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joining a point B of one of these orthogonal arcs with a point C of the 
other, and we bound the modified sector by the orthogonal arcs HB 
and CK and the tangent arc BC. 
If a parabolic sector is bounded by the base curves k and k\ it is 
always possible to choose between them a finite number of base 
curves k n in such a way, that each k p and k fi +\ can 
be connected, inside the sector but outside the leaves lying in it, 
by an orthogonal arc. By those orthogonal arcs and the arcs of base 
curves joining their endpoints we bound the modified sector. The 
simple closed curve c obtained in this way has a direction of tangents 
varying everywhere continuously, with the exception of a finite number 
of rectangular bends. To a definite sense of circuit of c\ which we 
shall call the positive one, corresponds in each point of c' a definite 
tangent vector, and for a full circuit of c' that tangent vector 
describes a positive angle 2 jt. 
We shall now consider two successive parabolic sectors, n 1 and 
jr 3 , of which (for the positive sense of circuit) the first is positive, 
therefore the second negative, and we suppose them to be separated 
by a hyperbolic sector f. On the orthogonal arcs belonging to the 
boundary of n 1 the given vector then forms with the tangent vector 
an angle ^2 n — (measured in the positive sense), on the orthogonal 
arcs belonging to the boundary of jr a an angle ^2 n -{- ^ jr. 
The transition takes place along the tangent arc belonging to the 
boundary of f, by a negative rotation over an angle nr of the given 
vector with respect to the tangent vector. 
The same remains the case if we suppose at, to be negative, jr, 
to be positive. 
But if we suppose f to be an elliptic sector, then the transition 
under discussion takes place along the tangent arc bounding f, by a 
positive rotation over an angle n of the given vector with respect to 
the tangent vector. 
As now the total angle, which the given vector describes for a 
full circuit of c\ is equal to the total angle which the tangent vector 
describes plus the total angle which the given vector describes with 
respect to the tangent vector, the former angle is equal to jr (2 -f- n x — » 2 ), 
where n x represents the number of elliptic sectors, the number of 
hyperbolic ones. 
Let further j be an arbitrary simple closed curve enveloping P, 
but enveloping no other singular point, then we can transform & into 
y by continuous modification in such a way, that at every moment 
