By the time Joseph was of an age to attend school, his parents had moved to Belfast so it was in that city that he attended the Royal Belfast Academical Institution. At this time he was :-
A shy, delicate [and] precocious boy ...After leaving school, Larmor continued his education in Belfast, studying for his B.A. and M.A. at Queen's University, Belfast. In 1877, having graduated from the Queen's University, he went to St John's College, Cambridge, where he studied the Mathematical Tripos. In 1880 he graduated as Senior Wrangler (the top First Class student) and he was first Smith's prizeman. It is interesting to note that J J Thomson, who like Larmor would make an important contribution to the understanding of the electron, was Second Wrangler (taking second place in the Mathematical Tripos examinations to Larmor).
After graduating Larmor was elected a Fellow of St John's College. Soon after this, still in the year 1880, he returned to Ireland when he was appointed as professor of Natural Philosophy at Queen's College, Galway. He spent five years, 1880 to 1885, teaching in Galway before he returned to St John's College, Cambridge, as a lecturer in 1885. He went on to become Lucasian Professor of Mathematics at Cambridge in 1903, the chair becoming vacant on the death of Stokes in February of that year.
Larmor's contributions came at a time when there were major revolutions in physics with the passing of classical physics to be replaced by quantum theory and relativity. His contributions can be seen as a bridge between the old and the new physics. In  and  Buchwald describes the impact of Larmor's major contributions. The main underlying idea was the principle of least action, which would be of fundamental importance in all Larmor's work throughout his career, and its implications were first set out by him in his paper Least action as the fundamental formulation in dynamics and physics which he published in the Proceedings of the London Mathematical Society in 1884. He published three papers all entitled A dynamical theory of the electric and luminiferous medium between 1894 and 1897. These papers presented his theory of the electron, which of course gained further weight in 1897 when J J Thomson experimentally identified the electron. Buchwald puts these papers by Larmor in context:-
Between 1873 and 1894 British and American physicists were proponents of a theory which they almost all learned directly from J C Maxwell's book Treatise on electricity and magnetism (1873). After 1897 only a few among them, including Heaviside, still adhered to that theory. During these three years (1894-97) the most basic principles of Maxwell's theory of electromagnetism were abandoned, and the entire subject was reconstructed on a new foundation - the electron - by Joseph Larmor in consultation with George FitzGerald. ... [He proposed that] the only source of charge is a particle, that the flow of such particles uniquely constitutes the current of conduction, and that the ether must be strictly separated from matter ...Warwick in  explains in detail how Larmor developed his theory. He summarises the process as follows:-
... Larmor initially tackled the problem of the earth's motion through physical optics and thermodynamics, but ... as he made contact with other Maxwellians beyond Cambridge - especially with George FitzGerald - he came increasingly to make electromagnetic theory fundamental to his work. Indeed, following the introduction of the electron, he began to approach the problem of motion through the ether as one in the electrodynamics of moving bodies. In this specifically electromagnetic context, Larmor confronted the problem of the null result of the Michelson-Morley experiment, adopted the famous FitzGerald-Lorentz contraction hypothesis, and became the first physicist to employ what are now called the 'Lorentz' transformations.Larmor wrote Aether and Matter in 1900 (renamed by Lamb Aether and no matter ) which was a winning entry for the Adams Prize at Cambridge in 1898. It incorporated much of the work of the three major papers of 1894-1897 we referred to above. Warwick writes in :-
His book of 1900, Aether and matter, Cambridge University Press, Cambridge, 1900, helped to establish a research school that guided the development of mathematical electromagnetic theory in Cambridge until the end of World War I.However Warwick  also writes:-
Today, however, Larmor is widely remembered by scientists for just two formulae and one theorem which, although correctly attributed to him, have been seen by historians of science as tangential to his main research interests. Indeed, none of the recent scholarly studies of Larmor's scientific work even mention the now famous formulae and theorem.We should take Warwick's lead and make sure that we mention those concepts to which Larmor's name is attached today. These are the 'Larmor precession', the 'Larmor frequency', 'Larmor's theorem' and 'Larmor's formula'. The first explains the splitting and polarisation of the spectral lines in a magnetic field. The Larmor frequency relates to electrons orbiting in a magnetic field and led him to postulate electrons as orbiting around some centre. He appears to have been the first to predict this behaviour. Larmor's theorem is a related result concerning how a certain transformation can negate the magnetic field for a charged particle subject to electric and magnetic fields. He was the first to calculate the rate of energy radiation from an accelerating electron and for this he gave Larmor's formula which gives the power radiated in terms of the electron's charge and acceleration. The formula breaks down for velocities close to the speed of light due to relativistic effects.
However, Larmor certainly did not rush to embrace the new ideas of space time which were being developed. He :-
... was decidedly conservative in his scientific views. It was difficult to ascertain how much he appreciated the new developments (especially quantum theory), because he was accustomed to adopt a pose which exaggerated his aloofness. He wavered much over Einstein's theory of gravitation. For a short time he became a full convert, but afterwards relapsed into opposition. In the end he rejected, not only the curvature of space, but even the standpoint of the earlier special theory of relativity.When George Stokes and William Thomson (Lord Kelvin) died, Larmor acted as an editor for their complete works. He also brought out a new version of Henry Cavendish's works in 1921, Maxwell had been the editor for the original publication. Larmor also put considerable effort into writing obituaries of Stokes (1903), Gibbs (1905), and Thomson (1908).
Larmor retired from the Lucasian Chair of Mathematics at Cambridge in 1932. He was succeeded in this position by Dirac. With his health deteriorating, Larmor returned to Ireland where he spent his final years at Holywood, County Down. He never married and was described by those close to him as :-
... an unassuming, diffident man who did not readily form close friendships and whose numerous acts of generosity were performed without publicity.D'Arcy Thompson wrote:-
Larmor made few friends, perhaps; but while he lived, and they lived, he lost none.He became a member of the London Mathematical Society in 1884 and he contributed much to the Society being a council member from 1887 until 1912. During his period on the council he was vice president in 1890 and 1891 and also served as treasurer of the Society for over twenty years from 1892 until 1914. He was president of the Society in 1914 and, in the same year, he was awarded the De Morgan Medal by the Society.
The Royal Society elected Larmor as a Fellow 1892 and he served as secretary from 1901 to 1912. The Royal Society awarded him its Royal Medal in 1915 and its Copley Medal in 1921. He was honoured by various universities who awarded him honorary degrees: Dublin, Oxford, Belfast, Glasgow, Aberdeen, Birmingham, St Andrews, Durham and Cambridge. He was also elected to membership of many learned societies including the Royal Irish Academy, the American Academy of Arts and Sciences, and the Accademia dei Lincei.
Knighted in 1909, Larmor served as MP for the University of Cambridge from 1911 to 1922. He made his maiden speech in Parliament in 1912 when, not surprisingly given his background, he supported the Unionists in a debate on Irish home rule. His main contributions in Parliament, however, were to give strong support to universities in particular, and education in general.
Larmor was active in college affairs, being a member of the council of St John's College for many years. In  a nice story is told of his involvement in College affairs:-
... he was conservative in temperament, questioning modern trends even in such matters as the installation of baths in the College (1920). "We have done without them for 400 years, why begin now?", he once said at a College meeting. Yet once the innovation were made he was a regular user. Morning by morning in a mackintosh and cap, in which he was not seen at other times, he found his way across the bridge to the New Court baths.
Article by: J J O'Connor and E F Robertson