64 pitcher: interrelations op 



sequences {Oij and {6,j occur we must show a function ,uo such that 



({flil . {bi\) :=, : 3 \c^] * (i2 > ii . =) . Ac,, < ylc,-,) . (Lci = 0). 



i 



(i . 3 . Aa, ^ Ac, . Abi ^ Acf). 

 {c,} . 3 . 3 {dii 9 (L cA = . L f^, = oc) . (L a.rf, = L M, = 0). 



Ml 9 Miu = f^Mu • 3 • M 



1 



The function /xi is available as the desired yuo- 



Proof of m'^'-.—We show that 9JU + [all (^■^'"•']. In the proof 

 of SQr'^ we saw that the function 6 there defined belongs to 9JJ^. We 

 will now show that 6 is not dominated by Wl and therefore cannot have 

 the property i^i50J. Suppose 6'^'"'. 



0.3. 3fjLiAdu ^ Aju,; (y). 

 M :n : 3^5 (i ^ V -i) ■=>• Mu = a. cos 2^+ 6.- sin 2^- 

 A(^a,cos2^ + 6,sm2-j 



= v/^f+T! A ( --.#4= cos i? + -4= sin ^ V* 



= l/^yqp^Asin(«, + g) -~^«, ^|. 



ai.3.3J,.9.l(^ai+%^)< " 



9 



:^i 



'"^ . 3 . ^T ^ V a^ + b: sin 2^ (i ^ v). 



0^'" . 3 . Va; + 6^ ^ (i > ij. 



I sm ^. 



i . . TT TT 



2 sin ^ 



3 ii 9 i ^ i'l . 3 . Vo; + 6' > 



* Oi and h," are never simultaneously 0. Otherwise m does not dominate d. -^aj + 6; 

 denotes positive square root of a] + fe^ 



