Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

OF SEVERAL COMPLEX VARIABLES. 411
where on the left under the integral sign the first summation extends to every combination
of (r-1) different numbers d, e,..., h, k from 1, 2,..., n, and the second summation
extends to the (n-r+1) different values from 1, 2,..., n which mn can have so as not
to be equal to any one of d, e,....,,; on the right under the integral sign the
summation extends to every combination of r different numbers b, d, e,..., h, k from
1, 2,..., n.
3. Of this result it will be sufficient to give a proof for the case r= 3, the general
case being similar.
We suppose then a finite (non-singular) portion Hn- of an (n - 2)-fold, which is given
by the equations
fi (x,..., xn) = 0, f2 (!x,..., xn) = 0,
to be bounded by a closed (non-singular) (n - 3)-fold H,_3 given by
fi (i,..., xn) =O, f2 (x,..., xn)=0, f3 (Xi,..., )=.
We can imagine H,_2 divided into cells in a manner before indicated, the satellite points
of P, whose coordinates are (x1,..., oe), being denoted by Pk whose coordinates are
(x1 + dkXi,..., X + dkX.), k =, 2,..., (n - 2).
In general the differentials dkXr are arbitrary, save that the determinants of (n - 2) rows
and columns formed from their matrix must not all be zero; but we shall ultimately
find it convenient for our purpose to suppose that of the differentials
dn-_2x, dn-2x2)... dn-2xn
all but three, say all but dl-2xb, dn-2xe, dîn-2Xh, are zero; the ratios of these three will
then be determined from
-f dn-2zb +f dn_2Xe + -- dn-2h = 0,
axb axe aXh
af dn-2_ + -f dn-_2Xe + f dn- h = 0;
Xb axe, aXh
it is clear, in fact, that we can draw on Hn-2 through every point P a one-fold (or
curve) along which all the coordinates except Xb, Xe, Xh are constant; taking then any
point P and taking (n-3) of its satellite points P1,..., Pn3 arbitrarily, we can draw
such a curve through P and each of P1,..., Pn3,, and take for the satellite point P,_2
a point near to P along the curve through P; we thus arrange the cells into 'strips,'
each strip having (n-2) curves, such as those through P, P,..., Pn-_, as edges.
4. A set of (nz-3) neighbouring points Q1,..., Qn- in which the curves drawn on
Hn-2 through P...,.. P- intersect the (n-3)-fold Hn- may then be taken as the
satellite points on Hns of the neighbouring point Q in which the curve through P
intersects Hn-3; we have thus a possible basis for the division of Hn- into cells, which
it will later be convenient to adopt. We assume that the curve on H_-2 which is drawn
52 —2