  Considering the fact that the present universe might have been formed out of
a system of ficticious self-gravitating particles, fermionic in nature, each of
mass $m$, we are able to obtain a compact expression for the radius $R_0$ of
the universe by using a model density distribution $\rho (r)$ for the particles
which is singular at the origin. This singularity in $\rho (r)$ can be
considered to be consistent with the socalled Big Bang theory of the universe.
By assuming that Mach's principle holds good in the evolution of the universe,
we determine the number of particles, $N$, of the universe and its $R_0$, which
are obtained in terms of the mass $m$ of the constituent particles and the
Universal Gravitational constant $G$ only. It is seen that for a mass of the
constituent particles $m\simeq 1.07\times 10^{-35} g$ the age of the present
universe,$\tau_0$, becomes $\tau_0 \simeq 20\times 10^9 yr$, or equivalently
$R_0 \simeq 1.9\times 10^{28} cm $. For this $m$, the total number of particles
costituting the present universe is found to be $N \simeq 2.4 \times 10^{91}$
and its total mass $(M \simeq 1.27916 \times 10^{23} M_{\odot})$, $M_{\odot}$
being the solar mass. All these numbers seem to be quantitatively agreeing with
those evaluated from other theories. Using the present theory, we have also
made an estimation of the variation of the universal gravitational constant $G$
with time which gives $({\dot G \over G}) =-9.6\times 10^{-11} yr^{-1}$. This
is again in extremely good agreement with the results of some of the most
recent calculations. Lastly, a plausible explanation for the Dark Matter
present in today's universe is given.
