It thus seems doubtful what precise meaning can be attached to the mean of such discordant values. On forming groups corresponding to those used in this paper there are still clear discordances; but the whole question must be reserved for further examination.

Nothing has yet been said of the constant C indicated by the mean values of the columns. It will be seen that it is persistently negative, the mean value from Table I. being -0.25, and from Table II. - 013. These values are consistent, and indicate a steady drift of -0°019 or - 1'14 for every 10° of advance across the disc. Carrington found a phenomenon of this kind, and thought its reality so improbable that he set it down to some error in reduction. His value was 1''5 for 30° of rotation, or 18' for a whole rotation; and his direction was towards the North Pole, i.e. in the direction opposite to that here found. It thus becomes important to examine whether this general drift shows any clear indication of change during our period (1886-1901). For this purpose the material must be rearranged in consecutive years, as in Table IV.

TABLE IV.

Drift in Latitude between +65° and - 65° from central meridian.

The unit in the table is 001: the subscript figures represent the number of spots included in the mean.

The mean values of the drift, weighting all spots equally, are as follows:

There are some indications of change, but the small number of observations in the terminal years makes it difficult to pronounce with certainty. The reduction of the material recently made available should go far to settle the point. Meanwhile we may sum up the indications by taking means for four years as follows, Carrington's result being rendered comparable and added :

[NOTE, added December 28.-- A provisional reduction of the results for 1874-1885 shows that the drift is distinctly positive in those years, being smaller at the ends and large in the middle. An oscillation of about 26 years' period with maximum northerly drift about 1854, 1880, 1906, and maximum southerly about 1867, 1893, would roughly fit the facts observed hitherto.]

On the Mean Distances of the Groombridge Stars.
By A. S. Eddington, B.A. M.Sc.

In a former paper * I have given some calculations as to the mean distance of the stars of Groombridge's catalogue for the purpose of comparing the distances of the two star-drifts, with which that paper dealt. The results obtained were, I believe, sufficient to show that the two drifts must be at, roughly, the same mean distance from us, but there seemed no means of judging how closely the actual numerical results could be relied on. Some further reflection has, however, shown that the method has advantages (as regards freedom from systematic error) which were not at that time noticed; it also incidentally furnishes an additional test of the two-drift hypothesis. I have accordingly developed the results more precisely in this paper.

The principle of the method of determining the mean parallax of a large group of stars is well known. It depends on determining the apparent angular displacement corresponding to a known linear displacement. For example, in Professor Kapteyn's researches the parallactic motion has been mainly used as the known linear displacement; if that is assumed to be the same for all the different groups of stars discussed, the corresponding angular motion is proportional to the mean parallax of the group. In the present case the mean peculiar speed (i.e. the mean individual motion, irrespective of direction, which remains after all drift or parallactic motion has been abstracted) is taken for a similar purpose. If its linear amount is assumed to be the same in all parts of the heavens, the angular amount will be proportional to the mean parallax of the stars. We must, of course, be prepared for the possibility that neither the parallactic motion nor the mean peculiar speed is strictly constant in different parts of the sky or for different classes of stars. The determinations of "hV" (or the ratio of the drift motion to the mean peculiar speed) made in the previous paper throw some light on this question; they appear to favour the assumption of constant mean peculiar speed, but the test is at present a rather rough one.

As the proper motions of the stars include not only their peculiar motions but also the drift-motions, the mean peculiar motion cannot be found without mathematical investigation. The theory developed in the previous paper enabled this to be done. There are two stages in the calculation :-(1) The constants of the drifts must be determined from the numbers of stars moving in the different directions (without regard to the magnitudes of their proper motions); and (2) from these constants the theoretical relation between the mean peculiar motion and the whole mean motion can be found.

For example, consider that division of the sky which I have Monthly Notices, vol. lxvii. p. 34, referred to hereafter as Systematic Motions." The portion especially dealing with mean distances is pp. 55-57.

*

called Region B. In fig. 1, A is for this region the theoretical curve of the kind considered in the former paper, in which the radius vector is proportional to the number of stars moving in the corresponding direction; B is the theoretical curve in which the radius vector is proportional to the mean proper motion of the stars moving in the corresponding direction. The two curves are rather similar; the bi-lobed character of B is readily recognised, although the elongations and minima are much less pronounced than in A.

A

B

C

D

FIG. 1.-Diagrams for Region B.

A and C-theoretical and observed curves, showing total number of proper motions in different directions.

B and D-theoretical and observed curves, showing mean proper motion in different directions.

It should be mentioned that in drawing B the two drifts have been assumed to be at the same mean distance; as the two lobes of the curves correspond respectively to stars of the two drifts, it is clear that if one of the drifts is nearer than the other, this will be shown by the lobe of the curve B, which corresponds to that drift, being exaggerated in size compared with the other. It is in this way that the mean distances of the drifts can be compared. The curves C and D are those derived from observation, and are to be compared respectively with the theoretical curves A and B. Although our main purpose in this investigation is to discuss the distance of the stars of the two drifts, attention may be called to the clear way in which, in this and in other regions, the mean proper motions support the two-drift hypothesis. The evidence of the observed curves C and D respectively in favour of that hypothesis must be regarded as, at least to a large extent, independent; it could hardly be expected, a priori, that the mean proper motions would be greatest in the two directions in which the total number of proper motions is greatest. It may be noticed that the two drifts are slightly more prominent in the observed curve D than in the theoretical curve B; some of the other regions considered show this rather more strongly, and it appears to be a general result. I have not been able to arrive at any satisfactory explanation of it.

For the other regions, reference must be made to Table I. In it will be found, for each of the seven regions covered by the Groombridge catalogue, the mean proper motion of stars moving in each of twelve directions at intervals of 30°. (For forming the mean proper motion in any direction, stars moving in directions within 15° on either side of it were used; thus each direction corresponds to a 30° sector.) Curves similar to Din fig. I could readily be constructed from these data, and we should find that they are clearly bilobed, indicating two drifts. Region D constitutes an exception; for, since it contains the apex of Drift II., that drift is not very apparent in it. The same thing may be seen by inspecting the numbers in the table: in each column there are two maxima and two minima well indicated, and a comparison with the Tables II. to VII. in "Systematic Motions" shows that they correspond with the maxima and minima in the numbers of stars moving in the different directions.

In calculating these means, it is necessary to adopt some rule for rejecting excessive proper motions. The difficulty is to find a limit which will affect the two drifts equally. The rule which