The detection of the two satellites of Mars by Professor Hall with the great equatorial at Washington may be considered the most interesting recent achievement in pure discovery. Phobos, the inner of these, is the only satellite in the solar system which revolves about its primary in less time than it takes the primary to rotate on its axis, and hence the only satellite which rises in the west and sets in the east.

The accuracy of the values of the masses of the planets obtained from discussions of orbits of comets and asteroids varies greatly according to the size of the inequalities produced. If the principal inequality is large and the body has been carefully observed through a long period by a large number of observers, the result ought to be accurate and practically free from personal error. But the most favorable circumstances seldom occur, and it was not till the discovery of the satellites of Mars that a means was offered for the accurate determination of the mass of this planet. No satellites of Venus and Mercury have as yet been discovered, and the values at present assumed for the masses of those planets are very un

certain.

In 1788, just one hundred years ago, Laplace published a complete theory of Jupiter's satellites. This theory is still the basis of the tables now in use. Souillart's analytical theory of these satellites appeared in 1881. His numerical theory was completed only within the last year, and tables therefrom still remain to be formed.

Titan, the largest satellite of Saturn, was discovered by Huygens in 1655; soon afterwards Cassini added four to the number; a century later two more were added by Herschel; and, finally, Hyperion was discovered by Bond in 1848. Bessel made a careful investigation of the orbit of Titan and therefrom obtained a value for the mass of Saturn which since that time has been generally employed in the determination of the perturbations produced by that planet. The general theory of the Saturnian system which he commenced, he did not live to finish. It may be found in its incomplete condition in the Nachrichten. "This memoir," as Professor Hall remarks, "is still the most comprehensive investigation we have of the differential equations of this system and of the various forms of the perturbative function arising from the figure of the planet, the ring, the action of the satellites on each other, and the sun."

Our knowledge of the motions of the satellites of Saturn, with the exception of Titan, was very meagre until quite recently. This system of satellites is, in many respects, the most interesting in the solar system, and its form is quite analogous to that of the primary system. The number of members is the same, and Titan plays very much the same role in the one system that Jupiter does in the other. Since its erection, the great equatorial of the Washington Observatory has been chiefly devoted to observations of the numerous satellites of the outer planets. Professor Hall has published a discussion of his observations of all the satellites of Saturn except Hyperion, but omitting a discussion of the motion of the node of Iapetus. The mass of Saturn obtained by Professor Hall from micrometric measures of position angles and distances is smaller than that obtained by Bessel. The difference is perhaps due to the difference in personal errors in the measurements of the satellites when in different positions with regard to the primary. This error is increased by the fact that when the position of the satellite is compared directly with the primary, one of the micrometer wires must bisect the disc or we must measure from one of the edges of the planet. In the case of Saturn, the difficulty is further increased by the presence of the rings. To avoid these difficulties, Otto Struve has suggested the comparison of the positions of the satellites with one another instead of with the planet. Within the last two or three years Hermann Struve has made such a series of observations, with the great refractor of the Pulkowa Observatory, resulting in a mass of Saturn practically the same as that of Bessel.

A difficulty in the determination of a correct theory of the motions of Saturn's satellites is the fact that there are a number of cases of approximate commensurabilities in the ratios between their so-called mean motions. For example, the mean motion of Iapetus is almost exactly one-fifth that of Titan, Dione one-half that of Enceladus, and Thetus one-half that of Mimas. But the most interesting case is that of Hyperion, whose mean motion is very nearly three-fourths that of Titan. In this case, there is the additional difficulty that their distance from one another is only about one-seventh as great at conjunction as at opposition. The apparent eccentricity of Hyperion's orbit is principally due to the perturbations produced by Titan. As a consequence, Hyperion is always at apo-saturnium when in conjunction with Titan. The values for the mass of Titan obtained from discussions of the motion of Hyperion by Mr. Hill and myself agree closely with one

another, and with that obtained by Hermann Struve from the motion of the node of Iapetus. The values previously obtained by Newcomb and Tisserand are evidently too small.

Our knowledge of the motions of the satellites of Uranus and Neptune depends almost entirely on the observations made at Washington. Quite accurate determinations of the masses of these two planets have been obtained from them. The theory of their satellites offers no other points of special interest. The large secular motion of the plane of Neptune's satellite to which Marth has called attention needs confirmation; so that we may say that with the exception of the lunar acceleration there is little evidence that any of the satellites of the solar system move otherwise than in exact accordance with Newton's law of gravity.

The discovery of the asteroids began with the century. Only four were detected by the end of 1807, and it took forty years to double this number. Since that time, however, their discovery has progressed with ever increasing activity, until the number of these diminutive bodies has reached two hundred and seventy-eight, and is now increasing at the rate of nearly one hundred to every decade. The asteroids individually possess little interest of their own; but, on account of their motions, and on account of the assistance which their observation offers in the solution of other problems, they have already played an important part in the history of astronomy. The discovery of Ceres led to the composition of the Theoria Motus, and a demand for a more accurate knowledge of their motions was the origin of Hansen's Auseinandersetzung. Several of the asteroids approach Jupiter sufficiently close to offer a valuable means for the determination of the mass of that planet. On the other hand, another of the asteroids is at times actually nearer to the sun than Mars, but unfortunately the inclination of its orbit is very great.

