**Francis Murnaghan**was known as Frank. His parents, Angela Mooney and George Murnaghan, were Roman Catholics although George had been educated at a Protestant school. They were Irish, but George had emigrated to the United States where he had made money building houses and as the keeper of a livery stable in St Louis, Missouri. Angela and George had three children while living in the United States, but they returned to Ireland in the late 1880s and purchased a farm, Lisanelly, near Omagh, Co. Tyrone. After their return they had a further six children, and Frank was the seventh of the nine. George Murnaghan took an active part in political life, serving on Tyrone County Council and becoming a Nationalist Member of Parliament, and left the running of the farm to his large family.

Frank was educated at the Christian Brothers School, a Catholic school in Omagh. He graduated in 1910, entering University College, Dublin, in that year. Three years of study saw him achieve a first class degree in mathematics in 1913, and he remained at University College, Dublin, for one further year to take a Master's Degree. He was taught applied mathematics at University College by Arthur Conway who held the chair of Mathematical Physics there. Murnaghan was awarded a National University of Ireland Travelling Scholarship to study abroad but, with the outbreak of World War I, his initial idea of studying in Germany became a non-starter. He discussed with Arthur Conway possible alternative places at which he could study, and Johns Hopkins University in the United States was suggested. Harry Bateman had been appointed there in 1912 and his interests in partial differential equations fitted perfectly with Murnaghan's interests at the time.

Arriving at Johns Hopkins University in Baltimore, Murnaghan began doctoral studies working on differential equations which arose in the study of radio-active decay. His official supervisor was Frank Morley who was Head of the Mathematics Department but he was much influenced by Bateman. In 1916 Murnaghan submitted his thesis *The Lines of Electric Force Due to a Moving Electron* and was awarded his Ph.D. Following this he was appointed as a lecturer at the Rice Institute in Houston, Texas. This was a newly opened institution funded through the will of William Marsh Rice, with the first classes being held in 1912 (it did not become Rice University until 1960). Murnaghan helped supervise the studies of the first Ph.D. produced by the Rice Institute, namely Hubert Evelyn Bray whose thesis *A Green's Theorem in Terms of Lebesgue Integrals* was submitted in 1918, the year Murnaghan left.

From the Rice Institute he returned to Johns Hopkins University where he was appointed Associate Professor. While he had been at the Rice Institute, Murnaghan had met Ada Mary Kimbell and in 1919, the year following his return to Johns Hopkins, they married. They had a son named Francis Dominic after his father, and known as Francis Dominic Jnr, and a daughter Patricia. Their son had a career in the legal profession, while Patricia studied mathematics at Bryn Mawr and became Head of the Mathematics Department at a girl's school in Baltimore. In 1928 Murnaghan became a naturalized citizen of the United States but [5]:-

In fact he returned to Ireland for a number of extended visits throughout his life.... he never lost his Irish heritage or deep feeling for Ireland.

In 1928 Frank Morley retired as Head of Mathematics at Johns Hopkins, and Murnaghan was appointed as Professor and Head of the Mathematics Department. He held this post until 1948 when he retired after a disagreement with the President of Johns Hopkins University, and went to São Paulo, Brazil, where he was appointed as the first Professor of Mathematics at the Instituto Tecnologico de Aeronautica. He returned to the United States on several occasions but in 1959 he retired from his position in São Paulo and, back in the United States, continued to work on mathematics. In particular he was a consultant on ship and aircraft design for the Department of the Navy, David Taylor Model Basin, in Bethesda, Maryland.

Let us now examine the mathematics which interested Murnaghan. The first comment to make here is that he was interested in a wide range of mathematical topics from within both pure and applied mathematics. Perhaps it is easiest to understand how he saw the study of mathematics and its applications by quoting from the Preface to his first book *Vector analysis and the theory of relativity* (1922). He wrote:-

This text was based on lectures he gave in the summer of 1920 at Johns Hopkins. Other excellent works on the applications of mathematics followed. In 1929, in collaboration with Joseph Sweetman Ames, a physicist colleague at Johns Hopkins, he published... it is to the physicist rather than to the mathematician that we must look for the conquest of the secrets of nature. ... this makes it a pleasure and a duty of the mathematician to adapt his powerful methods to the needs of the physicist and especially to explain these methods in a manner intelligible to anyone well-grounded in algebra and calculus.

*Theoretical mechanics: an introduction to mathematical physics*. This was another classic, reprinted by Dover in 1958. In 1932, in collaboration with Harry Bateman and Hugh Dryden, he produced an impressive National Research Council report

*Hydrodynamics*which was again reprinted by Dover in 1956.

Over the period up to 1936, in addition to the major texts we have already mentioned, Murnaghan undertook research and published papers on a wide variety of topics such as electrodynamics, relativity, tensor analysis, elasticity, dynamics, aerodynamics, quantum mechanics, and celestial mechanics. Of course this meant that he was deeply involved in solving differential equations, and indeed he also wrote papers on this topic. After 1936 he became interested in a pure mathematics topic.

His great interest in pure mathematics was in the theory of group representations, but he makes it clear that his interest in this topic is motivated by the importance of its applications. He wrote *The theory of group representations* (1938) and he states his aims in the Preface which are to give an:-

This text became a classic and was republished by Dover Publications in 1963. He also published... elementary and self-contained account of the theory of group representations with special reference to those groups which have turned out to be of fundamental significance for quantum mechanics, especially nuclear physics.

*The orthogonal and symplectic groups*in 1958 which arose from a series of twenty lectures he gave in Dublin in 1957. A further important text on this topic was

*The unitary and rotation groups*(1962) which concentrated on representations of unitary and orthogonal groups.

Although we have mentioned six books by Murnaghan, this is less than half his total of books since he published 16 in all. We also mention *Introduction to Applied Mathematics* (1948) which is a text for graduate students. It covers topics such as: vectors and matrices; Fourier series; boundary value problems; Legendre and Bessel functions; integral equations; the calculus of variations and dynamics; and the operational calculus. After he retired from his position in Brazil and returned to the United States he published *The calculus of variations* and *The Laplace transformation* , both in 1962. The first of these is a short book of less that 100 pages written for engineers and scientists, while the second consists of 19 lectures on such topics as: the Fourier integral; the Laplace integral transformation; the differential equations of Laguerre and Bessel; properties of special functions; asymptotic series for an error function, and for certain Bessel functions.

The final book he published was *Evaluation of the probability integral to high precision* which was a report for the Department of the Navy, David Taylor Model Basin. Murnaghan's own summary of its contents is as follows:-

As a teacher, Murnaghan certainly appears to have had great style. He is described by Zund [5] as:-The 'converging factor' for an asymptotic series representing a function f(x)is that number by which the(n+1)st term of the series must be multiplied so that the result of adding this product to the sum of the first n terms will be f(x). This report describes the determination to high precision of this factor for the asymptotic series representing the probability integral. Tables of this factor to63decimal places are included for n ranging from2to64.

In [2] Lewis gives a delightful quote from Russell Baker (who went on to become a prize-winning journalist) who attended Murnaghan's calculus lectures in 1942:-... a born teacher who had a seemingly insatiable desire to communicate his enthusiasm for mathematics.

Dr Murnaghan was a spirited rosy-cheeked gentleman with silvery locks, who spoke with a pronounced Irish brogue. his energetic classroom manner was like the theatrical performance of an Irish character actor with a fondness for pixie roles.

**Article by:** *J J O'Connor* and *E F Robertson*