This Tommy Armour 1.68 Mesh ball was most likely produced in 1927 by the Great Lakes Golf Company. According to Kevin McGimpsey in The Story of the Golf Ball, Great Lakes made a Tommy Armour mesh ball that year. McGimpsey also records that, in 1935, the Worthington Golf Ball company offered an "Armour" mesh pattern ball and "Tommy Armour" dimple pattern but not a Tommy Armour mesh pattern.
This ball appears to have never been used.