**Louis Melville Milne-Thomson**'s mother was Eva Mary Milne, the daughter of the Revd J Milne, and his father was Colonel Alexander Milne-Thomson who was a physician and surgeon. Louis was the eldest of his parents' sons. Milne-Thomson entered Clifton College in Bristol in 1906 as a classical scholar and in his final year at the College he won a scholarship to study mathematics at Corpus Christi College, Cambridge. Entering Cambridge in 1909 he took part I of the Mathematical Tripos in 1911, achieving a First Class, and graduated with distinction as a Wrangler in 1913.

Milne-Thomson was appointed as an assistant mathematics master at Winchester College in 1914. This is one of the oldest of the famous public schools of England, founded in 1382, and is situated in Winchester, Hampshire. On 12 September 1914, shortly after he took up his appointment, Milne-Thomson married Gertrude Frommknecht; the marriage produced three daughters. After teaching for seven years at Winchester College, Milne-Thomson left in 1921 to take up an appointment as professor of mathematics at the Royal Naval College in Greenwich.

The most important topic that Milne-Thomson undertook research on near the start of his career was compiling tables. His first publication on this topic was undertaken jointly with an established table constructor, L J Comrie. Their *Standard Four Figure Mathematical Tables* (1931) was used for many years. A year later he published *Standard table of square roots* and *Jacobian Elliptic Function Tables*. The second of these was written in German but it was published in English in 1950. S C van Veen, reviewing the tables, writes:-

The chief problem in using elliptic functions has always been the lack of suitable numerical tables. Therefore the collection of tables, compiled by the author is an excellent manual, which supplies a long-felt want. ... A comprehensive collection of formulae is included, and many numerical examples are given in order to illustrate the use of these tables. The formulae have been carefully chosen with a view to facilitating calculations which may present themselves. Formulae special to particular branches of knowledge are not included, with the exception of some conformal transformations which cover ground common to several sciences.

Of course the main mathematical tool used in constructing tables was the method of finite differences and in 1933 Milne-Thomson published his first textbook, *The Calculus of Finite Differences*, a text in which he set out to explain to students the techniques which he used in table making. He taught these methods to his students in the Royal Naval College in Greenwich and they found his clearly written text a great asset. The book became a classic student text and the original text was reprinted in 1951.

His next text was also to become a classic but it marked a change in direction in Milne-Thomson's research interests. This text was *Theoretical Hydrodynamics* which was first published in 1938. A second edition of the work appeared in 1950 and contained some additional material, partly based on his own work during the intervening years. He writes:-

Apart from rearrangements and new methods of presentation this edition differs from its predecessor in three important particulars: the introduction of the circle theorem[Milne-Thomson(1940)], whereby the disturbance of a given two-dimensional flow by the introduction of a circular cylinder can be written down without calculation; the corresponding theorem for the sphere[P Weiss(1944)]; the addition of a chapter on the flow of compressible fluids.

Three further additions of new material were added for the third edition which was published in 1956, then still more material was added for the fourth edition in 1960 and again for a fifth edition which appeared in 1968. Similarly *Theoretical Aerodynamics*, first published in 1948, went though a number of editions with the fourth appearing twenty years later in 1968.

In 1956 Milne-Thomson reached the age of sixty-five years and retired from the Royal Naval College in Greenwich. He then took up various posts as Visiting Professor at institutions throughout the world: Applied Mathematics at Brown

University, Rhode Island; the US Army Mathematics Research Center at the University of Wisconsin from 1958 to 1960 where he worked on plane and antiplane elastic problems; the University of Arizona from 1961 to 1970 where he led a very active group of research students; the University of Rome in 1968; the University of Queensland in 1969; the University of Calgary in 1970; and the University of Otago in 1971. By 1971 he had reached the age of eighty and felt that he really did want to retire and went to live in Sevenoaks, Kent.

Let us now look briefly at some papers which Milne-Thomson published between 1940 and his final paper in 1972. In *Hydrodynamical images* (1940) he finds the flow about any two-dimensional cylinder. In 1948 he published *Applications of elliptic functions to wind tunnel interference* while in 1957 he wrote a review paper *A general solution of the equations of hydrodynamics* which M G Scherberg reviews as follows:-

To write a review having at least the brevity of this elegant use of mathematics was not easy. The paper treats compressible or incompressible fluids of constant viscosity and finds the density, velocity and stress distribution in terms of an arbitrary function and a second rank arbitrary tensor.

In another paper *Some hydrodynamical methods* written in the same year Milne-Thomson gives us a flavour of his views on mathematics:-

Mathematics is about the logical consequences of assumed propositions, nowadays called axioms. Thus all mathematics is one. The fancied distinction between 'pure' and 'applied' is a modern and false dichotomy unknown to Euler and Cauchy.

Another topic which interested him was stress. For example he wrote *Consistency equations for the stresses in isotropic elastic and plastic materials* (1942), and *Stress in an infinite half-plane* (1947). He gave two lectures in Madrid in 1951 on the elements of finite elasticity theory, the first lecture covering the topics of deformation tensors, stress, equations of motion, and energy. He published a monograph *Plane elastic systems* in 1960 and it was noted in a review that:-

[

t]he particular clearness of the treatment as well as the mathematical accuracy are to be noticed.

Two years later Milne-Thomson published a sequel entitled *Antiplane elastic systems*. His final paper *Some aspects of antiplane stress* was on similar topics. Here is his own summary of its contents:-

Important problems in the linear theory of the equilibrium of beams of isotropic elastic material have usually been approached by the so-called semi-inverse method, whereby a system of stresses or displacements is guessed and subsequently verified. It is, however, possible to look at these problems in such a way that guesswork is eliminated and reliance placed solely on the boundary conditions. Thus in the case of a cylinder suspended with its axis vertical under gravity, the mere fact that the lateral surface and the lower face are unloaded suffices to solve the problem. Again in the case of torsion the presence of a couple about the axis suffices to give the distribution which leads to the solution. In our opinion too much attention has in the past been given to displacements and strain coefficients and not enough to stress. It is stress, not strain, which should be stressed!

Milne-Thomson received many honours for his achievements. He was elected a fellow of the Royal Society of Edinburgh on 6 March 1933. He was also elected to the Royal Astronomical Society and the Cambridge Philosophical Society. In 1952 he was made a CBE. Finally let us record that he listed sailing and foreign travel as his favourite recreations.

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