900 



in the domain («) the absohite vahies of the terras, both in the 

 first and in the second scheme are not greater than those of the 

 corresponding ones in the second scheme for .v =z .v^ -\- a ; from this 

 it follows again that for all suchlike values of a: the elements of 

 both schemes form an absolutely convergent aggregate, the sum ot 

 which is independent of its grouping. 



We ap[)ly this result in such a way as to combine all the terms 

 of the same power of D u {D u) into one single tei-m of that same 

 power; this way of grouping produces in either case the resulting 

 series required. This series therefore converges absolutely and 

 uniformly in the domain («) for all functions u belonging to {^) ; for 

 if a function belongs to {{i), there is also a somewhat greater circle 

 {{t) to which it belongs. In other words the series is complete in 

 the domain (or), with a corresponding domain that at most is equal 

 to (ji). Thus point 2 of the proposition has been proved. 



The theorem of this |)aragraph has consequently been established 

 entirely, but the object we proposed in the beginning of N°. 23 

 has not yet quite been reached. The theorem really states that the 

 resulling series is complete and also gives some indications as to 

 the dependence between domains corresponding to each other with 

 respect to that series, but it says nothing about the way in which 

 the coefficients may be calculated. Now we did not intend to point 

 this out here, since it has already been done by Bourlet, who put 

 the result produced by the scheme (56) in a definite, elegant form. 

 We might therefore keep silent about it if from our previous con- 

 siderations it followed as a iTiatter of course that the form in 

 question is correct; in fact in that case we could only repeat what 

 is found in Bodrlet's paper. But in order to arrive at the under- 

 standing of the correctness referred to, a further explanation seems 

 to be necessary and we shall therefore for a moment call the 

 attention to this point. 



In determining the series P, Bourlet uses the operative function, 

 introduced by him; i.e. an expression ƒ (a*, 2) determined by 



. /(■^.^) = «o + ^ + ^ + ...., .... (57) 



from which the transmuting series, applied to the function u is 

 derived, by replacing 2;'" by Z)'«?< ; at the same time it holds that the 

 formal ?//^ derivative of this series with respect to z, is the operative 

 function that answers to the n^^ derivative of the transmutation P, 

 as it is easy to see. The operative function, originally only a ^ymóö/, 

 has for a sei-ies that is complete in a domain («), the property that, 



