( 65 ) 



then also <p o (,r -f- h) — g> o . o (.»•) is approximated by 



1 IS 



y« A 0') )/3 >) (* 4- A) - </> a ^ A oo to , 



which last expression remains <^ f , however great p may become, 

 so that the first can neither be }> s, ; so the set of «// second diffe- 

 rence quotients is uniformly continuous and there is a finite conti- 

 nuous second differential quotient. 



Theorem 7. If there is among the second difference quotients of a 

 finite continuous function a uniformly continuous fundamental series, 

 in which both «-increases decrease indefinitely, the function possesses 

 finite and continuous first and second differential quotients. 



For, according to the above given proof of theorem 6 the whole 

 system of the second difference quotients proves to be uniformly 

 continuous, and out of the above given proof of theorem 4 this 

 system proves to possess for indefinite decrease of the two «-increases 

 one single finite continuous limiting function ƒ " («) to which they 

 converge uniformly. Let t' be the maximum deviation from this limiting 

 function of the second difference quotients, whose «-increases are smaller 

 than A', and A' s , and let us regard the system £ of all <p n («) whose 

 A <[ A', , then all difference quotients with «-increase <^ A', of the 

 system £ lie between /" («) -f- r' and ƒ" («) — t' , from which may be 

 deduced easily, that the system "£ is uniformly continuous, so that now 

 first according to the proof of theorem 1 a finite continuous first 

 differential quotient exists and then according to the proof of theorem 

 4 a finite continuous second differential quotient. 



Analogous to the preceding are the proofs of the following more 

 general theorems : 



Theorem 8. If there is among the n th difference quotients of a finite 

 continuous function a uniformly continuous fundamental series, in 

 which all «-increases decrease indefinitely, then the function possesses 

 finite and continuous first, second, up to the n {][ differential quotients; 

 each » th differential quotient is here first the only limit for indefinitely 

 decreasing «-increases of the p th difference quotients to which limit 

 a uniform convergence takes place, and then differential quotient of 

 the (p — l) st differential quotient. 



Theorem 9. If a function possesses a finite continuous n ,h differential 



5 

 Pro ceedings Royal Acad. Amsterdam. Vol. XI. 



