Green’s Dictionary of Slang

Beilby’s ball n.

[ety. unknown; the identity of Mr Beilby is unknown but a number of suggestions exist. The most obvious is that Beilby was a well-known sheriff; a second is that Beilby is a mispronunciation of Old Bailey, the court in which so many villains were sentenced to death. The third, and that espoused by Partridge, is that Beilby refers to the bilbo, a long iron bar, furnished with sliding shackles to confine the ankles of prisoners and a lock by which to fix one end of the bar to the floor or ground. Bilbo comes from the Spanish town of Bilbao, where these fetters were invented]

the reification of judicial hanging.

In phrases

dance at Beilby’s ball (v.)

[mid-18C–early 19C] to be hanged; also ext. with ...where the sheriff plays the music or ...where the sheriff pays the fiddlers.

shake one’s trotters at Beilby’s ball (v.) (also shake one’s trotters at Bilby’s ball, shiver… ) [trotter n. (1)]

[mid-18C–early 19C] to be hanged.