857 
which in the point «, is determined by a series of the form (1)'), 
In future we shall say of a function which is regular within and on 
the circumference of a certain circle, that it “belongs to that circle” 
(after PiNCHERLE); its radius of convergence is in this case greater 
than the radius v of that circle. We shall further, if we speak of 
the domain (7), mean by it the closed set of points within and on 
the circumference of a circle with centre v, and radius 7; this 
circle we shall briefly indicate as the cirele (7) and if we mean a 
centre other than 2,, we shall indicate this specially. 
With regard to the complete transmutation thus defined Bourrer 
states his theorem XI, which follows here: 
La condition nécessaire et suffisante pour que la transmutation (1) 
fournisse une transmuce pour toute fonction régulière dans un domaine 
de rayon @ autour du point w,, est que la série 
N 
ay (x5) a, (#5) 
| —+t =, Sear ake (A) 
(2—-w,)? 
soit convergente pour toute valeur de z telle que \z—v,| = 9. 
That this condition is necessary is proved by BourLrr by applying 
the transmutation (1) to the function 
Wb 2) =a, (2,) + 
; 
0 
Z a 
] 
TP hase) eas Me By ee aoe CE) 
& v 
in which z is a constant, such that |z—wx,|= 9. This part of his 
proof, so far as [ am aware, is correct. 
‘IT object to the second part of his proof. BourLer says: If the 
condition is fulfilled 7’ may be represented by the identity : 
: iene 42) 
Tu = —, |—— w(z,, 2) dz, 
RJ 2-2, 
in which the integral is taken along the circumference C of the 
circle (©). 
This, however, is incorrect, if the function w has o evactly as 
radius of convergence. Yet, according to the theorem, such a function, 
being regular within (9), ought to have a transmuted in a, ; for by 
the expression “régulière dans un domaine de rayon 9’ Bourrer means, 
as he expressly states in a note before : “développable en une série. . ., 
1) BourLer’s exact words are: “Complete dans un domaine de rayon e”. This 
may give rise to the misunderstanding as if was meant that a transmuted function 
existed in the whole domain (e), which is not true in general. From the reasoning 
by which Bourrer arrives at the theorem XI to be mentioned directly follows 
only that, under the conditions mentioned there, there exists a transmuted in the 
single point vy. | have therefore preferred to say “complete in a point’, reserving 
the name “complete in a domain’ for another case (cf. NO, 4). 
Proceedings Royal Acad. Amsterdam, Vol. XIX. 
