How we cite our quotes:
Both are my kinsmen. (2.2.6)
Puzzling over how to act, and whose side to take, York struggles between two different systems. One depends on hierarchy and loyalty to the king, who has no equal. The other is family. It's significant here that, at least according to the second system, Richard and Henry Bolingbroke are equals.
Were I but now the lord of such hot youth
As when brave Gaunt, thy father, and myself
Rescued the Black Prince, that young Mars of men (2.3.2)
Although York invokes his power as the king's representative in the line before these, he's really appealing to Henry Bolingbroke not as a king but as an uncle. The key here is to emphasize the military might of the family. The Black Prince is compared to Mars, the Roman god of war. Gaunt, by wishing he could be as strong as he was in former years so as to properly punish his nephew for returning early, is incidentally showing the fact that the monarchy is militarily weak. Richard may be young, but he hardly has the reputation of a "young Mars" that his father had.
And by the honourable tomb he swears
That stands upon your royal grandsire's bones,
And by the royalties of both your bloods –
Currents that spring from one most gracious head (3.3.4)
Northumberland's interpretation of Henry Bolingbroke's vow to the king actually does the opposite of what it should. Instead of stressing Bolingbroke's status as Richard's subject, Northumberland emphasizes all the ways in which they're descended from the same man, and therefore equal.