(Latin, comes-stabuli) means “Master of the Horse.” The constable of England and France was at one time a military officer of state, next in rank to the crown.
Quoth Hudibras, Friend Ralph, thou hast Outrun the constable at last; For thou hast fallen on a new Dispute, as senseless as untrue.
Butler: Hudibras, i. 3.
The constable arrests debtors, and, of course, represents the creditor; wherefore, to overrun the constable is to overrun your credit account. To pay the constable is to give him the money due, to prevent an arrest.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894
More on Constable from Infoplease: