120 
for an arbitrary simplicial approximation’) a (9,) of g, and an 
arbitrary simplicial approximation a (o,) of @,. 
Let A, and K, be in Sp, two spheres lying outside each other, 
a(o,) a simplicial image of g, lying inside A, a(g,) a simplicial 
image of 0, lying inside A,. The looping coefficient of «(g,) with 
respect to a(g,) is then zero; for, by transferring K, with «(g,) 
outside A, to infinity, we can vary this looping coefficient only. 
continuously, thus not at all. 
We can now transform «@(9,) continuously into a(g,) by causing 
the. base poinis*) of « (o,) to describe continuous -paths, and we can 
choose for these base point paths such broken lines that in none of 
the intermediary positions of « (@,) an(i—1)-dimensionat element limit of 
a(o,) has a point in common with «(o,), neither an (n—h—2)- 
dimensional element limit of « (o,) has a point in common with a (9), 
whilst those intermediary positions of «(9,) which correspond to the 
angles of the base point paths, have no point in common with a (Q,). 
Then for this variation of @(oe,) the looping coefficient of a (@,) 
with respect to «@(g,) increases by a unit as often as an element 
e, of a(o,) is traversed by an element 7, of @(9,) positively, i.e. in 
such a way that the volume product of e,, #,, and the direction of 
motion of the traversing point is positive according to the above 
definition. 
If on the other hand we understand by na (g,) resp. na (g,) a two- 
sided (n—h)-dimensional net fragment®), limited by a(g,) resp. a(g.) 
and crossing a(9,) only in a finite number of points, belonging neither 
to an (4—1)-dimensional base limit of a(o), nor to an inner (2—/—1)- 
dimensional base limit of na (e,) resp. na (e,), whilst such a crossing 
is called positive, if in the crossing point the n-dimensional indicatrix 
composed of (—1)* « the positive indicatrix of a (g,) and the positive 
indicatrix of na (e,) resp. ne(g,) is positive, then for the above- 
mentioned variation of a@(o,) the algebraical sum of the number of 
positive and the number of negative crossings of a (o,) and na (g‚) 
increases likewise by a unit each time that a(o,) is traversed by 
«a (o,) positively. 
From this ensues that the looping coefficient of 9, with respect to 
o, can also be defined as the algebraical sum @ {t(9,), na(Q,)} of the 
number of positive and the number of negative crossings of an arbitrary 
simplicial approximation a(9,) of @, and an arbitrary (n—h)-dimen- 
1) Mathem. Annalen 71, p. 102 and p. 316. 
2) ibid., p. 317. 
5) ibid., p. 316. 
