mean magnitude by three classes, the introduction of blue, green, and white stars, has not essentially altered the numerical relationship between magnitude and colour. There are several points about this which deserve fuller consideration. Magnitule is often taken as a true variable. When dealing with the relationship of magnitude to parallax in the former paper it was shown that magnitude used as a numerical variate, unless we convert it into light units, does not give very satisfactory results, and that, further, the curves of means such as we find for stellar characters are generally widely different from straight lines. In the present case we cannot calculate the correlation coefficient r, because there is no quantitative scale of colour; we can, however, find the correlation ratio on the assumption that magnitude is a true scalar quantity, which, of course, it is not. We find numerically 7='17±·01. The results given on p. 456 of the former paper (for visual magnitude) lead to ŋ='13±05. Thus, again, the two results are alike within the range of probable error. But we note at once that if magnitude be used as a true quantitative character we shall not get results for C, and 7 which are really comparable. Indeed, a little thought will show that a mean magnitude is something having little physical meaning, and the ratio of two mean magnitudes, which is the nature of 7, may often be deceptive.* We must accordingly anticipate erratic results when magnitude is used as a numerical variate, and we shall endeavour to adopt it only as an index to classification. η Breaking, however, for a moment through this good rule we may note that the mean magnitudes of the colour arrays are— We see at once from this result that, within the limits of probable error, the mean magnitude of the white stars is the mean magnitude of the whole group of stars, or this group may be omitted when we determine the influence of deviation from mean colour on deviation from mean magnitude. The presence or absence of white stars does not affect the contingency. In the accompanying Diagram I.† a graph has been drawn of the variation of mean magnitude with * The mean magnitude of two stars of magnitudes 5 and 10 is 7'5, but the magnitude of the star which has their mean amount of light is 5'74. The mean magnitude of two stars of magnitude 14 and 1 is also 7 5, but the magnitude of the star with their mean amount of light is 175. The ratio of the means found in the first manner is unity, in the second manner more than 3. Yet we perpetually see mean magnitudes compared and the difference between them asserted to be of significance or non-siguificance for the character of stellar groups. This diagram was taken out of its place in this paper and published, without the data on which it is based, in a letter to Nature, Oct. 17, 1907. colour. To obtain a reasonable scale of colour, we have used that of the solar spectrum, plotting first a curve of mean magnitude, and then replacing it by a curve of star luminosity, the scale of luminosity being that in which the unit of luminosity is that of a star of magnitude 10. We have naturally omitted the white stars. (If the white stars be omitted from the Harvard data the value of the contingency is 297, a value in absolute accordance with that found for the Cape stars in the earlier paper.) The graph shows at once a somewhat striking result, the distribution of luminosity among the coloured stars gives a curve remarkably similar to that of luminosity in the solar spectrum if we shift the solar curve towards the violet end of the spectrum. How far is the method of determining magnitude influential in this matter? We have at present in hand the Potsdam colour observations, and hope shortly to publish further results on this point. (ii) Colour and Spectral Class. As far as we are aware, no classifications of stellar colour according to parallax or proper motion have yet been made. Mr. Franks has recently published three tables giving the classification of colour according to spectral class. These tables, as they stand, are actually contingency tables, but the smallness of some of the groups and the extremely laborious process of working out 7 x 12-fold contingency tables has led us to concentrate the material in rather larger colour groups, keeping, however, the spectral classes the same. We have taken as our colour groups (a) O, (b) YG1, Y, OrY1 (practically the "white" group), (c) Y2, (d) OrY2, Or2, (e) Ys, OrYs, Or, (f) OrR3, R3, thus making six colour groups or a 7 x 6-fold table. We owe to Dr. A. Loc. cit., pp. 539-541. * Scale of Luminosity Lee, of the Biometric Laboratory, University College, London, the calculation of the constauts for Mr. Franks' three tables, with the following