FORMAL KEALIZAHILITY THEORY — II 591 



finite and not zero at po . The exponents of (p — po) in the elements of 

 \V-i(p) are therefore exactly the exponents oi q — qo in the elements of 

 Wi(q). From 5.29, then 



S(W2 , Po) = *S:(TFi , qo). 

 This with (1) and the definition 5.3 proves the theorem. 



5.32 Definition: Given any po , let p = T(q) be a non-singular bi-rational 

 transformation such that qo = T~\po) 9^ <». We define *S'*(Z, po) by 



aS*(Z, Po) = UriZ, Po). 



5.33 Lemma: S*{Z, po) is independent of the T chosen to define it. 

 Proof: Consider q = T~ (p) and r = U~ (p), each such that po is 



mapped on a finite point. Then by definition 



Sr{Z, Po) = S(Z, , qo), 



Sv(Z, Po) = S{Zi , To), 

 where 



go = T~\po), To = U~\po), 



Z,{q) = Z(T(g)), 



Z,{r) = Z(U(r)). 



Now r = U~\T(q)) = V(q), say, and /o and qo are finite. Hence by 5.31 



Sy(Zo , ro) = S{Z, , To) = Su(Z, Po). (2) 



But by definition 



Sy(Z, , n) = S(Z, , V-\ro)) = S(Z, , qo) (3) 



where 



But 



Hence 



Zz{q) = Z.iViq)) 

 Z.iViq)) = ZiU(U-\T(q)))) = Z(T(q)) = Z,(q). 



S(Zs , qo) = S(Z, , qo) = St(Z, po). 



This, with (2) and (3), prov^es the lemma. 



5.34 Theorem: Theorems 5.2G, 5.27, and 5.29 hold for S* without the 

 restriction that po be finite. 



