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plete account of the present status of the subject. More emphasis has been laid on those portions which are under discussion at the present time and on problems, at present unsolved, which need discussion and solution.

PART I. THE SOLAR SYSTEM

The order of treatment is as follows: The moon, the eight major planets, their satellites other than the moon, the asteroids or minor planets, comets. The general view of the subject now as in the past, has been to consider the consequences of the law of gravitation, the extent to which it accounts for the observed motionsleading to the discovery of other possible influences and prediction for future observation and comparison with theory.

The Moon. The gravitational motion has been worked out sufficiently to satisfy all observational needs of the past and probably of some centuries in the future, and the results are fully embodied in tables constructed to furnish the moon's position without excessive labor. The observational data are the daily Greenwich observations (weather permitting) since 1750, isolated series of observations, eclipses, occultations since the beginning of the sixteenth century, and occasional ancient records of eclipses and occultations during the past forty centuries. These have led to the establishment of the following differences from a purely gravitational theory:

(a) An apparent secular acceleration of the moon's mean motion of about 4"5* per century, per century, combined with an acceleration of the earth's mean motion about the sun ("acceleration of the sun") of a little over 1" with probable errors, according to Fotheringham,1 to whom the latest figures are due, of about 0".5. The former has frequently been attributed to a slowing down of the earth's rate of rotation due to tidal friction: the new work of G. I. Taylor2 and H. Jeffreys3 has rendered this explanation very probable both qualitatively and quantitatively, especially as it also accounts for most of the sun's acceleration.

(b) A long-period term of some 275 years period and 13" amplitude in the mean longitude, obtained from observations extending *This is the coefficient of t2 in the expression for the longitude generally misnamed the acceleration. The true acceleration is therefore twice this amount. 1 M. N. R. A. S., 80, p. 581.

2 Phil. Trans. R. S., 220, p. 1.

3 Ibid., 221, p. 239; M. N. R. A. S., 80, 309.

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over about the same time. Numerous hypotheses have been advanced to account for this deviation, but none of them rest on any secure physical basis nor have they received independent testimony.1

(c) Fluctuations which are evident in the observations of the past 170 years and well defined during the last 70 years. In the former time their principal period seems to have decreased from some 70 years to about 40 years, with an amplitude of some 3" or 4". In 1914 E. W. Brown2 pointed out that similar fluctuations of much smaller amplitude could be traced in the motions of the earth and of Mercury: these fluctuations were confirmed by Glauert3 who found them also in the longitude of Venus. The latter also showed that they could all be moderately well accounted for by changes in the rate of rotation of the earth. No cause is assigned for these changes and their magnitude, amounting sometimes to as much as a loss or gain of 0.07 in one year in the rotation of the earth considered as a clock, makes the acceptance of the hypothesis difficult. It hass been uggested that this hypothesis might be tested by observations of the eclipses of Jupiter's satellites, which at present seem to furnish the only possible means for the purpose.

The Major Planets. The theories and tables by Newcomb and Hill seem to satisfy present needs, except perhaps those of Jupiter and Saturn, into which some small errors have crept but are now in process of examination by Innes. The sufficiency of the adopted theories is well shown by the theoretical and observed secular changes of the perihelia, nodes, eccentricities and inclination. The only large outstanding difference, that of the perihelion of Mercury, is fully accounted for by Einstein's addition to the Newtonian law, although one or two others need to be kept in mind as being perhaps in excess of their actual errors. It may be mentioned that the Einstein addition causes an increase of about 2" in the centennial motions of the lunar perigee and node1 but this is just at the limit of accuracy of Brown's theory and probably much beyond detection by observation for many decades to come.

Attempts made to discover a supposed trans-Neptunian planet by its perturbations on Neptune or Uranus have been unsuccessful. The errors of the latter though considerable when the tables of Lever

1 See E. W. Brown, Amer. Jour. Sc., Ser. 4, 29, p. 529.

2 Brit. Assoc. Report, 1914, p. 320.

3 M. N. R. A. S., 75, p. 489.

'de Sitter, M. N. R. A. S., 77, p. 172.

5 P. Lowell, Lowell Obs. Trans., 1.

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rier or Gaillot are used, have become very small with Newcomb's tables, and the observations of Neptune are not sufficient for the purpose.

From time to time, numerical relations of the masses, distances, etc., like those contained in Bode's law, appear but have not so far given theoretical results. A curious fact concerning the distribution of the poles of the planetary orbits, noted by H. C. Plummer,1 deserves mention.

Satellites. Neither the Uranian nor Neptunian systems present many points of interest to the theoretical astronomer, on account of their distances from the sun and Earth. The four inner satellites of Jupiter, partly on account of the librational relation between three of them and partly because of the possibility of testing the constancy of the rate of rotation of the Earth by observations of their eclipses, have been again considered by R. A. Sampson, who has worked out their theory and has published tables. The outer satellites present several features of mathematical and physical interest. In Hyperion and Titan, satellites of Saturn, there is another case of libration worked out to a limited extent by Newcomb. Since the issue of Vol. IV of Tisserand's Mécanique Celeste in 1896, which contains a full account of the work to that date, the orbit of Phoebe (Saturnian system), and of the sixth satellite of Jupiter have been worked out by Ross.2

Asteroids. Nearly one thousand of these bodies are now known. From the gravitational point of view, they possess the greatest mathematical interest, on account of the large perturbations produced in their orbits by Jupiter.

