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Boss, of course, deals only with a selected group of stars, but we can test the result more or less closely on the Yale observations, which give total proper motions for magnitude classes, and magnitudes for total proper motion classes. We owe to Dr. Lee the correlation ratios for the two cases; they are respectively
Correlation ratio: proper motions for magnitude classes, n='43'04 magnitudes for proper motion classes, n=2205
The average relationship between the two characters is thus again about 33. Thus we may safely conclude that the deviation
Magnitude and Proper Motion in R.A. (p positive).
Magnitude and Proper Motion in R.A. (p negative).
Magnitude and Proper Motion in Declination (p' positive).
Magnitude and Proper Motion in Declination (p' negative).
from independent variation in the distribution of magnitude and proper motion is not very far from 35. It is therefore possibly slightly higher than the value found for magnitude and parallax, which is about 30. It will, we think, be evident that a selection by proper motions would have comparatively small effect in modifying the relation found between magnitude and parallax. Some rough estimate of the order of the change can be formed by considering what would be the absolute coefficient of correlation resulting from a partial coefficient of 30 obtained by selecting stars of a single proper motion only; this will probably give an upper limit. We have
P12= √1 − ('35)2 √1 − 723
Here 723 is the correlation between proper motion and parallax. In the former paper for the 72 stars this was found to be about 4 (p. 449); from the Yale data (see below) it is under 38. Assuming it=38, and putting P1230, we deduce r1239, say. We think, therefore, we may safely assert that the relationship between parallax and magnitude, if no selection by proper motions had taken place, is not likely to prove as great as 4, and probably lies considerably under this value.
We would suggest, therefore, that the lowness of the magnitude and parallax result obtained in the former paper is not due to any special selection of parallax stars by their proper motions. This could not largely influence the relationship of magnitude and parallax, because proper motion is only moderately correlated with parallax, and still less with magnitude.
(iii) Before we leave the subject of the correlation of magnitude and proper motion, it is worth while noting that it has also been investigated in an entirely different manner. In the above inquiry we have found the correlation ratio for the total proper motion and the magnitude in the case of the Yale stars; we have. further calculated the contingency coefficients between two sets of groups of magnitude and proper motion in R.A. and declination respectively from the Boss Catalogue stars. But for other purposes the proper motion in declination, μ, and the proper motion perpendicular to a declination circle, μ cos 8, had been found and taken out. Now we are concerned only with the size of the proper motions, and not their sense, at present. We divided, however, our data into two halves for the purpose of checking results. The stars from the equator to the North Pole, 298 in number,* have been taken in one group, and those from the equator to the South Pole, 323 in number, in a second.
Further, to avoid lengthy analysis, we dealt separately with proper motion parallel to the declination circle (') and perpendicular to it (u cos 8). Thus we have four series to discuss. The classifications of magnitude were as follows: under 15, 1'5 and under 2.5, 2°5 and under 3'5, · · 6'5 and over. The proper motions are of such a wide range that no grouping was adopted, but the means and standard deviations found by the laborious processes of adding and of squaring. No stars were omitted, although one is sorely tempted to omit those two or three of abnormal proper motions, as undoubtedly abnormal on any probability test. Any selection of this or any other kind must, however, be for our present purpose dangerous, and accordingly we have allowed equal weight to all the available stars in Boss's list. † The following results were obtained :—
6 variable stars in Boss's Catalogue were of necessity omitted.
+ It may be of interest to note that, working by correlation methods on the Boss stars I find for the apex of the Sun's way R. A. = 275°, d= +28°5, but the requisite conditions for a random distribution are hopelessly unfulfilled. The correlations directly provide a method of approaching the problem of multiple drifts which I hope to deal with later.-K. P.
81.99 ± 6.66.
The complete constants for the whole of the 298 stars are
Mean μ cos 8
The complete constants for the whole of the 323 stars are―
At first sight it might appear that the northern group of stars had a larger proper motion than the southern, but, considering the probable errors, it is doubtful whether any stress can be laid on this. Thus we have for southern and northern groups—
Only in the third case is the difference more than twice its probable error, and it may just be that proper motions in declination in the southern hemisphere are more variable than in the northern, but much larger numbers would have to be used really to demonstrate this.
If, on the other hand, we compare the results for proper motion along and perpendicular to the declination circle, we have
Difference of mean and mean μ cos d
= I 16.84 ± 10.20
Difference of S. D. of μ' and S. D. of μ cos 8 27.36 ± 7.22
Difference of mean μ and mean μ cos d
Difference of S. D. of ' and S. D. of μ cos 8=75'02 ± 7°59
Three of these differences are more than twice their probable errors; and looking at the values as a whole, it seems not improbable that proper motion in the declination circle is larger and more variable than proper motion perpendicular to it. The point deserves fuller consideration, especially in its relation to the position of the apex of the Sun's way. We shall return to the matter later, from another standpoint.
If we now examine the means of the magnitude arrays, we notice at once some very remarkable points. Regarding the mean values of both proper motion components, we note that, with one exception to be discussed below, the deviations from the mean values of the means of the arrays are small as compared to the corresponding variabilities, but these deviations are remarkably regular.
Taking the southern stars as typical, we find (Diagram V.) a continuous rise in proper motion for both components until stars of about 45 magnitude, then the proper motion falls again and becomes still smaller for the faintest stars. The same rule is apparent in the northern stars also, except that in the group of extremely bright stars we have a very large proper motion, which again, however, when we remember size of mean and S. D., is not so big as it appears. Still it is definitely significant, and marks the group of brightest northern stars as significantly different from the corresponding group of southern stars. The Diagram V. will bring out the remarkable character of these magnitude and proper motion curves.
If the determining link between magnitude and proper motion were parallax, surely we might anticipate a uniformly decreasing proper motion with increasing magnitude? Yet we see in all four cases the same phenomenon-the low value of proper motion for the stars of 15 to 3.5 magnitude, its growth to a maximum, and then its ultimate fall. It is true the variations are small, but their comparative regularity is very great. It is difficult to see what special selection of these standard stars could have led to this result. It is one that will be tested on much larger masses of material, but Boss's Catalogue was selected on account of the high