574 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



In a similar way, we compute from 



1 = Z''\v)y'\v) = Z''\v)Y''\p) -^^- HY''\v) + vFY''\v) (15) 



V 



that 



HY^^\0) = = Y'-^\Q)H, 



m ,-1^ (16) 



FY^'\oo) = = Y^'\oo)F. 



Now Y^^^ (p) is finite at and =0 , so we may expand it in a power series 

 about either point. Let these be 



Y''\p) = F"^(0) + pYi'^O) + Oip\ 



Y^'\p) = Y''\o.) + 1 rl^)(oo) + (i) ^^'^ 



p \py. 



Putting the appropriate one of these into (13) and taking a limit at or 

 CO Ave get, by using (14), that 



1 



1 = -o AG + AY'riO) + Z{0)Y''\0), 



1 = 2BG + BYi'\^) + Z(oo)F^'\oo) 



A relation (18') ^^^th factors commuted is also true. 

 We may also put (17) into (15) and get 



1 = Z^'\0)Y^'\0) + HYi'\0), 

 1 = Z^'\o.)Y^'\o.) + FY['\o.), 



(18) 



(19) 



and also a commuted form (19')- 



Right multiply the first line of (19) by A and the second by B, and 

 use (14). This gives 



A = HYi'\0)A, 



(20) 

 B = FY['\oo)B. 



Left multiply the first line of (18') by H and the second by F. This 

 gives, by (16), 



// = -2 HGA + HY['\0)A, 



0)0 (21) 



F = 2FGB + FFl'^(oo)5. 

 Using (20) in (21), we have the relations 



