Green’s Dictionary of Slang

a glass holding one third of a quartern (i.e. ¼ pint).

1823		‘Jon Bee’ Dict. of the Turf, the Ring, the Chase, etc. 70: ‘Drain, of gin’ — the third person in a quartern of gin, when the glass is too large for ‘three outs.’.
1824		J. Wight Mornings in Bow St. 273: [He] immediately called for ‘a quartern of gin of three outs,’ with which he offered to treat or as a Corinthian would say, to ‘sluice the ivories’ of the gentlemen present.
1832		Satirist (London) 9 Dec. 394/3: As far as it [i.e. a quartern of gin] would go, in a three-out glass.
1837		Dickens ‘Seven Dials’ in Slater Dickens’ Journalism I (1994) 72: A couple of ladies, who have imbibed the contents of various ‘three-outs’ of gin and bitters.
1859		G.A. Sala Gaslight and Daylight 73: A quick, sharp, jerking twist for the spirit tap [...] a dextrous tilt of the ‘two,’ or ‘three out’ glass required.
1909		J. Ware Passing Eng. of the Victorian Era 245/1: Three out brush (Public-house). A glass shaped like an inverted cone, and therefore something like a housepainter’s brush, especially when dry. The glass holds one-third of a quartern – a quartern being just half of half a pint.