p. 306, give this observed excess as +40".1 1".4, per century, this. being generally eferred to as the "advance of the perihelion of Mercury." According to the theory of relativity, Mercury should travel out an advancing orbit, the amount of the advance being calculated from Einstein's well-known equation. The constants in this equation are accurately known, for it can now be said that the velocity of light is closely determined owing to Michelson's recent fine measurements, the value being 299796 kilometres per second in vacuo (Michelson in his "Studies in Optics," 1927, p. 137, intimates that the error is reduced to 1 in the last digit, which indicates a remarkably great achievement). The writer has attempted to calculate this constant from 237 239 239 240 66 THE ROTATIONS OF THE ORBITS OF SIX PLANETS, AND COGNATE PHENOMENA. By F. H. LORING. The advance of Mercury's orbit has long been known, as was the fact that it was slightly in excess of the theoretical value due to the perturbations of the other planets, which effect has been fairly accurately calculated from astronomical observations. This remarkable discovery was made by the French astronomer Leverrier, and he found the extra advance to be 38 seconds of arc per century. This was confirmed by a later and more precise determination by Newcomb. H. N. Russell, R. S. Dugan, and J. Q. Stewart in their "Astronomy" (which appeared in 1927, being a revision of "Young's Manual of Astronomy "), vol. I, The calculation is here given practically in full. The other constants are taken from Astronomy," referred to above. The Einstein equation is Sw=2473a2 / C2T2 (1 − e2). numerical factor. = 31.006276680. T = Mercury's sidereal period in mean solar days which is 87.96926, but this must be converted into seconds before squaring. It should be explained that the sidereal period, or astronomical period, is the time taken by a planet to revolve round the sun, as measured from its crossing of the point of a fixed star (one so far away as to be stationary for astronomical observational purposes) as seen from the sun. This period is expressible in terms of solar days (see table). To convert the sidereal period of revolution into seconds, the value in days given (see table) has to be multiplied by the number of seconds in the day, namely, 86400; the variable in these cases not being the second or the day as such, but the number of days including a fraction thereof. Therefore 87.96926 × 86400 = 7600544.064. This value squared is So... This value (see explanatory notes) as obtained by the above equation when using the foregoing values has to be multiplied by (365.25636042)/ (87.96926) to obtain the year value, but not yet seconds of arc. The foregoing division gives the value 4.1520908. Returning to the main calculation, the value obtained by this modification has to be divided by 0.000004848136-taken as the length of an arc of one second when the radius is unity, which is: 2/(number of seconds in a circle); or, π/648000—-the answer being the advance of the perihelion of Mercury's orbit in one year in seconds of arc. As the value is given per century, this year value has to be multiplied by 100. Carrying out the complete computation, the final value is 42."8951. Referring to the values above, it will be obvious that it is not necessary to make the computations with quite so many signifi Semi-Major Axis of Orbit. cant figures, but the rounding off can be done by cancelling out when the above values are entered on each side of the solidus, before performing the final multiplications. It must be remembered that the calculation of the perturbations, referred to at the outset,* is a highly intricate procedure, and even if no small errors are present, the determination by Newcomb in 1895 of + 41."6 ± 1."4 or selecting the revised value 40."1-is not sufficiently different from the Einstein value here calculated to affect the validity of the theory of relativity. No great importance need be attached to plus and minus signs after determinations; for, as a rule, they are of too limited a significance to convey any meaning as to the precision of the value. It should be noted that the use of kilometres in the equation instead of centimetres is quite in order, because the km. values occur on each side of the solidus. The question as to the accuracy of the various constants entering into the equation has to be considered in judging the closeness of the final value. P. R. Heyl (Proc. Nat. Acad. Sci., 1927, XIII, 601) has redetermined the gravitational constant which came out very near to the previously accepted value. In fact it almost looks as if the true value could be taken as 2/3 × 10- C.G.S.U., with a minimum of error. This value might introduce a slight modification of some of the astronomical values directly or indirectly involved. In the B.A.A. Handbook Boys' value is given as 6.66 × 10-8 in c.G.S.U. Whatever minute changes may be made in the future the values have a relative relation that cannot