Barrington was a farmer and land valuer. He was educated at Trinity College where he gained an M.A.. He wrote reports on the flora of Lough Ree, Lough Erne, Ben Bulben, Tory Island and the Blaskets all published by the Royal Irish Academy but most of his scientific papers are on birds. His best known work is The migration of birds, as observed at Irish lighthouses and lightships including the original reports from 1888–97, now published for the first time, and an analysis of these and of the previously put together with an appendix giving the measurements of about 1600 wings London : R.H. Porter  Only 350 copies of this 667 page work were printed.
Barrington was a Fellow of the Linnean Society a Member of the British Ornithologists' Union, and of the British Association Committee for obtaining Observations on the Migration of Birds at Light- houses and Lightships formed to study bird migration. He was also interested in mammals.
- John Wilson Foster, Helena C. G. Chesney (1998) Nature in Ireland: a scientific and cultural history, p. 269 "In the introduction to The Migration of Birds as Observed at Irish Lighthouses and Lightships he wrote that 'few, perhaps, ... and by Major RF Ruttledge, but none carried out the work so comprehensively as Richard Barrington had done."
- Bulletin of the British Ornithologists' Club: 1916 -"Richard M. Barrington, so well known for his excellent work on the migration of Irish birds, died on September 15th, 1915, near Dublin. "
- John Wilson Foster, Helena C. G. Chesney (1998) Nature in Ireland: a scientific and cultural history, p. 269 "Barrington had an interest in travel and climbed in Switzerland and Canada. He visited Rockall in 1896 with Praeger, Harvie-Brown and others on an expedition partly financed by the Royal Irish Academy and partly by Barrington and Harvie-Brown."
- Irish Naturalist 24: 193.
- Journal of Botany 1915:364 
- James Britten & Gerorge S. Boulger 1931 "Bibliographical Index of Deceased British and Irish Botanists" Ed. 2, 22 
- Some Irish naturalists: a biographical note-book
- Habitas Partial bibliography