At Brechin High School in 1955 the Mathematics class for Highers contained six pupils, four of whom went on to take Sixth Year Mathematics (at that time comprising Geometry, Elementary Analysis and Dynamics): Alex Craik, Ian Currie, Marion Greig and Don Shaw. The mathematics teacher was David McGregor, and in the sixth year the headmaster, Sam Anderson, taught Geometry and Elementary Analysis, while Arthur Paterson, the science teacher, taught Dynamics.

I was a student in perhaps Jim Tatchell's first lecture course in St Andrews. As always, his lectures were remarkably clear and precise. He was an analyst of distinction.

I should like to thank Dr T Brooke Benjamin for his guidance and helpful criticism throughout the course of this work.

In the presence of an air stream, a uniform liquid film on a horizontal flat plate may be unstable to small disturbances, and waves may arise. In this paper the hydrodynamic stability of thin liquid films is examined both experimentally and theoretically. The experiments concern water films thinner than those which have been examined in the past. It is found that, when the film thickness is sufficiently 'small', a previously unknown type of instability occurs. The theoretical analysis explains this surprising phenomenon. Due to interaction of the mean airflow and small disturbances of the liquid-air interface, normal and tangential stress perturbations are produced at the liquid surface. It is shown that small wave-like disturbances become unstable when the joint influence of the component of normal stress in phase with the wave elevation and the component of tangential stress in phase with the wave slope is sufficient to overcome the 'stiffness' of the liquid surface due to gravity and surface tension. It is found that the destabilising role of the tangential stress component is dominant for very thin films, and that instability may occur whatever the velocity of the air stream, provided the film is made sufficiently thin.

I am grateful to Dr T Brooke Benjamin for some helpful comments. This work was done during a visit to the University of California, Institute of Geophysics and Planetary Physics, La Jolla. This visit was supported by the National Science Foundation, by the Office of Naval Research and by a travel grant from the Sir James Caird Trust.

I am grateful to Dr R E Kelly, Dr R E Davis, Professor J W Miles and Dr H Huppert, with whom I had numerous helpful discussions. This work was performed during a visit to the University of California, Institute of Geophysics and Planetary Physics, La Jolla, California. The visit was supported by the National Science Foundation, by the Office of Naval Research, and by a travel grant from the Sir James Caird Trust.

I am grateful to Dr T Brooke Benjamin for helpful discussions throughout the course of this work, and to Prof J W Miles and Mr F I P Smith for comments on an earlier draft of this paper. The manuscript was completed during a visit to the University of California, Institute of Geophysics and Planetary Physics, La Jolla, California. This visit was supported by the National Science Foundation, by the Office of Naval Research, and by a travel grant from the Sir James Caird Trust.

Part of this work was performed while one of us (Frank Smith) was the recipient of an S.R.C. research studentship. The computations were carried out on the IBM 1620 computer of the University of St Andrews.

I intended staying [in St Andrews] for only a few years; but have remained for forty! ... There was no electronic computer of any kind in St Andrews; nor were there any Xerox copiers. Almost all letters, teaching material and research papers were written by hand and typed by a departmental secretary. Though the computer has revolutionised how mathematics is done, there have been many other advances too.

... paper in the 'American Mathematical Monthly' ... entitled "Prehistory of Faà di Bruno's formula". This formula gives the $n$th derivative of a composite function $f(g(x))$, in an analogous way to Leibniz's formula for the $n$th derivative of a product. It is particularly relevant to discrete mathematics, for two reasons: first, it applies to general formal power series over commutative rings with identity, no issues of convergence need to be considered; and second, the coefficients that arise are slightly disguised Stirling numbers (much as Leibniz's formula involves binomial coefficients). Francesco Faà di Bruno (1825-1888) is perhaps the only mathematician to be also a Roman Catholic saint. He was a student of Cauchy and friend of Hermite. He returned to Turin and devoted himself to work among the poor, for which he was canonised in 1988. In his paper, Alex Craik describes several authors who had anticipated the formula to varying extents, and awards the palm of discovery to Arbogast in 1800. I recommend the paper to you.

In his Foreword, Freeman Dyson comments on the scant recognition given in Japan to Fukagawa, and the lack of interest among Japanese academics, that first drove him to publish his work in English with help from Dan Pedoe. But, in more recent years, there has been a resurgence of interest in traditional Japanese mathematics. Many Japanese mathematicians are now knowledgeable about, and proud of, their distinctive heritage. ... Is it not now the British, rather than the Japanese, who neglect their cultural heritage?

Alex was respected by staff and students as a fine lecturer and a thoughtful, supportive and very modest colleague. In 1983 he was elected Fellow of the Royal Society of Edinburgh, in 1987 he was awarded a Personal Chair at St Andrews and from 1993 to 1995 he was President of the Edinburgh Mathematical Society.