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.

428 MR BAKER, ON THE THEORY OF FUNCTIONS
Hence, if f(7r,..., rp) be single-valued, finite and continuous for the whole interior of
the (n- )-fold of integration, the integral is zero when (t1,..., t) is outside, and,
when (t,...,tn) is inside the (n-l)-fold, it is equal to /(7,..., Tp), as we see by
supposing the (n - 1)-fold of integration to be deformed to
(Xi- t )2 +... + (xn- t)2= r2,
and then taking r to diminish without limit.
Now consider, in the region of convergence of the series b, a (multiply-connected)
closed region, bounded by (i) part of the (n-l)-fold Z surrounding the (n- )-fold
b =O which has already been described, (ii) part of a closed (n - )-fold S described
in the region of convergence of (b, the part being limited by the (closed) (n - 2)-folds in
which S is intersected by E; and take
/(r) = log,
a'Tr
where r is one of 1, 2,..., p. Then when (t,,..., tp) is interior to the (multiply-connected) region above described, we shall have
()= /J +i (1 -a ) ( t)+ *} dSn-,
where the integral is taken over the two partial (n-l)-folds denoted by (i) and (ii).
The part (ii) of this integral is finite for all the positions of (t,,..., t,) under consideration; consider the limiting value of the part (i) as the (n - )-fold 2 is taken
nearer and nearer to the (n - 2)-fold I, namely by the decrease of the quantity denoted
above by e. By what has been stated above we may ultimately put
K*2s-1+ itS=-(^S) l 2 e, ' f(f)=, dS=n-l = eddSdn&_;
then, if (ti,..., t,) have some fixed position at finite, not infinitely small, distance from
the (n - 2)-fold I, we obtain, for this part of the integral,
- 1 d f (f edSn,-, (: (, e -2s atzs) ç (t}
O J (JE) { =l() )t ( -_ i 28-1) ~ (x ) )}_ n;
[s=i a - a, )
now the direction cosines of the (n - 2)-fold I are (~ 9) such that
U2. -i. V2s-i - U-2S-1 V2r-i + i (2.r-1 V2S - U2, Vr -1)
s2r-1, 2s-1 + I~ 2'-1, 2S - h2
= 2 (I2r-1 + iv2r-1) (u2S-1- iv2S-1);