Colloquium publications.

136

THE MADISON COLLOQUIUM.

fff(x, y, z) d 2 = lim f(xk, yk, Zk)_Ak,
the element of surface, Ask, being taken as an essentially positive
quantity.
Allied with this integral is the other surface integral:
ffjAdydz + Bdzdx + Cdxdy,
where A, B, C are continuous functions of x, y, z. Here, the
sign to be attached to the differential factor, dydz, etc., requires
special definition, and can be assigned in terms of the sense of an
indicatrix which moves continuously over the surface, or in
terms of the signs of the Jacobians
a(y, Z) a(z, x) a(x, y)
a(s,t) ' (s,t) ' (s, t) '
where s, t are parameters by means of which x, y, z are expressed.
Such integrals, suitably generalized for manifolds of order k
in space of m dimensions, have been applied by Picard and
Poincare* to analytic functions of n complex variables. Poincare develops conditions that the value of such an integral,
extended over a closed manifold, be invariant of slight deformations of the manifold, and hence also of large variations, provided
the manifold retains its character and does not sweep over a
point in which an integrand is discontinuous or the conditions
in question cease to hold. Cf. also I, ~ 8.
Residues. The value of any such integral Poincare calls a
residue. Only bilateral manifolds come into consideration,
since an integral extended over a unilateral manifold evidently
vanishes.
As a first application of the foregoing, consider Cauchy's
integral formula. In the form in which it stands above, the
* Picard, C. R., 96 (1883), p. 320; ibid., a series of papers in vols. 102-3
(1886). Poincar6, C. R., 102 (1886), p. 202; Acta, 9 (1887), p. 321, where
reference to Jacobi and Marie in connection with these integrals is made.