( 792 ) 
we call it an even group, and we represent it by two signs; if it 
lies between two inner ares of different curves, we call it an odd 
group, which we represent by one sign. 
The number of the groups must of necessity be finite; thus be- 
longs to each « a row of signs containing an even number of signs 
p or d, of which not more than two equal ones follow on each other. 
Let us now first consider a value a,, for which the curve ’,, possesses 
in the immediate vicinity of each of the dividing points and dividing 
arcs points on both sides of 4. On either side of such a value a, 
there is a segment of values of «, to each of which belongs the 
same row of signs as to ¢,. For when a converges to a,, the 
corresponding set of dividing points and dividing arcs converges 
uniformly to the whole set of elements belonging to a, 
Let us next regard values a,, where 
if in the immediate vicinity of the 
d- elements which do not possess (he 
just-mentioned property, the boundary 
| nee aes of y., belongs exclusively to kz, or 
ve N exclusively to k',, (see fig. 2). For 
i Jo values « differing sufficiently little 
/ from that a, the corresponding set of 
Fig. 2. elements uniformly differs indefinitely 
little from a part of the set of elements corresponding to a, (which 
part can be different for different «’s). However, in such a part 
only even groups can be non-represented, and farther even groups 
belonging to that part are approximated by even groups and odd 
ones by odd ones. 
Thus for the two regarded species of values «a, within a certain 
vicinity each row of signs is obtained out of the row of a, by 
suppression of a certain number of pairs of equal successive signs. 
We shall now understand by the reduced row of a given row of 
signs that one, which is obtained out of it by checking off a pair 
of equal successive signs, and repeating this process, until it is no 
longer possible. *) 
Thus e.g. of pom td pil p p 
the reduced row of signs is p. 
1) That this reduced row of signs is determined unequivocally is evident e.g. as 
follows: Let in a plane with rectangular system of coordinates p mean: semi- 
revolution about the point (+ 1,0), and d: semi-revolution about the point 
(— 1,0), and let an arbitrary row of signs represent the product of the opera- 
tions indicated by the signs. Different reduced rows of sign then furnish different 
results, and an arbitrary row of signs is equivalent lo its reduced one. 
