Conway receives his early education at St. Peter’s College, Wexford and proceeds to enter old University College, Dublin in 1892. He receives his BA degree from the Royal University of Ireland in 1896 with honours in Latin, English, Mathematics, and Natural Philosophy. In 1897, he receives his MA degree with highest honours in mathematics and proceeds to Corpus Christi College, Oxford, becoming University Scholar there in 1901. Also in 1901, he is appointed to the professorship of Mathematical Physics in the old University College and holds the Chair until the creation of the new college in 1909. He also teaches for a short time at St. Patrick’s College, Maynooth.
Conway marries Agnes Christina Bingham on August 19, 1903. They have three daughters and one son.
One of Conway’s students is Éamon de Valera, whom he introduces to quaternions which originate in Ireland. De Valera warms to the subject and engages in research of this novelty of abstract algebra. Later, when de Valera becomes Taoiseach, he calls upon Conway while forming the Dublin Institute for Advanced Studies.
Conway is remembered for his application of biquaternion algebra to the special theory of relativity. He publishes an article in 1911, and in 1912 asserts priority over Ludwik Silberstein, who also applies biquaternions to relativity. This claim is backed up by George Temple in his book 100 Years of Mathematics. In 1947 Conway puts quaternions to use with rotations in hyperbolic space. The next year he publishes quantum mechanics applications which are referred to in a PhD thesis by Joachim Lambek in 1950.
Conway continues his scholarship in the field of mathematics and theoretical physics, and makes a special study of William Rowan Hamilton. With John Lighton Synge, he edits the first volume of Hamilton’s mathematical papers and, with A. J. McConnell, he edits the second volume of Hamilton’s mathematical papers. Conway is also active in college life, being appointed Registrar, a position he occupies until his election as president in 1940. He retires in 1947 from the presidency of UCD. In 1953, some of his writings are edited by J. McConnell for publication by the Dublin Institute for Advanced Studies.
He is elected President of the Royal Irish Academy from 1937 to 1940.