594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



V = T(oo) both of Z(p) and Z~'(p) are finite. Let 



ZM) = z{r{q)). 



Then 



ZX\q) = Z-\T(q)). 



Let Wi{q) be the normal form of Zi(q), with diagonal elements 



ek(q) 

 ^kk(q) 



in lowest terms. Since Zi(q) is of rank n, none of these vanish identically. 

 We first claim that 8(Z) — 5(Zi). The poles po of Z are exactly the 

 points 



Po = T(go) 

 where go runs over the poles of Z\ . At each pole, 



S*(Z, po) = S*r(Z, po) = S*(Z, , go) 



by 5.35. Hence 8(Z, po) = 5(Zi , go) and the result follows by addition. 

 Similarly, then, d{Z~') = 5(Z7'). 



Next we assert that 8(Zi) is just the degree of the polynomial 



For 5(Zi , go) is the exponent of (g — go) in this polynomial, and the 

 zeros of this polj^nomial are exactly the poles of Zi{q). 

 We observe that if 



Tri(g) = A(q)Z^(q)B(q), 



then 



WT\q) = B-\q)Zl\q)A-\q). 



This then is the result of polynomial operations on Z7 (g), and has 

 diagonal elements 



^ (1) 



Clearly by arranging these in reverse order, we have a normal form. 

 This^is 5.13. Hence the functions (1) are the diagonal elements of the 

 normal form of Z7\g). The argument above applied to Z7^(g) then 

 shows that 8{Z^^) is the degree of 



ei(g) • • • e„(g). 



