937 



nuous within the interval of completeness (0,/l). That /■? is, moreover, 

 monotonously increasing together with a, since a {a) cannot decrease 

 as a is increasing, has already been remarked in earlier parts of 

 the present paper. 



Let us finally consider the case that the functions a,,, (.r) do not 

 coincide with their natural majorants, and let us write the latter 

 with the usual lines over them. We proved in N" 23 that the identity 



a {(() = a («) (95) 



is valid under the following two conditions: l^t . If tjie uniformity 

 supposition of N". 4 is satisfied ; 2"^ if the quantity <7(o) is a continuous 

 function of a within the interval (0, A). The latter has now been 

 proved to be always the case, so that we may infer that the equality 

 (95) is only a consequence of the same uniformity supposition from 

 which we derived in N°. 4 the extended theorem of Bourlkt. 



37. In connection with the latter considerations it may be con- 

 venient to observe that the uniformity supposition of N". 4 can be 

 replaced for either of the two purposes mentioned by one of some- 

 what wider compass. If the reasoning in K°. 23, leading to the 

 identity (95) be carefully examined, it appears that another, viz. 



1 



A {a) = Urn \A,n («)1 '« == a {x) (96) 



where /!,„(«) is the maximummodulus of a,n[.v) on the circumference 

 of («) can be derived from the continuity of a {u) only, ^) and since 

 the latter is always realized, the same holds for formula (96) so 



1) We found namely that on the circumference of an arbUrarij circle [v.') < (a) 

 and concentric with {y.) 



«'«(«)< T' 



u — « 



thus 



a{u') <A{a); 



further 



a {«) > A {(() , 

 so that 



a {(c) z= A{ft), 



by the further assumption that a (x) should be a contirmoiis function of y.. Mean- 

 while, now that this continuity has appeared to hold universally, the question 

 arises if it is possible to show the latter identity in a direct manner, without 

 having recourse to the continuity of a {x). This may in fact be done as follows. 

 Let us again suppose the quantity am (a) to be divided in the manner exhibited 



