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.

MR BAKER, ON THE THEORY OF FUNCTIONS, ETC. 409
PART I. PRELIMINARY.
Formal proof of the general Green-Stokes theorem.
1. In Euclidian space of n dimensions we can take near to any point P whose
coordinates are (x1,..., xa) the n points
P1 with coordinates (x1 + d1x,..., xS +dixn),..............................,...,.............
P, with coordinates (x1 + dnX,..., x + dnx,),
it being supposed that the determinant, M, of n rows and columns, whose (r, s)th element
is drXs, is not zero. At each of the points P1,..., P,, we can similarly take n independent
consecutive points, those at Pr being Pri, Pr2,..., Prn; at each of these points of two
suffixes we can take n others of three suffixes, and so on. Making the convention that
the sth satellite point of Pr, namely P,.,, is the same as the rth satellite point of P,, or
Psr, or in other words that the suffixes shall be commutative, we can associate the determinant M with the 'cell' which is defined by the 2n points
P, Pl,.., nP n) P12,... > Pin,)., P12... )
whose suffixes consist of all the combinations of not more than n different numbers from
1, 2,..., n. We may suppose space of n dimensions to be divided into such cells, and
call the absolute value of the determinant M the element of extent of the space, denoting
it by dSn.
Similarly if we have in n dimensions a space of (n-r) dimensions, defined suppose by
r equations
fi (X1,..., f) =0, fr (Xl,..., Xj)-=0,
with a certain number of inequalities, we can associate with every point P of this space
('n-r) satellite points, P1,..., P,_-., also lying in this space, the coordinates of these
points being denoted by
x + dkx, *..., Xn + dk k 2,..., (n-r),
and with each of these (n-r) others, and so on; and so we can suppose the space of
(n- r) dimensions divided into cells, each defined by 21-'' points; with each of these cells
we can as before associate an element of extent for this space, which we denote by
dSn_-.; this is defined as the positive square root of the sum of the squares of all the
determinants of (n-r) rows and columns which can be formed frorn the matrix of
n columns and (n-r) rows
[dkxi, dkX2,..., dkXn, = 2,..., (n-r),
or, what is the same thing, as the positive square root of the determinant of (n - r) rows
and columns which is formed by multiplying this matrix into itself, row into row, in
the ordinary way.
VOL. XVIII. 52