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 ASTRO N O M Y from time to time pass between us and the sun, and therefore be visible as minute points on the disc of that body. But nothing of the sort has been brought to light by the photographs of the sun which have been constantly taken in recent years, nor by the observers who, during the last half century, have so assiduously watched the sun for spots. There is another difficulty in the way of accepting this explanation. A mass of bodies sufficiently large to produce the observed motion of the perihelion of Mercury would affect both the other elements of Mercury’s orbit and the motions of Venus, but it is shown by the most refined discussions of the observations that these effects are not produced. The most recent surmise on the subject is that the law of gravitation may not act exactly according to the inverse square. A very simple hypothesis propounded by Professor Asaph Hall is that in the expression for the mutual gravitation of two bodies of masses ni and which, according to the Newtonian law is of the form the exponent 2 should be increased by a very minute fraction. The value of the exponent which would produce the observed effect is 2.000 000 1612, so that the discrepancy is removed if we suppose the attraction to be of the form zavow ooo leA The effect of this modification would be insensible except in the motion of the pericentres of the heavenly bodies. The only cases in which it could be made evident by the century and a half of observations yet made are those of Mercury, the moon, and Mars. The perihelion of Mars does actually seem to be affected by the corresponding increase, which is about per century; but this excess, though made very probable by the observations, is too minute to be conclusively established. In the case of the moon’s perigee the increase of motion would be 150" per century, a quantity too large to escape observation; but unfortunately the theoretical determination of this motion has not yet been made with such precision that it may not be affected by an error of this amount. The most refined determination is that recently made by Professor E. W. Brown, which does show a discrepancy of nearly the required amount; but the difficulties of the determination are such that a conclusive result has yet to be reached. We may sum up our conclusions on this point by saying that the discrepancy in the case of the perihelion of Mercury is well established, and that there is some reason to believe it a general rule that the motions of the pericentres of the moon and planets are somewhat greater than the gravitation of other bodies is competent to produce. Furthermore, it may be said that the simplest way of explaining the excess of motion is to assume that gravitation increases at a minutely greater rate than the inverse square. Many other modifications of the Newtonian law have been suggested, especially some of a form analogous to that of electro-magnetic action, but none of these consistently represent all the phenomena. (b) The other exception to which we allude occurs in the apparent mean motion of the moon around the earth, which has now been observed with an approach Tbe moon's to modern precision since the year 1675, and mean with less than modern precision from 1625 to motion. 1675. We have also eclipses of the sun and moon recorded by Ptolemy in the Almagest, or observed by the mediaeval astronomers, by which the mean longitude of the moon may be followed for more than 2500 years. No amount of research has yet reconciled the results of these observations with gravitational theory. To make clear the existing state of the question, we remark that the inequalities in the motion of the moon are of two classes—those produced by the action of the sun,

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which are always of comparatively short period, and those produced by the action of the planets, which in exceptional cases are of long period. If, in a period of twenty, thirty, or fifty years, the moon is found to be, in the general average, ahead of her computed place, or behind it, we may say with certainty that the deviation is not due to the action of the sun, because all the effects of this action would be compensated within eighteen years. It has been known for a century that deviations of this character, which are called deviations of long period, really exist in the motion of our satellite. In the middle of the 19th century Hansen announced that he had discovered two inequalities produced by the action of Venus, which completely reconciled these deviations. The theoretical computation of one of these inequalities has been repeated by several investigators since Hansen, and his result confirmed; but it has been shown that the other inequality has no existence in theory, and that it was the result of imperfections in the method employed by its discoverer. If any doubt could arise as to this conclusion, it is set at rest by the discovery that Hansen was in error in supposing that his two inequalities, singly or combined, would represent the observed deviations. Since 1870 the action of the planets on the moon has been exhaustively treated by several investigators with the special object of deciding whether their gravitation could produce any inequality of long period other than that of Hansen, but without result. The impossibility of any such inequality seems to be as well established as any proposition can be that relates to so complicated a subject. Another possible cause of apparent inequalities is to be examined, namely, variations in the earth’s axial rotation. What we actually observe is not the absolute motion of the moon, but the relation of this motion to the rotation of the earth on its axis, on which we necessarily depend for our measure of time. Let us now suppose this time of rotation to be increased by a very minute amount. Then the day will be longer by this amount. The motion of the moon in one day will then seem to be greater than it was, though in fact there has been no real change in it. If the rate of rotation is accelerated the opposite effect is produced, the day is shorter, the moon does not move so far in a day, and so seems to be retarded. The discrepancies in question can be explained by variations always less than a second in a year, which, however, accumulate year after year, so that before the end of half a century they might amount to twenty or thirty seconds of time. A decision between these two causes can be reached only by observations on other bodies. In general, the celestial motions go on so slowly that their ^ ^ amount during so brief an interval as twenty t^s^s of seconds cannot be certainly detected. Only the Mercury. moon, the planet Mercury, and Jupiter’s satellites move so rapidly that an accumulated error of this amount in our measure of time might be brought to light by them. In the case of Jupiter’s satellites we have to depend on the time of their eclipses, and the observations of these phenomena are so far from accurate that no conclusive result has yet been derived from them. But transits of Mercury over the sun’s disc have been observed with greater or less accuracy since 1677. The present state of the question is presented in the following tables. The first column gives the mean dates of eclipses or other observations of the moon, and the second the mean excess of her observed mean longitude over that computed from the theory of her motion about these dates. In the third column this excess is given in time, and shows how far we must suppose the actual earth to be in advance of a uniformly revolving earth in order to account for the apparent excess. S. 1.-93