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. 433
in the sense that it is possible to divide n-fold space into period parallelograms, the
interior of any one of these being given by the p equations
ti = X + 2ei, lX +... + 2oi, 2pX2p (i = 1,..., p)
where X is a constant and X,..., 2p are real variables each between O and 1, and to
regard the portions of the (n - 2)-fold lying within these various parallelograms as
repetitions of one of these portions. Then it can be proved, under a certain hypothesis,
that the value mn= 1 is sufficient for the convergence of the integrals. The hypothesis
is that the extent of the (n-2)-fold contained in any one such parallelogram is finite;
and the truth of this hypothesis is deducible from the mode in which the (n-2)-fold
of integration has been supposed to be defined.
Of this result, which is given by Poincaré, the proof is included in the investigation
below (~ 22); it may be remarked at once however that the formula obtained here is
not limited to the case of periodic functions; as we may see by taking a simple example.
We apply the formula when n= 4, to form the equation of the complex (n - 2)-fold
l1=7;
putting y = a + ib this is then the two-fold given by x = a, x2 = b. The matrix
dl x dix2 dlx3 dlX4
d2xl d2x2 d2x. d2x4
with the help of which the direction cosines may be defined, may be taken to be
0 0 dx, 0,
0 0 0 dx4
so that Kc.- O= except Ki2= 1, and dSn-2 = dxdx4. As the integral
(f dx dx4 - f2rd R rdir
J(a2 + b2 2 + x +2)2 J O Jo (D + r2)2
vanishes when,R R0 are infinite we infer that it is sufficient to take n= O, and
therefore
Hm = p() t)- |0), H,,+ = (x\t) - (\1 0) + (t a) ~;
then (~ 12) we obtain, for
af/H. aaHm W
=V o JL.2 b -l -3 2 + (515 + K4) a -t - i ) dSn-2i. ( aHM m f.Hm
= (31 + 32) - + - - )Z K34. [Hr)} dSn-2,
the respective values?3=0
=- fId3dx4 {[(x x -tï )2 +i(2 -t ) [+ x-ix }
VOL. XVIII. 55