538 Mr Hargreaves, The Harmonic Expression of the 



when p — s is even and ^ 0, 



= 

 when p — s is even and < 0, 



|p.cos*M-l) 4(p+s+ZTh 

 0-p)(s-_p+2)...(s+|)) 



when p ~ s is odd. 



On the right-hand of (6) the terms of the series which do not 

 vanish are those for which n — s is even, in which case 



1 n \y)— a n 1 /■ 



2 n i(n*-s)U(n + sy 



Taking p — s even, p — n is also even, and the expansion is 

 finite, terminating with n = p. Thus s being not greater than p, 

 n ranges from s to p, and 



7T \p (1 :- /X 2 )2 _ »=* 7T ■ (^ + 1). ||> [»- S 



2»+i 11 (p _ 5 )~|i (50 + s) n=s 21.31. .(p+n+l).2A...(p-n)\ n+s 



(_l)«"^»>f n + a • 



* 2 ^_ g) |[(^/ W^P/(^ 



or 



I = ' »=* (-l)* (n - <) .2P(2 W + l).|n- g |i(p- g )|Hl> + ^) 

 (1 fi*)t-^ n=8 2 ni^...{n+p+l).2A...(p-n)a(n-s) \h(n+s) 



(l-f*»)SP»*(/») (15). 



With _p - s odd, w - £> is odd and the series infinite. Thus 



[p.(l_^)f(-l)*^+ s+ZI|s ) 

 (s -p) (s -p + 2) . . . (s + p) 



, 7T(2W +l)|p.l. 3. ..(7l+p)(-l) i(n+P+ ~ ll " ] 



2.2 A. . .(p + n + 1) (n- p) (n- p + 2) .. .(n+p) 



= 2 



+ s |i(w — s) ^(rc + s) 



••• (1-^ 



= S 



Tr(2n+l)1.3...(n+p)(s-p)(s-p+2)...(s+p)\n-s 



■=• 2 ?l+1 2.4...(w+^ + l)(w-^)(w-p + 2)...(7i+/)) |i(n-s) j £(« + *•) 



(i-/^iV0") (16). 



