INTRODUCTION.

At most places on the North Atlantic the prediction of high and low water is fairly easy, because there is hardly any diurnal tide. This abnormality makes it sufficient to have a table of the mean fortnightly inequality in the height and interval after lunar transit, supplemented by tables of corrections for the declinations and parallaxes of the disturbing bodies. But when there is a large diurnal inequality, as is commonly the case in other seas, the heights and intervals after the upper and lower lunar transits are widely different; the two halves of each lunation differ much in their characters, and the season of the year has great influence. Thus simple tables, such as are applicable in the absence of diurnal tide, are of no avail.

The tidal information supplied by the Admiralty for such places, consists of rough means of the rise and interval at spring and neap, modified by the important warning that the tide is affected by diurnal inequality. Information of this kind affords scarcely any indication of the time and height of high and low water on any given day, and must, I should think, be almost useless.

This is the present state of affairs at many ports of some importance, but at others a specially constructed tide-table for each day of each year is published in advance. A special tide-table is clearly the best sort of information for the sailor, but the heavy expense of prediction and publication is rarely incurred, except at ports of first rate commercial importance.

The ‘Discovery’ wintered in 1902 and in 1903 at the south-eastern extremity of Ross Island, on which Mount Erebus is situated, in south latitude 78° 49′ and east longitude 166° 20′.

The station is near the west coast of a great bay in the Antarctic Continent, and the westerly coast line runs northward from the station for about 9° of latitude. To the eastward of the bay, however, the coast only attains a latitude of about 75° and follows approximately a small circle of latitude. Since the tide-wave comes from the east and travels to the west the station is not sheltered by the coast to the westward, and the continent to the eastward can do but little to impede the full sweep of the tide-wave in the Antarctic Ocean. It is true that Ross Island itself is partially to the east of the anchorage, but it is so small that its influence cannot be important. Of course the westward coast line must exercise an influence on the state of tidal oscillation, for regarding the tide-wave as a free wave coming in from the east, it is clear that it will run up to the end of the bay and then wheel round northward along the westerly coast. It would seem then that the situation is on the whole a good one for such observations. Of course their value would have been much increased if it had been possible to obtain other observations elsewhere.

Preface. Account of Operations.

A Committee appointed for the examination of the question of the Harmonic Analysis of Tidal Observations practically finds itself engaged in the question of the reduction of Indian Tidal Observations; since it is only in that country that any extensive system of observation with systematic publication of results exists.

[Early in 1883] I proceeded to draw up a considerable part of this Report, had it printed, and submitted it to Major Baird. I was not at that time aware of the extent to which Mr Roberts, of the Nautical Almanac office, co-operated in England in the tidal operations, nor did I know that he was not unfrequently taking the advice of Professor Adams. It was not until Major Baird had read what I had written, and expressed his approval of the methods suggested, that these facts came to my knowledge; but it must be admitted that it was through my own carelessness that this was so. I then found that Professor Adams decidedly disapproved of the notation adopted, and would have preferred to throw over the notation of the old Reports and take a new departure. The notation of the old Reports seems to me also to be unsatisfactory, but, seeing that Major Baird and his staff were already familiar with that notation, I considered that an entire change would be impolitic, and that it was better to allow the greater part of the existing notation to stand, but to introduce modifications.

WHEN the Syndics of the Cambridge University Press did me the honour of offering to publish a collection of my mathematical papers, I had to consider the method of arrangement which would be most convenient. A simple chronological order has been adopted in various collections of this kind, and this plan certainly has advantages; but an arrangement of papers according to subject may be more convenient. In the case of my own work the separation into well-defined groups of subjects was easy, and I have therefore adopted this latter method. I shall, however, give at the beginning of each volume a chronological list with a statement as to the volume in which each paper will be found.

This first volume contains papers on Oceanic Tides and on an attempt to measure the Lunar Disturbance of Gravity; the second will give my papers on Tidal Friction and on the astronomical speculations arising therefrom; the third will be devoted to papers on Figures of Equilibrium of Rotating Liquid and on cognate subjects; and the fourth will be on Periodic Orbits and on various miscellaneous subjects.

Throughout corrections and additions will be marked by inclusion in square parentheses.

The whole of my work on oceanic tides and the attempt made by my brother Horace and me to measure the attraction of the moon sprang from ideas initiated by Lord Kelvin, and I should wish to regard this present volume as being, in a special sense, a tribute to him.

Introduction.

The tidal oscillation of the ocean may be represented as the sum of a number of simple harmonic waves which go through their periods approximately once, twice, thrice, four times in a mean solar day. But these simple harmonic waves may be regarded as being rigorously diurnal, semi-diurnal, ter-diurnal, and so forth, if the length of the day referred to be adapted to suit the particular wave under consideration. The idea of a series of special scales of time is thus introduced, each time-scale being appropriate to a special tide. For example, the mean interval between successive culminations of the moon is 24h 50m, and this interval may be described as the mean lunar day. Now there is a series of tides, bearing the initials M1, M2, M3, M4, &c., which go through their periods rigorously once, twice, thrice, four times, &c., in a mean lunar day. The solar tides, S, proceed according to mean solar time; but, besides mean lunar and mean solar times, there are special time scales appropriate to the larger (N) and smaller (L) lunar elliptic tides, to the evectional (ν), to the diurnal (K1) and semi-diurnal (K2) luni-solar tides, to the lunar diurnal (O), &c.

