an Expression for the Numbers of Bernoulli. 



( d ) 2p ~Vi ir i- -i-g-8~(gg-ir 



*1» 



y"* 00 , /sin #\ ^ cos (2 p x — x) 

 \ x J cos # 



363 



(14.) 





-1 



Developing therefore (e -f 1) according to ascending 



powers of h ; subtracting the development from — , multiply- 



x 



ing by ^, and changing ^ to 2 /* ; we obtain 



h e h - e ~ h _ 2 P°> dx <»/ hsmx \ 2p 



e h -u e~ h ~ **J cos x Q>) 1 V x ) 



(15.) 



cos (2p x — x); 

 that is, effecting the summation, and dividing by h , 



1 e h -e~ h _ r 2 

 h J 1 a „~ h ~~ n 





X 



^a; sin # (1 — .A a? sin a? ) 



sin x cos 2x + h x sin 



1 — 2 h x 

 or, integrating both members with respect to h 



4 » 



(16.) 



/„ 



h dhl_ 

 A 



A 





/^tan^log^/i 



1 o o T o 



1 + h x sin 2 a? + h x sinx 

 a? - sin2^ + Zr x~ sin# 



(17.) 



It might seem, at first sight, from this equation, that the 

 integral in the first member ought to vanish, when taken from 

 h = to h = 00 ; because, if we set about to develope the 

 second member, according to descending powers of h, we see 

 that the coefficient of h° vanishes; but when we find that, 



on the same plan, the coefficient of h~ is infinite, being 



= — / d x, we perceive that this mode of development is 



here inappropriate : and in fact, it is clear that the first mem- 

 ber of the formula (17.) increases continually with //, while h 

 increases from 0. 

 4. Again, since 



= 4,(2^)- 4, W, tf*(*iW-TT^J • • (18-) 



/:+! 



e" - 1 



we shall have (for p > 0) the expression 



