Zwicky: "Dark Matter in the Coma Cluster"

In 1933, Fritz Zwicky wrote a paper on the evidence for Dark Matter in the Coma Cluster.
Firstly, what is the Coma Cluster?
The Coma Cluster, pictured below, is a galactic cluster of over 1000 galaxies located in the same region of the sky as the constellation Coma Berenices.  For all practical purposes, this acts a large group distributed particles, something that we and Zwicky can apply Virial Theorem to.  

http://upload.wikimedia.org/wikipedia/commons/7/7d/Ssc2007-10a1.jpg

Zwicky starts by stating the viral theorem that we know as:
\[2K=-U\]
Where K is kinetic/thermal energy and U is gravitational energy.
Zwicky then calculates the total mass of the system.  He postulates that:
Radius \(R=10^{24}cm\)
Mass of a Nebula \(M_N=10^9M_*\)
1 Solar Mass \(M_*=2\times 10^{33}g\)
There are 800 nebulae in the cluster.
Newton's Gravitational Constant \(G=6.7\times 10^{-8}\frac{cm^3}{gs^2}\)
The average velocity of a component of the system \(v=1.75\times 10^8cm/s\)
Therefore:
\[M=800\times M_N\times M_*=800\times 2\times10^{33}g \times 10^9= \boxed{1.6\times 10^{45}g}\]
This should be the mass of the whole system...

Next let's set up virial theorem.
\[2K=-U\]

\[Mv^2=\frac{3GM^2}{5R}\]

\[M=\frac{5v^2R}{3G}=\frac{(1.75\times 10^8cm/s)^2(10^{24}cm)}{6.7\times 10^{-8}\frac{cm^3}{gs^2}}=\boxed{7.6\times 10^{47}g}\]

We now notice a slight problem, this calculation for mass is about 400 times larger than our calculated observed mass.  Something must be missing...

We can see the size of the cluster, so it must be 400 times denser than what we can see.  Hence, it must be filled with...
DARK MATTER

Zwicky follows this calculation with a series of other variations of the viral theorem in a futile attempt to account for the missing matter, but he is unable to do so.  Thus the answer must be that the coma cluster is filled with...
DARK MATTER