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.

414 MR BAKER, ON THE THEORY OF FUNCTIONS
H2, and the direction cosine Kcr of the (n -2)-fold t =O, f,=O, which by definition is
given by
Kcrs = Jrs. V J2rs 1,
and is therefore equal to
sgn e-2. M rs ] /SM,
where sgn e2_- means + 1 or - 1 according as E,,-2 is positive or negative, has throughout Hn_- the sign of Mr,. Thus
KrsdSn-2 = Ms sgn (r /) = Mi, sgn (- df `) = M.
6. Next we consider any point Q of Hn_3, and its satellite points
(xl+dkXl,..., x+-dk X), k=1, 2,..., (n-3).
From the equations
fidk + + +. fldkX, = 0,
f2dkx +... +f2dkn = 0,
3dkX +.. +f3dkXn = 0, = 1, 2,..., (n-3),
we find as before
J12 _ __ Jrt
M123 rs- Mt., y,
where
"rst =
afi afi af8
axy ax' axt
xr' axs' axt
and Mrst is the value obtained by omitting the rth, sth and kth columns in the
matrix of n columns and (n - 3) rows
I dki,..., dkX, k= l, 2,..., (n-3),
and prefixing a certain sign to the resulting determinant. This sign is supposed to
be given, as for the two previous cases and as in general, by the following rule;
consider the determinant of n rows and columns whose first (n - 3) rows are formed
by the matrix just described, whose (n-2)th row is A1,..., An, whose (n -l)th row
is B1,..., Bn, whose nth row is C1,..., C,; then the expansion of this determinant is
lton,Ç ArBsCtlM,,st;
thus when r, s, t are in ascending order the sign to be prefixed to the determinant