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. 437
and therefore less than
r M (rn - rn) rm-1
n B rn+m
which, if ri = r (1 + e), is equal to
nB (1 + -
n BÊ e r ri
we can suppose the spheres chosen so that e does not become infinite; it is therefore
obvious that the integral is convergent.
It is tacitly assumed in this arrangement that the extent of the (n-2)-fold lying in
any finite n-fold extent taken entirely in the finite part of space is finite. This follows
from the method by which the (n - 2)-fold is supposed to be defined; for it can be
shewn that if (71,..., 7-.) be a power series, the extent of the (n- 2)-fold Q = O which
lies within a closed (n - l)-fold lying within the region of convergence is necessarily
finite*. This generalises the well-known theorem for functions of one variable, that a
power series cannot have an infinite number of zeros lying within a region which is
actually within its circle of convergence, that is, cannot have an infinite number of zeros
with point of condensation actually within the circle of convergence.
23. The investigation of ~ 22 applies to the integral (~ 13)
V = fHjm+dSn-2;
denote by (xi,..., x,) as before a point of the (n- 2)-fold, and by (t1,..., tn) a finite
point not upon the (n-2)-fold of integration; when R2 = x12 + + xn2 is large, that is,
for the very distant elements of the integral, and r2 t +... + t,2 is finite, we have
rm+2
Hn+i = R-n+m KM+2 (p) +...,
and it will (~ 12) be sufficient for the convergence of the integral that for any assigned
small quantity e it be possible to find a finite Ro such that the integral
fdSn-_
jRfn+mn
taken over the part of the (n - 2)-fold of integration, extending to infinity, for which
R > R, shall be less than e. We have in ~ 22 proved that this is so under the
hypothesis advanced.
24. The method just applied to the integral
fHn+i dSn-2
avails to justify the assumptions which have been made in regard to the other (n - 2)fold integrals considered in this paper.
* A sketch of a proof is added below, ~ 27.