clusters have been examined with respect to the magnitudes of R1 the contained stars, a mean value of the ratio can be had. Of R2 course the best value is only to be had from those clusters which have been most carefully studied, but these are now too few for our purpose. I have collected, in Table C (pp. 146–151), the results from the principal monographs and from the clusters of Sir JOHN HERSCHEL'S Southern sweeps,- in all 167 cases. The mean value of is 4.1. That is, in these 167 clusters, supposing the stars in each cluster to be equally bright per unit-area, the diameter of the largest star is on the average 4.1 times the diameter of the smallest star of each cluster. This depends upon the fact, which is well known, that clusters usually are composed of stars of about the same magnitudes. Under the fundamental assumption the mean difference of magnitudes is about 3.8m on HERSCHEL'S Scale. SUMMARY. The fundamental assumption that the brightness of the unit-area of all stars is the same has now been applied to the cases of 1st, stars of known distance, 2nd, binary stars, 3rd, clusters. The results reached are 1st, that, admitting this assumption, some of the isolated stars are as much as 271 times greater in diameter than others, 2nd, that the binary stars of like color are composed of two equal stars, while those of different color are composed of two stars one about three times the diameter of the other on the average. 3rd, that, on the average, the brightest star of any cluster is about 4.1 times as great in diameter as the smallest of the same cluster. There is nothing impossible, or it may even be said highly improbable, in these conclusions. They stand or fall with the hypothesis on which they are founded. We can push our conclusions a little farther. If we have two stars A and B whose brightness is B' and B" we have seen that B' B" is necessarily finite and for our present telescopes its greatest value is about 400,000. R" we have seen for isolated stars has a maximum value (Table A) 271, whence D271X633171,543. From this it would appear that on the fundamental assumption and using the best data now available, it is not likely that any visible star is more than 200,000 times farther from us than the nearest fixed star. B B There are strong reasons to believe, as I have elsewhere noted, that an indefinite increase of the aperture of our telescopes would not bring to view an indefinite number of stars of the fainter magnitudes, or that has a limiting value not far perhaps from 500,000. The co-efficient 271 (Table A) will be changed with further determinations of parallax but must always remain finite. Thus it would follow that our sidereal system is comparatively narrowly limited in its boundaries, since is always finite and small. D" The above conclusions follow when once the fundamental hypothesis is admitted as a basis for computation. I have thought that they should be presented in connection with that hypothesis to aid in forming an adequate idea of its justness and probability. U.S. NAVAL OBSERVATORY, 1880, August. |