114 
of the number of times that k, circulates around k,, and generally 
iL En 
defined as PE the variation corresponding to a circuit of #, of the 
x 
solid angle projecting /, out of a variable point of 4. 
A first objection to this definition is, that without further agreement 
it can be applied only to special categories of simple closed curves. 
For, as soon as e.g. a simple closed curve & intersects of a sheaf s 
all the rays contained in a certain finite solid angle, the solid angle 
projecting & out of the vertex of s, has no more a definite value. 
A second objection to the definition is, that it cannot be generalized 
to a notion of “looping coefficient in Spa of a two-sided closed Spr 
with respect to a two-sided closed Spy—n—1 not intersected by Spr.” 
In the following we shall give a definition for which these two 
objections bave been annulled. 
§ 1. 
On each of the two curves /, and k, we construct a scale of 
measurement®), and we consider the set /& of pairs of points consisting 
of a point of %, and a point of &,. A part of A determined by an 
element?) of &, and an element of &, we shall cali a parallelo- 
element. It appears as a continuous one-one image of a parallelogram. 
Each of these image parallelograms can be divided into four triangles 
with a common vertex inside the parallelogram and with their bases 
in the sides of the parallelogram. Accordingly we can divide each 
paralleloelement of Zl into four two-dimensional elements, and with 
this we attain that the whole set R is divided into two-dimensional 
elements which by their mode of being joined cause A to appear 
as a closed two-dimensional space. *) 
Let p be a paralleloelement of R, d, resp. d, the corresponding 
element of k, resp. &,, A, resp. 5, the negative resp. positive end- 
point of d,, A, resp. B, the negative resp. positive endpoint of ds, 
we then define the row of pairs of points (A,A,), (4, B,), (B,B,) as. 
a positive indicatrix of the partitional simplex *) of p determined by 
those pairs of points, and with the aid of it we fix the positive 
indicatrix of the four elements of A belonging to p°). In this way 
we determine of all elements of A the positive indicatrix, where for 
1) Mathem. Annalen 71, p. 98—100. 
2) ibid., p. 97. 
3) ibid., p. 98. 
4) jbid., p. 100. 
5) ibid., p. 101. 
