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.

440 MR BAKER, ON THE THEORY OF FUNCTIONS
of which the infinite part, for diminishing e and fixed a, is exactly loge. But as we
approach the (n- 2)-fold in the way here taken we can put 6 = heei0, (see ~ 16); so that
the infinite part of log is also loge. Thus in the limit the difference
_ — \x (\1 t) dSn_ - log s6
remains finite, as stated.
26. In this paper we have hitherto supposed the (n-2)-fold of integration to be
given à priori, by means of a succession of power series. Some remarks must be made in
regard to the problem in which this conception has arisen.
Suppose that a single-valued function F(r1,..., r) is known to exist for ail finite
values of T1,..., p, and to have no essential singularities for any finite values of
',..., Tp, namely can be represented in the neighbourhood of any finite point (r1(0),..., T(
in the form
F= o (T1-7,I..., Tp- 7-p() ). (71r- -r7)..., r r - (0))
where r0, o0 are ordinary power series (of positive powers) with a presumably limited
common region of convergence. If the series 0o, 4o have a common vanishing factor
at (1(0),..., T(0)), that is, are both divisible by another convergent series which vanishes
at (-i1(),..., pr)), this factor may be supposed divided out (Weierstrass, Werke, ii. (1895),
p. 151). There is then a region about (r1(),..., Tp0>)), within the common region of
convergence of 0k and S0, but not necessarily coextensive with it, such that, if
(C1 + 7r 0),..., Cp+ -p (0))
be any point in this region, and the series '0, b0 be written as power series with this
point as centre, by putting Tk-7k(~) = Ck +Uk, the resulting series in 'ui,..., up have no
common factor vanishing at uzl=0,..., p=O (Weierstrass, loc. cit., p. 154). This region
we may temporarily call the proper region of (T1(~),..., 7Tp()) for the function F. There
may be points within this region at which *o, <so both vanish without having a common
factor vanishing there, such points lying upon an (n - 4)-fold at every point of which
F has no determinate value. If the series o0, /0 as originally given have no common
factor vanishing at (r)),..., r7(0)) there will similarly be a region about this point at
no point of which have they a common vanishing factor. This region also we call the
proper region of (71(~),..., Tp~) for the function F.
By hypothesis there is then a proper region belonging to every finite point. We
assume further, what is not quite obviously a deduction from the former hypothesis,
that the whole of finite space can be divided into regions, each of finite extent, each having
the property of being entirely contained in the proper region of every point of itself.
The function F will then be represented in one of these regions K0 by an expression,
belonging to an interior point (),
F=,o
0o