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.

412 MR BAKER, ON THE THEORY OF FUNCTIONS
through P intersects the closed Hn-3 in an even number of points; and to shorten the
proof we shall speak only of two, say Q(o) and Q(i). Then if the differentials dn_2xb,
dn_2Xe, d-_2xh be always taken in the same direction along this curve Q'o)PQf(), the
expression
dn-2f3 = 'af dn-2Xb + '- dn-2Xe + 3' dn-2Xh
aXb aXe aXh
will have different signs at Q(0) and Q(i), and in fact, since f < 0 over Hn-, the expression
will be positive at the point, say Q(1), where the curve through P leaves H,n2, and negative
at the point Q(0), where the curve enters Hn-2.
5. Considering now any point P of Hn2, and its satellite points
(x1 + dkxl,..., X( + dkX2), = 2
in regard to which we do not until special mention is made of the fact introduce the convention that all but three of the differentials
(dn-_ lx *, dn-2_xn)
are zero, we have
f dkX +... f- dkXn = 0,
É^-é 80fi'
c=, 2,..., (n-2),,f2 dkx.. +. f dkx = 0,
and hence easily find
J12 _ tirs _
M12 - - M.S - = e say,
wherein
T - afi af
axr' ax,
af2 af2
ax,., ax
and Mr, denotes a determinant of (n - 2) rows and columns, obtained by taking the determinant which remains when in the matrix of n columns and (n - 2) rows
| dkX1,..., dkXn, k=1, 2,..., (n-2),
the rth and sth columns are omitted, and prefixing to this determinant the sign (- l)r+s-1
or (- 1)r+ according as r < s or r > s.
We require now to make it clear that we can suppose the sign of the ratios en- to
be the same for all points of the limited H,_2; for this purpose suppose
Pn-X, with coordinates (xw + dn-xz,..., x + dn-_xn),
and Pn, with coordinates (xz + dx,..., xn + dnxn ),