The papers contained in this fourth and last volume are too diverse in character to admit of such a complete arrangement according to subjects, as was made in the earlier volumes. I begin, however, with three papers on Periodic Orbits, a subject which has played a very important part in the recent developments of dynamical astronomy. The middle one of the three is by Mr Hough, and it is reproduced, by his permission, as explaining the sequence of ideas which led from the first to the third paper.

The proprietors of the Encyklopädie der mathematischen Wissenschaften have kindly allowed me to print here the English text of the article ‘Die Bewegung der Hydrosphäre.’ It is the long paper entitled ‘The Tides,’ which forms Part II. The section on the dynamical theory of the tides is by Mr Hough, but I wrote the rest. According to my general scheme this paper would naturally have belonged to the group of papers contained in Vol. I, but it was published after the appearance of that volume. The same remark also applies to the two recent papers on tides contained in Part IV.

Part III consists of miscellaneous papers arranged in chronological order. Some of them are obviously of slight value and are reproduced merely for the sake of completeness.

Mr Davison's interesting paper was, he says, suggested by a letter of mine published in Nature on February 6, 1879. In that letter it is pointed out that the stratum of the Earth where the rate of cooling is most rapid lies some miles below the Earth's surface. Commenting on this, I wrote:—

“The Rev. O. Fisher very justly remarks that the more rapid contraction of the internal than the external strata would cause a wrinkling of the surface, although he does not admit that this can be the sole cause of geological distortion. The fact that the region of maximum rate of cooling is so near to the surface recalls the interesting series of experiments recently made by M. Favre (Nature, Vol. XIX., p. 108), where all the phenomena of geological contortion were reproduced in a layer of clay placed on a stretched india-rubber membrane, which was afterwards allowed to contract. Does it not seem possible that Mr Fisher may have under-estimated the contractibility of rock in cooling, and that this is the sole cause of geological contortion?”

Mr Davison works out the suggestion, and gives precision to the general idea contained in the letter. He shows, however, that there is a layer of zero strain in the Earth's surface, and that this layer, instead of that of greatest cooling, must be taken to represent Favre's elastic membrane.

The Proportion of First Cousin Marriages to all Marriages

It is well known that when the Census Act, 1871, was passing through the House of Commons, an attempt was made by Sir J. Lubbock, Dr Playfair, and others, to have a question inserted with respect to the prevalence of cousin marriages, under the idea that when we were in possession of such statistics we should be able to arrive at a satisfactory conclusion as to whether these marriages are, as has been suspected, deleterious to the bodily and mental constitution of the offspring. It is unfortunately equally well known that the proposal was rejected, amidst the scornful laughter of the House, on the ground that the idle curiosity of philosophers was not to be satisfied.

It was urged, that when we had these statistics it would be possible to discover by inquiries in asylums, whether the percentage of the offspring of consanguineous marriages amongst the diseased was greater than that in the healthy population, and thus to settle the question as to the injuriousness of such marriages. The difficulty of this subsequent part of the inquiry was, I fear, much underrated by those who advocated the introduction of these questions into the census. It may possibly have been right to reject the proposal on the ground that every additional question diminishes the trustworthiness of the answers to the rest, but in any case the tone taken by many members of the House shows how little they are permeated with the idea of the importance of inheritance to the human race.

I am not aware that anyone has taken the trouble to work out Lesage's theory, except in the case where the particles of gross matter, subjected to the bombardment of ultramundane corpuscles, are at a distance apart which is a large multiple of the linear dimensions of either of them. Some years ago I had the curiosity to investigate the case where the particles are near together, and having been reminded of my work by reading Professor Poynting's paper on the pressure of radiation, I have thought it might be worth while to publish my solution, together with some recent additions thereto.

If a corpuscle of mass m moving with velocity v impinges on a plane surface, so that the inclination of its direction of motion before impact to the normal to the surface is ϑ, it communicates to the surface normal momentum kmv cos ϑ, and tangential momentum k′mv sin ϑ; where k is 1 for complete inelasticity, and 2 for perfect elasticity, and k′ is 0 for perfect smoothness and 1 for perfect roughness.

In the following paper the effects are investigated of the bombardment by Lesagian corpuscles of two spheres, which are taken to be types of the atoms or molecules of gross matter. The effects of the normal and tangential components of the momentum communicated by each blow from a corpuscle will be treated separately.

After I had read my paper on this subject in March last before the Statistical Society, Mr Arthur Browning (a Fellow of the Society) suggested to me another method of determining whether cousin marriages were injurious or not. This method was to discover whether the proportion of offspring of first cousins, amongst persons distinctly above the average, either physically or mentally, was less or greater than the general proportion given by my paper for persons in a similar rank of life.

Mr Browning and I agreed to carry out this scheme together; but we thought it would be well to delay extensive operations until we saw what success was attainable in a more limited inquiry. The results are so very unequal to our expectations that we do not intend to proceed further. The statistics are, however, of some interest as far as they go.

