151 
Of this theorem, which was enunciated by Poincaré‘) in 1899 
already, without proof however, and in a form expressing the rule 
of signs in a less simple manner, I shall here give the proof anew, 
editing it somewhat more precisely than in my quoted paper. 
I]. In the n-dimensional space (2,,...2,), which we shall denote 
by S, let the p-tuple integral 
p 
j= Pa a, (Ee ERD dite. vann a er eat) 
a 
be given, where the p's are continuous. 
Let Q be a two-sided p-dimensional net fragment’) provided 
with an indicatrix and situated in S, 6 a base simplex of Q with 
the vertex indicatrix “A A... AA, A an arbitrary point of o, 
AP a. the value of Pane, at A, ,v, the value of z, at A,. For 
every o we determine the value of 
i =, PEED © Glee 9 
P 
where 
ey eee | 
| 
lt ra 
sta, “) = ag | x En | ’ 
| | 
een 
et EN U 
and where, for different terms under the = sign A may be chosen 
differently; and we sum ‚p over the different base simplexes of Q. 
The upper and lower limit between which this latter sum varies 
on account of the free choice of the points A, we call the upper 
and lower value of (3) over Q. 
If we now subject Q to a sequence of indefinitely condensing 
simplicial divisions which give rise to a sequence Q’, Q",... of 
net fragments covering Q, then, as v increases indefinitely, the upper 
and lower value of (3) over Q”) converge to the same limit, which 
we call the value of (3) over Q. 
Let F be a two-sided p-dimensional fragment provided with an 
1) Les méthodes nouvelles de la mécanique céleste III, p. 10. The significance 
of the rule of signs here formulated, is apparent only after comparing former 
publications of the same author from the Acta Mathematica and the Journal de 
PÉcole Polytechnique in which the equivalence of the identically vanishing of (2) 
and the vanishing of (1) over every g was pronounced, 
*) Math. Annalen 71, p. 316. 
