Unfortunately there's hardly anything online for South Australia, but I've found what appears to be the missing link; there's a death in Queensland in 1944 for a Francis Law, son of Henry and Elizabeth Arabella Barnett:
https://www.bdm.qld.gov.au/IndexSear....m?type=deaths
And one of the various Ancestry trees has the birth of a Francis Law Barnett to Henry and Arabella in NSW in 1872. I haven't been able to find it on the NSW birth index, but they give an exact date and place of birth (3/1/1872 in Wentworth), so maybe they have a cert.
They also have a wife for Francis, but I haven't found a marriage yet.
I did check the Victoria BMD index, but it doesn't appear to give Arabella's middle name or parents' names:
https://online.justice.vic.gov.au/bd...SearchCriteria
and I haven't found anything on Trove so far:
http://trove.nla.gov.au