214 



n 



P—la 7'(/>+l) +.. + a, rC/'+V ] 

 '-in ' ' k ii-/c-|rlJ 



Heiice it follows from 



T(/'+i)= 7'(;'+i) — rr(;') + T()') + . . . -f ï'(/'i 1 



11— it II "^ n n— 1 n — ft+l 



that 

 P 



— .ik 



gi^ . 



Tlp+i) T'^P) (s —4,) + Tip) (s -»,) +. .+ Tip) , («,—«) 



r= /2 4- S. 



It is evident that the absolute value of S is less than 



1 3W1 4- I Tip) I + . . . + I Tip) , , I 



' n I ^ ' n—\ ^ ' ' ^ n-k+V 



2a 



^(;'+i> 



We now prove that we can find fi > k so that if r ^ [i 



— *— — <' then it follows that \S\ <' ~- if ?i > f^. 



I — fc 



For we have by hypothesis /m?i 



Tip) 



i=^' iA^) 



= 0. 



Since 



^(;.+l) i J^ p—l 



Ail') p 



If ?i ]> ft we have : 



we have lim . 



Tip) 



t 



A[p+^) 



= 0. 



WiP+^) Tip+^) 



Aip+^) ^ Aip+^) 



<f 



and since \ s — Sk\ -^^ — our theorem is proved. 

 ' ' ^ 3« 



Remark 3. 



A. Rosenblatt (Bulletin International de I'Academie des Sciences 

 de Cracovie", ser. A 1913 p. 612—620')) has proved the following 

 theorem : 



') Rosenblatt's memoir not being accessible to me, the reference above is taken 

 from an article of G. Doetsch, Mathematische Zeitschrift Bd. 11, p. 161 — 175. 



