110 On the Partial Fraction Problem. 



Differentiating (3) and (4) h times with respect to a' 2 , and 

 letting x 2 = — c 2 , we have 



Pi 1 ] 



A h = _2 (-l)«-V.- 2 „V ( "- 4) (j), 



and rvi 



Therefore 



2m a x n -« B _ ir /p^] x x 



±=° =2 Us f-nW*V p ^"-"Mff 



III. To separate 



S= ^ , where p< 2<?, 



into partial fractions. 



Now 



P P P P 



II (««+#) + H(a K — x) II(a K + a?)-II (« K — a?) 



n («.+.) -^ ^ + — T^ ' 



the first part being of even and the second of odd degree. 

 Hence we may denote 



v p 



II (a K -f x) + II (a K — x) 



«=1 _^=! = ^), 



and p p 



n (««+#) — n (a K — x) 



Therefore @(>2) . ( ^j 



