856 
Mathematics. — “Some considerations on complete transmutation’. 
(First Communication). By Dr. H. B. A. BockwiNker. (Com- 
municated by Prof. L. E. J. Brouwer). 
(Communicated in the Meeting of June 24, 1916.) 
1. In a paper “Sur les opérations en général et les équations 
différentielles linéares d'ordre infini’, which appeared in the Ann. 
de l'Ecole Norm. of 1897, C. Bovrtnr considered a very general 
category of additive functional operations, called transmutations by 
him. The name ‘additive’, or “distributive”*) they owe to the 
property that the transmuted function of a sum is equal to the sum 
of the transmuted functions. The transmutation is further called: 
Uniform, if it makes a given function pass into only one other; 
Continuous, if the limit of the transmuted function is equal to 
the transmuted limit of that function?) ; 
Regular, if it transforms a regular function into another likewise 
regular function. 
We have in this case always in view a certain circular domain 
with centre «= .w,, in which the functions w, to which the operation 
is to be applied, are regular. The meaning of the last definition is 
more exactly that the transmutation is called regular, if the result of 
it is a function v, which is also regular in such a domain. 
The result of the operation may often be represented by a series 
of the form 
Am (x) 
GN ee ae 
Tu =O UE = +... 
t 
/ 
1 (2) 
SN 
1. m. 
in which a, (2), a; (@)..-., dm (@),... represent functions of x perfectly 
determined by the given transmutation, and are regular in a domain 
with centre «,, whereas u‚u',.… ud ,.. are the respective deriva- 
tives of the function uw, to which the operation is applied. The 
series (1), which also occurs in PINCHERLE’s paper, is to the theory of 
operations, what the series of Maciaurin is to the theory of functions. 
We shall call a transmutation complete in a certain point 
e=2, of the complex plane, if a circle with centre z, and 
radius @ is to be indicated, such that each function, which is regular 
within that circle and on its circumference, has a transmuted function 
t The latter name is used by S. PINCHERLE in a paper on the same subject, 
Math. Ann. 49 (1897) p. 325—382. 
*) Cf. for a complete explanation of this term N°, 9 (29d Comm.), 
