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.

424 MR BAKER, ON THE THEORY OF FUNCTIONS
taken over the closed (n-3)-fold; assuming now, what is a similar assumption to those
already made, that this integral diminishes without limit when the (n- 3)-fold passes
to infinity, we have
s2r= - 2r-i, 2r d Xn-2 k=- ji, 2k -2 — M-2
J 2r-1 k=IX2d-i
which, since
KC12 + C34 + * + 4n-1, n = 1,
is the result stated, namely that
32r = -f a dS 2.
The corresponding form for s2,-i can be proved in a similar way, starting from
çs _~- [(~- ~ ~ SlT;Hm aHm a L^ 7
2r-1 = J (2r, 2 Ka r + -2r, 1 a + K2r1, n-l ) Sn-2.
__ xJHm aHm
Thus also (= a- - - da-S
and, if we put (Poincaré, Acta Math. t. xxII. p. 168)
V f= Hm+ dSn-2,
i2r =- 2r-l - - t
at2i7-1 Ct2r
so that V, which may be regarded as a generalised potential, of the (n - 2)-fold of
integration at the point (tl,..., tn), is the real part of <b, or differs therefrom by a
constant.
14. Supposing that the integral is taken from the point t = O = t2=... =tn, which
is supposed not to be on the (n - 2)-fold of integration, we may write
q) =/ (adT1 + dr2, +... + p-_dTp),
= I (2d1 - l dt2 +... + S2pdt2p-l - 2p-lt2p)
+ i f(S1dt1 + 82dt2 +... + 82p-dt,2P-1 + 82pdt2p);
of this, in virtue of the results of ~ 13, bearing in mind that Hm+,, and therefore
also V, vanishes for t =O,..., =t,=O, the real part is exactly V; the imaginary part is
il2 where