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a. There exists inside c a simple closed tangent curve q through | 
P. We can then choose c smaller, so that it meets p, thus containing | 
in its inner domain a tangent eurve p x which (in its pursuing ] 
direction) • runs from P to c, and another running from c to P, 
and we further look for such tangent curves inside c which cross 
neither q 1 nor <>,. Of the possible kinds of tangent curves mentioned .J 
at the conclusion of § 1 we shall agree about those, which enter into a 
closed tangent curve, to continue them along that tab gent curve until | 
they reach either P or c, and to stop there. Spirally converging to 
, an inner circumference cannot appear, as the other end of such a 
tangent curve would be separated from P as well as from c, and so 
Would determine a closed tangent curve, outside which P would be : 
lying, which is impossible. Neither can appear spirally converging to - 
an outer circumference, as P would have to lie in that outer circura- | 
ference and the spiral would necessarily have to cross q 1 and p s . 
b. There exists inside c no simple closed tangent curve through | 
P. Then inside c there exists no simple closed tangent curve at all, 
so that again spirally conveiging is excluded. 
. In any case, if we-agree not to continue a tangent curve, when 1 
it reaches P or c, we can distinguish the tangent curyes inside c, and 
hot crossing and p 2 if the latter exist, into three categories: 
1 st . Closed curves, containing P but not reaching c. 
2 nd . Arcs of curve , joining two points of c, but not containing P. 
3 rd . Arcs of curve which run from P to a point of c (positive 
curves of the third kind ).or from a point of c to P (negativecurves 
of the third kind); 
Of this third kind there must certainly exist tangent curves. For 
otherwise the closed sets determined by the curves of the first, -and 
by those of the second kind would cover the whole inner domain of 
c, thus would certainly, possess a point in common . through this point 
however a curve of the third kind would pass. 
So we can commence by constructing one curve of the third 
kind and we choose eventually for it. If possible, we then draw 
a second curve of the third kind not crossing the first and we choose 
eventually p, for it. Into each of-the-two sectors, determined in this 
way mside c, we introduce if possible again a curve of the third 
kind, not crossing the already existing ones, and chosen in such a 
way that it reaches a distance as great as possible from tl)p two 
curves of the third kind, which bound the sector, whilst, if the new 
curve terminates somewhere on one of the curves bounding the sector, 
we fiirther follow the latter curve. In each of the sectors, deter- 
mined after that m the inner domain of c, we . repeat if possible tips 
