How did Billee die?

Whe Billee fell and couldn't go on, Hal killed him with an axe and left him on the side of the path.

There came a day when Billee, the good-natured, fell and could not rise. Hal had traded off his revolver, so he took the axe and knocked Billee on the head as he lay in the traces, then cut the carcass out of the harness and dragged it to one side.