115 
two arbitrary elements having a side in common these indicatrices 
satisfy the relation prescribed for two-sided spaces *). 
So Ris a closed two-sided two-dimensional space. 
The set of the vector directions of Sp, forms likewise a closed 
two-sided two-dimensional space (of the connection of the sphere) 
which we shall represent by 4. The positive indicatrix of the spheres 
of Sp, (and with it at the same time the positive indicatrix of B) we 
determine by regarding them as boundary of their inner domain *). 
If we conjugate to each pair of points consisting of a point of 
k, and a point of &, the direction of the vector connecting the two 
points, we determine a continuous one-one representation a of R on 
B. To this representation belongs a finite integer c independent of 
the mode- of measurement of FR, and therefore also of the mode of 
measurement of £, and #,, which is called the degree of the repre- 
sentation, and possesses the property that the image of FR covers 
positively each partitional domain of B in toto c times ®). 
It is this degree of representation which we define as the looping 
coefficient of k, with respect to k,. 
By exchange of £, and &, we find that on one hand the indicatrix 
of F# changes its sign, but on the other hand each image point on 
B is replaced by its opposite point. So the looping coefficient of k, 
with respect to k, is equal to the looping coefficient of k, with respect 
to k,. 
We shall now show that for rectifiable curves the looping coeffi- 
cient of £, with respect to #, can be expressed by the formula: 
1 
iz ve prod (derd TA ae ant wo eek 
An 
This integral namely can be interpreted for rectifiable curves as 
follows: We construct in #, resp. &, a simplicial division *) z, resp. 
z,. To this corresponds a simplicial division z of R, whose base 
simplexes *) are determined in connection with the base ares °) of 
1) ibid., p. 101. 
2) ibid., p. 108. 
3) ibid., p. 106. 
4) ibid., p. 101. 
5) That here the base simplexes are found by division of a paralleloelement, 
not as l.c. by division of an element, has of course no influence on our reasoning. 
Moreover, after HADAMARD (comp. J. TANNERY, “Introduction à la théorie des 
fonctions d'une variable”, Vol. Ll, p. 463) a simplicial division of the parallelo- 
elements can be subdivided to a simplicial division of the elements. 
6) i.e, one-dimensional base simplexes. 
