of about 4 per magnitude, so that the increase in log N for 0025 mag. would be about 001. This would increase 100 stars to 100 23 (since log 100'23 = 1'001), so that a difference of atmospheric absorption indicated by visual observations would only make a difference of o 23 per cent. between the upper and lower halves of a plate. Photographic absorption would need to be twenty times as large to explain the observed difference, which is not likely. Sir W. Abney emphasises (Mon. Not., xlvii. p. 265) the necessity for knowing the "spectrum value" of the plates employed if we wish to determine the photographed absorption, but the limits between which it varies are not likely to be wider than from one to four times the visual absorption.

18. It occurred to me that we might get some indication of the photographic absorption by comparing in some way the plates taken at different hour-angles; for instance, in the formula given at the heading of each of our plates for determining magnitude from measures of diameters,

mag. = a - b√d,

the constant a would vary with the atmospheric absorption. But a little examination showed that the differences of Z.D. between plates taken otherwise under the same conditions were too small to afford a trustworthy indication of the phenomenon, which is swamped by a number of other larger variations. A single example will serve to show this. In Table XIII. are shown in the first three columns the Decl. and R.A. of the plate centre and the hourangle at the middle of the exposure; in the fourth column is given the Z.D. of the centre; and in the fifth the value of the constant a of the above formula. If these values of a are arranged according to Z.D. we find—

i.e. the paradoxical result that fainter magnitudes (by o°38 magnitude) are shown on the plates of 6° greater Z.D. Suspicion arose that the person who measured the plate might influence the result; and on collecting the results for the three measurers shown in the sixth column, it was found that their mean values of a were

but on applying corrections as in the last column the 16:43 only
became 16.50 and the 16.78 became 1675, so that the anomaly
was not removed. It seems better to determine the photographic
absorption by special observations rather than to seek it in the
plates already taken, in which its effect was consciously or un-
consciously minimised.

19. We may now turn to cause (y), the distribution of stars in
the sky.
That the density increases as we go northwards in the
Oxford zones is shown, for instance, in Argelander's Durch-
musterung. His numbers per square degree for zones + 28°, +29°,
+30°, +31°, are 14'9, 162, 164, 16'4; so that the differences per
cent. between the north half and the south half of plates in zones
with centres + 29°, + 30°, +31°, here discussed, would be 8'1, 1°3,
o'o; mean, 3*1. This is comparable with our 40 per cent., and
the difference is what we might expect, considering that Argelander
only goes to magnitude 10. But it is unsatisfactory that the
difference is chiefly in one zone. We may put this down to
accidental error, but the evidence is only sufficient to state a case
for inquiry. In Monthly Notices, vol. lx., plate 2 (opposite p. 16),
Mr. Bellamy has given a curve showing the approximate variation
in stellar density (for the Oxford zones) with Galactic latitude
somewhat as follows:-

N = No. of stars 800 460 310 230 180 155 135 120

Differences
Log N
Differences

2'90 2.66 2'49 2:36 2.26 2'19 2'13 2:08 2'04 2'02 0'36 0'17 0'13 ΟΙΟ 0°07 0'06 0'05 0'04 0'02

20. There are two important points lying away from the curve, but we will neglect these in the following brief and general remarks. The Galaxy cuts the parallel of 30° Dec. about 51 and 1916.

(A) At these crossing points the changes of stellar density are very large, but if they are symmetrical on opposite sides of the Galaxy there should be a rough conpensation. On one side the N half of a plate will be richer, on the other the S half. We must be prepared, however, to find this compensation break down if the Milky Way is unsymmetrical.

(B) In the neighbourhood of oh the Oxford zones are near the Milky Way, south of it, and nearly parallel to it. Thus to pass from the southern half of a plate to the north is to approach the Milky Way, and consequently to find more stars. In these R.A.'s, therefore, we may expect to find the main contributions to the discrepancy between N and S halves of a plate. What is the maximum contribution we can expect from them? Suppose they ran all the way from 20h to 5h (9 hours of R.A. out of the 24), parallel to the Milky Way and 15° from it. We see from Table XIV. that log N increases o'17 in 10° of Galactic latitude, or 0017 in 1°; so that 100 stars on the S half of a plate would become 104 on the N half, due to approach to the Galaxy. This is an increase of 4 per cent., which is about what is required; but it becomes inadequate when we take 9/24 of it in order to distribute it round the whole zone. This fraction 9/24 corresponds to uniform distribution in R.A., which is, of course, not in strict accordance with the facts; but weighting each hour by the number of stars in it gives a nearly equal fraction, since the rich Galactic portions are chiefly in the non-contributing R. A.'s.

