Yours, Etc.: Origins and Uses of 8 Sign-Offs
1590s (n.); 1650s (adv.), 1727 (adj.), from down (adv.) + hill. Meaning "a downhill skiing race" is from 1960.
To a worse condition; to the bad: all trends ineluctably downhill (1795+)
The final half of a prison sentence or a military enlistment (1930+ Prison)