irishpapermoney.com

The first Bank with the name ‘Belfast Bank’ was founded in 1787 by John Ewing. The three other partners were John Holmes, John Brown and John Hamilton so the bank became known as the ‘Bank of the Four Johns’. Charles Ranken joined the partnership in 1794. A 1 guinea note issued in 1794 survives. Irish private bank notes pre-1800 are rare and the example shown is beautifully preserved given its age.

A very nice example!
Did this Belfast Bank continue in some form to become the joint stock Belfast Banking Company?

Curiously, there is no obvious link between this bank and the joint stock Belfast Banking Company but there is a link with the joint stock Northern Banking Company !

The key figure here is John Hamilton who was subsequently one of the partners in the private bank of Hugh Montgomery who founded a bank called 'The Northern Bank' in 1809 which eventually became the joint stock Northern Banking Company in 1824.