24. If o(p, d, p,d, p,...dr—, p, dy+, p,--.- dm p) =r, be any function 
of p, that is linear in the differentials, then if d, denote differentiation corre- 
sponding to d,; p, while A affects the subscripts of the differential symbols that 
follow it, we have for integration through the m space, | 
’ om 
~ ~m -1 
(i d, + A Op =} Ao(p,d’, p,..-.Ump) 
over the 7’ pair of faces, where d’, p— + d; p is outward at each face. Thus 
summing for every pair of faces and noting that, by th 2’, = (— 1)"1.d; A =m 
A.d,%(p,d, p, d; p,..dm p), we have the following theorem connecting inte- 
gration through any m-space with integration over the (m-1) boundary of that 
space : 
Theor. 7. m (A. dy 9 (Py dy By dy Py» «dm p) 
ee “TA ¢ (p, dp, d's p,..: d'm p)- 
25. The elements d, p,...d, p are in the same order of arrangement 
throughout the integration; and at the boundary d’, p, .. d’m p are in the same 
order, with d’, p outward. 
Any space may be divided into these small finite spaces by the differential 
lines and the contributions to the boundary integral made by intermediate faces 
cancel each other, except where the value of © is different on two sides of the 
same face at the same point of it. In this case such intermediate boundary must 
be retained in computing the boundary integral. 
V. ALTERNATE PRODUCTS. 
26. Let? (1, po, -- > Pn) = Pi - Oo Po On Pn- 
We then have: 
i, ae — PePiy Tsp is- -) Siew 
$; Po 92) Pa, +++ On Po | 
9, Pn 2 Pn On Pn 
27. _ This determinant is multiplied out by ordinary rules, except that since 
the factors may not be commutative, or even associative, each product must appear 
in the order and association given by o. The expansions of Theor, 2’ are then 
expansions of this determinant in terms of its minors either by rows or columns, 
according as we employ place or variable substitutions. 
28. When two factors of the product 9 are commutative, for all arrange- 
ments of the variables, such factors are also commutative in A. The inter- 
change of the corresponding functional symbols alone has, therefore, the same 
