862 
not so, but that the upper limit of a, for the circumference of («) 
was a certain number a <a. According to the supposition made, 
the values which an arbitrary function a (), with rank m>m:, 
assumes on the circumference of (a) would all have a modulus 
smaller than 
(ad + er. 
We now suppose ¢ chosen so small that 
até <i a. 
Then an njinite number of functions a (7) satisfy the condition 
that in a certain point of the domain (a) 
an(e) | > (a + er. 
Among this infinite number of functions we select one for which 
m>m:. But this same function on the circumference (a) satisfies the 
condition 
dale) JS (epe) 
Since the maaiunum modulus of a function in a domain (a), lying 
within the domain of regularity, is found on the circumference of 
(a), the preceding two inequalities are contradictory and the propo- 
sition has been proved. 
If now the functions a, (x) satisfy the conditions mentioned above, the 
transmutation determined by the series (1), is complete in the domain 
(u), as the following generalisation of the theorem of BovrLer shows: 
If the series (1) is to produce in any point of a yiven circular 
domain (a) a transmuted for all functions belonging to a domain 
(0) concentric with (@), it is necessary and sufficient that the radius 
(0) is not smaller than the number B determined by the equality 
md B eee eee SA 
or equivalent to this: 
The circle (8) is the mintmumeircle which has the property that 
the series (1) produces for all functions belonging to it, a transmuted 
m any point of the given circle (a). 
Proof 1°. The condition, mentioned in the theorem, is necessary. 
Let us suppose, in order to prove this, that 9 is smaller than 3; 
we have only to indicate a function belonging to (e) and for which 
the series (1) does not produce a transmuted in a certain point of (a). 
Choose a number 7 such that 
Ore Fe 
1 
ty + re 
and consider the function 
u 
’ 
Lv 
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