616 



according to "the theorem of Mac-Laurin" would c.p. just hold if 

 1\ e£e; Tv were a normal transmutation, the question arises whether 

 this is the case. This question niust be answered in the affirmative. 

 For, if a function u of the functional field F{T) of T belongs <o (/?), 

 this is also the case with the product vu, so that T{v}() and conse- 

 quently also l\u^ T{vu) e.rists, according to our agreement in the 

 beginning of N", 19, in the domain («). From this it follows that 

 there is for the transmutation T^ a pair of fields to be indicated, 

 the numerical field of which is the circle (a), and the f imctional iield 

 consists of the functions that belong to the circle {[3). Since the 

 rational integral functions are also among these, 1\ satisfies already 

 the conditions sub 1" and 2", which we indicated in N°. 15 (3^"^ 

 communication) for a normal additive transmutation. But 7\ is also 

 contimioiis in the pair of fields mentioned; for, if u in the domain 

 (^) tends towards zero, this is also the case with the product vu 

 of u and the fixed function v belonging to d?}. But in that case 

 T{vu) tends to zero in the domain («) according to the property of 

 continuity of 7', and since ^'(vu) is identical with y\u the property 

 of continuity of T^ has been proved. 



If the expression Tvu is considered as the result of the transmuta- 

 tion l\-^{Tv), applied to the function «, the following establishment 

 of the expression (45) for the coefficients a',,, ^ 7("0(^y) of the series 

 P corresponding to 7\, is at once obvious. If the function in which 

 X"* is transformed by 7\ be denoted by ^',„ , we have 



and the application of the symbolic formula (24) according to which 

 we have 



a',„ = (5'-A-)"«, 

 leads at once to (45). From this the other representation (39) of 

 a'm may be found then, by first deriving from (45) that we have 



for 771 = 1 



r {v) = T {xv) — w T {v) (46) 



and further the recurrent relation 



T^'»){v)=T("^-'^) {xv) — .v T('") (v) ') (47) 



According to the "theorem of Mac Laurin" we have further in 

 the domain (a) 



00 



Tiv) ^--S^ an"^. . . . . . . . (48) 



j^^ n! 



1) The relations (46) and (47) occur in Pingherle's paper as definition of the 

 respective derived transmutations of T. 



