re: Archibald Adair, Bishop of Killala, was appointed after the execution of Bishop John Atherton, as Bishop of Waterford and Lismore, Ireland. He was forced to flee Ireland during the 1641 rebellion. He dc.1647 in Bristol, and I think that after his death, or thereabouts, Maxwell, Bishop of Ross in Scotland may have then been appointed Bishop of Waterford.
From "Scotch and Irish Seeds in American Soil" by Rev. J. G. Craighead :
Freed from the restraints of Wentworth, the Earl of Strafford, ministers (in Ireland) drew up a petition detailing their many grievances, both civil and religious, and praying for the enjoyment of liberty of conscience. This was presented to the Long Parliament. They had great influence in securing the conviction of Wentworth, Earl of Stafford, and in obtaining in 1641 a complete change in the government of Ireland. Two lords-justices were appointed, both belonging to the Puritan party, who received the hearty co-operation of the English Parliament.
The High Commission court was summarily abolished, many of the illegal penalties imposed by ecclesiastical courts were declared null and void, BISHOP ADAIR (Archibald Adair, Bishop of Waterford ?) and five Presbyterian ministers of the diocese of Down were restored to their offices, and the army which Wentworth, Earl of Strafford had collected in Ulster was disbanded. ................. Would the petition presented to the Long Parliament have been signed by Bishop Archibald Adair ?