121 
sional net fragment na (g), limited by an arbitrary simplicial approai- 
mation a(o,) of @,. 
That this algebraical sum is unequivocally determined by @, and 
can also be shown by a direct proof. 
If, namely, we have two different net fragments ma (o,) and 
n’a(Q,), limited by the same simplicial approximation a(g,), and if 
we represent the net fragment obtained out of n’a(e,) by inversion 
of the indicatrix, by na (g,), then na A and ite »,) form together 
a two-sided closed net), so that o {a (9,), na (9) + n'a (9,)} must be 
equal to zero, thus w {a (g,), n'a (@,)} = wo aan an na (Q,)}. 
If farthermore we have two different rte approximations 
a(g,) and a’ (e,) corresponding to one and the same mode of 
measurement of @,, two different simplicial approximations a (g,) 
and a’ (9,) corresponding to one and the same mode of measurement 
of 9,, and two two-sided net fragments na (e,) and na’ (e,), which, 
leaving their rims out of consideration, have the same base points, 
then for continuous transformation of « (9,) into a (@‚) we have: 
‘1 Sis 
w ja’ (9,), na (@,)} = w fa (9,), na (@,)}, 
and for continous transformation of a'(e,) into a (9): 
N 2 
Or, 
oo a’ (0), na’ (9) die a’ (o), na (9.)}. 
If finally we have two different modes of measurement u, and 
u, with corresponding indicatrices of @,, and two different modes of 
measurement gu, and w, with corresponding indicatrices of o,, then 
on account of the theorem, that a continuous one-one correspondence 
between two closed spaces possesses the degree + 1 ®, there exists 
a simplicial approximation a'(@,) corresponding to w',, covering a 
simplicial approximation a(9,) corresponding to u, with the degree 
one, and asimplicial approximation a'(9,) corresponding to u',, covering 
a simplicial approximation a (@,) corresponding to u, with the degree 
one, from which ensues immediately : 
w {a'(0,), nd’ (ep = w fa (9,), na (@,), 
with which the proof that the abovementioned algebraical sum 
depends exclusively on 9, and g,, is completed. 
In close connection with the looping coefficient is the notion of 
enlaced spaces recently introduced by Lususcun*). Two spaces enlaced 
- 
1) ibid, p. 316. 
2) ibid., p. 324 and p. 598. 
2) Ce 24 mars 1917, 
