Orr thrived under the excellent tuition given by McNeill and he was taught geometry from the textbooks of Richard Townsend (Chapters on the modern geometry of the point, line, and circle), and John Casey. In 1883 he was awarded a scholarship from the Royal University of Ireland. Queen's College, Belfast, was one of the colleges of the Royal University of Ireland and it was there that Orr studied. He was taught by John Purser. With Purser's teaching, Orr :-
... became famous for the brilliance of his answering at the examinations of the Royal University of Ireland ...He graduated from Queen's College, Belfast, in 1885 and, on 27 April of the same year, he was admitted as a pensioner at St John's College, Cambridge. He matriculated at St John's College in the Michaelmas Term (the first term) of 1885 as a scholar, meaning that he had won a scholarship. At St John's College he was taught by Joseph Larmor and the two became life-long friends with Orr obtaining much advice from his teacher. He graduated with a B.A. in 1888 being Senior Wrangler in the mathematical tripos, obtained first place in Part II in 1889, followed by the award of a fellowship in St John's College in 1891. The work which won him the fellowship was on extending Feuerbach's theorem about systems of circles. In the same year as he was awarded the fellowship, he was appointed professor of applied mathematics at the Royal College of Science for Ireland, in Dublin. This College had been created in 1867, growing out of the Museum of Irish Industry. It was based at 51 St Stephen's Green and it offered a range of science subjects linking classroom and laboratory teaching. Orr became one of several eminent professors who struggled to keep up their research because of heavy teaching loads.
In 1892 Orr married Elizabeth Campbell, known as Lizzie, a daughter of Samuel Watson Campbell, of Melbourne, Australia. The Campbell family were originally Irish, coming from County Down. William and Elizabeth Orr had three children, all girls: Elizabeth Lowry Orr was born on 18 May 1894 and baptised on 14 October 1894; Annie Gwendoline Holmes Orr was born on 19 June 1896 and baptised on 12 January 1897; and Edith Kathleen Orr was born on 7 June 1900 and baptised on 14 December 1900. All the baptisms took place at Eustace Street Presbyterian Church in Dublin, Dublin County, Ireland.
We learn something of Orr's problems in his new job from a letter he wrote from Rathgar, Dublin, to Joseph Larmor on 8 October 1895. In this letter Orr asks Larmor for information regarding Captain Abney's knowledge of mathematics. Captain William de Wiveleslie Abney (1843-1920) was a scientific photographer particularly interested in applications of photography to astronomy. He worked at the Department of Science and Art, South Kensington, London, and he served as president of the Royal Astronomical Society in 1893-95. One might wonder what Abney had to do with the Royal College of Science for Ireland, Dublin, but the reason was that this Irish institution was under the direct supervision of the Department of Science and Art at South Kensington, London. Orr tells Larmor that he has a meeting with Abney shortly regarding the syllabus he has to teach at the Royal College of Science, Dublin. Orr writes that he considers the subjects he has been asked to teach as being too much, both for himself and for his students. He wishes to make a case to Abney but is unsure how much Abney will understand regarding a mathematics syllabus. In the letter, he also complains vigorously about the hours he has to spend at the University and the amount of work he has to do. He tells Larmor that he is most aggrieved at the situation.
Orr's first publications were on hypergeometric series, Fourier double integrals involving Bessel functions which he followed up with a similar paper involving Legendre functions. In 1898 Orr published On the forced precession and nutations of a rotating ellipsoidal shell containing liquid in the Philosophical Magazine. His paper begins:-
The object of the following analysis is mainly to determine the difference between the precession and nutation of a spinning body like the earth subject to external couples such as those which act on the earth on the hypothesis that it is perfectly rigid and that it is a shell filled with liquid. The liquid is supposed to be homogeneous, incompressible, and frictionless, the shell to be rigid, and the inner and outer surfaces to be surfaces of revolution about a common axis. Owing to these suppositions the bearing of the results obtained on the question of the internal liquidity or solidity of the earth is probably remote. Results for the problem to be discussed here have been stated long ago by Lord Kelvin, but without any indication of the analysis by which he obtained them. It will appear that while the results here obtained for the precession and the nineteen-yearly nutation agree closely with Lord Kelvin's, those for the half-yearly and the fortnightly nutations differ.Orr was not the only one to find errors in Lord Kelvin's paper, for some years later Henri Poincaré also found the errors. Lord Kelvin wrote to Poincaré on 2 March 1901 from 15 Eaton Place, London:-
I am sorry to say it is too true that there are, as you tell me in your letter which I have received this morning, several mistakes in respect to magnitude and to sign in my statements regarding the nutation which would exist if the earth consisted of a rigid ellipsoidal shell filled with frictionless liquid. Ever since 1876 when that statement of results was published, I have been looking for time to go through the mathematical work again and publish it. If I had succeeded in finding the time, no doubt I would have corrected the errors myself; but in this I was anticipated by an Irish professor (W McF Orr, of the Royal College of Science, Dublin), in a paper published in the 'Philosophical Magazine' for December 1898. The corrections which you will find in this paper agree, I believe, with those you now give me. The subject is most interesting, and I hope yet to return to it, but meantime I hope you will publish your own work with your corrections of my errors. In my mathematical work I suppose for simplicity the ideal interior liquid to be frictionless. No assumption of viscosity in the liquid would be, to my mind, very interesting. It certainly could not show any way of escaping my main conclusion that the earth is on the whole an elastic solid of high rigidity; though probably or possibly not so high as to cause practically perfect resistance against change of shape by the tide-generating influences of the sun and moon. I thank you warmly for your kindness in writing to me and telling me of the errors you have found.In 1904 Orr published On Clausius' theorem for irreversible cycles, and on the increase of entropy, published in the Philosophical Magazine. Albert Einstein wrote a review  in which he wrote:-
The author shows that in the 'Treatise on Thermodynamics' Planck applies the concepts "reversible" and "irreversible" in a sense somewhat different from that which he defines them. Then he advances a series of objections that may be raised against various ways of representing the foundations of thermodynamics; especially noteworthy among these objections is that of Bertrand, i.e., that the pressure, temperature, and entropy are defined only for the case that at least sufficiently small parts of a system can be regarded as being in equilibrium; a similar objection is raised with respect to the heat supplied.In 1906, Orr submitted to the Royal Irish Academy the first of his two papers entitled The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid, this first part being subtitled A Perfect Liquid. The second part, submitted in 1907, was subtitled A Viscous Liquid.