materially change, so it is of interest to note the rotations of other planets than Mercury. The following table shows the computations made for 6 planets. To complete the survey some empirical relations are given at the conclusion. Planet. Eccentricity per Century 87.96926 224.7008 365.25636042 643.230 686.9797 JUPITER The asteroids make a break in the continuity of the scheme, as possibly at one time there was a small planet at the proper place which exploded, or suffered in its parts a series of explosions. This appears to be borne out by the studies of K. Hirayama (see Sci. Abs., 1923, xxvI, 943) who has shown that there are five asteroidal groups, with about 30 members in each group, which satisfy the dynamic condition that they must intersect at the point or region at which the explosion took place.* On this account Eros, which is one of the would remain very similar, having nearly, though not quite, the same period, eccentricity, and inclination. At first they would all pass nearly through the point of explosion, but the perturbations, due mainly to Jupiter, would gradually shift the orbits, and after a few hundred thousand years this would no longer be true. It can be shown, however, that these perturbations would not alter, in the long run, the mean distances of the planets or their inclinations to the plane of Jupiter's orbit. Moreover, although the eccentricities and longitudes of perihelion would be altered by perturbations, they would change in such a manner that the centres of their orbits, when plotted in space, would all be equidistant from a certain definite point on the line joining the sun with the centre of Jupiter's orbit." asteroids has to be left out of the present scheme. Eros, as insignificant as it is from the point of view of a planet, has afforded a very accurate determination of the solar parallax; but it is questionable whether such a minute turning as here given (1′′.5694) can ever be measured with a small enough error to be of service in connection with the subject of this article. On January 30, 1931, Eros will again be fairly near to the earth when further measurements will be made.* It must be remembered that the values are for 100 years. In proportion as the orbits approach a circle it becomes increasingly difficult to measure the advance of the orbit, and this has to be taken into account as well as the expected amount of the extra advance, or difference; and, besides, the perturbations due to the out lying planets (all others) make the problem complex, so that highly accurate measurements and computations comparable with the calculated values above given, are not at present possible. EXPLANATORY NOTES. Einstein's equation gives the answer in radians per revolution. At the finish, instead of dividing by 0.0000048481368, the value may be multiplied by the number of seconds in one radian, viz. :206264".80625 (the foregoing value into unity, i.e., reciprocal) but the answer will, of course, be the same. It is to be noted that 0.000004848136 is the sin of 1". The B numbers as applied to the distances of the planets and their satellites are less in error than was originally supposed, because it is necessary to consider some planets and some satellites as pairs competing for the same path, so that harmonic mean distances have to be taken. This is not so artificial as may seem at first sight, for by adopting Jeans' theory of planetary evolution along two arms of a spiral structure, the planets have to be arranged in alternate sets, or two sets, arranged side by side, but with the pairs just mentioned brought together-apart from the asteroids one pair in this case, and one planet under the other. When this is done the character * It is probable that as a result of these measurements the astronomical unit will be readjusted, for the measurements will be mainly concerned with the solar parallax. istics of all the planets harmonise very well in paired relationship, and a sort of periodic table of the planets thus constructed brings out these relations clearly. There is a blank opposite to or facing Mercury, but no B number for this place unless it is nought, or a starting value near to unity. This has, however, suggested that another planet between Mercury and the sun had existed but had exploded as in the probable case of the asteroids. No bodies have been observed crossing the sun's disc, so that they must be less than 30 miles in diameter if they exist. That such a planet could have set Mercury's orbit turning so that the effect is still in some evidence seems highly improbable from dynamic considerations. It is of passing interest, nevertheless, as there is a slight discrepancy in the case of Mercury, but this is probably due to experimental error. Returning to the spiral structure with alternate sets of B numbers, this procedure has the advantage of avoiding the overlap of the asteroids for greater separation distances are thus obtained. This then helps † Strictly speaking the Chamberlin- = the |