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.

436 MR BAKER, ON THE THEORY OF FUNCTIONS
quantity, say A. In regard to the shape of these closed (n-l)-folds the important
point is that the linear dimensions shall be always of the same order of magnitude in
all directions. In regard then to the n-fold extent, V, of these closed (n- l)-folds two
things are possible as we pass to the infinite parts of space. Either V may have a
lower outside value B finitely greater than zero, which case arises ini considering functions
having 2p sets of simultaneous periods. Or, the zero (n - 2)-fold may become so bent
and crumpled upon itself that at sufficient (not infinite) distance from the finite parts
of space it may be possible to find an n-fold extent V less than any assigned quantity,
which shall still contain an extent A of the zero (n - 2)-fold; or in other words, that
the volumes V may have zero for lower outside value as we pass off to infinity. When
this latter is the case it is conceivable, denoting by R the average distance of the
points of a closed (n -l)-fold from some finite point, that its n-fold extent V may not
diminish faster than some positive power of R increases, namely that there may be a
quantity m, not less than unity, such that
Rm-lV y B,
where B is a finite constant, for all values of R which are not too small.
Under this hypothesis it can be shewn that the integral
i dSn-2
J Rn+m
extended over the whole infinite (n -2)-fold, is convergent, R denoting the distance of a point
of the (n - 2)-fold from some finite point.
For suppose concentric spherical (n-l)-folds to be described, with centre at the
finite point from which R is measured, and consider the extent of the (n - 2)-fold
lying in an annulus bounded by two of these spheres, of radii r and ri (r > r). In
accordance with the hypothesis we can suppose the n-fold content of the annulus divided
into regions each containing a finite extent, say M, of the (n - 2)-fold, such that if V be
the n-fold extent of any such region
rm-l YV B,
where B is some constant greater than zero. Let k be the number of these regions,
which will be finite when ri is finite. Then
(r1_ - r ) rm-i - kB
n
as the total extent of the (n - 2)-fold lying in the annulus is 1cM, the contribution to
the integral
dSn_-2
J n+m
which arises from the annulus is less than
kM
rn+m '