Mathematics. — “On Mac Laurtn’s Theorem in the Functional 
‘'aleulus’. By Dr. H. B. A. BockwinkeL. (Communicated by 
Prof. L. E. J. Brouwer). 
(Communicated in the meeting of March 29, 1919). 
In the third communication of my paper “Some observations on 
complete transmutation’”’?) | proved a restricted validity of Mac LAURIN's 
theorem in the Functional Calculus for a normal additive trans- 
mutation. A normal transmutation was defined by me as follows: 
1. There is a functional field /(7'), the functions « of which 
belong to*) the very same circle (0), and for these functions the 
transmutation 7’ produces functions belonging to the very same circle 
(uv), concentric with (6). 
2. All rational integral functions are included in the functional field. 
3. The transmutation 7’ is continuous in the pair of associated 
fields F(T) and (a)*). 
From 2 it can be derived that to any such transmutation 7’ another 
transmutation P? formally corresponds, which is given by 
au aa Aus”) 
age EIR 2/ diene ml! 
heenescH a 
where the quantities u”) are the derivatives of the subject of operation 
u and the quantities a, functions of the numerical variable w, which 
by means of the formula 
Am = Sm EK Mm, Em—1 apo eS = ams, 4) +: AET: (2) 
can be derived from the functions 
Epa T (a!) [08 NOS) 
1) These Proc. Vol. XIX N°. 6 and 8 and Vol. XX N°. 3—7, to be quoted as l.c. 
2) A function belongs to a circle, if it is regular within and on it. The symbol 
(c) means a circle with radius c. 
3) See for the definition of continuity l.c. Vol. XIX N°. 8, where also definitions 
are given of the expressions functional field (F. F.). numerical field of operation 
(N. F.0.) (here the circle («)) and numerical field of functions (N. F. F.) (here 
the circle (c)). 
4) By mk the binomialcoefficient of order k of the number m is meant. 
