la — pa ta — pha, 0 
| ae | 
(p—1)! | fie, ti i anti Pa, faa ties QO |? 
oe plz © (Ce bese) sts . Be pez Pa, 
thus also as 
ar dina, NA + € 
so that the contribution of the term [Fa say dn dag, te 
the value of (1) over e, and e, can be expressed as 
; n OF, en 
e+ a. day... a 
y=p A Ow, pt » 
and the value of (1) over 8 is obtained in the form: 
> 
P OF; : 
Dy a PES ha one Of 5 
eee ae A Out, : 
Ps d 
(indicatria j, a, aeq. indicatriz «‚... Uy); 
where A represents a point of o which may be different for different 
terms under the > > sign. 
Hence it follows immediately, that, if Q is a two-sided p-dimen- 
sional net fragment situated in S, and A denotes the boundary of 
Q, while the indicatrices of Q and R correspond, then the value of 
(1) over Ris equal to the value of (2) over Q. 
IV. To complete the proof of the theorem formulated in I, we 
consider a category yp of simplicial divisions of g such that the 
aggregate of the base sides possesses for y uniformly continuously 
varying plane tangent spaces, and the ratio of the volume of a base 
simplex to the (p—1)"" power of its greatest coordinate fluctuation 
does not fall below a certain minimum for w. Let g',g",... be a 
sequence of indefinitely condensing simplicial approximations of g 
corresponding to w. If, ong we construct an approximating simplicial 
representation g°?” of g®, then, by choosing both u and r above a 
suitable limit, we can, in virtue of III, see to it that the values of 
(1) over ge” and g® differ from each other by as little as we please, 
whilst g®) is covered by ge” with degree one, so that the values of (1) 
over g'”) and g®”) are equal. Thus there exists a value of (1) over 
g for ww which, naturally, does not change if, instead of w, some 
other category of the same kind is chosen. 
