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To sum up our results for spectral class we may conclude as highly probable
That there is a very considerable relationship between magnitude and spectral classes, which will probably be found to be intensified if more satisfactory spectral categories can be determined. This relationship, however, is at present sensibly lower than the relationship between colour and existing spectral classes, which is very high. There are, further, fairly considerable relations, in part independent, between spectral class and both parallax and proper motion. It is thus probable that the chemical constitution of a star is not only a-sociated with a certain colour and brilliancy, but also with its spatial position (as determined by its distance from us), its motion in space, and its size (so far as size as well as intensity is an attribute of magnitude).
(5) Correlations with Magnitude. - In the present paper we have already seen that magnitude is related to both colour and spectral class. We may note here that the relation of magnitude to spectral class is not due to the relation of both to colour. On the other hand, the relationships between spectral class and both colour and magnitude are so high compared with that between magnitude and colour, that it is highly probable that the partial correlation between colour and magnitude for a given spectral class is negative. In other words, if the colour classes were arranged in order of the average magnitude of the stars contained in them, then, if we confined our attention to one spectral class only, say the Arcturus type, the order of magnitude for the colour classes within this spectral class would be reversed.
(i) Magnitude and Parallax.—In the former paper this relationship was deduced from 72 stars given for parallax by Newcomb. The deficiency of material was fully admitted, and the results as to correlation could only be considered as preliminary. They indicated, however, (i) that we should not expect to find a very high association between magnitude and parallax; (ii) that the correlation coefficient differing widely from both the correlation ratio and the contingency coefficient, the distribution was far from
normal; and (iii) the line of means was not approximately straight, but the parallax rose again with the fainter stars. The Yale memoir provides far better material of more than double the amount, namely, 173 stars. It is desirable to discuss how far this diverges from the previous results. The Yale authors have grouped their arrays of parallax in somewhat irregular magnitude classes and get the following results (Table ii. p. 201, loc. cit.):
The mean magnitude for the whole series is 6'16 ±11, with a standard deviation of 2'06 ± 07. The mean magnitude of the Newcomb series was 4'03 ± 22, with a standard deviation of 2'74±15. It contained on an average much brighter stars, but was a more variable sample in brightness. It will be seen that, as in the first sample, parallax tends to increase again with the faintest stars. Working out for the arrays of parallax corresponding to the above magnitude ranges, we have
The relationship, therefore, of magnitude to parallax is about equal to its relation to colour, and not more than half its relation to spectral class.
We can, however, approach the subject from another standpoint: we may inquire into the distribution of magnitude for given arrays of parallax. We have the following table deduced from our author's Table iii. (p. 202):—
The mean parallax of the whole series is o"0460 ± ̋0035, with a standard deviation of o"0664±"0025. In the earlier series the mean parallax was o"145"OII, with a standard deviation of o".134±007, indicating that we were dealing, on the whole, with a nearer set of stars, but with a greater variability in distance. It is obvious that very little can be judged from the mean magni*.3748; the authors have 3.8. The authors have "066.
tudes, especially if we exclude the first two categories of negative parallaxes.* We have included them because the authors do, but it is not easy to see why the 7 stars with the greatest negative parallaxes are to be looked upon as a class with a really lower quantitative parallax than those of the 29 group with a less negative parallax. Since the parallax cannot be below zero, what the results signify is that an average error of o"II was possible in the group of 7, and one of o"025, or about a quarter of it, possible in the group of 29 stars. Errors of observation being thus more liable in the group, it might even be reasonable to suppose members of this group to have, on the whole, higher true parallaxes than those of the 29 group; and the appearance of this group at the top with the high magnitude 734 is somewhat misleading. Leaving it, however, to form part of the series, we find
or we conclude that the determination of magnitude from parallax is considerably more inaccurate than that of parallax from magnitude. The association has indeed fallen below anything yet dealt with in the present paper. The inequality of the two values of ʼn shows that our distribution is very far from Gaussian. It is clear that in these circumstances contingency is the only method by which we can approach a unique measure of the relationship of parallax and magnitude, and accordingly the following contingency table has been prepared connecting the two characters. gives us
C1 = '32 ('05).
In the previous memoir for Newcomb's 72 stars we had
Contingency: Magnitude and Parallax.
* We would place considerable stress on this, because much of the impressiveness of the above table, both visually and when reduced numerically, lies on the place assigned to the group of largest negative parallaxes, with their mean magnitude 734. Now this consists of only 7 stars, and the mean inagnitude may be merely indicative of greater errors arising with fainter
With the exception of three stars, 2.3 2'0, and 2.8, all were above 3'0.
Thus practically the previous result and the present for the contingency agree within the limits of the probable error of the difference. Further, the contingency and correlation ratio results are also in close agreement, assuming we class by magnitude. Now the series we are considering at present is very different from the earlier series that series contained 20 stars brighter than magnitude 2, while the present contains none. It contained only 7 stars above 7.95, while the present series contains 36. It is perfectly correct that in both cases stars have been frequently selected because they had large proper motions, and so may have large parallaxes. But does this really affect the argument that there is comparatively small relationship between magnitude and parallax? In order to do so, it must mean that the partial correlation is much lower than the absolute correlation, and this can only happen if the correlation between magnitude and proper motion is very high. To this point we now turn, because some criticism of the earlier paper has been made on this ground.
(ii) Magnitude and Proper Motion.-Any amount of material is forthcoming, of course, on this point, but it seemed advisable to work with the best available proper motions. Accordingly we used the Catalogue of 627 Principal Standard Stars, 1904, by Lewis Boss. Omitting variables, we had 303 stars with positive, 317 with negative, proper motions in R.A.* There were 195 stars with positive and 426 stars with negative proper motion in declination. There was no necessity to separate these groups, as we are dealing only with the size and not sign of the proper motions, but it was desirable to separate into random groups in order to test the steadiness of the coefficients, and the positive and negative groups served as well as any others.
Four tables of contingency were formed, given as X., XI., XII. and XIII. below. We owe the working of the contingency coefficients of the first two tables to Mr. A. M. Pritchard, of the Hartley University College, Southampton, and the last two to Dr. Alice Lee. The statistical constants reached are as follows:
The mean degree of relationship between proper motion and magnitude is accordingly 32.
In Diagrams III. and IV. the means of arrays of magnitude for given classes of proper motion are plotted; it will be seen at once how little influence of an orderly kind a selection of stars by proper motions would have on their magnitude.
* One star out of the 621 non-variables has been overlooked in tabling the proper motions in R. A. It lies in group of magnitude 3 5-4'4.