Suppose an attractive particle or satellite of mass m to be moving in a circular orbit, with an angular velocity Ω, round a planet of mass M, and suppose the planet to be rotating about an axis perpendicular to the plane of the orbit, with an angular velocity n; suppose, also, the mass of the planet to be partially or wholly imperfectly elastic or viscous, or that there are oceans on the surface of the planet; then the attraction of the satellite must produce a relative motion in the parts of the planet, and that motion must be subject to friction, or, in other words, there must be frictional tides of some sort or other. The system must accordingly be losing energy by friction, and its configuration must change in such a way that its whole energy diminishes.

Such a system does not differ much from those of actual planets and satellites, and, therefore, the results deduced in this hypothetical case must agree pretty closely with the actual course of evolution, provided that time enough has been and will be given for such changes.

Let C be the moment of inertia of the planet about its axis of rotation;

1. r the distance of the satellite from the centre of the planet;

2. h the resultant moment of momentum of the whole system;

3. e the whole energy, both kinetic and potential, of the system.

In the following paper several problems are considered, which were alluded to in my two previous papers on this subject.

The paper is divided into sections which deal with the problems referred to in the table of contents. It was found advantageous to throw the several investigations together, because their separation would have entailed a good deal of repetition, and one system of notation now serves throughout.

It has, of course, been impossible to render the mathematical parts entirely independent of the previous papers, to which I shall accordingly have occasion to make a good many references.

As the whole inquiry is directed by considerations of applicability to the earth, I shall retain the convenient phraseology afforded by speaking of the tidally distorted spheroid as the earth, and of the disturbing body as the moon.

It is probable that but few readers will care to go through the somewhat complex arguments and analysis by which the conclusions are supported, and therefore in the fourth part a summary of results is given, together with some discussion of their physical applicability to the case of the earth.

Secular distortion of the spheroid, and certain tides of the second order

In considering the tides of a viscous spheroid, it was supposed that the tidal protuberances might be considered as the excess and deficiency of matter above and below the mean sphere—or more strictly the mean spheroid of revolution which represents the average shape of the earth.

The following paper contains the investigation of the mass-motion of viscous and imperfectly elastic spheroids, as modified by a relative motion of their parts, produced in them by the attraction of external disturbing bodies; it must be regarded as the continuation of my previous paper, where the theory of the bodily tides of such spheroids was given.

The problem is one of theoretical dynamics, but the subject is so large and complex, that I thought it best, in the first instance, to guide the direction of the speculation by considerations of applicability to the case of the earth, as disturbed by the sun and moon.

In order to avoid an incessant use of the conditional mood, I speak simply of the earth, sun, and moon; the first being taken as the type of the rotating body, and the two latter as types of the disturbing or tide-raising bodies. This course will be justified, if these ideas should lead (as I believe they will) to important conclusions with respect to the history of the evolution of the solar system. This plan was the more necessary, because it seemed to me impossible to attain a full comprehension of the physical meaning of the long and complex formulæ which occur, without having recourse to numerical values; moreover, the differential equations to be integrated were so complex, that a laborious treatment, partly by analysis and partly by numerical quadratures, was the only method that I was able to devise.

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.