  The Hylleraas coordinates $s=r_{1}+r_{2}$, $t=r_{1}-r_{2}$, $u=|{\bf
r}_{1}-{\bf r}_{2}|$ are the natural coordinates for the determination of
properties of the Helium atom, the positive ions of its isoelectronic sequence,
and the negative Hydrogen ion. In this paper, we derive a new expression for
integrals representing properties such as the energy, normalization and
expectation of arbitrary operators, as written in the $(s,t,u)$ coordinates.
The expression derived is valid for both \emph{finite} and \emph{infinite}
space. The integrals for the various properties are comprised in each case of
two components $A$ and $B$. The contribution of these components to the volume
of integration and the normalization of a wave function for finite space, and
in variational calculations of the ground state energy of the Helium atom
confined in a finite volume is demonstrated by example. We prove that when the
integration space is \emph{infinite}, the expression for the energy and other
properties employed by Hylleraas corresponds \emph{only} to that of integral
$A$. We further prove that for the approximate variational wave functions
employed by Hylleraas and other authors, the contribution of the term $B$
vanishes. This contribution also vanishes for the exact wave function. It is
interesting to note that the component $B$ to the integral is not mentioned in
the literature. A principle purpose of the paper, therefore, is to point out
the existence of this term.
