Hi Wendy,
http://www.linc.tas.gov.au/Click on Research Tab
Click on Family History
Click on Picture of “Convict Life”
Click on:
Indexes searchable by name
With the Tasmanian Names Index you can search and view many records, including those related to convicts
Now you can search for James HARRINGTON, at the top of this page.
There is a small house symbol, click on the dropdown box and select “Name”.
In the long, search box type in: James Harrington. (34 Results Found)
The following result (result no. 6) is the birth, of your James HARRINGTON Junior, on 15 Feb 1846, at Launceston:
Name Harrington, James
Record type Births
Year 1846
Click on Harrington, James.
Then click on, the link to view his handwritten, digitized birth registration:
Resource RGD33/1/23/ no 1136
http://stors.tas.gov.au/RGD33-1-23-p827j2kThe following result (result no. 21) is the birth, of your Timothy (?) HARRINGTON, on 14 Sep 1844, at Launceston:
Name Harrington, Given Name Not Recorded
Record type Births
Year 1844
http://stors.tas.gov.au/RGD33-1-23-p774j2kMary