I have adopted, which seems to meet this, has been to reject the fastest eighth moving in each direction. That is to say, if there are n proper motions in one of the twelve 30° sectors, I pick out and reject the n/8 highest of these, and take the mean of the remainder. The net result must be nearly the same as if the nearest stars of each drift, to the number of rather more than one-eighth of the whole, were rejected. This means a much more drastic rejection of high proper motions than is usually resorted to; but some such plan seems necessary if the more distant stars are not to be altogether outweighted.

Each of the mean proper motions provides us with an equation involving linearly the unknown mean parallaxes of the two drifts (see "Systematic Motions," p. 56, equation (1)). Thus for each region twelve equations were formed, involving these two unknowns; these were properly weighted and solved by least squares. Table II. contains the results. "d" and "d2" denote the distances of the drifts. For the meaning of "h" reference must be made to the mathematical analysis of the former paper; it is connected with the mean peculiar speed of the stars. It is, however, the relative rather than the absolute values of the distance that are of interest, so that the precise unit used is not of great importance; but, accepting Campbell's provisional value (20 k.m. per sec.) for the speed of the solar motion, and my own determinations of the driftconstants, the actual mean parallaxes may be obtained by dividing the numbers in Table II. by 500. (It must, however, be remembered that a great many near stars have been rejected, as explained above, so that the mean parallax has a rather artificial significance.)

TABLE II.

Mean Parallaxes.

I

I

(The quantities and tabulated are the mean parallaxes multiplied hd hday

by a factor whose value is probably about 500.)

The equality of distance of the two drifts is evidently confirmed. Region E is the only one in which there is any indication of a considerable difference between their distances. The probable errors are of interest, not only as indicating the reliability of the

* The restricted region G (excluding subdivision y) was used; see Systematic Motions, p. 52.

results found, but as a measure of the accordance between theory and observation, for they are derived from the discordance of the calculated and observed mean proper motions in the twelve directions. Region B, which was represented in fig. 1, shows the best agreement between theory and observation; but it may be remarked that, where the agreement is not so good, that is always due to the phenomenon mentioned above-that the mean proper motions show the effect of the two drift-motions more prominently than calculation leads us to expect.

Before we can rely on the values of the probable error given in Table II., it is necessary to consider whether a possible systematic error in the catalogue, from which the proper motions are derived, may not lead to errors of importance not included in the theoretical probable error. The conclusion is rather unexpected; a systematic error in the proper motions does not invalidate these results. The reason is this: a systematic error affects the mean proper motion, and also the total number of proper motions, in a particular direction; owing to the latter error, we arrive at incorrect values of the constants of the drifts; but it is not difficult to see that, if these incorrect values of the drift-constants are used in forming the equations of condition, this will compensate for using incorrect values of the mean proper motions. Looked at in another way, the analysis was designed to extricate the mean peculiar motion from the systematic drift-motions with which it is involved; it at the same time extricates it from any possible systematic error which is hardly distinguishable in its effect from systematic drift or parallactic motion. In order that this elimination of error may take place, it is necessary to use driftconstants derived from the observed proper motions in the particular region considered, not those determined from examination of the whole sky; this has been attended to.

an

Assuming that the two drifts are actually at the same mean distance, and accordingly combining their parallaxes given in Table II. with weights inversely proportional to the squares of their probable errors, we have the following table of mean parallaxes (multiplied by a factor) of the Groombridge stars in different parts of the sky.

The mean parallax steadily increases with the distance from the galaxy, a result which is in accordance with the generally accepted ideas of the distribution of stars, viz. that the increased number of stars in the low galactic latitudes is due to additional more distant stars being visible, and not to any crowding among the nearer stars. The table apparently indicates a gradual change'in the mean distance; but, as the regions are rather extensive in area, it is not definitely incompatible with the hypothesis that the differences are caused by a belt of distant stars almost limited to the galaxy.

On Ancient Eclipses. By P. H. Cowell, M.A., F.R.S.

I am much obliged to Mr. Crommelin for repeating my calculations in the case of six solar eclipses, and so setting the question of their accuracy beyond doubt.

The fact that emerges from the discussion of the eclipses is that there is an unexpected rate of change in what may be termed the "nodical year," or period of revolution of the Sun relatively to the Moon's node.

This abnormal rate of change may arise from secular changes in the motion of the Moon's node (other than that arising from the change of the eccentricity of the Earth's orbit round the Sun), or from secular changes in the motion of the Sun, or, of course, from a combination of both hypotheses.

The motion of the node, as found from observation, differs very slightly from the formula given for it by Professor Brown in M.N., Ixiv. p. 532. This formula is differentiated and the secular acceleration of the node obtained on the supposition that the eccentricity and inclination of the Moon's orbit are constant. We know no reason why these quantities should vary, and observation shows that if they do vary, the variations do not exceed certain limits. It happens, however, that these limits are sufficiently large to admit of a considerable part of the unexpected change in the nodical year being attributed to the motion of the node (at least as an alternative hypothesis), thus diminishing the part to be ascribed to the Sun.

