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starting at ?/, divide the part of t, approximated l»y R and K\ 
into successive “transformation domains”, separated by Schnitte in 
this circumference, so that in a circuit each following domain is the 
image of the preceding one; of the last domain of the series then 
in general only a part appears, but three at least must be complete; 
fartheron the derived set of this last partial domain must cohere 
with that of the first domain; and we may suppose the number of 
the domains to be finite, as otherwise we should at once fall back 
on case II. 
These domains of the circumference we 
can reduce by destroying in each of them 
the subsets of T, belonging to no other 
part of r. If e.g. in fig. 7 the first domain 
extends from a to b , the second from b to c, 
we can deprive the first of the arc pq, the 
Fig. 7. second of the arc rs. 
On the circumference of the rest set R, we have then still a 
division into domains with the same properties, as the original division 
of the circumference of T. By a displacement of the separating 
Schnitte between the domains (after which in general the first as 
well as the last domain will be a partial one) it may occur, that 
the same process of reduction is once more applicable to R and 
in this way it is repeated, until after a denumerable number of 
reductions a set R 0 is left, which by no displacement of the separating 
Schnitte can be made fit for further reduction. 
A domain d then spreads over a part of the circumference h of 
an arc of curve k in such a manner, that the domain itself as 
well as the rest of the circumference possesses the whole k as its 
derived set. 
Let the next domain d ' follow of that rest of the circumference 
of k a part e', before it leaves k, and let / be the part of h belonging 
neither to d, nor to e'. Let <?' be the image of e, and let us call g 
the segment of h approximated by k'. These notations are illustrated 
by two examples in fig. 8. 
We now distinguish the following two cases: 
I6«. f does not lie everywhere dense on k. Then e must lie every¬ 
where dense on k. 
For in the opposite case we might take the part of h enclosed 
between the end Schnitt of e and the end Schnitt of e* as a domain, 
which would be capable of reduction, and this is impossible. 
But if e lies everywhere dense on k, the image of k is a part of 
K so that we fall back on case II. 
