453 



meant that the continuity of Pu could be deduced from {hQ supposed 

 continuity of Tu, and that after this the identity of Pu and Tu 

 might be proved. But this idea would not do either, and it may 

 be convenient to set forth this somewhat more clearly, as it may 

 contribute to enable us to understand the rather restricted tendency of 

 the theorem X of Bourlet. This theorem is of great importance as 

 it states, broadly speaking, that an arbitrary transmutation may be 

 developed into a series of integral positive powers of the special 

 transmutation D ; in this respect it is to be compared with the 

 theorems of Taylor or Mac-Laurtn in the theory of functions. 



15. Before starting with the statement in question we shall for 

 the sake of brevity introduce a name for the transmutation we have 

 gradually begun to realize ; we shall call such a transmutation normal. 



A normal, additive transmutation is therefore a transmutation for 

 which the following holds : 



1. There is a functional field F{T), all the functions of which 

 belong to a circle {o), for which functions the transmutation produces 

 as result functions belonging all to a circle («). 



2. All rational integral functions belong to the field, 



3. The transmutation is continuous in the pair of associated fields 

 F{T) and («). 



A normal transmutation therefore is always regular, without the 

 reverse being necessary. It further follows from the definition in 

 connection with the property of cö?2^mM% of a C(?/>?/?/é?te transmutation 

 that any such transmutation is also normal. On the other hand, 

 however, so far I can see, it cannot be proved that the series, 

 which answers to a normal transmutation, has also necessarily the 

 property of completeness. 



The circle (a) may be equal to and smaller or greater than ''«) ; 

 the first is for example the case with the transmutation D~^, the 

 second and third with the operation of substitution So>, if to is 

 respectively equal to \ x and 2x ix^ = 0). 



We now develop an arbitrary function u of F{T) into a series 

 of powers 



W = Co + <^i ('^ — '^'o) 4- C2 ('^• — '^'o)' + 



If we write 



<Pm {x) = c^ + c^ (^—^0) + • • • 4- «w (.^— ^'o)'" • • • (25) 

 the fundamental series 



VoC^)' 9'i('«)' • • • '/'m(A'). (26) 



converges in the domain {g) uniformly towards the function u, 

 because u belongs to (0). 



