The mutual relative velocity of the two drifts is thus determined nearly independently of the assumed division of the stars between them, a result which has already been arrived at theoretically.

For a final solution, using (b) as a first approximation, I calculated for each region values of k and ẞ, assuming that the stars were equally divided between the drifts.

The values of k for the Regions I, II, respectively

1*35, 1*35, 1*36, 1°40, 1°44, 1'44, 1'43, 1*38,

VIII were

I'12, 1'12, I'II, 1'09, 1'07, 107, 1*08, 1*10.

Performing the analysis with these values, the constants of the drifts in the eight regions were found as follows:—

If we resolve these drift-velocities along and perpendicular to the plane of the ecliptic, the components perpendicular to the ecliptic derived from each region should agree.

The determinations of these components are

for Drift I, - '97, - *77, — 1*35, — 1'19, — '94, — 1′00, - '96, - *76, Mean - '99 for Drift II, 30, — '43, — '52, - '51, - '54, 48, 37, 39, Mean – '44

The accordance of these seems decidedly good when it is remembered that they are derived from regions containing on an average less than 200 stars.

The component drift-velocities in the ecliptic are more or less foreshortened according to the longitude of the region; I determined the mean values by a least-squares solution.

The slight difference between this result and that of solution (b) represents the effect of allowing for the variation of ẞ and k from region to region, instead of adopting a mean. The comparison between solutions (a) and (b) serves to indicate how the results would be changed if the stars were not evenly divided between the drifts. The constants most affected by such a change are the speed of Drift I and the direction of Drift II; the other constants are nearly independent of the assumption, and are therefore more reliably determined.*

From an examination of residuals, I estimate that the probable accidental error of the determination of hV, and hV2 is about ±°06, and the probable errors of the apices of Drifts I and II are respectively about 2° and 6° of a great circle.

Converting latitude and longitude into R.A. and Dec., the positions of the antapices may be compared with previous determinations as follows

The great R.A. of the antapices of both drifts found in the present discussion is rather hard to account for.

The velocities of the two drifts hv1 = 1.78, hv2 =0'59 are in excellent agreement with those found from the Groombridge stars hv1 = 1'7, hv2 =0*5.

For the velocity of one drift relative to the other, the Zodiacal stars give the value 194. From the Groombridge stars (by a leastsquares solution from the results of the separate regions) I have found the value 1'90. The determinations of the point towards which this relative velocity is directed (called by Professor Kapteyn the true vertex) are

Professor Schwarzschild's determination of the line of symmetry of motion may be added. This, although based on a rather different theory, is directly comparable with the above. He found,

* It should be understood that it is not impossible to determine a from the observations, but simply that when, as in the present case, a few stars are discussed, the value of a is liable to a greater uncertainty than some of the other results. Undoubtedly from the whole 1533 stars a fairly good mean value of a could be determined; but the task of computing it for sixteen regions separately, and taking the mean, would be laborious.

Further, from Professor Dyson's investigation, an R.A. of about 92° for this point may be inferred.

Thus the Right Ascension given by the Zodiacal proper motions is discordant as compared with the other determinations. The number of stars here considered is fewer, and the proper motions are perhaps not so well determined, but I do not think the discordance can be altogether attributed to this. Nor can it be traced to a local anomaly, for it seems to be indicated by the proper motions all round the ecliptic. I have verified by calculation that the Regions Ia to VIIIa agree with Regions Ib to VIIIb in leading to this high value of the Right Ascension.

Note on the evaluation of certain integrals required
in the analysis.

2π

(1) To calculate n ̧C1 = [**p cos (0 – 01)dė, for a single drift.

We may choose the initial line, along the direction of the drift, so that 01 = 0.

Tables of the Bessel Functions of an imaginary argument are given in Brit. Assoc. Reports, 1893 and 1896, and have been used in calculating Table I.

(the integrated part vanishes for r∞ and reduces to

n

2h2v,2

(1)

(2)

at the lower limit; the part remaining to be integrated is simplified by means of the identity (2)),

(3) The integral **

p cos 30d0, and corresponding integrals for any odd multiples of 0, can be found by a simple extension of

the method employed for

P COS Odo. The integrals for even

multiples of are more troublesome to evaluate, but the method employed for p cos 20d0 always succeeds.

2π

2

Tables of the two hypergeometrical functions, F (1/6, 5/6, 2, sin2 and F (- – 1/6, 7/6, 2, sin2 —-), between the limits iota equals

2

90 and 180 degrees. By C. J. Merfield.

In the method of Mr. R. T. A. Innes for the determination of the secular perturbations,* there are two hypergeometrical functions to be deduced. In an appendix † to this valuable paper, tables of the logarithms of these functions are given with the argument iota for each degree for the first quadrant.

Tables of these functions facilitate the application of this method in no small degree, and it seemed desirable to extend them, as in many future investigations it will be found that the angle iota will much exceed a quadrant. Taking an example, Eros §-Earth, it will be noted that the modular angle theta, the argument to the tables of elliptical integrals, reaches the value 60°, corresponding to iota 138° 18', and there will be many other cases in which it exceeds this value.

In the preparation of the tables here given the formulæ || (1.c.,

'Computation of Secular Perturbations," by R. T. A. Innes, Monthly Notices, vol. lxvii. 427.

+ Tables for the application of Mr. Innes's Method, by Frank Robbins. 7.c., 444.

The values of these functions for iota equals o° have been omitted in the tabulation by Mr. Robbins.

§"Secular Perturbations of Eros," by C. J. Merfield, Astr. Nachr., 4178-79, Band 175.

The values of these functions may be deduced from series. I have given the coefficients of twenty terms, Astr. Nachr., 4215, Band 176, p. 246.