22 
within W, then we start from the known expansion with known 
remainder of 1:(z—.2) in such a series. In the same way we may 
reach our present purpose, if we use the known expansion with 
known remainder of the just mentioned elementary function in a 
binomial series, viz. 
ni 
I x (ef) ..(@—B—m+41)  (w—A)...(@—B—n41) 1 
m 
mara (e—f)...(e—B—m) ee ee ee er es 
0 
Substituting this expression for 1/z—a in the integral (12) and 
choosing the path of integration so as to include, besides the point 
z=, the points z= 8, 8B+1,...,8-+n—1, we find *) 
n—1 
oi" Paro + Ba AE 
0 
where 
i 2) (v—6).…… (w—s—n+]1) dz 
(e—B)...(¢—B—n-+1) y= 
Tr (13) 
Zi 
Formula (13) is the ordinary formula of interpolation of Newton 
with a remainder added to it and valid for all complex «-values 
lying within W. 
If all points z=g8, BH1,...,8 And, are to lie within the 
integration-curve W, this curve will in general have to be modified 
with increasing ». It is required to choose W as fit as possible, that 
is to say: so that the remainder (13) tends to zero with indefinite 
increase of n, and that vet the aggregate of functions (x) for which 
this takes place, is as extensive as possible. If, now, the form (7) 
is taken as majorant-value of these functions, where the number a 
is, as yet, left undetermined and the number y, in order to have a 
definite case, is chosen zero (so that 8 > 0), it is found after a 
rather long but principally not difficult inquirement: 1. that the 
most favourable integration-curve is a circle with z= 7 as centre 
and n as radius so that it passes through the origin; 2. that for a 
a complex number may be taken lying on the circumference of the 
circle (1,1), with the specifications concerning the domains of validity 
already mentioned in the Statement of the above theorem. 
We may further observe that, in case the number ec in formula 
1) If a few points @, @+1,..., are excluded from the closed curve W, we 
obtain an expression the further examination of which leads to the so-called zero- 
expansions, which are treated in an elementary way by PINCHERLE (Rendic. d. 
R. Accad. d. Lincei, 1902, 2e Sem.) f 
