230 Methods for the Solution of Special Problems [CH. vm 



separate, and lying between 1 and + 1. Let us take these roots to be 

 i> 2, ...n. Then 



1 1 



- i) {P n (^Y (r - 1) (fi + 1) (A* - ,)* 0* - 2 ) 2 . . . (/i - a n ) 2 



+_O ............ (185), 



Z 



on resolving into partial fractions. 



Putting fjb = + 1 and 1, we find at once that a = ^, b = -J. 

 In the general fraction 



D ~~ (x a^(x o a ) . . . ' 



let us suppose all the factors in the denominator to be distinct, so that we 

 may write 



On putting 0? = ^, we obtain at once 



1 



^&l (Z 2 ) (ttj d' A ) \di $-4) ... 



1 



x 



(a 2 - aj) (a 2 - a 3 ) (a 2 - a 4 ) . . . ' 

 Now let &J and a 2 become very nearly equal, say a 2 = a a + c^a^ then 



1 



i 

 while c 2 = 



The fractions -^- + 2 



x 



,. . , 

 now combine into 



and on putting this equal to 



Ci C 2 



x a-L (x ttj) 2 ' 



it is clear that the value of c/ must be taken to be Cj + c 2 . Now 



1 f 1 1 



p I n . _ J _ __ _ - -- 



ddi |(a 2 - a s ) (02 - a 4 ) . . . K - a s ) (a a - a 4 ) . . . 



_ _ 



j (das \(x-a z )(x-at)..J x = ai 



-^) 2 ] 



D } x=ai > 



