933 



that a{a) has a finite value within it; and we shall prove that 

 discontinuity of a{c() loithin that interval is not possible. 



We divide all power-series of a„i(,u), for the different values of 

 m, inta two parts, the first of which contains a number of terms 

 proportional to m, say km ; thus we write 



CO 



«,„(«) = \7, i'm,n «" = Pm{ct) + Qm («) , . . . . (94) 







where 



km — 1 00 



P,n {(i) = \7i Cia,n <i" , Qtii («) = Jn C.,n,n <<" , 

 km 



and k is a number, independent of m, which is at our disposal. 

 To either of the parts Py^ and Q„i there corresponds, as to their 

 sum, an upper limit as exhibited in (93) ; these we shall respec- 

 tively denote by the names first limit and second limit, whereas we 

 may call total limit that corresponding to the whole series. We 

 may now again use the proposition stated in N'. 32 of the prece- 

 ding communication, which has already served us a few times in 

 estimating limits such as we have to deal with here. In virtue of 

 this the greater of the limits calculated for Pm and Q,„ separately 

 is equal to the total limit, and if the first two limits are equal, the 

 total limit is either equal to them or less: the latter, however, 

 cannot be realized here, since all the terms of the series are positive. 

 If now the total limit a {a) is zero in all internal points of (0,.4), 

 then a («) is also continuous in those points and there remains 

 nothing to be proved then. Thus we lake the case that there is a 

 point «1 in (0, ^4) for which the limit in question is a certain positive 

 number Pj. We may state then: There is for the point cc = a^ a 

 value k^ of the above number k such that the first limit is not less 

 than the second and thus equal to the total limit X^. For if we 

 suppose for a moment that the second limit were greater than the 

 first and thus equal to -^.j for an arbitral^ value of k, that limit 



/ « Y 

 would for a value of « in (0, A) greater than «j be at least I — I 



times as great as ).^ and thus, as k could be t^ken arbitrarily great, 

 the limit in question would be necessarily injinite, contrary to the 

 hypothesis. Thus there is a value k^ of the property mentioned. 



For a point « on the left of «, the second limit is for the same 

 value k^ of k no more greater than the first, so that the first limit for 

 k = k^ is again equal to the total limit in such a point. In fact, the 



