925 



(i,,m,a^,in,- ■ ' bj their natural majorants; and we infer from (23) that 

 the functions ^,,m,%^,m,- ■ ■ are in this case also replaced bj majorant- 

 functions'), though, these need not be the natural majorants of the 

 former functions. In order to avoid reproductions of formulae taking 

 up much room we propose to imagine for a moment that formula (84) 

 relates to the last-men I ioned case. But even so we are not justified 

 jet in considering the aggregate arising from the expansion of each 

 binomial of the series in (84') as an absolutely converging one, 

 if X lies in the domain («J, because it is not known whether we 

 may i-eplace — x by x in that expansion. We therefore make an 

 estimate of the magnitude of the sum 







which w^e may denote by the symbol {i -{- w)' , and to this purpose 

 remark in the first place that the quantities 5,„ for a complete trans- 

 mutation satisfy the same characteristic property as the coefficients a,n 

 of (he corresponding series: to be snfaller, as to their moduli, than the 

 ?/?>'' power of a number which is independent of vi. This follows 

 from formula (23). For from and after some value of k [k ^ E) we 

 have in a domain of completeness {«(),\ak\<C{a -f ^y^ where « depends 

 on a. and e may be chosen arbitrarily small. Then by (23) and 

 supposing VI y E 



E^ 1 m 



\^in\ < "% /t '"A- «"'-^' \Ok\^ Sic ink «'«-^" (« + f)^' 

 If 



< ^ ^ + \l ^ ^ 







< < ^ -1- (a -\- a 4- f)'« 



The latter right-hand member consists of a finite number of terms, 

 viz. E^\, which is independent of in\ thus it is sufficient, in 



1 

 order to calculate a majorant value for lim \ §„, |'" , to determine 

 the corresponding limit for each of the terms individually : the 

 greatest among these limits will, according to the lemma of N". 32, 

 be a majorant value as required. Now for each of the firsts terms 

 the limit in question is clearly not greatei- than ••< ; and for the last 

 term it is (i-|-« = l^; thus /^ is a majorant-value as was required. 

 We have now proved the following proposition: 



J) This, if an arbitrary point x^ is the centre of the domains, is valid only if 

 by 'i 111 be denoted the transmuted of (x — .r,,)"' • 



65 

 Proceedings Royal Acad. Amsterdam. Vol. XX. 



