This value of ẞ must now be substituted in (22),

Similarly we find P=174 expi 132°7.

The directions of the two drifts are accordingly

To find their velocities we make use of Table I., which gives the corresponding values of | P❘ and hv,

Thus 452 per cent. of the stars belong to Drift I and 54.8 per cent. to Drift II.

Finally, we can check the provisional value of k,

k = √(1 + α)51 + (1 - a)2 = 1395,

agreeing with the adopted value 1'40, so that no further approximation is needed.

The results obtained by this method agree closely with those previously obtained by the method of trial and error ("Systematic Motions," p. 50). I give below a comparison of the constants determined in the two ways for four of the Groombridge Regions; it affords some sort of indication of the reliability of both methods.

The weakest point in the solution is exemplified in the case of Region F; the determination of a (or B) is not very satisfactory. The difference in the two solutions in the case of Region F depends entirely on the division of the stars between the two drifts; if we had adopted the same value of a in the two cases, the remaining four constants found from (22) would have agreed almost exactly with those found by trial and error. The weakness of the determination of ẞ can be shown analytically.

Now P1- P2 is generally about o'7, but may be less, hence αβ aN is generally about 10, but may be more.

Thus in a region containing 500 stars, a change of 2.5 in Σ cos 30 or Σ sin 30 would produce a change in N of 005, and the corresponding change in ẞ or a might be ±·05.

The question arises whether the weakness of the determination of ẞ is a defect of the method of analysis or is necessarily involved in the nature of the observational data. I think there is little doubt that the latter alternative is correct; I have examined various other methods of determining ẞ, but all are rather insensitive. In the case of Region F, both the solutions given above are found to agree with the observed distributions almost equally well.

Thus it seems likely that even in an ideal solution we should have one equation very much weaker than the other four (the weakness may be more pronounced for some regions of the sky than for others). As the five constants depend on one another and on this fifth equation, they may all share in the uncertainty.

One or two considerations help to avoid this difficulty to some extent. We may be content to assume a=o, i.e. that the stars are evenly divided between the two drifts; all evidence seems to indicate that this is approximately true, and it is conceivable that there may be some physical reason for it. Or, instead of adopting the value of a found for the particular region, we may adopt a mean found from all the regions discussed; this will have a much smaller probable error. But the most fortunate circumstance is that we may determine the relative motion of the two drifts almost independently of a.

Equation (18) gives

is

If ẞ lies anywhere between-03 and +03, (182) may be put equal to 98 with an error certainly less than 3 per cent.; but this range of ẞ includes all values likely to occur. Only in the case of a very great disparity in the distribution between the two drifts could a value outside these limits occur. Thus P1- P2 nearly independent of ẞ or a. P is a sufficiently nearly linear function of hV for h(V,- V), to be also nearly independent of a. Thus although adopting a=o may lead to some error in the determinations of hV, and hV2, the error will nearly be eliminated from the determination h(V1- V2). This relative motion of the two drifts is the quantity which most interests us, especially as a systematic error in the proper motions does not affect its determination so adversely as it affects the determinations of hV ̧ and hV1⁄2.

1

Systematic Motions of Zodiacal Stars.

I have applied the theory given above to the discussion of the proper motions of the zodiacal stars. The proper motions were taken from the Catalogue of Zodiacal Stars, Astronomical Papers of the American Ephemeris, vol. viii., part iii. Excluding the stars of the Pleiades, this contains 1533 proper motions. I divided the zodiac into sixteen regions, each extending 221° in longitude by about 16 in latitude; these are denoted successively by Ia, IIa, . . . VIIIa, Ib, . . . . . VIIIb, the centre of region Ia being at the first point of Aries. As regions Ia and Ib are diametrically opposite to one another, the observed motions are in parallel planes, and the two regions may be treated together; similarly, the other regions can be treated together in pairs. Thus the number of regions is virtually reduced to eight, each containing from 150 to 250 stars.

The distribution of the proper motions as regards direction in the eight regions is shown in Table II. Opposite =0° in the

first column is given the number of stars whose observed motions are in directions between 0=355° and 0=5°, and so on. The numbers have not been smoothed. For stars in Ia, IIa, etc., 0=0° is in the direction of increasing R.A., and 0 = 90° in the direction of increasing Dec. For stars in Ib, IIb, etc., the reckoning of agrees with that in the opposite parallel planes Ia, IIa, etc.

It will be seen, by looking down the columns of the Table that the two streams are plainly marked in Regions I, II, III, and VIII. In the other four regions their directions (projected on the sky) are inclined at an acute angle, and the existence of the two streams is rather concealed. The success of the analysis in these four regions is on that account especially interesting. It may be noticed that this belt of the sky is not so favourable as the Groombridge region for showing conspicuously the separation into two streams; the centre of the latter region lies between the two apices, so that in it the streams are in nearly opposite directions.

Although, in the main, pairs of regions such as Ia and Ib were treated together, I thought it safer to calculate √Mk-L2 and L for Ia and Ib separately, and to take the mean afterwards. This was in order to avoid the possible effects of systematic error, by ensuring that the difference of motion of the two drifts found from the observations was a difference of motion of intermingled systems of stars, and not the difference in the apparent motion of stars in Ia from that of stars in Ib. Actually, however, the precaution might have been omitted. I found that in every region very nearly

the same result was obtained whether the two halves were treated separately or together.

For the solutions I used entirely the equations

I did not calculate N, or attempt to find ẞ from the observations. The final results given below depend, therefore, on the assumption (based on previous experience) that the stars are evenly distributed between the two drifts; but calculations are given which show to what extent the results obtained would need to be modified if the assumption is incorrect.

As a preliminary I made two solutions, (a) assuming

1 + B

=1'0 and (b) assuming

I

=

II; in both cases

k was assumed to be 1'40. These correspond to assuming that the stars belonging to Drift I are about (a) 55 per cent., (b) 50 per cent. of the whole.

The combined results from all the regions were