932 



• 

 the first mentioned, and therefore also continuous within (0, A), if 

 we further assumed, as we have done conlinuallj, that a certain 

 supposition were realized which we proposed to quote as the uniformity 

 supposition of iV". 4. The above mentioned simplification of statements 

 was a consequence of this identity of ai<() and a{<i). Again we 

 intimated in No. 23 that we should perhaps recur to the question 

 as to whether the continuity of a{(() represents the only possible 

 case. We now proceed to do so. 



We may for shortness of notation and without loss of generality 

 consider a neighbourhood of the origin; furthei' we may, as long 

 as only the natural majorants of a,n{.v) are considered, denote these 

 quantities without the lines above the letters which were used 

 hitherto; the same thing may be done in denoting quantities connected 

 with the first mentioned, as for example the left-hand member of (6'). 

 This latter attains in the domain («) its maximum value for the 

 real positive value .c = «, so that we have 



1 



a (a) .Jhii\a,,(a)y^ (93) 



iii=^<rj 



The supposition that the series P is complete in (a) implies that 

 all functions am belong to (<«). tl»at is, that they are regular within 

 that circle and on its circumference. Let R be the upper limit 

 of the radii « for which this holds. Then we may put the question 

 as follows: If the function «,„(«) of the ;vr// variable « foi' all integral 

 positive values of m can within the interval (0, A*) be developed in 

 a power-series of a with real positive coefficients, to inxestigate the 

 question as to whether the function a{(<) defined by (93) is continuous 

 in the same interval. 



Since a{a) evidently increases together with a, we may at once 

 infer the following-. 1. If a{u) be finite for a certain value of « in 

 the interval (0, R), then a[ic) is also finite for all smaller values of 

 «, belonging to the same interval. 2. If a{<i.) be infinite for a certain 

 value of a in the interval (0, R), then a{n) is also infinite for all 

 greater values of « in (0, R). Thus there is in (0, R) a point A 

 forming the section between those values of the "interval for which 

 a[(() is finite and those for which it is infinite. The point A may 

 coincide with « =: or with ((=R; in the first case the corre- 

 sponding transmutation is not complete in any domain of the origin, 

 however small, so that this case need not be regarded. (An example 

 is furnished by a,n^=m\ x"\ where R = <X), but ^4 = 0). 



We therefore take the case that there is a certain sub-interval 

 (0, .4) of (0,/^), which may eventually coincide with the latter, such 



