359 
— 1, according to which we distinguish domains of the first, the 
second, and the third kind. 
If for the representation «*’, which may be denoted henceforth 
by af, all domains d, are of the first kind, we have attained our 
aim; for then we have transformed « continuously into a represen- 
-tation of w in a single point of u’. So we further confine ourselves 
to the case that among the d, there are domains of the second or of 
the third kind, and we will suppose that there occur moreover 
domains of the first kind. Then there is certainly a domain d, of 
the first kind adjacent to a domain ds of the second or third 
kind. The domain formed by d, and ds together, may be indicated by 
ds, the sphere deduced from d,s by contraction of its rims into 
points, by d,s. We then can modify the univalent continuous repre- 
sentation of ds on uw’ determined by ey continuously into a conti- 
nuous one-one representation of ds on uw’. The variation of the 
image points of those rims of d,s which originate from d, necessarily 
implied by this modification, can once more be followed in the man- 
ner described above by a continuous modification of the representa- 
tion determined by az of those residual domains of d,s which origi- 
nate from d,, furnishing us with a representation a’ distinguishing 
itself thereby from «af that a domain of the first kind and a domain 
of the second (resp. third) kind have been united into a single 
domain of the second (resp. third) kind. 
By repeating this operation as many times as possible we arrive 
after a finite number of steps at a representation al), distinguishing 
itself thereby from ay that all the domains of the first kind have 
been absorbed by domains of the second and of the third kind. 
If there are for the representation a), which may be denoted 
henceforth by a, domains of the second as well as of the third kind, 
we consider a domain d. of the second kind separated by a simple 
closed curve z,, from a domain d. of the third kind, and we repre- 
sent the domain formed by d, and d, together, by d,,, and the 
sphere deduced from d,, by contraction of its rims into points, by 
d;,. Moreover we represent by P, the image point of 7,, for ay, by 
P, the opposite point of P, on u’, and we modify the vepresenta- 
tion of d;, determined by a, into a representation of d;, in the 
single point P,, by diminishing the polar distances measured from P, 
continuously and proportionally to each other to zero. The variation 
of the image points of the rims of d.. necessarily implied by this 
ite 
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