Green’s Dictionary of Slang

ballinocack n.

[? Irish baile, a town + cack n.² (1); thus lit. ‘shit-town’]

the anus.

1785, 1788, 1796

Grose Classical Dict. of the Vulgar Tongue n.p.: Tipperary fortune, two town lands, stream’s town, and ballinocack; said of Irish women without fortune.