( 796 ) 
The element P, cannot coincide with R; for then it would be 
evident when describing the boundary of G, as a whole, that all the 
elements of J/ had their images on #', on the same outer arc of JJ. 
Neither can Q, coincide with R. 
Now the image of FR lies on k', between’ P, and Q,, thus within 
the are P, R as well as within the arc RQ, of k,. So B gets for 
the domain / the sign d and for the domain // the sign p. 
We must now distinguish three cases: the special inner arc of k'z 
namely appears as such either in / or in // or (if it contains &) 
in neither of the two. We shall continue the proof only for the 
first case; the other two can be treated analogously. 
By the point A the row of signs of G, is broken up into three 
successive parts, which we shall call e, 6 and rt; then the row of 
signs of I becomes 
o dt, 
and that of II 
Pp 6. 
The reduction of the row of signs of II, ie. of po, having to 
give pd, the reduction of o must give d. | 
Thus by substituting in an arbitrary row of signs a 5 for a d, 
its reduced row of signs remains unchanged. 
However, pd being the reduction of the row of signs of I, i.e. of 
edt, it is also the reduction of gort, ie. of the row of signs of 
G,, the property which we had to demonstrate. 
So we can choose Ya, in such a way, that its reduced row of 
signs is not pd, and, starting from that, we can continue the succession 
of y.’s arbitrarily until in this way after a certain segment of values 
a the row of signs would change for the first time for «,,. However, 
we can then again choose Ya, in such a way that its row of signs 
is not pd, and we can keep going on in this way. If an au is 
- ) 
reached as the limit of the a,’s, then if the case of exception appears 
also for that, it can be treated similarly; we can choose y, thus, 
. u 
0) 
that its reduced row of signs is not pd, and we can continue the 
succession of y.’s with this property till a value «, ay is reached. 
0) 
The whole set of es must, however, be denumerable, and finally 
ap is reached with a y.,,, whose reduced row of signs is not 
pd, thus whose row of signs itself is not p d. 
_On the other hand: of & Y¥z,, On whose boundary lies no invariant 
