572 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



By their definitions, 



for i = 2, 3. By hypothesis, Z{p) and 



Zip)-' = Y(p) 



are both finite everywhere on p = fco. By its construction, Z''^\p) is 

 finite at p = and p = ^o , 



4.63 We claim now that each Y'''\p) is finite at p = and oo , ^ = 2, 3. 

 Proof: We need consider only F^"^(p) since F'^'(p) differs from it by 

 something which vanishes at p = and p = <» ((5) above). Let 



Y''\p) = Y(p) + ^-E + pQ 

 P 



where f (p) is finite at p = and p = <» . Since F^'^p) is PR (4.43), E 

 and Q are real and symmetric. 

 Using the form (4) above for Z^^\p), 



1 = Z''\p)Y''\p) = Z{p)np) + BE+AQ 



+ p(Z{p)Q + BYip)) + p'BQ (7) 



+ -{Z(p)E + Anp)) + \AE. 

 P P" 



Multiplying through by p, p, -^ , - and taking limits as p — » 0, 0, <» , <» , 



p^ p 



respectively, we obtain 



AE = ^ 



z(o)je; + Afco) = 0, 



(8) 

 BQ = 0, 



Z(o.)Q + 5f(oo) = 0. 



We can also write a formula like (7) with the factors in reverse order, 

 and obtain the analogous forms to (8) in which the factors are com- 

 muted. Let us call these commuted relations (8'). Multiply the second 

 relation (8) on the left by E and use the first relation of (8'). We obtain 



EZ{^)E = 0. (9) 



Working similarly with the last two relations in (8) and (8'), we get 



QZ(oo)Q = 0. (10) 



