6 Mr Glaisher, On aformulaofCauchysfor the [Nov. 6, 



f-{cc) being supposed to be an even function of x, viz. denoting this 

 integi-al by B^^^, and putting 



^2.= f oc'J{x)dx, 



J 



then the formula is 



{n+l)n (n + 2)(n+'[)n(n-l) 

 A» = ^o+ 1^2 ^'^ 1.2.3.4 ^■" 



+ ~Y—An-. + An-; (1); 



but it is curious that neither Cauchy, nor, as far as I know, any of 

 those who have quoted his result, seem to have noticed the corre- 

 sponding formula for the case when the arbitrary function is 

 uneven ; which may be thus enunciated. 



If ^ (x) denote an uneven function of x, and if 



P..-^ = jy^""^ (^) ^^. Q.n-^ =/f ^""^ (^ - ^) ^^' 



then 



(n+l)nin-l) 



"*2n-l "-^ i^ 12 3 ^ 



(n+2){n+l)n{n-l)in-2) 2n-2 



1.2.3.4.5 ^5 •••-!- I -^2n-3-^-^in-l"-\^J' 



§ 2. This formula can be proved exactly as Cauchy proved (1) ; 

 for using ^ (x) to denote an uneven function of x and f [x) an even 



function of a; ; we find by taking x = -, , that 



and that generally 



j'"x'"-'(j>(x-^dx = -j'^x-"'-'<})(x-^dx (4); 



while by taking x =x' so that to the limits oo and of « 



correspond the limits oo and — oo of x, we have 



