LS 
products appearing in this way for the different pairs (*,,%,) we 
1 al 
take the sum JS, our integral is to be regarded as — X the limit of 
n 
S for infinite condensation of z, and z,. 
Let us on the other hand represent each pair of points consisting 
of a point of a plane simplex determined by z, and a point of a 
plane simplex determined by z,, by the endpoint of a vector with 
fixed origin O, and having the size and direction of the vector con- 
necting the corresponding pair of points. Then for infinite condensation 
of z, and z, the ratio of the element of S corresponding to x, and x, 
to the value of the solid angle projecting out of O the simplotope repre- 
senting the simplexes z, and x,, approaches indefinitely to unity, and 
so does the ratio of the element of S corresponding to x, and x, to 
the part of B, filled for the simplicial approximation of @ corre- 
sponding to z, by the “base simplotope” resulting from 2, ond @,. 
As farthermore on account of the evaluability of 9, and o, the 
sum of the absolute values of the elements of S for infinite conden- 
é S 
sation of z, and z, cannot exceed a certain finite value, — conver- 
5 . n 
ges indeed to the looping coefficient defined as the degree of the 
representation «. 
§ 3. 
Let us now consider in Sp, two sets of points 9’, and vy’, which 
have no point in common and are successively a continuous one- 
one image of an /-dimensional two-sided closed space 9, and of an 
(a—h—1)-dimensional two-sided closed space g,, then for these all 
the considerations of the,former § remain of force. Let farthermore 
e," be a second continuous one-one image of o,, and let o,” be a 
second continuous one-one image of e,, then there exists a quantity 
7 with the property that if the distance of two corresponding 
points of 9,’ and oe," as well as the distance of two corresponding 
points of eg, and g,” is smaller than 4, the looping coefficient of 
0,’ with respect to ge,” is equal to the looping coefficient of @,' with 
respect to o,. 
From this ensues that in Sp, the looping coefficient of an A-dimen- 
sional two-sided closed space ge, with respect to an (n—/—1)-dimen- 
sional two-sided closed space g, not intersecting @, is equal to the 
value of the integral 
1 
fre prod; (di, ,di,, 7 
kn 