The boating eights, who race at Oxford and Cambridge in May, are a picked body of athletic men. There are twenty boats at Oxford, and thirty at Cambridge, in the “first and second divisions”; and their crews are 400 men, exclusive of coxswains. We accordingly sent circulars to the stroke-oars of these fifty boats, during their preparatory training, begging them to ask the members of their crews whether their parents were first cousins or not. Where there were several brothers rowing in the eight, they were only to be counted as one case; and cases of refusal to answer were also to be marked.

The following paper contains an account of experiments and observations on the formation of ripple-mark in sand. The first section is devoted to experiments on the general conditions under which ripple-mark is formed, and especially on the mode of formation and maintenance of irregular ripples by currents. In the second section it is shown that regular ripple-mark in sand is due to a complex arrangement of vortices in oscillating water; and the last section gives some account of the views of certain recent observers in this field, and a discussion of some phenomena in the vortex motion of air and water.

First Series of Experiments

A cylindrical zinc vessel, like a flat bath, with upright sides, 2 feet 8 inches in diameter and 9 inches deep, was placed on a table, which was free to turn about a vertical axis. Some fine sand was strewn over the bottom to a depth of about an inch, and water was poured in until it stood three inches deep over the sand. After some trials of simply whirling the bath, in which no regular ripple-mark was formed, I found that rotational oscillation with a jerking motion of small amplitude gave rise almost immediately to beautiful radial ripples all round the bath. If the jerks were of small amplitude the ripples were small, and if larger they were larger.

In Chapter ii. of Book ix. of the Mécanique Céleste, Laplace considers the transformation of the orbit of a comet when it passes a large planet. His object is to show that the action of Jupiter suffices to account for the disappearance of Lexell's comet after 1779.

He remarks that if a comet passes very near to Jupiter, it will throughout a small portion of its orbit move round the planet almost as though it were unperturbed by the sun, and that both before its approach to and after its recession from the planet it will move round the sun almost as though it were unperturbed by the planet. The nature of the orbit of the comet will usually be much transformed by its encounter with the planet. It is clear then that there must be some surface surrounding the planet which separates the region, inside which the comet moves nearly round the planet, from the region in which it moves nearly round the sun. Such a surface is to be found by the comparison of the ratio of the perturbing force to the central force in the motion round the sun with its value in the motion round the planet. There is a certain surface at which this ratio will be the same in the two cases, and this is the surface required for the proposed approximate treatment of the problem.

Bartholomeu Diaz, the discoverer of the Cape of Storms, spent sixteen months on his voyage, and the little flotilla of Vasco da Gama, sailing from Lisbon on July 8, 1497, only reached the Cape in the middle of November. These bold men, sailing in their puny fishing smacks to unknown lands, met the perils of the sea and the attacks of savages with equal courage. How great was the danger of such a voyage may be gathered from the fact that less than half the men who sailed with da Gama lived to return to Lisbon. Four hundred and eight years have passed since that voyage, and a ship of 13,000 tons has just brought us here, in safety and luxury, in but little more than a fortnight.

How striking are the contrasts presented by these events! On the one hand compare the courage, the endurance, and the persistence of the early navigators with the little that has been demanded of us; on the other hand consider how much man's power over the forces of Nature has been augmented during the past four centuries. The capacity for heroism is probably undiminished, but certainly the occasions are now rarer when it is demanded of us. If we are heroes, at least but few of us ever find it out, and, when we read stories of ancient feats of courage, it is hard to prevent an uneasy thought that, notwithstanding our boasted mechanical inventions, we are perhaps degenerate descendants of our great predecessors.

The object of the author was to devise a form of slide-rule which should be small enough for the pocket, and yet be a powerful instrument.

The first proposed form was to have a pair of watch-spring tapes graduated logarithmically, and coiled on spring bobbins side by side. There was to be an arrangement for clipping the tapes together and unwinding them simultaneously. Two modifications of this idea were given.

The second form was the logarithmic graduation of several coils of a helix engraved on a brass cylinder. On the brass cylinder was to fit a glass one, similarly graduated.

To avoid the parallax due to the elevation of the glass above the other scale, the author proposed that the glass cylinder might be replaced by a metal corkscrew sliding in a deep worm, by which means the two scales might be brought flush with one another.

In the final part of his work on Celestial Mechanics which has lately appeared M. Poincaré devotes some space to the consideration of the orbits discussed by Professor Darwin in his recent memoir on Periodic Orbits [Paper 1]. From considerations of analytical continuity M. Poincaré has been driven to the conclusion that Professor Darwin is in error in classifying together certain orbits of the form of a figure-of-8 and others which he designates as satellites of the class A. “Je conclus” says Poincaré “que les satellites A instables ne sont pas la continuation analytique des satellites A stables. Mais alors que sont devenus les satellites A stables?”