(C) For the remainder of the zone contributes nothing, or goes the other way. We have spoken of the crossing points in (A) above; from R.A. 5-6, for instance, the increase northwards is rapid for the first half hour, but is balanced by an equally rapid diminution when the central line is crossed. Reserving for future investigation the effects of possible asymmetry, we may put aside this crossing region as contributing nothing. Proceeding to greater R.A.'s, we are now north of the Galaxy, and thus stars decrease in number as we go northwards. The quantity N-S will thus be negative. But the zone +30° is a small circle of the sphere, and presently its sharper curvature brings it perpendicular to the Galaxy, when the difference N-S will be zero, and then positive again. It is positive for about an hour only (near 121) and then is negative again up to 19h (say), when we get the other crossing point (19h-20h).

21. These facts can, of course, be expressed in tabular form, and are given below in Table XV.; but it seemed desirable to call attention to them also in general terms, because general considerations seem to show that we cannot fairly expect, by improved knowledge of the details, to explain the whole of the difference N-S by actual distribution of the stars in Galactic latitude alone. The figures given below are only approximate: they could doubt

less be improved, for instance, by using Professor Kapteyn's results in No. 18 of his publications, which have become available by his kindness in sending an advance copy since Table XV. was constructed. But the change would not be great. One of the most important details left outstanding is that of the symmetry or want of symmetry of the Galaxy on opposite sides; but Professor Kapteyn's paper does not touch this point, since he has not yet. undertaken any discussion of distribution in Galactic longitude.

22. Percentages are given in Table XV. rather than total numbers of stars, in order to avoid undue influence from the Galaxy. It will be seen that the observed percentage is from 20h 5 to 23h5 +9, and from oh.5 to 4h5 it is +8; the mean for the whole 9 hours being +87, as against a theoretical +3'1. For the portion 6.5 to 18h5 the observed N-S percentage is +22, as against a theoretical -0.5. Hence the mean observed N-S is throughout in excess of the theoretical; or, in other

words, only a part of it can be set down to star distribution; the rest must be due either to atmospheric absorption or unsymmetrical performance of the object-glass. But these latter causes do not vary with the R.A. If we estimate their combined effect as 2 or 3 per cent. (being guided by the figures for 6h.5 to 18h.5, where the effect of stellar distribution is small), we get 6 or 7 per cent. for the observed effect of stellar distribution in R.A. 20h5 to 45 instead of about 3. This suggests that the rate of diminution with Galactic latitude has been considerably underestimated in those regions. A scrutiny of the figures shows that this is quite possibly the case, the reason being that the rate of diminution varies considerably in different Galactic longitudes. But a complete discussion of this point cannot be given here.

S$ 1, 2. Introductory.

Summary.

S$ 3, 4. Method of measuring the average star density on each quarter-plate, at different distances from the centre.

$5. Definition of A, which indicates the distance of maximum density from the centre. Table of values of star density and total number of stars for different values of A.

§7. List of observed values of A, in order of date of exposure of plates, in vols. i., ii., and iii.

SS 8-11. Variations of focal length indicated by changes of mean A may be due to discontinuous disturbances of the plate, but are more probably due to seasonal changes in focal length.

But there is certainly one discontinuity, at the erection of the new dome in 1900. Before this the plate was satisfactorily normal to the telescope axis; afterwards it was apparently tilted in R.A., greater R.A.'s being nearer the O.G.

SS12-14. In consequence of this tilt, the side of greater R.A, is richer in stars; but the detailed study of the effect of tilt is reserved for a separate investigation.

§ 15. But there is an excess of stars in the N half of the plate, which is 4 per cent. richer than the S half, and this cannot be due to tilt.

§ 17, 18. Visual atmospheric absorption would give an excess of o 23 per cent. in the N half: photographic absorption may be greater, but its value cannot be well determined from the Oxford plates, which were taken at nearly constant Z.D. A special investigation of it is desirable.

SS 19-21. The distribution of the stars in Galactic latitude alone explains a part, but cannot explain the whole of the 4 per cent. excess in the N half of the plates. The general nature of the distribution in Galactic longitude is such as to explain another portion of the excess, but quantitative estimates cannot yet be made.