Norhamshire was a detached part of the County Palatine of Durham in the north of England and so under the jurisdiction of the Bishop of Durham with its own courts and administrative centre at Norham Castle. It comprised a triangle of land south of the river Tweed; two of the points lying on the coast at
Tweedsmouth and Budle Bay with the third Tillmouth lying inland on the Tweed.
Together with the similar enclaves of Islandshire and Bedlingtonshire which were also in the possession of the bishops of Durham, Norhamshire was originally part of the endowment of the Bishop of Lindisfarne and passed to their successors the bishops of Durham. These territorial enclaves were
sometimes collectively known as North Durham.
Norhamshire lasted until 1844 when the temporal privileges of the Bishop of Durham were abolished and Norhamshire was incorporated within the modern county of Northumberland, but the name Norhamshire is still used to refer to the area which now forms a ward within the unitary authority of Berwick-upon-Tweed.
http://www.keystothepast.info/durhamcc/k2p.nsf/k2pGlossaryList?readform&letter=N
http://www.berwickonline.org.uk/census/norhamshire.htm