896 



1. lite resultant T is a normal transmutation with a pair of 

 fields of which the N. F. is the circle (a), and the F. F. consists of 

 the f unctions belonging to a circle [q] arbitrarily little greater than {1^). 



2. The series P belonging to T is complete in («)" ivith a corre- 

 sponding domain ivhich is at most equal to (/?). 



We suppose («) and (y) not to be the maximum domains of com- 

 pleteness of l\ resp. P^. Then, since corresponding circle radii, 

 according to oiii' supposition, increase and decrease continuously with 

 eacii other, there corresponds to an3^ number q surpassing /? by 

 an aibitiarily small amount, a number (»', such that the radius which 

 for the series P^ corresponds to «, i.e. y, is less than ^', and the 

 radius that for the series P^ corresponds to q', and is consequently 

 greater than /i, is less tlian q. The transmutation P,, and con- 

 sequently also 1\, according to premise 3, is in that case normal, 

 if we take as N. F. 0. the circle {q') and as N. F. F. the circle (^) ; 

 in the same way P^ and 1\ are normal, if we take as N. ¥. 0. 

 the circle («) and as N. F. F. the circle ((>')')• If ^* is a function be- 

 longing to {{)), then, in connection with wiiat was just mentioned, 

 V = 1\ (u) is a function belonging to ((>'), and tn = 1\ (v), or 



a function belonging to («). The transmutation therefore produces 

 for all functions belonging to {q), a transmuted that is regular within 

 and on the circumference of the circle («)• Since among these 

 functions there are as a matter of course the integral rational ones, we 

 have, in order that the conclusion that 7' is normal may be admitted, 

 only to investigate the question of continuity. This does not cause 

 any difficulty. For according to what has just been observed in con- 

 nection with premise 3, 1\ is continuous in the F. F. of functions 

 that belong to (q'), and the N. F. («) ; i.e. corresponding to any 

 positive number t, chosen arbitrarily small, there is an amount ti, 

 such that 



I IV ^ T^iv) I < T, in the closed domain («), 

 if 



\v\ <^t] , in the closed domain {q'). 



Again premise 3 includes that T^ is continuous in the F. F. of 

 functions belonging to (q), with respect to the N. F. {q') ; i. e. : 



'j If we state in future, without any additional observation that we take a circle 

 (r) as N. K. F., we shall mean by it that the F. F. consists of all the functions 

 belonging to the circle in question. This is the case with which we are principally 

 concerned and it is therefore easy to have for it a shorter expression that cha- 

 racterizes the F. F. 



