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accuracy of his results and the manageable number of stars provided. We may look at the point from a second aspect. The lines of means are curved, and accordingly the coefficient of correlation is not the proper measure of the association between magnitude and proper motion. It is provided by the correlation ratio n. We
It will thus be seen that the amount of correlation even for the northern stars is only small. Working on the 173 stars of the Yale memoir, we have found the correlation ratio for proper motion in arrays of limited magnitude; it is:
n = '22±05.
Considering that the Yale stars are all northern stars, but that in this case the total proper motion has been taken, and not the component proper motions, we see that the agreement, notwithstanding the wide difference of material, is excellent,-well within the limits of the probable errors.
While the correlation is slight for the northern stars, it is practically of little or no service, as far as selection goes, for the southern stars, or we must conclude that
A selection of stars by their proper motions must have very little influence indeed on the relation between magnitude and parallax.
It will accordingly require some very stringent form of investigation to demonstrate that the form of the parallax-magnitude curve can be sensibly influenced by a proper motion selection.
As far as the present data extends, we would venture to suggest, therefore, that the criticism that the stars dealt with are specially selected is not really a valid one, for this selection would not largely influence the relationship of magnitude and parallax. We must conclude that the present results confirm the early ones, namely, they show that magnitude is a quantity much more closely associated with chemical condition (spectral class) than with distance; indeed, the association with colour is almost as great as the association with distance.
(6) Correlations with Parallax.—We have already dealt with the association between parallax and magnitude, colour and spectral class (pp. 422, 423, 428, and 432). The remaining feature is proper motion. This is provided by the Yale observations. Accepting the authors' proper motion classes, the arrays of parallax give us (Table I.),
Correlation ratio, parallaxes for proper motion classes, n=36'05
* This result is for total proper motion in a great circle, but whether we use component proper motions in ares or again the values in R. A. and & makes but little difference in the correlation values.
and again (Table III.),
Correlation ratio, proper motions for parallax classes, The previous work on the 72 stars gave
Correlation coefficient, proper inotion in R. A. and parallax, in declination and parallax, r= 41±07
Considering the size of the probable errors, the present results are reasonably in accordance with the former, or we may take it that the correlation between parallax and proper motion is not far from linear, and of magnitude about 40. Thus we have the following scheme of relationship :
Parallax and proper motion
Parallax and spectral class.
Thus, while the distance of a star is sensibly related to its proper motion, this relationship is not really more significant than the relation to spectral class; and it is quite possible that if the spectra were assorted according to parallax, it would be possible to form spectral categories which would give a far higher association between parallax and spectral class than between the former and proper motion. The parallax stars have not yet been dealt with as to colour.
We have seen that, on the basis of the Yale stars, the correlation between parallax and proper motion is nearly linear. It is accordingly of considerable interest to obtain the line giving the mean parallax for a group of stars with a given total proper motion p in a great circle. Let be the mean parallax, p the mean proper motion, c, the standard deviation in parallax, σ, the standard deviation in proper motion of the whole group of the Yale stars. Then we have =0"0460 ± 0035, p=0" 6763'0205,
(p − p),
= 1000 + 15P.
If we neglect the " as negligible in the case of parallax, we conclude that the mean parallax of an array of stars of given proper motion is one-fifteenth of that proper motion.* This is absolutely identical with the statement made by Newcomb, largely on theoretical grounds: †
* The 70 odd stars dealt with in the earlier memoir, if we compare total proper motion in a great circle and parallax, give the correlation of parallax and proper motion as 58 and the regression coefficient. This is as close to as we could expect from the material.
The Astronomical Journal, vol. xxii. p. 169, 1902.
"That is, if we measure the parallaxes of all the stars having a given proper motion, we may expect the mean result to be about of the proper motion."
The interesting point of this result is that the value which was given on theoretical grounds by a brilliant astronomer, and appears then to have passed unquestioned, should have met with disapproval when actually found from observations by the statistician. As we have just indicated, Newcomb's is but little more or less than our statement that the correlation between proper motion and parallax equals 4. Newcomb's
where μ = our p, is only a regression line without the constant term, which in this case, if it be not actually zero, is certainly very small. As far, then, as our reduction of the Yale data goes, it tends to confirm Newcomb's theory of stellar distribution; it also shows that our values of the parallax and proper motion correlation and of the parallax and proper motion standard deviations were reasonable values.
We find from the Yale data that the mean proper motion P for stars of a given parallax π is given by the regression line
This does not agree with Newcomb's relation (loc. cit., p. 168), and, of course, only applies to the range of stars in the Yale data. These, however, do not appear to satisfy Newcomb's equation.
(7) Parallactic Motion.-Still another method of approaching the parallax and proper motion correlation may be deduced from the Sun's motion. Let v = the velocity of the Sun, p the distance of a given star, and i the index v/p. Then, if be the parallax, p the proper motion of the star, and a the radius of the Earth's orbit,
where and i may be measured in seconds of angle.
Since v/a is a constant, we have
or the correlation of parallax and proper motion is the same as the correlation of this index and proper motion. Suppose that represents the standard deviation of any variable x, then
Now room is the slope of the best fitting line to the curve in which the mean value of i is plotted to a given proper motion class of stars.
The value of m has not been found at present because of the
labour of determining for small groups of proper motions, but some idea of its value for large groupings of proper motions can be obtained from the papers by Oscar Stumpe* and by Messrs. Dyson and Thackeray.† These give us
It will be seen that Stumpe worked with stars of far larger proper motion than Messrs. Dyson and Thackeray. We have only been able to give the presumed means in the case of the first set of stars as the actual values of the total proper motion in a great circle are not tabled. Assuming the means to approximate to the values given, we find for the first series
and for the second series
using the method of least squares to determine _m=r1pσ¿/σp,—i.e. the correlation process. It will be seen at once that the two sets of observations give no close agreement; the ranges of stars dealt with are very different. If we take m= 88 we find
Astronomische Nachrichten, Bd. 125, ss. 385-426.
+ Monthly Notices, vol. lxv. pp. 429.
The graph of the Dyson-Thackeray and Stumpe data shows that their first three points form a more or less continuous curve with his four points, but that if our estimation of the mean proper motion of the stars above o" 20 proper motion be at all correct, the fourth Dyson-Thackeray point does not lie on this continuous curve. This is possibly due to a considerable underestimation of the mean value of the " over o".20 group. If we omitted this fourth point, we should get an initial slope m, to the regression curve of less than 5, a value considerably nearer that required to reconcile the parallax and Sun's motion observations discussed below. The actual regression lines im = "7395 p-o'*004 (Dyson-Thackeray)
differ significantly. from linear.
This is due to the fact that the regression curve is far