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Henry and Arabella Barnett also had three others in Wenworth NSW

6244/1869 BARNETT HERBERT W HENRY ARABELLA WENTWORTH
5010/1864 BARNETT THOMAS H HENRY ARABELLA WENTWORTH
21127/1874 BARNETT OLIVER BRISTOW HENRY ARABELLA WENTWORTH
19236/1872 BARNET RONALD F HENRY ARABELLA WENTWORTH

Wentworth is at the junction of the Darling and Murray Rivers in far western NSW - at that time it was an important River Port. However, people there had far more contact with Adelaide and Melbourne than with Sydney.

It would appear that Henry and Arabella married in Victoria in 1863-64.

Oliver B Barnett married Mary E F Gainer in Broken Hill NSW in 1905. - They had two children Henry C in 1906 and Pearl E in 1908 in Broken Hill.

My ancestry sub has run out, but you might be able to pick up Thomas, Herbert or Oliver in the electoral rolls. I could not see them in the NSW Deaths Index or the Ryerson Index.

Di