Hi Dot,
In answer to Pm yes Annie Elizabeth is a ROEBUCK.
http://www.cairns.qld.gov.au/content/cemeteries/Person.aspx?no=28771as you can see the plot numbers are adjoining Annie plot 440 and Harriet plot 441 which is why I thought they must have been connected.
Rotten leg cramp woke me so will work on with this.
I found Annie Elizabeth in the Intestacy Returns as born Queensland, died Cairns, Townsville Court 15 May 1915, CIV 1158 and 30 Sep 1915, CV Page 919.
1915 electoral Roll gave Harriet's eligibility as
ROEBUCK Harriet Mary Domestic Duties date of claim - 30 July 1908 Lake Street
one entry she is listed on a page for "Left, Dead, Disqualified" as "Left"
1905 she is the only one at that address"independent means"
1915
ROEBUCK Annie Bourbong St Domestic Duties date of claim 25 May 1915 (Only Roebuck at that address)
1913 ROEBUCK Annie Elizabeth is listed as Domestic Duties - Florence Street.
Other ROEBUCKS at Florence St in 1913 - Abel - Carpenter, Emma - Home Duties, Mark - Lab
1903 Abel is at Farrelly St Charters Towers. (PO Directory)
Getting more confused by the minute so looking at your 1881 UK Census to try to put it all together I see William 39, Harriet 44 and Dora Amy 11.
Looking at Queensland immigration there is an "Amy" of the right sort of age (15) arriving in 1884 aboard the
LINLITHGOWSHIRE Presumably with her father William but he is a few years too young and there is also Albert aged 16 and I don't see an Albert in the census with the family. (though there are Alberts in Yorkshire)
Abel 39 and Emma 38 seem to arrive in 1886 aboard the
DORUNDA with Louisa 21, Annie 16, and Mark 6. (So looks like there may be more than 1 Annie in Queensland)
There are 2 Harriets on the passenger lists but neither fit with age
http://www.archives.qld.gov.au/downloads/Indexes/immigration/Roberts_Herbert-Ross_James.pdfGoing back to 1871 we have William 29, Harriet 34, Harriet 10, Sarah A 3 and Dora 1 so perhaps Harriet put her age down and that is her arriving in 1884 aboard the
CHYEBASSA aged 36 with Sarah A aged 15. Perhaps William and Harriet did arrive separately both putting their ages down.
(We find a lot of people did put there ages down in order to meet immigration criteria.
Unfortunately Queensland BMD are down yet again so I can't look for an Annie Elizabeth born in Queensland to try to fit to the Cairns burial for Annie Elizabeth to try to work out how she may connect to Harriet.
Cheers Kris