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.

410 MR BAKER, ON THE THEORY OF FUNCTIONS
2. In what follows we call the aggregate of all the points of a space of (n - r)
dimensions, limited or not, an (n - r)-fold. We also use ( quantifies, called the direction
cosines of the normal to the (n-r)-fold; let the (n- 1)-folds
fi =,..., f= 0
be always supposed taken in the same order, given by the suffixes; let b, d, e,...., h, k
be any r of the numbers 1, 2,..., n, no two of them equal; then the ratio of the
Jacobian
T ( fi, **,f r)
Jb,d,..., )
Jb,...,hk a (xb, x..,k)
to the positive square root of the sum of the squares of all the possible () such
Jacobians is denoted by Kb,d,e,...,h,k, and is one of the direction cosines in question; we
suppose in general the suffixes taken in their natural ascending order; from each of the
(n) direction cosines - 1 others can be formed by permutation of the suffixes, every
interchange of two suffixes causing a change in sign in the direction cosine.
We have then the following theorem:
Suppose that a finite portion of the (non-singular) (n - r + l)-fold given by
fi= 0,...,/ f = 0,
is completely bounded by a closed (non-singular) (n - r)-fold given by
fi=,..., fr- 0, / = 0,
and that throughout the limited portion of the (n-r+ 1)-fold we have fr< 0; let P be
any function of xl,..., xn for which it is supposed that itself and its first differential
coefficients are finite and continuous (and single-valued) throughout the space considered;
then
f r( a ap aD
K(-1 Cde...hk bX + (- l2 Kbe...hk a- + + Kbd...h X dSn-r+i
aXb îax aXk
= Kbd... h P. dSn-,
wherein the second integral is taken over the complete closed (n -r)-fold, and the first
integral over the enclosed portion of the (n - r+ 1)-fold; in the first integrand there are
r terms, the suffix in any one of them consisting of (r-1) numbers in their natural
ascending order.
If we introduce (r) functions such as P, and make the rule that an interchange
of two numbers of a suffix shall entail a change in the sign of the function, we can put
the result in a clearer form
ffXKde.....aPde.hkm - =Sfn_+l bd......hkSn-r.