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. 415
by which Mrst is formed is (- l)+~+s+t-l. Hence, taking the satellite point Qn-2, of Q,
upon fi = O and f= 0, but not upon f3= 0, we have
E =rsi _-_rs2 i - rsn
_n-3 = Mi.l = M = M- = M...
s Mrsa Mrsn
Jrsi dn-2x1... + J rsndn-_xn
M.rsi dn_2-x1 +... + + drsn dn-2xn
here the numerator is
afi afl
axr ' 8s '-2,
af2 af2 d,
ax ' axs '
and therefore, since d_2f = 0, dn-2f, = O, is equal to
Js dn-2f3;
the denominator can be seen to be exactly equal to M;rs thus we have
en-3 -= M - n —J3
and so obtain
KiCst = Jrst - I /J2rst |
Mrst (Jrs
d8n^,-3 VMrs 2
the element of extent dSn_3 being by definition equal to \/ZM2,st1.
7. At this point it is convenient to recall the connection which has been established
above (~ 4) between the division of HIn- and Hn11 into cells. With that arrangement
the J,.s and Mr now arising in the consideration of iCrSt may be regarded as identical
with those that arose just previously in considering Kcrs. We proceed to utilise this.
Consider the determinant of n -2 rows and columns
di P-St, dx,..., dxi x,
dl_-2Pst> dn-2li,..., dn-2_n
wherein (x + dxx,..., x+ dn_,x) are satellite points of (x), upon H,2,, the columns containing xr, xc, xt are omitted, and Pst is a function which is single-valued, continuous
and finite upon Hn- and Hn_, and possessed of single-valued finite differential coefficients.
The suffix of Prst is not necessary for our present purpose; but it is convenient, as
enabling us to define functions derived from this one by interchange of two elements