The latitude of any place is the mean of the altitudes of a star at its two transits above and below the pole, and the meridian is the great circle through the zenith bisecting the small circle described by a circumpolar star. Thus both the latitude and the meridian depend on the instantaneous axis of the earth's rotation, the observed latitude being the altitude of the instantaneous axis, and the meridian a great circle passing through that axis and the zenith of the place of observation.

But the meridian may also be taken to mean a great circle fixed in the earth passing through the geographical pole and the place. We may call these two circles the astronomical and geographical meridians. The relationship between the two meridians may be illustrated by a figure which I leave to the reader to draw for himself. If C be the geographical pole, I the instantaneous axis and P the place of observation, then IP is the astronomical meridian and CP the geographical meridian. The Eulerian nutation of an absolutely rigid earth would be such that I would describe a circle round C in a period of 306 days; if we allow the earth to yield elastically to centrifugal force the period of the circular motion is augmented and observation shows that the period of 306 days becomes one of about 430 days; lastly it appears that in actuality the simple circular motion is perturbed by other inequalities.

The ordinary stereographic projection of the world in two hemispheres is utterly worthless as giving a true impression of the whole; for the linear scale at the margins of the circles is twice that at their centres. Its only merit is that there is no angular distortion. Mercator's projection gives a still more fallacious impression, except as regards the equatorial regions.

It appears to me therefore that there is a want, in the school-room and lecture-room, of some map which shall give a more truthful representation of the globe than the above, and which yet shall not be so expensive and cumbrous as a globe.

A gnomonic projection on to the faces of a regular icosahedron is but very slightly distorted, although a slight amount of angular distortion is here introduced. I have been told that at the recent Geographical Congress at Paris, some such projections as this were exhibited, and that they were of old date. Mr Proctor has also made star-maps by projection on to the faces of a regular dodecahedron; but in 1873, when the idea occurred to me of using this projection, I was not aware of the fact.

If the icosahedral projection be developed and arranged as a band of ten triangles round the equator, with saw-like edges of five triangles in the north and five in the south, a very fair representation of the globe is given.

The medal of the Royal Astronomical Society is this year awarded to M. Henri Poincaré, member of the Academy of Sciences of Paris. As your President, the agreeable duty of presenting the medal to him devolves upon me, but before I do so I must endeavour to lay before you the grounds upon which the Council has made this award.

M. Poincaré's researches have been so diverse in character, and have been carried out with such a wealth of knowledge, that I feel but little confidence in my fitness to perform this arduous task; yet I cannot but rejoice that my tenure of this chair should have furnished me with the opportunity of paying the homage which is due to him for his great achievements in the field of mathematics.

A large part of his work is concerned with the development of the science of pure mathematics, and of this side of his activity I am quite incompetent to speak. But in awarding our medal we naturally think of the value of the contributions by the proposed medallist to our science. Now, although many of M. Poincaré's investigations have perhaps already found, or will at some future time find, their application in the problems of dynamical astronomy, yet it is not necessary to search for cases of possible applicability to astronomy to justify our award.

Introduction

The existing methods of treating the Problem of the three Bodies are only applicable to the determination, by approximation, of the path of the third body when the attraction of the first largely preponderates over that of the second. A general solution of the problem is accordingly not to be obtained by these methods.

In the Lunar and Planetary theories it has always been found necessary to specify the motion of the perturbed body by reference to a standard curve or intermediate orbit, of which the properties are fully known. The degree of success attained by any of these methods has always depended on the aptness of the chosen intermediate orbit for the object in view. It is probable that future efforts will resemble their precursors in the use of standard curves of reference.

Mr G. W. Hill's papers on the Lunar Theory mark an epoch in the history of the subject. His substitution of the Variational Curve for the ellipse as the intermediate orbit is not only of primary importance in the Lunar Theory itself, but has pointed the way towards new fields of research.

The variational curve may be described as the distortion of the moon's circular orbit by the solar attraction. It is one of that class of periodic solutions of the Problem of the three Bodies which forms the subject of the present paper.

If we make any observation, for example the transit of a star, a definite numerical result is obtained. To say that that result is liable to errors of observation, is only correct from one point of view. It is true that the result does not really correspond with the time at which the star crossed the meridian, yet it is an accurate representation of a certain very complex event. Undoubtedly, the principal feature in that event is the time of crossing the meridian; but there is also involved in it various properties of the instruments, the atmosphere, and the observer himself, &c. The object of the observation is, of course, to get a result which shall represent that principal feature, after the elimination of the minor features. The comparative simplicity of this, and of many other observations, permits us to unravel the complex event into its constituent parts, and to estimate each numerically. One part consists of corrections of all sorts. But when all these have been made, there still remains a result representing a complex event, viz. the transit of the star, together with unknown properties of the circumstances of the observation. These unknown properties form the subject-matter of the theory of errors of observation; but it is only because there is a consistent theory of the principal phenomenon, that the most probable line of demarcation can be drawn between the two parts of the complex event.

