186 BELL SYSTEM TECHNICAL JOURNAL 



Now since the denominator of the expression on the right is always 

 positive, and all of the P's are less than unity, the magnitude of each 

 of the product terms is greater than unity if p is greater than unity 

 and less than unity if p is less than unity. Since, however, the magni- 

 tude of the complete product must be unity, the value of p must 

 be unity. 



After dividing through by the factor 1 + X, the remaining function 

 is a reciprocal equation in X and may be written as an equation in 

 ^ = X + (1/X). Since the magnitudes of the roots in X are all unity, 

 the roots in p must all be real and be in the region — 2, + 2. 



The degree of the polynomial in p is {n — l)/2. It may be shown 

 further that if {n — l)/2 is even there are an equal number of positive 

 and negative real roots, if the degree is odd there is one more positive 

 than negative root. 



The equations in p for various values of {n — l)/2 are 



-- 



- (1 - 2i + 22) = 

 2 - (2 - Sx + 23)^ 



+ (1 - 2i + 22 - 23) = 

 = 4, p' - (l - 24)^3 _ (3 _ 2i + 24)^2 



+ (2 - 2x + 23 - 224)^ 



+ (1 - 2i + 22 - 23 + 24) = 0, 



where the 2's are the symmetric functions of the P^'s, that is, 



2i = Pi2 + P22 + . . . P^„_i)/2, 



22 = P^Pi^ + • • • + P^(n--3)/2P^(n-l)/2- 



The equations in p may also be written in trigonometric form as 

 follows: 



— - — = 1, cos^^ + 21COS2 = 



r ^ fl 



= 2, cos^^ + 22COS2^ + 21COS2 = 



= 3, COS 2 ^ + 23 COS 2 + 2i cos 2 e + 22 cos | = 



9 7 5 3 



= 4, cos ^ ^ + 24 cos - e + 2i cos ^e -\- Xa cos ^ d 



+ 22COs|= 0. 



