d. ..n)- 



which, b}' means'of the formulaj in the Number of the ' Philosophical Magazine ' last 

 quoted, is readily identified with (1). 



The form of (4) affords the reason why the usual methods fail to give the value 

 of the integral, as it shows that the result is not generally expansible in integer 

 powers of a. Generally, therefore. 



But when n is a positive or negative odd integer, it is enough to talie only the ter- 

 minating series, and ignore the other altogether ; a more complete explanation of the 

 reason for this than is given here can be gathered from the paper in the ' Philoso- 

 phical Magazine.' If « = a positive or negative even integer, tlie series for U or V 

 becomes infinite, and then one of the series involves log a as a factor (see Euler, 

 Gale. Integ. vol. ii. chap. vii.). Even the partial discussion of this case must be 

 omitted in this abstract. 



0)1 the Function that stands in the same Relation to Bernoulli's Numbers that 

 the Gamma-function does to Factorials. By J. W. L. Glaisher, B.A., 

 F.R.A.S. 



It is always a matter of some interest to regard a series of const.ants as particular 

 values of a continuous function, which function can usually be exhibited as a de- 

 finite integral. The problem is of course indeterminate, as through a series of 

 Soints at finite intervals from one another an infinite number of curves can be 

 rawn ; but, as in the case of the Gamma-function in its connexion with the factorial 

 1 . 2. .. .r, there is usually but one curve, which, in an analytical point of view, 

 stands in this relation. It seems, therefore, worth while to investigate the function 

 connecting Bernoulli's numbers ; and this is readily effected as follows. 

 Denoting by B„ the wth Bernoulli's number, we have 



2(1.2. 3. ..Sn) f J_,l. 1 



= 4«l t'"- 

 Jo 



^r^-2jT( j_-in( 



ie-""+e-'''' +...)di 



■ in 



i 



'^t'-'-'di 



2wt 



-1 



