10 Mr Glaisher, On a formula of Cauchys for the [Nov. 6, 



Since 



Jo « 



it follows, by addition, that 



^A ^"-^ (e-^ + e-^^ + e''+ &c.) cZ^ = ^. + |;. + ^ + &c., 



viz. that 



r C^) ;„ e^- e- - 2r(n) j, ^"" ^^^^^^ «= c?;. = (^l - ^J ^„, 



CY 1 1 1 1 P 



where ^„ = p + 2^ + gUi + ^ + &c. 



Now S = — ^^ — ^^ • 



■^^ow A32« 2r(2n + l)' 



^^ being the w*^ BernouUian number, so that 



f ^'"-' cosech oj c?^ = 2r (2w) ("l - i) 



2r(2?i + l) 



(2-''-l)7r^" 

 2?i "' 



and we thus find from (2) that 

 I cc^"-^ cosech [a; \dx 



= |j (2^-l)5ya»-^+ ^^^^"^^ (2*- 1) ^.ttV-^' 



+ ^^^"^-^y-^^^2--l)^3vrV- 



... + (2 -1)77 l/„_,a+- 2^^— — , 



the general term being 



n{n^-V)...[n^-{r-Vf] ,^^,_ -, n 2. d „_r 