The discovery of new asteroids should be encouraged in the hope that one or more may be found having inequalities in their motions produced by Mars and the earth of sufficient amount to render observations of them valuable for the determination of the mass of Mars and of the earth, the mass of which is, strange to say, not nearly so well known as those of Jupiter and Saturn. The masses of the latter are obtained from the observed distances of their satellites, while in the case of our own satellite the distance is the one quantity which it is impossible to observe.

The number of asteroids is so great that they have been the fre

quent subject of statistical investigation. Perhaps the most important earlier investigations of this sort were those of d'Arrest and Newcomb, which threw doubt upon the hypothesis of Olbers, that the asteroids were the fragments of an explosion of a larger planet. The systematic grouping of their nodes and perihelia which exists, was shown by Newcomb to be the effect of perturbation. Α clearer light has been thrown upon this subject, more recently, by Glauser and Newton. As a result of the action of Jupiter, the orbit of each asteroid has a motion about the orbit of that planet. These motions being unequal, there is a tendency toward a uniform distribution of the nodes on the orbit of Jupiter. Glauser shows that the observed grouping of the nodes on the ecliptic is a subsidiary result of such a uniform distribution, which, however, is slightly disturbed by the action of Saturn. Newton had previously found that the centre of gravity of the poles of two hundred and fifty-one asteroid orbits, considered as points of equal weight, lies within half a degree of the pole of Jupiter's orbit, and in fact regarding this centre of gravity as the pole of the mean plane of the asteroid orbits, that this mean plane lies nearer to the plane of Jupiter's orbit than to the orbit plane of any individual asteroid. On the other hand, if weights be given derived from the observed magnitudes of the asteroids, the position of their mean plane as found by Svedstrup differs greatly from that of Jupiter.

Kirkwood concludes, from an inspection of a table of the mean distances of the asteroids, that those parts of the asteroid zone in which a simple relation of commensurability would obtain between the period of a minor planet and that of Jupiter, are distinguished as gaps or chasms similar to the intervals in Saturn's ring. This much can, at least, be said that eighty-five per cent of the asteroids have mean motions greater than twice and less than three times that of Jupiter, and the mean motions of none approximate closely either of these, the two simplest ratios possible. The next simplest ratios lie beyond the limits of the zone; that is, there are no asteroids having mean motions nearly equal to or less than three halves that of Jupiter, and none nearly equal to or greater than four times that of Jupiter. There are other cases, however, in which the mean motions approximate very closely to commensurabilities which, although more complicated than those just mentioned, are still comparatively simple.

The labor of determining the general perturbations and computing tables of an asteroid is as great as in the case of a major

planet. It is no wonder, therefore, that tables have been prepared for scarce a dozen of these small bodies, and that these are already out of date. An examination of the Berlin Jahrbuch will show how much labor is required to determine their motions with sufficient accuracy to identify them, to prevent them from being lost, and to distinguish them from any new ones which may be detected. So far as we are yet aware, their motions are in exact accordance with Newton's law of gravity. Any divergence from that law, if any exist, will require many years, perhaps centuries, of careful and patient investigation to determine.

Many comets being visible to the naked eye, records of them extend back to very early times. As the number of persons interested in astronomy has increased, and especially since the discovery of the telescope, the number of these records has continually enlarged. As, however, the ease with which comets are detected does not, beyond a certain limit, increase with the aperture of the telescope employed, as in the case of asteroids, the rate of their discovery has not increased as rapidly as that of the minor planets. Before 1867 there were only eight periodic comets which had each been observed at at least two appearances. Since that time three comets of short period have been discovered, all by Tempel, which have been observed at at least two appearances. More recently, five comets of short period have been discovered by others, which, with perhaps one exception, had not apparently been observed before, and the time for whose next appearance has not yet arrived. The exception is Findlay's 1886 VII which may be the same as de Vico 1844 I. Quite recently returns have also been observed of two comets of long period, those of Olbers and of Pons.

Of well-known comets of short period Encke's, which has the shortest period of any, possesses the greatest interest to the student of celestial motions, since it was from a discussion of the orbit of this comet that Encke detected evidence of the existence of a resisting medium which produces an acceleration in the comet's mean motion. A more recent investigation by von Asten confirmed Encke's hypothesis so far as the observations from 1819 to 1868 were concerned, but showed that this acceleration did not exist after 1865. Backlund, who took up the problem after the death of von Asten, finds that while the latter result was due to the use of erroneous formulæ in the computation of the terms of the second order for the period 1865 to 1868, the acceleration of the mean motion is nevertheless diminishing. Backlund ends his