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. 417
latter integral, over H,_,, is equal to the integral from which we started which is taken
over Hn-2; and this is what was desired.
We can then by a summation infer that
I S Kt Prst dSn-3 = K r, s. d
Jrs, t.rs t
which is a convenient way in which to state the result. There are () terms in the lefthand integral, and (n - 2) () =3 () terms in the right-hand integral.
A similar argument will be found to lead to the general result stated in ~ 2.
8. If we put, in the case for which the proof has been carried out,
_ P.st
Xrs =- C -t
t axt
we have >X = 0,
s a$s
as n necessary conditions that the integral
f|rs Krs Xrs dSn-2)
taken over a finite portion of an (n - 2)-fold may be capable of being represented as an
integral over the closed (nt-3)-fold bounding this portion. If these conditions are satisfied,
functions P,st satisfying the equations
aPrst
Xrs = Y
t axt
must be found, in order that the expression rnay be possible; but it is necessary that
the functions Prt so found should be finite on the (n -3)-fold (cf. ~ 28).
The equations
ax,.,
have been given by Poincar (Act Math. x. (1887) p. 37) from a somewhat different
have been given by Poincaré (Acta Math. Ix. (1887), p. 337) from a somewhat different
point of view. We can as an application generalise Cauchy's theorem to the p-fold
integral
j.....J (e,..., p) cie... p
where:,..., p are complex variables. For example, for n = 4, the integral
~~ SfVOL. XVIIT. 53
VoL. XVIIIT. 53