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.

OF SEVERAL COMPLEX VARIABLES. 413
to be satellite points of P of which P,,- is on fi= O but not on 12=0, and Pn is not on
either of the two f= 0, f= 0; then we have
en-2 M ~ - ' M fl
Jri d _n- xi +... + Jrn dn Xn
ri i dn-i Xi +.+ Min dn-ixn'
odf, - ~f2d f1
m/r
where M, denotes the value obtained by taking the determinant left when in the matrix
of n columns and (n - 1) rows
dkxi,..., dkXn, k=1, 2,..., (n-1),
the rth column is omitted and prefixing the sign (-l)n+r to this determinant; hence as
dn,,_f = we have
n_-2 MXr
ax, _ ax
~Ar D 1r
-M =..... '
M I
dfi xnX +...+ fi dx n
Mi dn x + @ @+ Mldnxnl
dnfi
where M is as before the determinant
| dkXi,..., dkxn, k= 1, 2,..., n.
Thus on the whole we have
J12 Jrs dnfi. dn-if
en-2 M12= - -" = 8 -M"
We now mace the assumptions (i) that for all points P of Hn- the satellite point Pn is taken
in space on that side of the (n - l)-fold f = O0 for which fi > O, so that dnfi is constantly
positive, (ii) that the satellite point P,,-_ is taken on fi = O on that side of f, = 0 for which
f2 > O, so that dn-if2 is positive, (iii) that the satellite points P1,..., Pn-2 are constantly taken
on f' = O, f = O in such a relation to Pn_ and Pn that M is constantly positive over Hn-2.
Under these assumptions each of the ratios Jrsl/Ms. maintains a positive sign over