This question contains what I can only describe as a schoolboy error. To cut a long story short: the football club known as Third Lanark never played in the town of Lanark. It owed its name to the fact that Lanarkshire is the traditional county that included the city of Glasgow.
Lanark Racecourse was indeed situated (to quote Wikipedia) in the small town of Lanark, in Scotland's Central Belt, 25 miles (40 km) from Glasgow. Reputedly founded by King William the Lion of Scotland (1165–1214), it closed in October 1977.
Third Lanark Athletic Club was founded in 1872 by members of the 3rd Lanarkshire Volunteer Rifle Corps. Founder members of the Scottish Football Association in 1872 and the Scottish Football League in 1890, the club went out of business in 1967. Just six years earlier it had finished in third place in the SFL.
The 3rd Lanarkshire Volunteer Rifle Corps was one of four volunteer corps formed in 1859, as French naval expansion caused 'invasion panic'. Inspired by the first Scotland v. England international match, played in 1872, Third Lanark Athletic Club played its first games at the regimental drill ground on Victoria Road, Glasgow. In 1875 they moved to Cathkin Park, in the Glasgow district of Crosshill.
In 1903, Partick Thistle FC moved out of their Hampden Park ground to a new home, which they also named Hampden Park. This became Scotland's most famous football ground. Meanwhile Third Lanark took over the original Hampden Park, renaming it New Cathkin Park, and this was to be their home for the remainder of their 95–year history. Following their demise the ground was taken over by Glasgow's Parks Department, which maintains it to this day as a public space where football is still played.
You can read more about the Rifle Volunteers, including a short section concerning the football club, here.
© Macclesfield Quiz League 2020