1112 
to any arbitrarily given number Ò, a function d(z) in the F.F. for 
which af the same time 
M;<d and | TO(x,) | =r, 
the latter for some point rz, of the N.F.O. 
The operation 7’ can never be continuous now; from whatever 
initial function w,(“) we may start, corresponding to any arbitrarily 
given number 0 there is a function u(s) such that at the same time 
Mi Sa 20 and | Tu (a,) — Tu, (#,) | >t. 
In order to see this we need only choose for u the function 
u, (a) + Ò (we) and apply the additive. property of 7. 
We may also express the proposition of this number in connection 
with the second definition of continuity as follows: Jf an additive 
transmutation is to be continuous, it is necessary and sufficient 
that the function v(«)—= Tu(«) converges in the N.F.O. to zero, 
if this is the case with u(#) in the N.E.F. 
After these general considerations we return to the complete trans- 
mutation. 
12. We called a transmutation complete in a circular domain 
(@) of centre w,, if there exists a circle (y), concentric with («) 
and therefore also a minimum circle (8) determined by the for- 
mula (7), such that the series (1) by which the transmutation 
is determined, produces for all functions belonging to that 
circle a transmuted function in the whole domain (a). Thus, a 
transmutation, of which we say that it is complete in a domain (a) 
is in consequence of this without more determined in a numerical 
field of operation, viz. (a), while we may take as associated functi- 
onal field: any group of functions belonging to a domain (g) as 
meant just now, not smaller than (3). The numerical field of the 
functions is in this case the circle (g); the upper limit in this 
field of the modulus of a function wu of the F.F., which we have 
represented by J/, in the general considerations, is now equal to 
the maximum modulus M (9) of « on the circumference of (@). 
For the complete transmutation the following proposition of con- 
tinuity holds: 
A transmutation T, which is complete in a numerical field of 
operation («), ts continuous in any FF. formed by the functions 
belonging to a circle (©) greater than the domain (8) corresponding 
to (a). 
We suppose v —=g8 + d,(e <0). In the second part of the proof 
of the proposition in N°. 4 it was found that in an arbitrary 
