In Hughes and Cresswell A New Introduction to Modal Logic (1996 ed.) page 271, they attempt to establish the incompleteness of the system K + G1 + BF (where K is L(P->Q)->(LP->LQ), G1 is MLP->LMP, and BF is the Barcan Formula) which they had previously established would have to be characterized by convergent frames (ie, (w1Rw2 & w1Rw3) -> Ew4:(w2Rw4 & w3Rw4) ) if it were complete.
So to establish the incompleteness of KG1 + BF it will suffice to find a non-theorem which is valid on evey convergent frame -- or, what comes to the same thing, a consistent wff which cannot be satisfied on any convergent frame.
They then provide a long and complicated wff which they state without proof does indeed prove the system's incompleteness. (Edit: I found the full statement of the proof in "Incompleteness and the Barcan Formula" M. J. Cresswell. Journal of Philosophical Logic, Vol. 24, No. 4 (Aug., 1995), pp. 379-403. He uses the same problematic phrasing in that article as well.)
My question is, how do those two things amount to the same thing? A non-theorem of KG1+BF which is valid on every convergent frame doesn't sound like a consistent wff which cannot be satisfied on any convergent frame. For one, the former is valid on every convergent frame, while the latter isn't valid on any. What am I misunderstanding?