Besides the question here raised by Poincaré a second immediately presents itself. After explaining the disappearance of the stable orbits A it is necessary also to give a satisfactory account of the origin of the unstable orbits A. These questions had occupied my mind prior to the publication of M. Poincaré's work, and the present paper contains in substance the conclusions at which I had arrived in connection with them.

It will be seen that the difficulties which have occurred in following up the changes in form of Darwin's orbits arise in some measure from the omission to take into account the orbits described in the present paper as “retrograde,” and the failure to recognize the analytical continuity between these orbits and the direct orbits.

The present investigation was undertaken at the request of Sir Ernest Shackleton; the expense of the reduction was defrayed by him, and this paper was communicated to the Royal Society by his permission. It will ultimately be republished as a contribution to the volume of the physical results of the expedition.

The first section, describing the method of observing, is by Mr James Murray. The second section explains the reduction of the observations and gives a comparison between the new results and those obtained by the Discovery in 1902–3. The third section is devoted to the discussion of certain remarkable oscillations of mean sea-level and to speculations as to their cause and meaning.

On the Method of Observing the Tides

Early in June, 1908, preparations were begun for the erection of a tidegauge, the most important feature of which was to be a recording apparatus made from a modified barograph. Owing to various delays and mishaps it was not before the middle of July that the gauge was completed in its final form, and the continuous record begun, which was carried on for more than three months, subject only to the loss of half an hour weekly, while the paper was being changed.

Dr Mackay undertook the erection of the instrument, the apparatus was devised by the joint suggestions of Messrs David, Mackay, Mawson, and Murray, while Mr Day did the more delicate part of the work, namely, the alteration of the barograph.

Fig. 1 represents two Peaucellier's cells, of which O1, O2 are the respective fulcra, which are fixed; A, P and B, P the poles. Let the moduli of the cells be respectively proportional to two charges of positive electricity at O1 and O2. Thus O1A = m1/O1P, O2B = m2/O2P, so that, if O1A + O2B is constant, P traces the equipotential line of m1 and m2 placed at O1 and O2 respectively. The constancy of O1A + O2B may be attained thus: Let the pivots at O1, O2, about which the cells turn, be two needles; fasten a piece of pack-thread to A, pass it round O1, round another needle E (driven into the drawing-board), round O2, and fasten the other end to B. The broken line in Fig. 1 represents this thread. Then, if P is moved so as to keep the thread taut, it describes one of the equipotential lines. If the needle E is shifted to F and G, we get other equipotential lines.

Fig. 2 represents the arrangement of the thread where one of the charges is negative. In this and the succeeding figures, fixed points are marked with crosses, and the needles are exaggerated so as to show the disposition of the strings, and the bars of the cells are omitted, leaving only the tracing point P and the other poles marked.

THE papers here collected together treat of the figure and of the movement of an actual or an ideal planet or satellite. I have failed to devise a short title for this volume which should describe exactly the scope of the subjects considered, and the title on the back of the book can only be held to apply strictly to three-quarters of the whole.

The first three papers fall somewhat further outside the proper meaning of the abridged title than do any of the others, for they are devoted to the mathematical solution of a geological problem. The second paper is indeed only a short note on a controversy long since dead; and the third is of little value.

The discussion of the amount of the possible changes in the position of the earth's axis of rotation, resulting from subsidences and upheavals, has some interest, but the conclusions arrived at in my paper are absolutely inconsistent with the sensational speculations as to the causes and effects of the glacial period which some geologists have permitted themselves to make.

At the end of this first paper there will be found an appendix containing an independent investigation by Lord Kelvin of the subject under discussion. He was one of the referees appointed by the Royal Society to report upon my paper, and he seemed to find that on these occasions the quickest way of coming to a decision was to talk over the subject with the author himself—at least this was frequently so as regards myself.

PREFACE

More than half a century ago Édouard Roche wrote his celebrated paper on the form assumed by a liquid satellite when revolving, without relative motion, about a solid planet. In consequence of the singular modesty of Roche's style, and also because the publication was made at Montpellier, this paper seems to have remained almost unnoticed for many years, but it has ultimately attained its due position as a classical memoir.

The laborious computations necessary for obtaining numerical results were carried out, partly at least, by graphical methods. Verification of the calculations, which as far as I know have never been repeated, forms part of the work of the present paper. The distance from a spherical planet which has been called “Roche's limit” is expressed by the number of planetary radii in the radius vector of the nearest possible infinitesimal liquid satellite, of the same density as the planet, revolving so as always to present the same aspect to the planet. Our moon, if it were homogeneous, would have the form of one of Roche's ellipsoids; but its present radius vector is of course far greater than the limit. Roche assigned to the limit in question the numerical value 2·44; in the present paper I show that the true value is 2·455, and the closeness of the agreement with the previously accepted value affords a remarkable testimony to the accuracy with which he must have drawn his figures.