What is the ruling regarding the oath of divorce sworn by the father to the mother, stipulating that if my sisters leave the house without his permission—and this was his intention on that day—then they did leave two days later without his permission? Is this considered a divorce, and is expiation obligatory upon him?

If your father intended to prevent your sisters from going out without his permission on that specific day only, then his oath is limited to that. If they did not go out on that day without his permission, then his divorce does not take effect, and he is not obligated to offer expiation. However, if they did go out on that day without their father's permission, the fatwa (religious edict) states that the divorce takes effect. Some scholars, like Shaykh Ibn Taymiyyah, hold the view that if a person swears by divorce intending to prevent something and not intending to effect a divorce, then if they break their oath, they are obligated to offer an expiation for an oath (kaffarat yameen), and their divorce does not take effect.