A couple of ideas appear to be on Banquo's mind at the start of Act III in Shakespeare's Macbeth.
First, he suspects Macbeth of treachery, of assassinating King Duncan:
Thou hast it now--King, Cawdor, Glamis, all
As the Weird Women promised, and I fear
Thou played'st most foully for't....
Second, he wonders whether or not the part of the weird sisters' prophecy that pertained to him will come true, also:
...If there come truth from them--...
May they not be my oracles as well,
And set me up in hope?
He's wondering if his heirs will rule Scotland, as the witches predicted.
Ironically, the one thing Banquo should be thinking about he isn't. He does not realize that Macbeth is plotting against him and Fleance with the same thought in mind--the prediction that Banquo's heirs will rule. Since gaining the crown, Macbeth's ambition has deepened, and having the throne for himself is no longer enough. He now wants to create a dynasty in his own name, and Fleance stands in the way. In a matter of seconds following Banquo's speech, Macbeth will ascertain from Banquo that he is riding this afternoon, and that Fleance will be going with him. That will prove to be the opportunity Macbeth is looking for.
At the beginning of Act III, Banquo recalls the prophecy of the three witches and wonders how it has all come true for Macbeth. He remembers that the witches told Macbeth that he would be the Thane of both Cawdor and Glamis and eventually the king and this has come true by this point in the play. Banquo also thinks that Macbeth has not done righteous acts to get these new promotions and questions whether or not Macbeth has done ill deeds in the process. Banquo then reminds himself that the witches also said that Macbeth would have no successors to the throne and that the future kings would be of Banquo’s line. He hopes that this will come true, as the prophecy has come true for Macbeth, and that he is not hopeful for nothing.