(see below the actual naming pattern)
With the Kenny family that is still residing in Kennystown, County Wicklow, Ireland, this practice has been used to some extent.
The present family still there, are using family names.
Denis and Bridget Kenny are the parents to Bartholomew, Michael, Denis, Richard, Mary, Elizabeth, Patricia, Bridget and Rita, not necessarily in this order.
In Kennystown, there seems to be at least one Bartholomew in every generation for many years. My 5th great grandfather’s name was Bartholomew, married to Anne Breen. Their son, Thomas and wife Ellen, came to Smith Falls, Ontario, Canada in 1851 with their children, Bartholomew, Richard, Edward, James and Anne.
Sons:
First son after
father's father
Second
son after mother's father
Third
son after father
Fourth
son after father's oldest brother
Fifth
son after mother's oldest brother
(Sometimes father's 2nd. oldest brother)
Daughters:
First daughter after
mother's mother (or father's mother)
Second daughter after father's mother (or mother's mother)
Third
daughter after mother
Fourth
daughter after mother's oldest sister
Fifth daughter after father's oldest sister
(Sometimes
mother's 2nd. oldest sister)