Orr is best remembered today by applied mathematicians through the Orr-Sommerfeld equation that is an eigenvalue problem which models 2-dimensional modes of disturbance in a parallel shear flow. Michael Eckert writes in :-
[In 1909] Arnold Sommerfeld, professor of theoretical physics in Munich, introduced what became known as the Orr-Sommerfeld method to account for the instabilty of flows at the Fourth International Congress of Mathematicians in 1908 in Rome. He was not aware of William McFadden Orr's more extensive work. Other mathematicians and physicists also did not take note of Orr's work. Even in Great Britain Orr's contribution seems to have been ignored for some years. Horace Lamb, the author of the well-known textbook on hydrodynamics, still did not mention Orr's publication in 1910 when Sommerfeld asked him about new publications on turbulence. In the fourth edition [published in 1916] of 'Hydrodynamics', however, Lamb acknowledged that the stability equation for plane Couette flow was given by Orr, and afterwards independently by Sommerfeld. This seems to be the first time when both Orr and Sommerfeld appeared together.Orr was elected a fellow of the Royal Society of London in 1909. Two years later, in October 1911, the Royal College of Science for Ireland moved to new premises in Merrion Street. At this time the College had 141 students. It had been aiming to expand from its modest size but the outbreak of World War I in 1914 was a blow. On 11 May 1916, Orr wrote to Larmor about the Easter Uprising in Dublin and its aftermath. Orr suggested that the government treated the rebels too leniently and says that most of the Protestants he has spoken to agree with him. After the World War I ended in 1918 the political troubles in Ireland added to the problems that the College was facing and in September 1922 the closure of the College was announced. However, University College Dublin offered the Royal College of Science accommodation and for two years Orr was one of the professors teaching in the University buildings on St Stephen's Green. Attempts were made to return to Merrion Street but, on 4 May 1926, University College Dublin offered to appoint most of the staff from the Royal College of Science. Orr accepted this offer and became a Professor of Pure and Applied Mathematics at University College Dublin. He continued to hold this position until he retired in 1933.
James Richard Timoney attended Orr's lectures at University College Dublin in 1928. Timoney, known as Dick, worked with Edmund Whittaker in Edinburgh before returning to University College Dublin in 1932 where he became a colleague of Orr. In the 1970s Timoney was Head of Mathematics at University College Dublin. He writes:-
We were taught mathematical physics in first year honours by William McFadden Orr. ... He was a great gentlemen but very stern in every sense. He would sit with his pocket watch out waiting for 9 o'clock to start the lecture on the dot. If you came in after 9 a.m. you got a 'late' on the role. ... He was heard to say that research in the College of Science consisted of solving a quadratic equation which had not been solved before. We learned little form him for most of the time was spent criticising the bad treatment of Newton's laws. We did not know very well what the laws were but we knew that Ernst Mach was the man to be respected.Another student, J S Strettan, relates an anecdote relating to Orr giving students a 'late' on the role :-
One morning he taxed me with being late. ... I pleaded that the College clock was five minutes fast by the Ballast Office clock. He got my estimate of the time, went off on his bicycle to the Ballast Office and checked it. The following morning he apologized and his apology was rendered all the more handsome by being made in the presence of his class.In the early 1920s Orr had a lengthy period of inactivity in research. In a letter to Larmor on 28 March 1925 he says he has started research again but is worried about his health, both physical and mental, and says he is not sleeping. He lived for less than a year after resigning his chair in 1933. He died at Nobles Hospital, Douglas, Isle of Man but was buried in Mount Jerome Cemetery in Dublin. Shortly after his death, F Gowland Hopkins, President of the Royal Society of London, gave an address at the anniversary meeting on 30 November 1934. He said :-
It would be an interesting subject for a psychologist to study as to how the creative faculty and the critical faculty can exist simultaneously in the one individual. Certainly the late William McFadden Orr possessed both in the highest degree. ... he led a life of unremitting labour in various branches of Mathematics. His original contributions to Bessel Functions, Fourier Analysis, Stability of certain Liquid Motions, etc., all contained results which seem destined to be of permanent value. As the years went on, his critical faculties became more pronounced and he was able to elucidate many difficulties and rectify many conceptions, even in the writings of distinguished mathematicians. These qualities made him of the highest value as a referee for learned bodies. ... much of his life work was unseen by the general mathematical reading public ...Let us end this biography by quoting from Arthur Conway :-
Anything connected with bicycles was of interest to Orr, who was up to the end a great cyclist. In his Tripos year at Cambridge he had carried off with great ease all the events at the Cambridge University meeting of that year. We in Dublin will long miss his lean, bearded figure going though the streets on a racing bicycle without an overcoat even in the coldest weather.
Article by: J J O'Connor and E F Robertson