912 



powers of a letter a that have to be multiplied analytically before 

 tiie proper meanings are substituted. In other words, the proper 

 expressions will always be linear functions of quantities involving 

 the same letter a and different indices k. 



To begin with we observe that the functional theorem of Mac- 

 Laurin, treated in the 3"' communication leads to a generalization 

 of the symbolic formula (23) ^ 



5,„=T(.t-)=:(a; + a)"' (23) 



which expresses the transmuted %m of the rational integral functions 

 a?"» in terms of the coefficients Om of the series P answering to the 

 normal transmutation T. Formula (23) is valid in any circular 

 domain to which belong all functions a,» and §„, ; the existence of 

 such domains is one of the characteristics which make a transmuta- 

 tion normal, according to the definition we gave in N°. 15. 



When the series P is complete in the domain {a) then, according 

 to the just mentioned theorem 



T\i = Pu ^ > wi 



a,„M'«)(^) 



m! 







for functions u which belong to the domain {^) corresponding to («). 

 The right-hand member may apparently be denoted by the symbol 

 u (x -\- a), provided we interpret this in the following way : sub- 

 stitute for the symbol the power-series in the letter a which answers 

 to the function u{.i'-\-a) if that letter means a complex number. 

 This power-series is unique, since x is a point in the domain («) 

 and n a function belonging to (,?) and thus certainly to («)• ') We 

 therefore obtain the symbolic formula ^ 



Tu{x) = u{a; +' a), (67) 



valid in [a) and of which (23) forms a particular case. ') 



i) Considerations of uniqueness were really of use already when in the S^^ 

 communication we put for the formula 



èm = yi 



mi: x"^~^' ajc 



the symbolic formula (23); in fact, if the expansion of {x-\-d)m in a power-series 

 according to a were nol unique, special reference should be made to the fact that 

 the series in the right-hand member is meant and no other. But no one thinks 

 of uniqueness in the development of a binomial, nor did we in writing our 

 3'''' communication. Nevertheless, in the light of the present general developments, 

 in which the uniqueness of a power-series forms the principal part, it seemed 

 convenient to us to mention this point. 



2) We have to take care that in the first term of the expansion the factor a^ 



