  The problem of N interacting spins on a lattice is equivalent to one of N
clusters linked in a specific manner. The energy of any configuration of spins
can be expressed in terms of the energy levels of this cluster. A new
expression is obtained for the probability of occurrence of any configuration.
A closed form expression is obtained for the partition function per spin in
terms of the energy levels of this cluster with the degeneracies being a
function of temperature. On physical grounds it is suggested to be the form of
the exact partition function per spin. The partition functions of Ising-like
systems all have a common form. For the 3D Ising model seven functions need to
be determined to describe the partition function completely. The key to
understanding second order phase transitions and critical phenomena lies in the
temperature dependence of various degeneracies. It is necessary to develop new
techniques to determine the partition function that account for this
temperature dependence, as it would represent the underlying physics correctly.
