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.

430 MR BAKER, ON THE THEORY OF FUNCTIONS
a distance e from I, approaches indefinitely near to I, the limit of e log (e'), when e
and therefore (e') vanishes, is zero. The part (iii) of the integral is equal to
27ri f[(K + i2) (a, - iZ a,) p (x +...) S
taken over only one side of the (limited) diaphragm P; for the values of log 4 at two
near points on opposite sides of P differ by 2ri.
Consider now the real part of this integral, namely
27r /C a +.-C2 a,)dSby the theorem of Part I. of this paper we can replace this by an (n - 2)-fold integral
taken over the (n - 2)-fold which forms the boundary of the diaphragm; this (n-2)fold lies partly on S and partly on S; the (n- 2)-fold is
(2 + 34 + * +... Ken-1, +n) (x | t) dSn-2,
as is immediately obvious on applying the theorem. If we now suppose that the
diaphragm is so chosen that the bounding (n - 2)-fold is a complex (n-2)-fold (~ 9),
we can infer that, when (ir,..., Tp) is within the region considered, log 4 (r) differs
only by a finite and continuous function from a function whose real part is equal to
-2 f (x t)dS2,
where the integral may be supposed to be taken only over the part of I which lies
within S; for we have seen (~ 9) that for a complex (n - 2)-fold
^i12 + C34 + *. + ^Cn-i, n = 1.
The theorem to be proved can then be immediately deduced.
17. Incidentally we have remarked in ~ 16 that if a finite portion of an (n-1)fold be bounded by a closed complex (n - )-fold, then, under certain conditions of
continuity and single-valuedness for the function U, we have
UdSn-2 = J'1 a - -2 a +... dSn-,,
the first integral being taken over the closed (n - 2)-fold, and the second over the
bounded portion of the (n - l)-fold.
We now extend this idea to the (n - 2)-fold I, given by the aggregate of the
series ). We imagine this (n - 2)-fold, which is defined only for finite space, to be
completed into a closed (n -2)-fold by means of a complex (n - 2)-fold at infinity;
and, as before, we assume tentatively, that the part of the integrals under consideration
which is contributed by the portion of the (n-2)-fold of integration lying at infinity
vanishes (see ~ 22).