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The coefficient of mean square contingency for this long series was C1='43(02). Thus, while it is not as large as that found for the Draper Catalogue stars given in the Yale memoir, it still points to a fairly close relationship between spectral class and magnitude. It will, of course, be obvious that the order of mean magnitude of the spectral classes given above is not that of Pickering's class numbers, so that it may be presumed that the characters which led to his order of classification are not those which lead to order of magnitude. If we attempt to place our magnitudes in a continuous curve, we note that (i) more than onepair might be interchanged without affecting the limits fixed by the probable errors of random sampling, and (ii) we have no spectral scale upon which to plot the magnitudes or luminosities. The latter difficulty may be overcome in a manner which has proved of some service in other branches of statistical inquiry, namely, by assuming that the total spectral frequencies of each class are portions of a normal curve of frequency, and then plotting on verticals through the means (as deduced from a well-known property of the Gaussian curve) of the corresponding areas. Assuming that the groups, being merely qualitative, may be interchanged in order, it rarely happens that more than one arrangement gives a continuous curve. We have found Diagram II. by this method, plotting both magnitude and luminosity* curves. It will be seen that fairly smooth curves are reached in this manner; they would probably have been better had the U stars had greater frequency, and the V stars been dealt with in smaller groups. The diagram suggests, without of course proving, the likelihood of an ascent to a maximum luminosity from two different ends of a spectral classification.

It appeared probable that a classification of spectral characters based more admittedly on temperature considerations than those of either Secchi or Pickering would lead to a still high correlationship between magnitude and spectral character. Accordingly, appeal was made to Sir Norman Lockyer, who, with great kindness, provided a key index linking Pickering's classes with his own temperature classification. On further consideration, however, it

The luminosity of the star of the 10th magnitude is taken as the unit and Pogson's value of the constant adopted.

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Curves of Magnitude and Luminosity plotted to means for a normal frequency scale.

DIAGRAM II.

seemed probable that somewhat better results would be reached by working with Sir Norman Lockyer's own classified catalogue,* although this involved supplying from other sources (Harvard Photometry, where possible) the magnitudes of the recorded stars. We have from this catalogue formed Table V., giving spectral class and magnitude of 448 stars.

TABLE V.

Contingency: Stellar Magnitude and Spectral Class
(Lockyer's Classification).

Magnitude.

Class. Under 15. 15-2'5. 2'5-3'5. 3'5-4 5. 4'5-5'5. 5'5 and over. Totals. Means.

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Here 15-25 signifies 15 and under 2.5. In finding the means, the group "under 1'5' 23 was broken up into three groups.

The mean magnitude of the stars in this list was 3'39±0.4, and their standard deviation 1158 ± 026. They are thus much brighter and less variable than the stars of the two earlier series. The contingency coefficient for this grouping is

C1 = '54(03).

We see accordingly that the association of magnitude and spectral class has been sensibly increased by using Sir Norman Lockyer's classification. The correlation is now about half way between that provided by Pickering's date and the 98 Yale stars.

* Catalogue of 470 of the brighter Stars classified according to their Chemistry at the Solar Physics Observatory, South Kensington.

Looking at the column of means, and rearranging it on Lockyer's

plan,

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we see that the average magnitudes are greater on the side of Descending Temperature," and less on the side of "Ascending Temperature," but the order of magnitude cannot be said to fit closely the classification order. The order is chaotic on the "Ascending Temperature" side; and although, with the exception of the Markabian class, it is orderly on the "Descending Temperature" side, the justification for the order does not lie in magnitude. When more material is available, it would seem that a spectral classification according to mean magnitude might be suggestive. All that we can venture to say at present is that magnitude is very far from being independent of spectral class, but that the reason why certain spectral classes have differentiated magnitudes is not evident in any of the classifications here dealt with.

(iii) Spectral Ciass with Proper Motion and Parallax.-We now turn to the relationship between spectral class and proper motion. In this case we have only dealt at present with the table provided in the Yale memoir, but there exists plenty of further data which we hope to work up. The correlation ratio for spectral class and the proper motions of the 98 stars given in that case is

n='39±'06.

Spectral class is thus shown to be definitely associated with proper motion. The association is only about half that of colour, and sensibly less than that of magnitude, still it is quite a considerable relationship. We might, a priori, anticipate that the association was an indirect effect of the correlation of spectral class and parallax, but it is easy to prove that this is at least only partly The correlation ratio between parallax and spectral class deduced from the same material is

the case.

='36 ±.06.

This, again, is quite a considerable amount of relationship, but it is not greater but practically equal to the relationship of spectral class and proper motion.

Now if we have three characters, 1, 2, and 3, and if r12 be the correlation of the first and second without regard to the third, then

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is the "partial" correlation coefficient of the first and second characters for a constant value of the third. In other words, if the relationship between 1 and 2 be solely due to the relationship of both to 3, we should expect that within the array of 1 and 2 corresponding to a constant value of 3, the correlation would be zero between I and 2, or P120. This means that 71213 723 is the test for the correlation of 1 and 2 being solely due to their relationship to 3. Applied to our particular case, if the correlation between spectral class and proper motion be solely due to the relation between both and parallax, we should expect the correlation coefficient of spectral class and proper motion to be equal to the product of the coefficients of parallax with spectral class and parallax with proper motion. Now we have seen that the relationship of spectral class with proper motion and parallax is about equal. Accordingly, we should expect a relationship between proper motion and parallax not very far from perfect, or the correlation coefficient about unity. This is very far indeed from the case; it does not exceed 3 to 5. It follows, accordingly, that the relationship of spectral class to proper motion is not an indirect effect of chemical constitution being a function of spatial distribution. It is, of course, partly due to this result, but, to judge from the Yale data, there is a sensible relationship between chemical constitution and intensity of stellar motion in space.

We may form some notion of the amount of this, as follows. Judging from the values of the correlation ratios, we may assume the correlation coefficients to be not very different from 4 for spectral class with proper motion or with parallax. The correlation between proper motion and parallax is also not far from 4 (see § 6). Hence we find for the value of the partial correlation of spectral class and proper motion for constant parallax

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Thus, in rough numbers, on the data at present available, about 25 p.c. only of the relationship between spectral class and proper motion is due to parallax. It is desirable to place in a table (Table VI.) the mean magnitudes, proper motions, and parallaxes, as given by the Yale results grouped as dealt with here. It is perhaps unnecessary to observe that far more extended frequencies are needful before the numbers given can be finally accepted.

While scarcely any weight at all is to be placed on the results for classes K, M and Q, we still see that for the remainder, while the orders of spectral class for parallax and proper motion are more alike than they are in either case to that for magnitude, they still differ considerably from each other.

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