1105 
smaller than |@ (0)|, the substitution is not in any arbitrarily small 
neighbourhood of O determined by a series of the form (1) *). The 
substitution S,4.2, however, treated above, for which w (O0) = 0, is 
expressed by such a series for any function w,with O as ordinary 
point in a certain domain round Q. For functions with the singular 
point a—-++1, the domain of the series (1) belonging to Se 
consists of the continuum within an oval which is symmetrical with 
regard to the real axis, and cuts it in the points 
e—=—t(V¥5 +1) ande«=}4(V5—1)... .« (16) 
This oval is obtained as the locus of the points where az or |." 
is equal to |v—1). The circle with radius 
4(y5—1)=—0,6... 
is therefore the greatest circular domain with O as centre where the 
series in question converges. The singular point «—1 is moved 
here to the two points (16), from which it follows that also for 
Snas self, the greatest circular domain of operation around O is the 
one with the radius 4(“’5—1), in agreement with what has been 
. If, however, the function 
— 
u has the singular point # == — 1, the domain of the series (1) is 
the continuum within an oval into which passes the first mentioned 
when it rotates 180° round the imaginary axis, so that the same 
circle as considered just now, with radius }(“’5—-1) forms the 
circular domain of that series round O as centre. But the singular 
point &=— 1 is moved to the intersections of the oval with 
the circle with radius 1, so that now the circular domain of 
operation of St, round QO, is that: same circle; we observed 
said above with regard to the function 
this above with regard to the function . In general the point or 
wu 
the points towards which a singular sot s of a function w is 
transposed, always lies, for the operation of substitution, on the 
circumference of the domain where the corresponding series con- 
verges. For, that circumference is determined by the equation 
lo (a) — al = |s—al, 
which is among others satisfied by the points for which 
we) = 8 
which exactly form the removed singular points. If a, lies in the 
domain where |w(v) —z | < | s — «!, the circular domain of operation, 
with centre x,, of the series of the form (1) corresponding to Sy, 
has a radius equal to the minimum distance of that point to the 
circumference; hence, if one of the removed singular points lies at 
1) Cf. further treatment of the substitution in N® 17, 
