Jinkabout Mill, Bo'ness location is shown on this map.
http://maps.nls.uk/view/74963114 It is some distance from the mouth of the Avon as stated. It probably milled for The Duke of Hamilton estate as well as local farmers. The carse was well known for growing grain with vast fields recorded to the west and north of this location. In competition was the mill south of the wall near Westquarter to the West of this location. That mill was owned by the Earl of Zetland's Kerse Estate.
RCAHMS website states: Archaeological Notes
NS97NW 84 9474 7975
'A large building two storeys high used as a corn mill having two large wheels propelled by water. Close to the mill is the dwelling house with offices attached, all in good repair; there is a vegetable garden and small farm attached. The whole is in the occupation of John Henderson and the property of the Duke of Hamilton.'
OS Name Book (Linlithgowshire), 1856.
Falkirk Collections has an image of part of the site:
http://collections.falkirk.gov.uk/search.do;jsessionid=EC13980B3E18FCC6371B79CEB420F78E?id=178344&db=object&view=detailThis record also appears on another thread;
William Meek, probably the son of William Meek & Janet Weir, born June 19, 1784 in Jinkabout Mills, Bo'Ness Parish, Linlithgowshire, married Margaret Angus, date and place unknown. They emigrated to Quebec about 1820. His brothers, John and Thomas, also came to Quebec initially but later moved on. I would appreciate learning more about them and their ancestors in Scotland.
At one time the area was a choice for the establishment of the world famous Carron Iron Works but the lack of suitable water supply and road, river and rail access saw it discarded.
Happy hunting.