895 



a («') < a («) 

 Further we have evidently 



a {a)> a («). 



We will now put the case that a («) is a continuous function of 

 («) in the interior of the interval (0, A), where as before we in- 

 dicate by A the upper limit of the ^«-values, for which a («) is 

 finite; we shall perhaps recur to the question as to whether this is 

 the only possible case. From the combination of the two last 

 equations it follows then that 



7(a) — a («) (54) 



For the series P, arising from P by substituting for the coefficients 

 a,n their majorant-functions, the number ^ corresponding to u is 

 now determined, according to formula (7) (J'*^ communication), by 



~^=zn + a{it) 



Thus according to (54) we have also 



/? = « + « («) ^ ^ (54a) 



We arrive consequently at the lemma: The series P has the 

 property of completeness in any domain (a) in which P possesses 

 the same property, such that the domain corresponding to {«) is the 

 same for both series, at least on the simplifying supposition we 

 have made here, that /? increases continuously with ix. 



24. We can now prove the following theorem -. I f the components 

 2\ and T^ of the compound transmutation T ^= I\ 1\ satisfy the 

 conditions: 



1. T^ and 2\ are normal transmutations; 



2. The series P, helonging to 2\ is complete in a circular 

 domain {u) ivith corresponding domain (f) ; the series P^ helonging 

 to Ti is complete in (y) loith corresponding domain (^) ; 



3. As pairs of associated fields, except the tiuo pairs that must 

 have been understood in stating the condition under 1, may be 

 considered : for J\ any pair of associated fields of P,, for 1\ any 

 pair of associated fields of P, ') ; then the folloiuing statements are 

 valid : 



1) We remind the reader of the fact that, if a series P, which is complete in 

 a domain (^) with corresponding domain (-»j), is to be called a normal transmutation, 

 as a F. F. associated with the N. F. {^) possibly only the group of functions, be- 

 longing to a circle being a little greater than (i^) comes into consideration ; for it 

 appears from the example given in N*'. 13 (2"<' communication) that the continuity 

 may be disturbed if corresponding to (^) as a N. F. 0., (?j) is assumed as a N. F. F. 



63* 