results:-- Correlation of Colour and Spectral Class. -25° to S. Pole C1 = '71 C1 = '74 C1 = '71 432 Southern Stars from 928 Northern Stars from 1360 combined results . These results are of much interest and in very close accordance. There is no substantial difference between northern and southern stars in the relationship of colour and spectrum, but, as we might have anticipated, there is a very high relationship between the two characters. The relationship is more than double that which we have found between magnitude and colour. We shall see later that spectral class is far more clearly associated with magnitude than colour is. Accordingly, we may sum up our first results on colour as follows:-The colour of a star depends to some extent on its magnitude, but in a far more marked manner on its spectral class. It seems unnecessary to reproduce Mr. Franks' southern and northern star tables. His total star table is given here, with the groupings adopted, and in brackets the excess or defect of each group from the independent probability expectation. Hydrogen stars a Carinæ Solar stars 41 Y2. OrY2, Or2. Y3, Or Y3, OrR3, Totals. 282 14 77 (+63'62) (+42′95) ( − 36°37) ( − 34'74) ( − 33°18) (−2·28) Totals 296 The great regularity of the differences certainly speaks for the excellence of the spectral and colour classes adopted by Mr. Franks. (4) Correlations with Spectral Class.-We have seen in the pre ceding section that the association of spectral class and colour is markedly high, giving the mean square contingency coefficient C1='71. Data for the correlations of spectral class with magnitude, proper motion, and parallax are given in the Yale Observatory memoir already cited.* It is true that the number of stars is small, but the total of 98 is larger than we were able to deal with when considering parallax in the first memoir. Further, some of the spectral classes have so small a frequency that, for statistical purposes, it was absolutely needful to group certain classes together. The classes actually used in the notation of the Draper Catalogue Practically, no weight can be placed on the last two results. Now parallax and proper motion are true quantitative characters, and n is accordingly the proper constant to calculate in these cases. It has also been found for magnitude. We again owe the whole numerical work to Dr. Lee. The results come out as follows:: Correlation Ratio of Spectral Class and Parallax, n='36 ±'06 Proper Motion, ŋ='39±'06 Magnitude n='68±'04 All these results are significant compared with their probable errors, but before we discuss them it is well to ascertain whether the last result will be confirmed if we drop the idea of magnitude as a scalar quantity and proceed only by using it as an index to classification. Table III. is a contingency table for magnitude and spectral class for these 98 stars : TABLE III. Contingency: Spectral Class and Magnitude Class. F and G. H and I. K, M, and Q. Totals. 6 Magnitude. A. E. 363 In working this table we have grouped in magnitude -5 to 15 and 15 to 3'5, or dealt only with a 6 × 5-fold table. The material is too sparse for finer classification. This table gives us C1 = '69(07). It accordingly fully confirms the value obtained by the η method. It is, perhaps, worth noticing that it is in the first place the K, M, Q group, and then in the next place the F, G group, which contribute most to the value of the mean square contingency, i.e. these are the classes which have most markedly differentiated magnitudes. We see that the relationship between parallax and spectral class is only a little more than half that between magnitude and spectral class. Or we conclude that, on the basis of the Yale stars, the present chemical constitution of a star is of considerably' greater importance than its distance in determining its magnitude. It must, however, be remembered that 98 is rather a limited number to base definite statements upon; and as the point is of considerable interest, it is important to consider the matter from other aspects. Before the Yale memoir came to hand we had already formed Table IV., giving the magnitudes and spectral classes of over a thousand stars. It is based on the data provided in vol. xxviii., Harvard Observatory Annals. Pickering's 22 spectral classes would give a contingency table of unworkable magnitude, and accordingly we have grouped his spectral classes as follows, making a 7 x 8-fold table: = Our Z Pickering's groups 1 to 5; Y=6; X=7 to 11; V = 13 to 16; U17 to 20; W = 12; T 21 (does not occur); S=22. The mean magnitudes determined from a 5 grouping of the classes arranged in order of magnitude are as follows: # Magnitude 4 to 5, for example, contains all stars of 4th and less than 5th magnitude. |