Their two main branches were those of "MacWilliam Eighter" in southern Connaught, and "MacWilliam Oughter" to the north of them, in what is now Mayo.
In 1603 "the MacWilliam Oughter," Theobald Bourke, similarly resigned his territory in Mayo, and received it back to hold by English tenure.
In 1781 John Bourke, a Mayo man, believed to be descended from the line of "MacWilliam Oughter," was created Viscount Mayo, and four years later earl of Mayo, a peerage still extant.
After the murder of William de Burgh, 3rd earl of Ulster (1333), the Bourkes (de Burghs) of the collateral male line, rejecting the claim of William's heiress (the wife of Lionel, son of King Edward III.) to the succession, succeeded in holding the bulk of the De Burgh possessions, what is now Mayo falling to the branch known by the name of "MacWilliam Oughter," who maintained their virtual independence till the time of Elizabeth.
The one appropriated Mayo as the Lower (Oughter) M`William, and the earldom of Mayo perpetuates the memory of the event.