455 



now also hold in the domain («)• l^^or shortness we write ct' — ,T^=:y, 

 and indicate the coefficients in the various derivatives by means of 

 accents. We then obtain the following schemes 



F {ff „I (.'»)) = «0 (Co + • • • + Cm y"' ) + 



-f a„ (cl"' + . . . 4- din 2/'»-") + 



+ 



P{m)z= a„(c„ + ... +c,«^'") + a„ (c^_^i y«+i + .••) + 

 + «1 (c'o + • • • + C'm-l y"'-') + «1 (C'm3/'" +.-.) + 



+ 

 + 



I (»«i I / ("0 , ^ 



■ + 



Now, it is first possible to find, corresponding to a given arbitrary 

 small number 8, an integer Q, such that we have in a point x of 

 the domain («) at the same time 



\P{u)-P,{u)\ <8 j 



in this P„(?ó) represents the sum of the first n -\- '\ terms of the 

 series Pu. If this has been done the only thing left is to compare 

 PnU and P(4)m{^i) for values of n and m>Q. It is sufficient to think 

 m>n, for if we choose 7i^m, the contribution which the terras in 

 PnU, for which ii^m, produce for the difference between P,tL and 

 P<pm{x) would be smaller than 2e, because we have already m>Q. 

 If we therefore choose m>n, we have for the difference between 

 Pu{u) and P(pm{^) the following scheme 



Pn{u)—P{if,n{x)) = a, (c,„+i y"+i + . . .) 4- 



+ ^ ^ (31) 



4- a„ (c"'_„+i v'«-"+i +...) + 



