To drain something completely, to exhaust all the possibilities of something. The verb to milk, meaning to obtain milk from a cow, was in the early sixteenth century transferred to getting money from someone. In subsequent centuries it was expanded to stealing messages from a telephone or telegraph wire, obtaining maximum audience laughter or applause, and similar unrelated endeavors. Today it appears in such locutions as, “His thesis on Wordsworth’s mentions of flowers milks the subject dry.”