He tells us that in 1902 his salary was £6 10s. Od. per year, and in 1907 it was £24 per year. His secondary education was begun by the headmaster, A R Smith, who gave him a lesson each morning from 8 to 8.45 before the school opened at 9. During the winters of 1902-4 he attended evening classes at the Widnes Technical School, studying art, mathematics and physiography. From 1903 to 1907 he only taught for half of each week; the other half he attended classes at the Widnes Secondary School. This was an excellent school, and it was here that the mathematical bent of his life was determined.

The following investigations arose out of an attempt to obtain an expression for the free tidal motion of a rotating flat sheet of water in the form of a circular sector. This was suggested as a subject for research by Prof Lamb. No progress, however, was made with the original problem except in the case of the semicircle, and in this case only when the angular velocity of rotation was small compared with the free periods of the relative motion of the water. Even in this restricted case the resulting expression was so complicated as to make interpretation very difficult, if not impossible, and consequently it is not given below.

After a statement of the general conditions (which were first investigated by Lord Kelvin), and their application to the case of a nearly circular sheet of water, a discussion is given of the limiting forms of forced tides as the period of the disturbing force tends to become infinite. It appears that these limiting forms are, for rotating flat sheets of water of uniform depth, always different from the forms given by the equilibrium theory. The limiting form is calculated for a rectangular sheet, and the result may, perhaps, be of interest, as the complete determination of the tides on such a sheet has not yet been obtained. Some partial solutions are then given, and, finally, the free symmetrical motion is obtained for a particular small circular sea on a sphere, as the next approximation after that which regards the sea as flat.

This problem has been discussed by Schwarzschild and Professor Nicholson, but their numerical results are not in complete agreement. The analysis of the latter is much the shorter, and it appeared likely that his results were nearer the truth. Dr Bromwich, however, was in possession of an independent treatment which showed that this was not the case as regards the magnitude of the maximum pressure. He informed me of this, and suggested that I should make a fresh calculation, very kindly showing me his results. I have worked through all the cases considered by Professor Nicholson, together with some additional and intermediate ones, and the results are substantially in agreement except as regards the maximum pressure. The present note contains the results referring to this maximum, making the determination much more precise than was done by Schwarzschild.

He found that being a lecturer was much harder than being a student, either undergraduate or postgraduate. He gave twelve lectures per week, two each morning for 6 days per week. During his first year as a lecturer he had no time for research except during vacations, but he found time to direct the postgraduate work of A T Doodson for the degree of M.Sc. and thus began a collaboration which continued until Doodson's death.

As the equations of tidal motion have been solved in only a few cases, which are very restricted when compared with the actual conditions of terrestrial tides, it is of interest to try to obtain some approximation to the long period tides by means of a discussion of the limiting forms of these tides as the period of the disturbing forces tends to become infinite.

Although a separate body, separately governed and funded, there was a very early association with Bidston Observatory. It was intended as a research institution with both theoretical and practical aspects of tidal dynamics as topics of study.

This paper is concerned exclusively with the principal lunar semi-diurnal harmonic constituent of the tides which is denoted by $M_{2}$. The primary object is to show how the fundamental dynamical equations of the tides may be used to obtain a knowledge of the distribution of the surface-elevation from such observational data as are available. The dynamical equations, as formulated in §4, connect the elevation-gradients with the currents and the external forces, including those of friction. From a knowledge of the currents and a hypothesis for the frictional forces the elevation-gradients can be calculated.

When the elevation is also known the directions of the co-tidal and co-range lines, and also the distance apart of neighbouring members of these lines, can be calculated. Such conditions are fulfilled for coastal stations, and it is remarkable that, in spite of the great attention that has been paid to co-tidal charts, these simple calculations do not appear to have been previously made. But if the elevation-gradients can be calculated along a line which passes through one or more points at which the elevation is known, it is clear that methods can be devised by which the elevation can be calculated all along the line. Again, such calculations do not appear to have been previously made.

The association between the [Tidal] Institute and Bidston [Observatory] gradually strengthened, with William Plummer joining the governing board of the Institute. Significantly a tide-predicting machine was installed at Bidston at the end of 1924. Tide-predicting machines are devices that can be 'programmed' with the harmonic tidal constants for a particular port and then proceed to provide predictions or hindcasts for any desired date. As they required very high precision engineering, very few were in existence and several foreign governments asked the Institute to supervise the construction of machines for their own use.

This book by a professor emeritus in the University of Liverpool and a young professor who has made notable advances in the study of the tides is a good example of what can be accomplished when the experience of a great teacher is combined with the brilliance of an ardent investigator. The subject is well presented ...

In 1928, William Plummer died and this may have been the catalyst for the amalgamation of the Observatory and Tidal Institute, which took place on the 1 January 1929. This was a pivotal moment as the next decade brought a considerable increase in oceanographic research to Bidston. With Joseph Proudman as Director and Arthur Doodson as Associate Director, the Liverpool Observatory and Tidal Institute developed new methods for the analysis of tides and their prediction, and a second tide-predicting machine was acquired.

... it was Proudman's idea that the department should be converted to one of physical oceanography. At that time there was much less provision for marine physics in this country than for marine biology. In addition to widening his own research interests to include oceanic circulation and the distribution of temperature and salinity in oceans and seas, Proudman initiated observational work in the Irish Sea.

Prof Proudman's book is written from the mathematical standpoint, but its results and implications are in terms which should be understood by scientists engaged in marine, as well as in physical, oceanography. Before his appointment as Professor of Oceanography at Liverpool University twenty years ago, he had been professor in the department of Applied Mathematics for some fourteen years. His book, therefore, has a particularly sound hydrodynamical background. Results applicable to practical oceanography are deduced from the basic equations of dynamics.

It is appropriate, I think, that I should lecture on a subject connected with the tides of the ocean, for Sir George Darwin himself gave a great deal of his attention to the subject. From about 1882 until the end of his life in 1912, he was generally regarded as the greatest living authority on ocean tides.

The National Academy of Sciences awarded one of its highest honours, the Alexander Agassiz medal, to an English researcher on the mathematics of the oceans' tides, Prof Joseph Proudman, F.R.S., director of the Liverpool Observatory and Tidal Institute. Since Prof Proudman was unable to be present in person, the medal was accepted in his behalf by Sir Alfred Egerton, secretary of the Royal Society of London.