590 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



p-sphere. Denote its inverse by 



q = T-\v)- 

 Given a rational Z(p) the matrix 



Z,{q) = Z{T{q)) 

 is rational in q. 



For any po such that T~ (po) 7^ <» , we define 



5.31 Theorem: If po and T~ (po) are both finite, 



SriZ, Po) = S(Z, Po). 

 Proof: Let Wi{q) be the normal form of 



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

 We have 



W,{q) = A(q)Z,{q)B{q). 

 Consider 



TF2(p) = Pri(r-'(p)) = A{T-\p))Z{p)B(T-\p)). 



Here the pre- and post factors of Z(p) are rational, finite, and non- 

 singular at Po . Hence by 5.26 



S{W, , Po) = S(Z, Po). (1) 



Let 5o = T~\po). It is then easily computed that the inverse trans- 

 formation T~^(p) takes the form 



aip - Po) . f. 



q - qo = T-—^ f-ny a ^ 0. 



Dip - Po) + 1 



Any given diagonal element of Wi{q) is of the form 



{q - qo)'R(q), 



where e may have any sign, and R(q) is rational, finite, and not zero 

 at go • The corresponding diagonal element of 11^2 (p) is then 



(p - p»)' G(p-p.)+i ) «■(?'• 



where Ri(p) = R(T~ (p)), and the factor multiplying (p — po)'' is again 



