116 
k, and 4, in the same way as we have determined above the elements 
of R in connection with the elements of 4, and #,. Each base arc 
of z, resp. z, we replace by the corresponding “chord”, i.e. by the 
straight line segment with the same endpoints. Let #, be the chord , 
corresponding to the base are B, of /,, x, the chord corresponding 
to the base are @, of &,, r the distance of their midpoints, then 
2, and x, regarded as vectors determine together with a vector of 
size r-2 in the direction of the straight line connecting their mid- 
points, a certain volume product. Of the volume products appearing 
in this way for the different pairs (#,,%,) we take the sum S; our 
1 
integral is to be regarded as cs X the limit of S for infinite con- 
A oe 
densation of z, and g,. 
Let us on the other hand represent each pair of points consisting 
of a point of a chord of &, and a point of a chord of &,, by the 
endpoint of a vector with fixed origin OQ, and having the size and 
direction of the vector connecting the corresponding pair of points. 
Then for infinite condensation of z, and z, the ratio of the element 
of S corresponding to #, and #, to the value of the solid angle 
projecting out of QO the parallelogram representing the chords %, 
and z,, approaches indefinitely to unity, and so does the ratio of the 
element of S corresponding to x, and x, to the part of 5 covered 
for the simplicial approximation’) of @ corresponding to z, by the 
“base parallelogram” resulting from 8, and ~,. 
As farthermore on account of the rectifiability of 4, and £, the 
sum of the absolute values of the elements of S for infinite con- 
densation of z, and z, cannot exceed a certain finite value, — 
v 
converges indeed to the looping coefficient defined as the degree of 
the representation e. 
On the other hand for rectifiable curves holds also the defimtion 
of the looping coefficient as a variation of a solid angle mentioned 
in the beginning, and we easily see also this definition to be equivalent 
to the expression (1). 
+) 
Let now in Sp, be given a two-sided closed /-dimensional space 
o, and a two-sided closed (2—/--1)-dimensional space e, not cutting 
Nal 
e,, each provided with a positive indicatrix. We make 9, as well 
Nd 
1) Mathem. Annalen 71, p. 102. 
