54 Mr. J. W. L. Glaisher on a New Formula 



^.^.(i)Let 



a 2n+l a 2n+l 



a„ = 



n 1.2...(2n+l) T{2n + 2) ' 



then a-$=l, and 



which is true. 

 (ii) Let 



then 



and 



the true result. 



I 



00 sin ax 7 it 



x - 2 



an a n 



a-i _ 1 



f 



2v« 



2» n 1n 



If we take «.= j-^ = j^-^, we have 



r 



cosased%=0, 



viz. sin oo = 0, the value we should expect to find by any process 

 that gave a result at all. 



2. Divide (1) by 1+x* 2 and integrate as before: the right- 

 hand side 



IT 



= j \a cos 2 6 - A# sin 2 6 cos 2 + A 2 « sin 4 cos 2 0—...)dd 



Jo 



-??Yl_l La . 3 I La2_^. ? 1 iA3o- ^ 

 2V2 4*2 + 6'4*2 8 ' 6 ' 4 ' 2 + * * 'J a ° 



_ ?? \/l+A — 1 _ w; 1 



~2 A °~~2 l4VE fl °; 



so that 



f 



~ jq^r dx=-{a Q -a± + a-a» + ...). . (3) 



Take 



« 2?i a 2n 



a n — 



1.2...2» r(2n + l)' 



