Sir William Rowan Hamilton on Fluctuating Functions. 297 



/x=^ 'Po2(„)"\ daR^_^^„f^; (r) 

 which, when p is a cosine, reduces itself to the form, 



/r = - ^,Z C ^« cos (2^r^ . "^^^j/^, (b ") 



X being > a, < ft, and h — a being not > tt ; and easily conducts to the known 



expression 



f 1^:. »C' ^ (2w— l)7r(a — ^) 

 /x = ^ 2^„) , "^^da cos ^ '—f^ /„, (cO 



which holds good for all values of x between — I and -j- 1- By supposing 7^ ■=■ 

 y"_a> we are conducted to the expression (y'") ; and by supposingy^ = — y_„, 

 we are conducted to this other known expression, 



„ 2 ,„ . (2n— l)7r^c' . (2w— l)7ra 

 /x = ^2(„)iSin^ __^^^c^asm-5^ ^f^—fah (dO 



which holds good even at the limit x -=1, by the principles of the seventeenth 

 article, and therefore offers the following transformation for the arbitrary func- 

 tion /"< : 



f 2_ 00^ ,xnC' J • (2/i— l)7ra 



/,= --2(„)i(-l)"J^^asin^ ^[-^—fa- (eO 



For example, by making^ = a*, and supposing ^ to be an uneven integer num- 

 ber ; effecting the integration indicated in (e ^), and dividing both members by f, 

 we find the following relation between the sums of the reciprocals of even powers 

 of odd whole numbers : 



in which 



[^•]*z=^(^•-l)(^•-2). . .(e_A;+l); (g") 



and 



-.*=2Q V>(2^-ir*; (hO 



thus 



1 = w, = 3w., — 3. 2. 1. 01, =z 5«.2 — 5. 4. Bw^ + 5. 4. 3. 2. 1 Wg, (i'') 



VOL. XIX. 2 Q 