The process of reduction consists of the determination of the mean height of the water at each of 24 special hours, and subsequent harmonic analysis. The means are taken over such periods of time that the influence of all the tides governed by other special times is eliminated.

Account of the experiments.

We feel some difficulty as to the form which this report should take, because we are still carrying on our experiments, and have, as yet, arrived at no final results. As, however, we have done a good deal of work, and have come to conclusions of some interest, we think it better to give at once an account of our operations up to the present time, rather than to defer it to the future.

In November, 1878, Sir William Thomson suggested to me that I should endeavour to investigate experimentally the lunar disturbance of gravity, and the question of the tidal yielding of the solid earth. In May, 1879, we both visited him at Glasgow, and there saw an instrument, which, although roughly put together, he believed to contain the principle by which success might perhaps be attained. The instrument was erected in the Physical Laboratory of the University of Glasgow. We are not in a position to give an accurate description of it, but the following rough details are quite sufficient.

A solid lead cylinder, weighing perhaps a pound or two, was suspended by a fine brass wire, about 5 feet in length, from the centre of the lintel or cross-beam of the solid stone gallows, which is erected erected there for the purpose of pendulum experiments.

Introduction.

Extensive use of the tide-gauge has only been made in recent years, and by far the largest number of tidal records consist only of observations of high and low water (H. and L. W.). Such observations have usually been reduced by determining the law governing the relationship between the times and heights of H. and L. W. and the positions of the moon and sun. This method is satisfactory so long as the diurnal inequalities are small, but it becomes both complex and unsatisfactory when the diurnal inequality is large. In such cases the harmonic notation for the tide is advantageous, and as, except in the North Atlantic Ocean, the diurnal inequality is generally considerable, a proper method of evaluating the harmonic constants from H. and L.W. observations is desirable.

The essential difference between the method here proposed and that followed by Laplace and his successors is that they introduced astronomical considerations from the first and applied them to each H. and L.W., whereas the positions of the sun and moon will only be required here at a single instant of time. In their method, the time of moon's transit, and hence the interval, was found for each tide; the age of the moon, and the moon's and sun's parallaxes and declinations were also required. An extensive table from the astronomical ephemeris was thus necessary, and there still remained the classification of heights and intervals according to the age of moon, and two parallaxes, and two declinations. The classification could hardly be less laborious, and was probably less mechanical, than the sorting processes employed below.

Introduction.

The object of the present article is to show how the best use may be made, for scientific purposes, of a short visit to any port.

We refer to the article “Hydrography” [Admiralty Scientific Manual] for an account of the method of observing the tides, and shall here assume that the height of the water above some zero mark may be measured, in feet and decimals of a foot, at any time, and that the zero of the tide gauge may be referred by levelling to a bench-mark ashore.

Something of the law of the tide might be discovered from hourly or half-hourly observations even through a single day and night, but to discover the law at all adequately it is necessary that the observations should embrace at least one spring tide and one neap tide. For the full use of the methods given below, the observations should be taken each hour for 360 hours, or 720 hours. A longer series must be regarded as a new set of observations, and the means must be taken of the results of the several sets.

It has been usual to recommend observations of the times and heights of high and low water, but hourly observations are far preferable, the hours being reckoned according to mean time of the port.

We shall, however, begin by a sketch of the treatment of observations of high and low water, and shall then give more detailed instructions for hourly observations and the formation of a tide table.

RECORD OF WORK DURING THE PAST YEAR.

A large number of tidal results have been obtained by the United States Coast Survey, and reduced under the superintendence of Professor Ferrel. Although the method pursued by him has been slightly different from that of the British Association, it appears that the American results should be comparable with those at the Indian and European ports. Professor Ferrel has given an assurance that this is the case; nevertheless, there appears to be strong internal evidence that, at some of the ports, some of the phases should be altered by 180°.

CERTAIN FACTORS AND ANGLES USED IN THE REDUCTION OF TIDAL OBSERVATIONS.

[These are given at the end of the last Paper.]

ON THE PERIODS CHOSEN FOR HARMONIC ANALYSIS IN THE COMPUTATION FORMS.

Before proceeding to the subject of this section, it may be remarked that it is unfortunate that the days of the year in the computation forms should have been numbered from unity upwards, instead of from zero, as in the case of the hours. It would have been preferable that the first entry should have been numbered Day 0, Hour 0, instead of Day 1, Hour 0. This may be rectified with advantage if ever a new issue of the forms is required, but the existing notation is adhered to in this section.

Shortly after the meeting of the British Association last year (1881),a the instrument with which my brother and I were experimenting at the Cavendish Laboratory, at Cambridge, broke down, through the snapping of the wire which supported the pendulum. A succession of unforeseen circumstances have prevented us, up to the present time, from resuming our experiments.

The body of the present Report, therefore, will merely contain an account of such observations by other observers as have come to our knowledge within the past year, and it must be taken as supplementary to the second part of the Report for 1881. The Appendix, however, contains certain theoretical investigations, which appear to me to throw doubt on the utility of very minute gravitational observations.

The readers of the Report for 1881 will remember that, in the course of our experiments, we were led away from the primary object of the Committee, namely, the measurement of the Lunar Disturbance of Gravity, and found ourselves compelled to investigate the slower oscillations of the soil.

It would be beyond the scope of the present Report to enter on the literature of seismology. But, the slower changes in the vertical having been found to be intimately connected with earthquakes, it would not have been possible, even if desirable, to eliminate all reference to seismology from the present Report.