461 



If these equations are added according to columns, the sum of the 

 coefficients for the k^'^ column is equal to J -|- (1 — 1)^', that is 1 for 

 every column, and we see how in this way the binomial coeflicients, 

 though increasing towards the right, balance each other. By this it 

 is explained how it comes that the series arisen after the summation 

 in question 



et: (1— ,f'«+l) 



1 iV 



has still a limit for all values of .v in the domain («) for )n = <x> , 

 whereas by summation according to horizontal roivs when they are 

 infinite, a limit does not exist for all such like values. We finitely 

 arrive at 



lijn P {ffm {x) ) = 



l — x 



agreeing with the right-hand member of (33). Thus the general 

 formula (28), according to which lim P<p"^ {,v) in the ivhole F. F. of 

 T, therefore eventually also outside the F. F. of P, must exist and 

 be equal to Tii, has been proved directly in this special case. 



17. As a second example we consider the substitution. Let to (x) 

 or shortly to be the function that is to be substituted for x, and 

 let us take again as N. F, of T=zSo> a neighbourhood of 0; 

 we have then to suppose that is an ordinary point of lo (x) and 

 that the N. F. of So> is a domain within, the circle of convergence 

 {A) of to, say a circle with radius a <^ A. In order to see which 

 group of functions have a transmuted in («), we consider the cor- 

 responding domain i2 of (or), which is obtained by conformal repre- 

 sentation of the relation x' = to [x), and is a domain round the point 

 .i' zr: to (0). Let the maximum modulus of this domain be o, then 

 Sr,> produces for all functions belonging to the circle (ö") a uniquely 

 determined transmuted v{x) in the domain («). These transmuted 

 belong to the circle {a); for it follows from the equation 



V {x) — aSco {u {x)) = u [to (.;;)] , 

 on account of the well-known theory about functions of functions 

 that v[x) is regular in the domain {a) including the circunjference. 

 The conditions under 1 and 2 for a normal additive transmutation 

 are therefore satisfied. aS^, is further continuous in the pair of fields 

 in view: if u{x) tends towards zero in the N. F. F., that is the 

 domain (a), this is a fortiori the case in the domain Si, which lies 

 within (tJ); thus i){x)=zu\ui{x)\ tends towards zero in the N,F.O. (tt). 



The transmutation *Sw therefore is normal. For the construction 



