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. 419
thus if we denote au/axs and av/axS respectively by us and v, the direction cosines of
the (n-2)-fold, defined in ~ 2 ante, are given by
Kr, s = (UrVs - UsVr)/h,
where h is the positive square root of
2 (L,.vs - utsV8)2 = (12 +... + Un2)2 = (12 +... + Vn2)2;
', s
now we have
Zt2r-1 V2s-1 -- - U,2s-1 Va? ' - 'vr s + V2s U2r = - '2r-1 U2s + U2rU2s-I,
U^2r-_V2s - U2sV2r_1 - V21'U2s1 -1 V22Sl-22r = U2r-1U2S-1 + U12îU2s,
so that
UG2r U2s- -- U2r-1 '2
2r-1, 2S-1 = r, 28 = —9.,n
2r-1,2S-1K2,2S 2 U12 +..+ + u
U'r_-i'L2s-1 + U 2r 2s
2r —1, 2s =-C2,2S-1 = _... + n2,
and
K12 + <34 +.. + KCnl,n = 1.
These relations are of importance to us. They of course require modification at any
singular points of the (n - 2)-fold; the present paper is so far incomplete that the consideration of the effect of the singular points is omitted; the final results obtained are
expressed in a form which is believed to be unaffected by the existence of such singular
points.
10. Consider now a limited portion of an (n - 2)-fold, bounded by a closed (n - 3)-fold.
Denote by xi,..., x, the coordinates of a point on the (n - 2)-fold or on the (n- 3)-fold,
and by (t,..., tn) the coordinates of a finite point of space not on either of these, the
corresponding complex variables being as before given by
= t2r-t + it2, r=1, 2,..., p.
Let L1,..., Ln and R2,..., Rn be single-valued functions of xi,..., xn and of ti,..., tn
which are continuous and finite, with their differential coefficients, so long as (x1,..., xn)
is upon the (n - 2)-fold or (n - 3)-fold under consideration, and the point (t1,..., tn) is
in finite space and not upon the (n - 2)-fold or (n - 3)-fold; further suppose that these
functions are such that
aLM aRi
at (i, s=a, 2,..., n),
and
aR aR2, Rn.
+- +... +' + 0.
axl ax2. x.
Consider the n integrals
r = f(irîL +... + KrnLn) dSn-2, (r = 1, 2,..., n)
53-2