In that portion from Cassiopeia in the opposite direction to near 19h. of R. A., in Aquila, ratio = 3.70. These remarkable results are derived from the D. M., and will be yet more striking if corrected by half the difference between it and the S. D., as we have done for the sky generally. They will then be 4.27 and 3.95, respectively. As might be expected, the regions of greater star density have generally, though not always, the higher ratio. The highest of all is in a patch south of Gemini, between 6h. and 7h. of R.A., and near + 5° of declination. Here it amounts to 5.94, showing that there are eighty-six stars of magnitude 9.0 to every one of magnitude 6.5. The D. M. does not stop at magnitude 9, as the above numbers do, but extends to 9.4, while the S. D. extends to magnitude 10. For these magnitudes Seeliger finds a yet higher ratio. This is, however, to be attributed to the personal equation of the observers, and need not be further considered. The only available material for estimating the ratio of increase above the ninth magnitude is found in the Potsdam photographs for the international chart of the heavens, which extend to magnitude 11. These are published only for a few special regions. Five of the published plates fall in regions not far from the galactic pole. I have made a count by magnitudes of the 312 stars contained in these plates. An adjustment is, however, necessary from the fact that the minuter fractions of a magnitude could not be precisely determined from the photographed images. The results are practically given to fourths of a magnitude, although expressed in tenths. But it is found that the numbers corresponding to round magnitudes and their halves are disproportionately more frequent than those corresponding to the intermediate fourths. For example, there are only 19 stars of magnitude 10.7 and 10.8 taken together; while there are 49 of 10.5. Under these circumstances I have made an adjustment to half-magnitudes by taking the stars of quarter-magnitudes and dividing them between half-magnitudes next higher and next lower. The number of stars of the several magnitudes is then as follows: It is difficult to derive a precise value of the starratio from this table, owing to the small number of stars of the brighter magnitudes, which are insufficient to form the first term of the ratio. Assuming, however, that the ratio is otherwise satisfactorily determined up to the ninth magnitude, we find that there is but a slight increase from the ninth up to the tenth. The number of the eleventh magnitude is, however, nearly three times that of the tenth and nearly double that of 10.5. Another way to consider the subject is to compare the total number of stars of the fainter magnitudes with the number of lucid stars corresponding, which, in the general average, will be found in the same space. We may assume that near the poles of the galaxy there is about one lucid star to every ten square degrees. The five belts included in the above statement cover about thirteen square degrees. The region is, therefore, that which would contain about one star of the sixth magnitude. An increase of this number by somewhat more than 100 times in the five steps from the sixth magnitude to the eleventh would indicate a ratio somewhat less than 3; about 2.5. But the comparison of the photographic and visual magnitudes renders this estimate somewhat doubtful. Besides this, it is questionable whether we should not reckon among stars of the eleventh magnitude those up to 11.5, which would greatly increase the number. It is a little uncertain. whether we should regard the limit of magnitude on the Potsdam plates as 11.0 or 11 plus some fraction near to one half. Altogether, our general conclusion must be that up to the eleventh magnitude there is no marked falling off in the ratio of increase, even near the poles of the galaxy. I have not made a corresponding count for the galactic region, but the great number of stars given on the plates show, as we might expect, that there is no diminution in the ratio of increase. The question where the series begins to fall away is, therefore, still an undecided one, and must remain so until a very exact count is made of the photographs taken for the international photographic chart of the heavens, or of the Harvard photographs. There is also a possibility of applying a photometric study of the sky to the question. The background of the sky itself is by no means black. The question to be investigated is whether a considerable fraction of the apparently smooth and uniform light of the nightly sky comes from countless telescopic stars, perhaps from stars too faint to be found on the most delicate photographs, or whether it is mostly reflected by our atmosphere from the stars. It may seem questionable whether the latter is the case, because the fraction reflected in a clear atmosphere is not supposed to exceed one tenth the total amount of light of the stars themselves. On the other hand, the seemingly blue colour of the sky might seem to indicate reflected light, since the average colour of all the stars is white rather than blue. The subject is an extremely interesting one and requires investigation before a definitive conclusion can be reached. CHAPTER XIX STATISTICAL STUDIES of PROPER MOTIONS How charming is divine philosophy, Not harsh and crabbed as dull fools suppose, And a perpetual feast of nectared sweets Where no crude surfeit reigns.-MILTON, HE number of stars now found to have a proper THE motion is sufficiently great to apply a statistical method to their study. The principal steps in this study have been taken by Kapteyn, who, in several papers published during the past ten years, has shown how important conclusions may be drawn in this way. We must begin our subject by showing the geometrical relations of the proper motion of a star, considered as an actuality in space, to the proper motion as we see it. The motion in question is supposed to take place in a straight line with uniform velocity. Leaving out the rare cases of variations in the motion due to the attraction of a revolving body, there is nothing either in observation or theory to justify us in assuming any deviation from this law of uniformity. The direction of a motion has no relation to the direction from the earth to the star. That is to say, it may make any angle whatever with that direction. |