If your Darby Cox was renting/leasing property in his own name at the time of Griffith's (1847 to 1864 depending on the area) then he would probably appear - but not everybody did. e.g. A son living on the property of his father (or vice versa), or uncle, would not appear separately. Certain occupations often dont appear - e.g. soldiers, coastguards, policemen, staff working in estates/big houses and living in provided accomodation etc
Tithe records dont cover all areas - e.g. it doesn't cover towns, or exempted properties.
Note: both ancestry and familysearch only have limited collections of pre-civil registration records for Ireland.
Shane