Phase transitions from ising model

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Main Question or Discussion Point

I'm having trouble understanding how changing parameters in the ising model leads to first order phase transitions. I understand how the intersection of
[itex]\frac{kT}{2nJS} (\eta - \frac{g \mu_o H_o}{kT} ) = B_s(\eta)[/itex]
where [itex]B_s(\eta)[/itex] is the Brouillion func leads (in the absence of a field H) to one stable state for high T and low T, and how if T is in between this there are three solutions, leading to symmetry breaking. What I don't get is what happens with a magnetic field. Say T is chosen such that we've got three solutions still. The linear component will shift, so one solution will be more stable than the other, but how can you tell which one?
 

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