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. 441
wherein %, 0o have no common factor vanishing at any point of K0; as we pass to
a contiguous region K1 we need a representation belonging to a point (T(1>,...) interior
to K1 of the form
By considering the equality
S0o S
in the region common to the proper regions of (i().,..., (0)) and (r7( >,..., 7Tp() we
are then able to deduce that aill the points for which.0 = O are also points for which
1= 0, and conversely.
We thus build up the idea of a zero (n - 2)-fold for the function F, and an infinity
(n- 2)-fold. If the former be represented by e =0, and the latter by P =0, the function
F can be represented in the form
= e,
where X is an integral function; and e, <) have no common zero other than points
belonging to an (n- 4)-fold at every point of which F is indeterminate.
27. Note to ~ 22. If an n-fold space bounded by a closed (n - )-fold be taken
actually within the region of convergence of a power series in the complex variables
1,.... ep, say b (e,..., tp), where n = 2p, the extent of the portion of the (n- 2)-fold
given by 0 = O which lies within the (n-l)-fold is finite. For consider the points of this
portion for which y2=72,..., p =, where 7y,..., yp are certain definite values; these
points are given by the equation in el, s (el, 2,..., p) = 0, wherein e is capable only of a
limited range of values determined by the (n-l)-fold; as this range is included within
the region of convergence of the l:-power series b (el, 7y2,..-, 7p), there cannot be an
infinite number of values of ev within this range for which (e,, 72,..., p) =. Thus on
the portion of the (n - 2)-fold + (e1, e,..., p) = 0O lying within the (n - 1)-fold there exists
only a finite number of values of el corresponding to given definite values of 2,..., *.
Let dS-_2 be an element of the (n - 2)-fold b =0; we have
dSn-2 = 1 J~2dSn-2 +... + |Kn-, n dSn-2,
the integrals being taken over the portion of the (n - 2)-fold which lies within the
(n - l)-fold; to prove that fdS_2 is finite it will be sufficient to prove that every one
of the integrals on the right is finite; we prove that the first of them is finite. Take
upon the (n - 2)-fold, = 0, (n - 2) independent sets of differentials given by the rows
dixi, dlx., dx3, O, 0, 0,,...
dx, xd2x, 0, dx4, 0, 0,...
d3x1, d3x., 0, O, dx0,,...
d4x1, d4x2, 0, 0, 0, dx,...
VOL. XVIII. 56