938 



that for this the uniformity supposition is superfluous. The latter 

 however served us to prove further that also the identity 



a{«)z^ A{(() . . . . . . . . (97) 



is valid. But now we may observe that for the latter the following 

 uniformity-supposition, wider tiian that of N". 4, is sufficient: 



A. Corresponding to an arbitrarily chosen number e, as small 

 as we please, there is an integral number iV^ such that at all points 

 X of the closed domain {n) 



\o-m{x)\ < {a {(() -^ eY' for m>A\ .... (98) 



We shall not explain any further that this supposition is sufficient in 

 order to deduce the equality (97): it is easy to see. From (97) 

 however and the identity (96), which has appeared to be valid in- 

 dependent of any particular hypothesis, the equality (95) may be 

 derived, so that the function a («) is equal to a (a) and thus conti- 

 nuous within the interval (0, A) under the single condition denoted 

 by A. 



But it cannot yet be inferred from that condition that we have 



<?=/?. If the latter is to be true as well we must have 



^ =^ ^ =z (c -f- a (a) =1 a -\- a {«) , 



that is the number i? corresponding to « must be determined by 

 formula (7), copied at the end of the preceding paragraph. This 

 formula, was obtained in N°. 4 and based upon the uniformity- 

 supposition of that paragraph. But on examining the proof of the 

 completeness-theorem we gave there it will appear that the formula 

 is a consequence of the following supposition only : 



by (94) and, for a certain value of a, the integral number k so chosen that we 

 again have (writing in this case P instead of P) 



1 

 a {(i) = lim [P,„ (ft)] '« 



Now, for every value of n, Am (a) > Cm,n a", if c„i,n be the modulus of the coef- 

 ficient Cm,n in the power-series of am{x). Let Cm,fj ccp be the term of maximum 

 value in Pm(a); the number p will in general vary with m, but we always have 



Cm,p y-P > Pm (.y.) : km and thus also ^'« (x) > Pm (a) : km. From this it follows, 

 1 



since lim. m '" =1, 



HI =00 



1 1 



A {(c) = iïm[^„.(")]'" ^ ^'i^ \Pm {(()Y' = « (") , 

 and thus, since A (x) cannot be greater than a (a), we must have A (a) = a (a). 



