There is a birth reg on Familysearch for a William Byrne.
William
Byrne
Birth
1870
Wexford, Ireland
Quarter and Year -1870
Reg Dist. Wexford
Volume 4
If you do a search, you will find several baptism records with various parents for the same year of 1870 and +/- 2 years, and same surname spelling.
Did he marry in Australia, and any idea of year arriving.
Steve.
