906 



the latter quantitj' indicates the radius of the smallest domain that 

 may be taken as N. F. F. together with («) as N. F. 0. 



That, besides the symbol of inequality, the symbol of equality 

 may occur here, is proved by the special case in which o)::=c-\-£c, 

 c being a constant. Thus, in this case the majorant-value, indicated 

 by our theorem for the number corresponding to « for the resulting 

 series P, at the same time provides the correct value, from which it 

 appears that the theorem does not necessarily give a majorant-value 

 which is too large. 



For the sake of completeness we mention that we have here for 

 the two operative-functions /, = ƒ, = (f> 



ff ::= > "I = e^K^" - x) , 







As the resulting transmutation is also a substitution with to ^ (a-) as 

 the function of substitution, we must have 



which is really given by Bourlet's formula. 



27. Third .example. We take 1\ = *Soj, 7\ =: Z>~^ consequently 

 7^ = S^„ D"^. As both com|)onents are transmutations already treated 

 in the preceding examples it may sutlice to elucidate a few points. 

 For the three numbers 'f, y, i^ ^^e may evidently take here : ^f, y, 2y, 

 if y is again determined by (60). We shall not further explain the 

 first part of the theorem, because the transmutation SD-^ is not 

 to such a degree known as a simple transmutation that in our 

 veritication of the point we might refer to known things : the test would 

 consequently be more or less a repetition of the general proof. But 

 we may verify point 2. of the theorem, that is to say, determine 

 either by means of Booklet's formula or in another way, the i-esulling 

 series F, and ascertain whether it satisfies the statement contained 

 in that point. 



We therefore proceed to prove that the number /?i, which, for 

 the series P^, corresponds to a, is smaller than 



2y = 2«-f 2|a>(.^■,„)— .t-,„| ...... (64) 



We may easily determine the series P here in a direct way, 

 that is to say by means of the transmuted §k of the function x^. 

 Apparently we have 



(O^'+l 



k-{-l 

 If this result is substituted in formula (24), we arrive after some 

 reductions at 



