Using the standard saddle-point method, we find an explicit relation f
or 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 theo
ry, and show that the ordinary integral equation of the density functi
on is not adequate to find the solution. There exist, however, another
equation which restricts the parameters. So one can find the free ene
rgy in the strong region and show that the theory has a third-order ph
ase transition. (C) 1998 Elsevier Science B.V.