( 793 ) 
We can now resume our results concerning the two regarded 
species of values a, as follows: 
For each of these values a, the reduced row of signs is invariant 
within a certain vicinity. 
We have now still to investigate a third species of values a,, 
namely those for which dividing points appear, in whose immediate 
vicinity 4, and 4",, do not intersect each other, whilst the boundary 
of y., belongs on one side of that dividing point to /:,, , on the other 
side. tO A’, - 
Such a dividing point may be endpoint of a recurrent arc, i.e. of 
an are along which 4, and 4",, coincide, but in a direction, which 
corresponds for one of the two, but not for the other, to the sense of 
circuit of ya. 
We shall now confine ourselves first to such values a, of the 
third species, as for all dividing points characterizing them as such 
possess that recurrent are, and that in such a manner that in the 
immediate vicinity of it #,, and #, come on either side of each 
other (comp. fig. 3, where PQ is the recurrent are and to the 
left of P the curves k,, and 4, can 
be supposed to wind an infinite 
number of times round about each 
other, when approaching to P). 
For values « differing sufficiently 
little from a, the corresponding set of 
elements uniformly differs indefinitely 
4 little from the sum of a part (with 
the same properties as for values a, 
Fig. 3. of the second species) of the set of 
elements corresponding to a, anda part of the recurrent arc. On account 
of the absence of an invariant point on that are here again the 
reduced row of signs must within a certain vicinity of a, be the 
same as that of a,. 
x There remain still the values «, of the third 
a beslis Rok 
‘ species having dividing points characterizing them 
as such, which either possess no recurrent arc, 
or such a one that in its immediate vicinity the 
curves k,, and #,, lie only on one side of 
/ \ each other (See fig. 4). 
/ | 7 We here see clearly, that the proof for the 
Pf invariability of the reduced row of signs does 
a hold for a segment of values a following on 
Fig. 4. a,, but not for one preceding «,. 