Long period inequalities constitute the chief difficulty in all the gravitational problems of the solar system, on account of the further approximations needed to obtain the required degree of accuracy of the numerical values of the comparatively large coefficients. The older methods went ahead without reference to them and carried out the approximations as they were needed, as for instance in Hansen's method used by G. W. Hill for Jupiter and Saturn in which a very long period term with a large coefficient causes most of the trouble. In the newer methods initiated by Gyldén and his followers, among them Backlund and Brendel, an attempt is made to introduce such terms as early as possible so as to diminish the ultimate work of computation.

1 M. N. R A. S., 76, p. 378.

2 Harvard Annals, 53, VI.

There are two types of oscillation which differ in their mathematical treatment and in their physical results. The ordinary long-period type is that in which a forced oscillation has a period near that of a free oscillation. But when the two periods become almost exactly the same, the free oscillation is compelled to take the period of the forced oscillation, and there is then a new oscillation of finite period about this position: the latter is called a libration. It is well known that though the periods of the asteroids have a considerable range, there are none certainly known whose periods are exactly 1/2, 1/3, 2/5 that of Jupiter, while there are considerable numbers with periods a little different from these fractions. It is obvious that the resonance has some relation to the distribution, but so far all mathematical investigation has failed to show any reason for the gaps: there is no evidence of instability in the deductions from the equations of motion. Attempts to search for the cause in cosmogonic speculations or in a resisting medium have been made, but a more complete investigation of the gravitational effects is needed. The problem is similar to that of the divisions in Saturn's ring, connected with perturbations produced by the satellites. The question is complicated by the fact that several cases of libration exist without apparent instability, e. g., amongst the satellites of Jupiter, of Saturn, in the Trojan group of asteroids whose mean period is the same as that of Jupiter and best known of all, in the rotation of the moon which has the same period as that of its revolution around the earth.

Numerous statistical investigations have been carried out, but little has been deduced from them, except in the way of confirming known perturbative effects.

Closely related in importance to the foregoing purely theoretical considerations is the practical problem of suitable numerical methods for the representation of the motion of the asteroids. Hansen's and the Gyldén-Brendel methods have been referred to. Of fundamental importance for practical purposes are also the methods inaugurated by Bohlin for the group determination of the perturbations of planets which have a mean motion nearly commensurable with that of Jupiter. Bohlin's method rests on Hansen and has been followed by Von Zeipel, Leuschner, and D. F. Wilson, by the former in application to the group 1/2, by the latter to the group 2/5. Bohlin's developments are general for all groups, with special application to the group 1/3. A feature of these methods is the use of elements which are similar to the elements ordinarily known as

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mean elements. In all of these methods the mean motion, eccentricity, and inclination may lead to complications. The success of any method depends upon the possibility of meeting the complications arising from critical values of these three elements separately and jointly. This has not been attempted in practice to a degree of precision which would reveal any departures from the motion under the Newtonian law of the type of the motion of the perihelion of Mercury, although for planets with moderate inclination and eccentricity and a mean motion not nearly commensurable with that of Jupiter, Hansen's method appears entirely suitable.

The principal aim of astronomers at the present time is to represent the motions with sufficient approximation to serve purposes of identification and observation. Even with this limitation of accuracy the difficulties are considerable. Leuschner published preliminary results of his experience with the Watson asteroids.1 Unpublished later results of the planets 10 Hygiea and 175 Andromache verify his conclusions that the revised tables of von Zeipel for the group 1⁄2 will give the most satisfactory results for all known planets of this group. It would seem therefore extremely advisable to have tables computed on Bohlin's or similar plans for other than the three groups for which they are available. The Trojan group, however, does not appear to lend itself to treatment by Bohlin's method without certain modifications. The perturbations of this group are being successfully dealt with by E. W. Brown, and Wilkins has represented the observations of 884 Priamus by his own treatment to within 10" of arc from one opposition to the next by including second order perturbations of the first degree in the eccentricity, inclination, and deviation from the center of libration. This method will more than answer practical requirements from one opposition to the next, while Brown's developments are intended to represent positions at any time.

Brendel classifies the planets after Gyldén as ordinary, characteristic and critical planets according to the ratio of their mean motions to that of Jupiter, the critical planets being those of close commensurability. Application of his method has been made by Brendel to one hundred planets with mean motions from 800 to 852. The elements are instantaneous but not osculating and perturbations greater than 3'.4 within fifty years are included so as to reproduce the geocentric places within 20' for one hundred years. 1 Proceedings of the National Academy of Sciences, 5, pp. 67-76, March, 1919. 2 A. N. 208, 233.