Divination by means of arrows.
Shipley's Dictionary of Early English (1955) classes belomancy under aeromancy as one of many methods of divination involving appearances in the air, including but not limited to these gems:
aichomancy: divination by sharp points
aleuromancy: divination by dough
anthomancy: flowers (she loves me, she loves me not)
axinomancy: divination by balanced hatchet
cleromancy: divination by dice (not unlike praying for that d20, eh?)
dririmancy: divination by dripping blood
Mostly, I envision the following scenario whereby the practitioner of belomancy, seeking to know whether his brother is going to die or not, shoots a random volley of arrows into the sky in the general vicinity of his sibling. Although, aichomancy also sounds fun:
"Tell me the future!" Poke. Poke.
"You're going to get punched in the head if you don't stop poking me."