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. 435
where C is the area of such a circle, may be finitely greater than zero for all values of R
greater than a certain assignable Ro.
We proceed to shew that under this hypothesis the infinite series formed by the
sum of the negative (2 + m)th powers of the zeros of the function is an absolutely
convergent series. The case n = 1 is that of the latter of the two kinds of functions
considered in ~ 20.
Let concentric circles be described with centre at a finite point of the plane; consider the greatest number of zeros of such a function which can lie in the annulus
between two such circles of radii r and r' (r' > r), the circles being supposed to be drawn
so that no zeros lie actually upon them. By the hypothesis, if r be taken great enough
(and finite), the annulus may be divided into regions each containing a finite number,
say M, of zeros, such that if C be the area of every such region
rOm-i C B,
where B is some quantity greater than zero. Let k be the number of these regions,
which is finite so long as r' is finite. Then
7 (rV2 _r) r m-i _ kB;
as there are kM zeros in the annulus, the sum of the moduli of the inverse (2 +m)th
powers of these zeros is less than
kM
r2+m,
which in turn is less than
7rM (r/2 - r rm r 1
B r2+m
which, if r' =r (1 + ), is equal to
o-B (1 +)M( +)(r r);
we can suppose the successive circles drawn so that e remains constant; then the sum of
the moduli of the inverse (2 + m)th powers of all the zeros of the function which lie
beyond the circle of radius r0 is less than
l l t r he
and can be maade as small as we please by taking ro large enough. This proves the
convergence of the series.
22. Pass now to consider an integral function of p coinplex variables, and consider
the (n-2)-fold over which the function vanishes, this being supposed to extend to
infinity. Imagine closed (nz-l)-folds to be described everywhere convex, and as far as
possible, for the sake of definiteness, of spherical form, with the condition that the
extent of the zero (n-2)-fold contained in any one of them shall be some definite
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