The formula +4′′ (T − 18′25)2 – 1"o takes numerical values + 1"o in 1755 and 1895, and 1"o in 1825, with lesser values numerically at intermediate dates. A second of arc is not an impossibly large quantity to attribute to systematic errors, or even to unknown long-period terms, and it is clear from Professor Newcomb's table (Ast. Const., p. 22) that residuals of over 1" cannot be avoided by any formulæ.

Hence it must be understood that a possible change in the eccentricity and inclination of the Moon's orbit is put forward as

an alternative suggestion. There are no modern observations capable of discriminating between various hypotheses, and, conversely, it is unnecessary to discriminate before approaching questions of chronology. All that is necessary in that case is to verify that certain empirical formulæ fit the records in a way that cannot reasonably be considered accidental.

It is a curious reflection that Oppolzer's formulæ should have been put aside merely because they contain impossible mean motions. Because they are based on so much coincidence and because Oppolzer's numerical accuracy is beyond reproach, there must be some solid foundation beneath the superficial blemishes of impossible mean motions. If we discard Oppolzer's position of the lunar perigee as not sensibly affecting the general agreement with the eclipse records, and if, wherever a mean motion needs correction, we apply to Oppolzer's formulæ some multiple of 20T+T2, a quantity that vanishes twenty centuries ago, we get to results closely resembling mine; and I attribute the small differences to the fact that Oppolzer nowhere claims to have satisfied but merely to have improved the solar eclipses, and, I believe, he obtained his formulæ from the lunar eclipses.

Let the supposed variation of the principal elliptic term in the Moon's longitude be +2" 22 in a century (i.e., a parts in 10,000); and let the supposed variation of the principal term in the Moon's latitude be "8y in a century (i.e., y parts in 10,000), and let z" be the (sidereal) secular acceleration of the Sun's longitude, then

the right-hand side being determined by the eclipses, and modern observations indicating that x is less than o'2 and y less than 03. In this way the solar eclipses are compatible with a secular acceleration of 2" for the Sun.

If the underlying cause be tidal friction, there is nothing improbable in the suggestion that slow changes are caused in the eccentricity and inclination of the Moon's orbit.

The present tables would expose on an average about one-tenth part of the Sun's diameter, and these 200" are reduced to less than 50" by a single empirical term in a large number of cases. The suggestion that this is mere chance appears to me untenable. When the observations of the eighteenth century present a difficulty of one second, what is one second against two hundred? The one second may be evidence that a particular hypothesis requires modification, but that is all. I believe equation (i) exhausts every geometrical possibility.

The Perturbations of Halley's Comet in the Past.-First Paper. The period 1301 to 1531.

By P. H. Cowell, M.A., F.R.S., and A. C. D. Crommelin, B. A.

We commence this paper by expressing our great indebtedness to Dr. Smart, F.R.A.S.; to Mr. F. R. Cripps, of 22 Hornsey Rise Gardens, N.; and to Mr. Thomas Wright, of 39 Cringle Road, Levenshulme, Manchester; who are really entitled to be considered as joint authors, since they have carried out by far the larger portion of the mechanical quadratures, the results of which are given below. Without their co-operation the completion of the calculations would have been indefinitely delayed; and they are again offering their help in carrying them back to a still more distant date.

The calculation of the perturbations of this comet has already been carried back to 1531 by de Pontécoulant; the identity, of the two apparitions before that (those of 1456 and 1378) is universally admitted; before that date Dr. Hind has given (M.N., x. p. 51) a list of conjectured identifications, some fairly certain, like those of 451, 760, 1145, others admittedly vague and uncertain. It occurred to us that new light might be thrown on the question by carrying back the calculation of the perturbations as far as possible, and seeing whether a sufficiently accurate correspondence existed between the conjectured and calculated dates. It is fairly evident that, to bring the labour of computing a large number of revolutions of the comet within reasonable limits, some abbreviations must be introduced, and the following was the plan adopted.

(1) For the purpose of computing the perturbations, the comet in each revolution was assumed to move in an ellipse of constant eccentricity (experience having shown that the perturbations in eccentricity are always small), and with a major axis corresponding to the observed period.

(2) The perihelion and node were supposed to change uniformly from revolution to revolution, the rates of change being deduced from the accepted elements of the comet from 1531 to 1910; subsequently these rates were modified by the use of M. Laugier's discussion (C.R., vol. xxiii.) of the apparitions of 451 and 760, for which fairly definite statements of position are found in the Chinese annals. M. Laugier has been able to represent these observations exactly by elements differing but little from those at the present time. Curiously enough, he does not print his elements, but he gives the longitudes and latitudes calculated from them, and the elements must be very approximately as follows