THE SECULAR ACCELERATION OF THE MOON AND THE LENGTH OF THE SIDEREAL DAY.—Laplace showed that the secular diminution of the eccentricity of the earth's orbit ought to produce in the longitude of the moon a term proportional to the square of the time, and which he determined as + 10″t2, where t is expressed in centuries. Adams and Delaunay have reduced this term to + 6″˙11i2. From a discussion of eclipses Airy concluded that the coefficient of acceleration is as much as 12″ or 13″; and accepting this, the question arises as to the cause, other than that indicated by Laplace, which will account for the difference of 6″t2. This forms the subject of a paper by M. Tisserand in Comptes rendus, No. 20, 1891. Prof. Darwin found that the tidal action between the earth and the moon was sufficient to furnish an apparent acceleration equivalent to the required complement. The accompanying decrease in the earth's rotational velocity produces an apparent acceleration of 3″.8t2in the case of Mercury, an amount which may make the longitude of the planet vary by as much as 15″ in a couple of hundred years. Since the observed transits of Mercury extend over more than two centuries, M. Tisserand has discussed them with the idea of determining whether the term 3″.8t2is really indicated by them. He finds, however, that the extreme transits are not so well represented with the new term as without it, although the difference is not very great. This result, therefore, is unfavourable to the idea as to the variability of the sidereal day, or at least to a variation sufficient to reconcile the result of Airy's research with the calculations of Adams and Delaunay. This being so, it is concluded that the increase in the length of the day, produced by tidal action, has nearly the same value as the diminution which results from the contraction of the earth caused by secular cooling, and that, on account of the compensating action of the two effects, the length of the sidereal day remains very nearly invariable.