In 1781, he was elected a fellow of Trinity College. He was
Donegall Lecturer of Mathematics (1790-1795), and in 1794, he was the first to hold the office of
Donnellan Divinity Lecturer. In 1795, he was appointed Archbishop King's Lecturer in Divinity, succeeded to a senior fellowship, and also became the third
Erasmus Smith's Professor of Mathematics (1795–1799). In 1799, he exchanged the Erasmus Smith's professorship of mathematics for the
Erasmus Smith's Professor of Natural and Experimental Philosophy. In 1789, he published the mathematical treatise
Euclidis Elementorum Sex Libri Priores, Cum Notis (Dublin University Press), whose 10th edition appeared in 1833. On resigning his fellowship in 1806, Elrinton was presented by his college to the rectory of Ardtrea, in the
diocese of Armagh, which he held until December 1811. He resigned, having been appointed to the provostship of Trinity College. During his tenure in this office, he was the acting manager of almost every public board and a supporter of charitable institutions. Elrington was advanced on 25 September 1820 to the diocese of Limerick, and on 21 December 1822, he was translated to Leighlin and Ferns. While on his way to attend Parliament duties in
London, he died of paralysis at
Liverpool on 12 July 1835. He was buried under the chapel of Trinity College Dublin, where a monument with a Latin inscription was in his memory. Another memorial was erected by his clergy in the cathedral church of Ferns. Elrington was an active member of the
Royal Irish Academy and other literary and scientific societies. The Elrington theological essay prize was instituted in Trinity College in 1837. A portrait of the bishop was painted in 1820 for his brother, Major Elrington, by
Thomas Foster; engraved by
William Ward, it was published in 1836 by Graves & Co. There was a marble bust in the
Library of Trinity College Dublin. Elrington was knowledgeable, but contemporaries found him inflexible, rigid, and narrow-minded. Students, while praising him for his learning, found him personally obnoxious. ==Works==