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. 427
To the proof given by Poincaré we may add the two following, both of which make use of
some results in Poincaré's paper.
(a) Denote the (n-2)-fold of integration by I; let (x1,...,,,) be the coordinates of
a varying point on I, so that (xi,..., xn) are functions of (n - 2) parameters; then
if (x11,..., xn') be current coordinates, and e a fixed small quantity, the envelope of
the spheres
(X/ - X)2 +...+ (x- xn)2= 2
is an (n-l)-fold S, surrounding I, of which, when (x1,..., x,,) is not a singular point of I,
the points are given by
i = + (ui cos 0 + vi sin 0), (= 1, 2,..., n),
where 0 is a variable quantity, so that (x1',..., x.') are functions of (n - 1) parameters;
here ui denotes u//axi, vi denotes av/axi and h is the positive square root of I12+... + un2
The point (x1',..., x,/n) lies on one of the single infinity of normals which can be drawn
to the (n - 2)-fold I at (X1,..., xI), and is at a distance e from I. The direction cosines
of the normal to 2 at (x1',..., x') are the quantities (uz cos 0 + vi sin 0)/h; the element
of extent of Z at (x/',..., x') is dSni = edOd&S2, ultimately, squares of e being neglected,
where dSn_- is a corresponding element of extent for I. If. = x2-1 - ix2, yr = X'2r + ix2r,
we have
()1 ' " ()' - h 2'
where (P,.)' is the conjugate complex of 9~/9r, and therefore equal to 21._- i ~ -v2_1; and,
what is permissible to the first order of small quantities, ' is written for
1 (e'l- e1) +. + ~P (ep - p).
With these results we combine now the following, which is a particular case of
a theorem of Kronecker's. Let f(ri,..., Tp) be a single-valued function finite and continuous upon a certain closed (n - l)-fold, whereof 1i,..., K, are the direction cosines;
consider the integral
f(,, p) (cl + iK2) (-L - ) ( 1 t) +.. + ( -l + e) ( - Ïi.- p (w I t)} dS,,,
where (x1,..., x,) denotes a varying point upon the (n - l)-fold. By Green's theorem
it is immediately clear that this integral is unaltered by any deformation of the
(n-l)-fold of integration which does not involve a crossing of the point (t1,..., tn) or
of any point where f(7-1,..., Tp) ceases to be finite, continuous and single-valued. For
the condition for this is simply (Part I. of this paper)
(a a a
+i f(, *.*.) ) - i p ( i t) +..0,
namely (a2 + 2~ +.) ( ) = 0.
54 2