148 MR. S. S. HOUGH ON THE APPLICATION OF HARMONIC 



This expression for A* may be somewhat simplified if we separate it into its com- 

 ponent partial fractions; we thus find 



s 2 (n - s) (n + s) T (n - I) 2 2n - 1 1 



= " ) [n*{n(n- 1) -a] ~ n*{n(n + 1) + <r}J 



(n + 2) 2 



3) 



+ 2) 2 __ __ 2n + 3 1 



!)( + 2 ) ~ "} ( + 1) 2 { ( + 1) + *} J 



1 ["^ tt 2 -s 2 Qi + iy - s 2 1 



( + 1) + <r} to- 2 """ w 2 (2 + 1) (71 + I) 3 (2n + 1)J 



2) 2 (>t - s 



" 7r (2/i - 1) (27i + l){n(n-l)-<r} (n + I) 3 (2n + 1) (2n + 3){( + 1) (n + 2) - a} ' 

 whence finally, remembering that or = 2ws/X, 



X- n (/i + 1) - 2as/X Qt - 1)- (M - s) Qi + s) 



Z 4^ "~"rf~(n + ]7 r " /r (27i - 1) (27, + 1) {(n - l)n - 2eos/\} 



(n + 2) 2 (TI - + 1) (n + s + 1) 



. ^zy;. 



(71 + I) 2 (271 + 1) (271 + 3) {(71 + 1) (71+ 2) - 20JS/X} 



The relation (28) will hold for all values of n equal to or greater than s, provided 

 we suppose that C*_2 = and C*_! = 0. If we put for brevity 



(n s+ l)(n- s + 2) 



JL,t 



it may be written 



* " (27i + 1) (2n + 3) {(n + 1) (71 + 2) - 2ws/\}> 



f- (30), 

 , __ (7^ + 5+ 1) (n + s + 2) _ 



= (2 + 3) (2w + 5) {(71 + 1) (n + 2) - 2a,s/\} 



" - ~* p* A S P-' I nf P* 



2 Jj n-y\>n-t *v'n "T yn^n+Z- 



4o; 2 2 

 Replacing T' by its value in terms of C' n , / [equation (22)], we obtain 



p., T.p., I .p, W 



where x'_ 2 , y' are defined by (30) and 



T < %" /on\ 



L' = T^-T -f AJ, (32), 



A*i being defined by the equation (29). 



On putting s = it may readily be verified that the equation (31) reduces to the 

 equation (23) of Part I, The manner in which such an equation may be utilized for 



