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.

438 MR BAKER, ON THE THEORY OF FUNCTIONS
There remain certain assumptions in regard to (n-3)-fold integrals, and in regard
to (n- l)-fold integrals.
We have assumed that if a finite portion of the (n -2)-fold of integration be
bounded by closed (n - 3)-folds, the corresponding (n- 3)-fold integrals
[(a i fm_-1. affm-i\.0
(i Cs, 2k-1 + Kr, S, 2k) (Hm i- aH -) dS-3
(J,.,K,=_=+i, ~,) \a2Âx-l ax2k /
ultimately vanish as these (n- 3)-folds pass to infinity.
This really follows from what has been demonstrated. The (n-3)-fold integral arose
as equal to an (n-2)-fold integral. In the course of the proof above it has been
shewn that this (n - 2)-fold integral is such that if taken over infinitely distant portions
of the (n - 2)-fold the corresponding contributions ultimately vanish. Thus it is legitimate
to regard the (n - 2)-fold as closed at infinity, namely by an (n - 2)-fold for which our
hypothesis (~ 22) remains valid. In which case the (n - 3)-fold integrals that arise
are mutually destructible.
We have considered also the (n - l)-fold integrals
v=f (1 Hm+- K2 aHm+i +..) dSn71,
V= aq ~,ax, a:~ ~x,~ + '"dS
= = ( -aHm+l + 2c am+1...)dSnltaken over the infinite (n - )-fold bounded by the hypothetically closed (n - 2)-fold
just considered. It is necessary to see that these are convergent. This follows because
the portion of either of these (n - )-fold integrals taken over the portion of the
(n - )-fold which lies at infinity can be replaced by an (n - 2)-fold integral taken over
a closed (n - )-fold lying entirely at infinity-and by the proof given above this
(n - 2)-fold integral ultimately vanishes.
25. Note to ~ 15. In the course of this demonstration we have utilised the fact
that as (ti,..., tn) approaches indefinitely near to the (n - 2)-fold of integration the integral
2w r
- (x t) dSn-2
becomes infinite like log mod. ), where = O is the equation of the (n - 2)-fold in the
neighbourhood. The following direct verification of this fact is of interest.
To a first approximation the points of the element dSn-, satisfy the following
equations, the origin of reckoning being taken at the point of the (n - 2)-fold,
'tIX1 + -U2X2 +... + UnXCn = 0,
vlx1 + v22X.. +... + Vnx, = 0;