724 BELL SYSTEM TECHS'ICAL JOURNAL 



To prove this theorem \vc start wiili the integral equations 



The second of these is in f(jrni for an immediate apphcalion of Borel's 

 tlieorem since 



1 1 \ 



p'Hip) ~ p ' piiipy 



The functions /i and /; of Borel's theorem then satisfy the equations 



pIHp) Jo 

 It follows at once that 



Ii(t) = 1 



whence bv Borel's theorem 



-r= r f,{t)e~P'dt, 



p "'O 



^,= rMDe-P'dl. 



g{t)=fjliT)d7 



Theorem III 



If h=h{t) and g=g{l) are defined by the operational equation': 

 h = \/II{p) 



g=p/mp) 



then 



dt 



g{t) ^JrHi) 



provided h{o) =o. 



The integral eciuations of the |)roblem are 



