(from http://www.somejamesons.com/)
Hugh and his wife, Jane, were both still living November 15, 1782, as appears from the following bond given by their sons, Alexander and Daniel, in consideration of a deed to them of his farm in Dunbarton, NH.
This map indicates the succession of battles and the sequence which run mostly from north to south. The Battle of Bennington, the Battle of Saratoga
With the surrender of General Burgoyne on October 17 (10 days after the final battle of Saratoga) the regiment was disbanded on October 27, 1777.

(somejamesons.com) "In June of 1753, Hugh Jameson and Robert Parinson were summoned into court in Portsmouth, NW, having been sued by Robert Gage of Coleraine, Ulster, Ireland, for not having been paid the monies due per the agreement by Jameson and Parkinson to do so, two years after arriving in America. Although both Hugh Jameson and Robert Parkinson denied owing any money both were found guilty on September 6, 1753 and ordered to pay in the amount of 18 pounds, two shillings and 4 sixpence, new tennor (sic) bills, as damages for both men, including court costs. This was apparently paid as nothing more is (recorded) of this case.
The court summons for Hugh Jameson & Robert Parkinson (Photo: Somejamesons)
"Hugh Jameson arrived in the new world apparently penniless. Documents associated with the above mentioned court case describe how he was nearly pulled off the boat at the very last moment as the ship was actually weighing anchor and about to leave port. Three men arrived unexpectedly demanding money from him. Unable to pay, a collection was taken by several of the other passengers on board to prevent that from happening.
Queen Anne (Photo: CoinLink)
Dunbarton with Jameson reference at orange spot (photo: SomeJamesons)
Jameson House (Photo: SomeJamesons)
Passage Payment Agreement:
We, Hugh Jamison and Robert Parkinson Doo bind
oursleves our ____ ____ ____ a figures in the fair
and just sum of six pounds thirteen shillings ____
lawful money of Great Britain to be paid unto Mr.
Robert Gage Mert (merchant): of Colerain in Ireland or his ______
to be paid in two years from the date here of
and after our arrival in Boston in America
as witness our hands and seals this August 4th ____
There is little to indicate why they left. In 1740-41, cold winters and poor harvests caused the Forgotten Famine during which 400,000 people died in Ireland. However, it is difficult to link that tragedy with the Jamesons' emigration five years
later.

William's parents emigrated to Ireland, or more precisely Ulster (Northern Ireland), in the early 1600's. "At the beginning of the 1600's, James I, of England, began a re-population of northern Ireland with mostly Scottish Protestants. The area was at the time largely occupied by the Desmonds, who had been defeated and depleted in various rebellions with the crown during the reign of Queen Elizabeth. This forced re-population started in 1611 and was an effort by the king to colonize this here-to-fore troublesome area with a more sympathetic and supportive people. The incentives were lucrative with large quantities of land available to each immigrant. Many Scots took advantage of this offer and is undoubtedly why the ancestors of Hugh Jameson found their way to Ireland. This population was largely increased with further immigration over the next 50 years as a result of persecutions by English Kings Charles I and Charles II in their effort to establish the Church of England in Scotland."
The family took up residence and William was born in Templemore Parish of Londonderry, Northern Ireland. The distance across the channel is not very far. Notice Scotland in the upper right hand corner of the map of Northern Ireland. William (1) was born in 1646.

"The Scottish colonists resided in Ireland over several decades during the middle 1600's. They lived somewhat autonomously, retaining their culture and traditions. Intermarriage of Scots with the Irish were exceedingly rare, so that the Scotch race remained nearly as distinct as it was prior to its settlement in this new land. They attended their own churches and continued life pretty much as it was known to them in Scotland... The Jamesons in general were said to have been a robust, active, muscular, and prolific race of people undoubtedly descended from the son of James or Jamie..."
"William Jameson was said to be a vigorous Scotchman, who was thought to have served in the defense of Londonderry during the siege of 1689, and probably the Battle of Boyne in 1690 with distinguished gallantry and such bravery that he was made free from taxation throughout the British dominion by William III, Prince of Orange."
The Battle of Boyne was a decisive battle in turning back King James II of England, Scotland and Ireland. He was Catholic and for him the war was an attempt to regain the throne so he could create sovereignty for Ireland, religious freedom and the right to own land for Catholics. No surprise, most of his soldiers were Irish Catholics.
Conversely, for William of Orange and his soldiers, the war was about maintaining Protestant and English rule against the fear that if James retook the throne he would take their land and or kill large numbers of them.