Account of Experiments

The pressure of loose earth against revetment walls is a subject which is frequently being brought in a practical manner under the notice of engineers. A considerable number of theoretical investigations have been published by Coulomb, Rankine, Lévy, Boussinesq, and others, on this subject, but it appears that there is a singular deficiency of experimental data for testing the accuracy of the results of theory. The following Paper contains an account of some experiments made with that view in the summer of 1877. Circumstances prevented the Author from carrying out his original intention of experimenting with various forms of apparatus and various materials. Although, then, the investigation is somewhat incomplete, yet it seemed to have been carried far enough to establish several propositions of interest with regard to the horizontal thrust of sand. As will be seen below, the nature of those conclusions was such that the renewal of the experiments seemed inexpedient. An important paper by Mr [afterwards Sir] Benjamin Baker, M. Inst. C.E., on the lateral pressure of earthwork, was read before the Institution of Civil Engineers in 1878, and excited a discussion of much interest. That paper, which has only recently been brought under the Author's notice, afforded the immediate inducement for bringing forward his results.

LITERATURE

{Separate memoirs are referred to in the text.}

G. B. Airy. “Tides and Waves,” Encyclopedia Metropolitana, 1845 (pp. 241–396). {Airy, Tides.}

A. W. Baird. Manual of Tidal Observation (pp. 54 and xl, Taylor and Francis, London, 1886). {Baird, Manual.}

A. B. Basset. Hydrodynamics (Deighton, Cambridge, 1888), Vol. II., Ch. XX., pp. 199–231.

D. Bernoulli. Traité sur le flux et reflux de la mer. Recueils des pièces, &c., Acad. Roy. des sciences Paris, Vol. IV. (1741), pp. 7, 53; reprinted in Lesueur and Jacquier's edit. of the Principia, Vol. III.

G. H. Darwin. “Tides,” Encyclopædia Britaunica (pp. 353–381), and article in Supplementary Volumes. {Darwin, Encycl. Brit.} Scientific Papers, Cambridge, 1907, Vol. I., Oceanic Tides and Lunar Disturbance of Gravity; Vol. II., Tidal Friction.

J. Eccles, S. G. Burrard, St G. C. Gore, E. Roberts. Details of the tidal observations…from 1872 to 1892, and…methods of reduction. Great Trigonometrical Survey of India, Vol. XVI. (Preface, pp. XV, Part I., pp. 383, Part II., pp. 152, 80 plates and charts, including drawings of the tide-predicting instrument, Dehra Dun, (1901).

W. Ferrel. Tidal Researches (pp. 268, Washington Government Printing Office), Appendix to Report of U. S. Coast and Geodetic Survey, 1874. {Ferrel, Tidal Researches.}

S. Günther. Handbuch der Geophysik, Stuttgart (1899), Vol. II., pp. 456–480.

A mere catalogue of facts, however well arranged, has never led to any important scientific generalisation. For in any subject the facts are so numerous and many-sided that they only lead us to a conclusion when they are marshalled by the light of some leading idea. A theory is then a necessity for the advance of science, and we may regard it as the branch of a living tree, of which facts are the nourishment. In the struggle between competing branches to reach the light some perish, and others form vigorous limbs. And as in a tree the shape of the young shoot can give us but little idea of the ultimate form of the branch, so theories become largely transformed in the course of their existence, and afford in their turn the parent stem for others.

The success of a theory may be measured by the extent to which it is capable of assimilating facts, and by the smallness of the change which it must undergo in the process. Every theory which is based on a true perception of facts is to some extent fertile in affording a nucleus for the aggregation of new observations. And a theory, apparently abandoned, has often ultimately appeared to contain an element of truth, which receives acknowledgment by the light of later views.

Mr [now Sir Norman] Lockyer writes in his interesting paper on Meteorites as follows:–

“The brighter lines in spiral nebulæ, and in those in which a rotation has been set up, are in all probability due to streams of meteorites with irregular motions out of the main streams, in which the collisions would be almost nil. It has already been suggested by Professor G. Darwin (Nature, Vol. XXXI., 1884–5, p. 25)—using the gaseous hypothesis—that in such nebulæ ‘the great mass of the gas is non-luminous, the luminosity being an evidence of condensation along lines of low velocity, according to a well known hydrodynamical law. From this point of view, the visible nebula may be regarded as a luminous diagram of its own stream-lines’.”

The whole of Mr Lockyer's paper, and especially this passage in it, leads me to make a suggestion for the reconciliation of two apparently divergent theories of the origin of planetary systems.

The nebular hypothesis depends essentially on the idea that the primitive nebula is a rotating mass of fluid, which at successive epochs becomes unstable from excess of rotation, and sheds a ring from the equatorial region.

The researches of Roche (apparently but little known in this country) have imparted to this theory a precision which was wanting in Laplace's original exposition, and have rendered the explanation of the origin of the planets more perfect.