indicatrix and situated in S, f a sequence of indefinitely condensing 
simplicial approximations P’, P",... of HF corresponding to a 
category w of simplicial divisions. If the values of (3) over P’, P",... 
converge to a limit which is independent of the choice of f so far 
as it is left free by w, then we call this limit the value of (3) over 
F for w. 
II]. We shall now occupy ourselves with the value of (1) over 
the boundary 8 of a p-dimensional simplex o provided with an 
indicatrix and situated in S. In doing so we take it that the indi- 
catrices of 3 and o correspond, that is to say, the indicatrix of an 
arbitrary (p—1l)-dimensional side of o is obtained by placing the 
vertex of o which does not belong to this side last in the indicatrix 
of 6 and subsequently omitting it. We begin by confining ourselves 
to the contribution of the single term 
P 
= 
les ee ee Oene: dea, 
to the value of (1) over 8. By a suitable simplicial division ¢ of 
the’ Spacen. 2 a, ) we determine a simplicial division of 3, 
ne 
ale): 
pl 
The family of those (n—p + 1)-dimensional spaces within which 
are constant, cuts the plane p-dimensional space in 
whose base simplexes correspond in pairs to those of (wz,,...# 
Vay 97 2 een 
which o is contained, in a family of straight lines which connect 
pairs of corresponding base simplexes of } into p-dimensional trun- 
cated simplicial prisms. If e, and e, are a pair of corresponding base 
simplexes of 8, d the concomitant truncated simplicial prism, /a line 
segment having components 74 ,...7, which leads from, a point 
Pv 
of e, to the corresponding point of e,, then the contribution of the 
term |F. oere , Utyz,.... de, _, to the value of (1) over e, and 
DE Pe 
e, becomes 
n OF. ty 
= Bee . Pale ° A Oa, { ze é, 
where A denotes a point of 5 which may be different for the diffe- 
rent terms under the > sign, and ¢ becomes indefinitely small with 
nespect tO stats seg j for indefinite condensation of &. 
p— 
INOW let BB, s B, be a vertex indicatrix of e, and ,r, the 
value of «, at B,, then the value of 1%, . eta .-- a) can be expressed as 
