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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 g, and e,, which is called the degree of the repre- 
sentation, and bas ihe property that the image of F 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 9, with respect to @,. 
Exchange of g, and ge, has only this consequence that the indi- 
catrix of Mè changes its sign in some cases, and that each image point 
on B is replaced by its Opposite point. So the looping coefficient of 
0, with respect to g‚ and the looping coefficient of 9, with respect 
to 9, are either equal or opposite. 
We shall now show that if e, and eg, are evaluable, i.e. if they 
have a definite finite /-dimensional resp. (%7—/—1)-dimensional volume, 
the looping coefficient of 9, with respect to 9, can be expressed by 
the formula : : 
1 r - 3 : 
zl Vol. prod. (tis digg TB) nn A eN 
where 4, represents the (n—1)-dimensional volume of an (n—1)- 
dimensional sphere described with a radius 1 in the Euclidean Sp, 
If namely eg, and eg, are evaluable, this integral can be inter- 
preted as follows: We construct in g, resp. 9, a simplicial division 
z, resp. z,. To this corresponds a simplicial division z of #, whose 
base simplexes are determined in connection with the base sim- 
plexes of ge, and ge, in the same way as we have determined 
above the elements of # in connection with the elements of 9, and 
e,. Each bas? simplex of z, resp. z, we replace by the plane simplex 
with the same vertices. Let %, be the plane simplex corresponding 
to the base simplex >, of g,, #, the plane simplex corresponding to 
the base simplex 2, of g,, r the distance of their centres of gravity, 
then x, and z,, the former regarded as an /-dimensional, the second 
as an (m—/h—1)-dimensional vector, determine together with a line- 
vector of size r'—” in the direction of the straight line connecting 
their centres of gravity, a certain volume product *). Of the volume 
1) The sign of this volume product we determine as follows: Afler having 
formed in the manner described above out of (—1)’ X the positive indicatrix of 4, 
and the positive indicatrix of 7, an indicatrix of a simplotope s parallel to xj and 
xyz, we add to the Jatter indicatrix the endpoint of a linevector described out of a 
point of s in the direction of the straight line connecting the centres of gravity of 
xa and x,. Tue sign of the #-dimensional indicatrix found in this way determines 
he sign of our volume product. 
