458 



ing that P is complete in the lohole N.B\0. («) of T. For if P were 

 only complete in a certain part («J of this N.F., we would simply, 

 in order to be able to apply the theorem, have to replace the first 

 pair of fields by another, of which the numerical tield is the part 

 («J mentioned. 



2nd \Ye have spoken of the part F-^ {T) that the functional field 

 F{T) of T has in common with that of F, formed by the functions 

 belonging to (/i). In this room has been left for the possibility that 

 not all functions belonging to Qi) form part of F{T). Should this 

 case, however, be realised, then the series P produces of course 

 at once the "analytic continuation" of T over the functional tield 

 formed by the functions in question. Any definition of T for these 

 functions that should lead to another result, would make of T a 

 transmutation, which would be no longer continuous iov {he extended 

 F.F, The reason for it is that a function of the F.F. added to it, 

 is the limit, in the domain (/i), of the fundamental series (26), all 

 functions in which form already part of the uncompleted F.F. 

 Consequently we have again for such a function in the domain («) 

 the equation (27) 



T(r/)„,<^)) = P(r/),„(a?)), (27) 



from which we deduce here, by passing to limits 



lim T{rf„{.v))^P{u) (28a) 



the counterpart of (28). If, however, T should keep its property 

 of continuity in the extended F.F., the left-hand member ought 

 to be made equal to Tu. For, if we have ^'^a, the function u, 

 which in the domain (/i) is the limit of the fundamental series (26), 

 is this a fortiori in (<>). And if we have ,'^<^^, and we want to in- 

 clude all functions belonging to (/?) in the new F.F. of T, the 

 numerical field of the functions is no longer [o] but {1^), so that now 

 in the latter we ha\'e u = lim y,„ {x). 



16. We now apply "Mac-Laurin's theorem" to a few examples. 

 We consider a neighbourhood of the origin 0, and the operation 

 7'= D~^, which transforms a function u into the integral of that 

 function, the origin ,i' = being taken as the lower limit of the 

 integration. This is a normal additive transmutation. For in the 

 first place it produces for all functions belonging to a certain circle 

 {a), functions belonging to the very same circle, with which the conditions 

 under 1 and 2 for a normal additive transmutation are satisfied 

 (with a=zG). It is further continuous in the pair of fields connected 

 with this; for, corresponding to any arbitrarily small number t. 



