William Williamson was born in Aberdeen, Scotland c. 1814. 
He arrived in Australia as Surgeon Superintendent on the emigrant ship Isabella on 18th October 1840. Three hundred and five grateful bounty emigrants came under his care -
Copy of Address To Doctor Williamson
We the Undersigned Irish emigrants being desirous to testify in the strongest manner our approbation of the humane and very proper conduct of Dr Williamson. of the barque Isabella, during our passage from Plymouth to Sydney, beg to return him our most grateful thanks for the many kindnesses conferred on us, and on our families, and to assure him that to his unwearied exertions and judicious arrangements in enforcing order, cleanliness, and regularity in the several apartments, we attribute the very healthy state in which we arrived at the place of our destination, and which elicited the approbation of the authorities here in so marked a manner, we therefore feel ourselves called upon to present Doctor Williamson with a silver snuff-box, as a very small token of the high estimation in which we hold his services. Sydney, October 29, 1840.
Copy of Dr. Williamson's Reply
Barque Isabella, Oct. 29, 1840.
Sir, - I have the pleasure of having received this day through your medium, an address accompanied by a silver snuff-box, from the Irish emigrants who lately arrived in this colony, on board of this ship, in token of and expressive of their gratitude to me for my exertions in behalf of their health and comfort during the passage. Through the same channel I beg to return my warmest thanks, and to bear testimony to the most willing obedience which was uniformly yielded to the rules and regulations established on board those vessels despatched by John Marshall Esq., for these shores, and to the observance of which is to be attributed under providence the healthy state in which you all landed. That you and they may attain to independence if not affluence in the land of your adoption, is the sincere wish of Your obedient servant, William Williamson, Surgeon-Superintendent.
For a time at least he resided at Morpeth in 'a neat stone cottage fit for a family and including outhouses' etc.
Departure from New South Wales
He departed Sydney for Hobart on 31 January 1843 on the steamer Seahorse.
In Tasmania he was employed as Medical Officer in charge of the Bridgewater Station. The Bridgewater Convict Station was established in 1828 to provide workers for the construction of a causeway across the Derwent River. Work on the causeway began in 1830. By the time Dr. Williamson arrived the causeway had been completed and Bridgewater was used as a probation station. 
William Williamson died on 9th January 1845 aged 29 after a short illness, universally regretted by everyone who had the pleasure of his acquaintance.
He was buried on 13th January in the Presbyterian Burying Ground, St. Andrews, in Patrick Street, Hobart.
 Australia Cemetery Index, 1808-2007. Ancestry.com