  Using the standard saddle-point method, we find an explicit relation for the
large-N limit of the free energy of an arbitrary generalized 2D Yang-Mills
theory in the weak ($A<A_c$) region. In the strong ($A>A_c$) region, we
investigate carefully the specific fourth Casimir theory, and show that the
ordinary integral equation of the density function is not adequate to find the
solution. There exist, however, another equation which restricts the
parameters. So one can find the free energy in strong region and show that the
theory has a third order phase transition.
