( 731 > 
P, but no other singular point, is enveloped by the modified curve. 
If we consider for each of the intermediary curves the total angle 
which the given vector describes by a positive circuit, then on one 
hand it can only have continuous modifications and on the other 
hand it must remain a multiple of 2 zt. Thus it remains unchanged, 
gnd we can formulate: 
Theorem 5. The total angle which, by a circuit of a simple closed 
curve enveloping only one point zero, the vector describes in the sense 
of that circuit, is equal to ^(2 4*^,— n,), where n^ represents 
the number of elliptic sectors, n 2 the number of hyperbolic ones, which 
appear when a vicinity of the point zero is covered with tangent cui'ves 
not crossing each other. 
In particular for source points, vanishing points and rotation points 
this angle is equal to 4- 2 n. , 
We now surround P with a simple closed curve x which can be 
supposed as small as one likes, and we leave the vector distribution 
outside x and on x unchanged, but inside x we construct a modified 
distribution in the following way: 
Let us first suppose that for a positive circuit of x the vector 
describes a positive ' angle 2nir. From an arbitrary point Q inside 
x we draw to x n arcs of simple curve fi lf fi t , ...fin, not cutting 
each other and determining in this order a positive sense of circuit. 
Let us call p x the arc of x lying between fi p and fi t> +\, and G p the 
domain bounded by fi p , v x and fip+i- Along fi t we bring an arbitrary 
continuous vector distribution becoming nowhere zero and passing on x 
into the original one. Then along & such a one passing on x and 
in Q into the already existing vectors, that along the boundary of, 
G l positively described the vector turns a positive angle 2tt. Then 
along fi 3 such a one passing on x and in Q into the existing vectors, 
that along the boundary of G, positively described the vector turns 
a positive angle 2jt, etc. 
As the angle described by the vector in a positive circuit of x is 
equal to the sum of the angles described in positive circuits of 
the boundaries of the domains G lt G„ - G„ it is finally evident, 
that also for a positive circuit of G n the vector describes a positive 
angle 2 n. 
In each of the domains G p with boundary x p we choose a simple 
closed curve c p not meeting x p , of which in a suitable system of 
coordinates the equation can be written in the form x^-\-y^ = r t . 
Inside and on c p we introduce a finite continuous vector distribution 
vanishing only in the point (o,o) p , which is directed along the lines 
