Colloquium publications.

50

THE CAMBRIDGE COLLOQUIUM.

the result obtained by carrying out in formal fashion the multiplication indicated when we put 2 = x + iy, 7 = z + it. In
order to make our intuition clear, we can imagine the two-dimensional locus as enclosed in a three-dimensional space, and the
integration as carried out over a surface in that 3-space. Further
considerations are necessary to determine the sense of the integration; these can be arrived at by making precise the relations to the enclosing 3-space (Poincare) or by considering the
continuity properties of Jacobians relating to curvilinear coordinates on the surface (Volterra).
Poincare showed that the necessary and sufficient condition
that the integral (49) be independent of the surface, i. e., depend
merely upon the closed curve of which it is the cap, is that
P + iQ be an analytic function of x + iy and z + it. In this
case, therefore, the quantities defined by (49) represent additive
complex functionals of curves in 4-space. Singular surfaces
(or singular curves as cut from them by the Poincare 3-space)
may be cut or looped by these curves.
Any two additive complex functionals of the type just defined
will be isogenous. In fact, if F[C] is one such functional and
F'[C] another, the condition to be satisfied is:
dF = f(xyzt),
in which the function f does not depend upon the manner of
letting the change a, given to C, approach zero. But as we see
from (49), we have always
dF' P' + iQ'
dF P+ iQ
It may be deduced in an equally simple manner that F[C] is
elementary. If in (49) we substitute fP and fQ for P and Q
respectively, where f = fi(xyzt) + if2(xyzt), it is merely necessary to write the condition that the new integral be independent
of the surface. This gives us the relations