How does Carton get this man to help him?

Carton tries to use what he knows about Barsad (the fact that he is currently employed by the Republican government under a false name but was formerly employed by the English government-which would lead the French government to believe he is a spy) to help free Darnay. He threatens to denounce Barsad, adding that he recognizes the man with whom Barsad was talking as Roger Cly.