Among the results cited from the researches of Pickering and Bailey are these: 1. The law of distribution is essentially the same for different clusters. 2. The bright stars and faint stars of a cluster obey the same law. Mr. H. C. Plummer also availed himself of the important researches of Dr. H. von Zeipel on the cluster Messier 3 (Annales de l'Observatoire de Paris, Tome XXV.), in which a method was developed for finding the law of distribution of the stars in space, from the observed law of distribution in the projection as we see it. Dr. von Zeipel effected this transformation by means of a theorem due to Abel. He subsequently compared his results for Messier 13 and Omega Centauri with the densities to be expected in a spherical mass of gas in isothermal equilibrium. In his paper of March, 1911, Mr. Plummer investigates the law of density for the clusters Omega Centauri, 47 Toucani, and the great cluster in Hercules (M. 13). By the use of von Zeipel's method he finds that in these three clusters there is a very good agreement as respects the law of density. In the accompanying table we give the ten points of Plummer's empirical curve of density, based on recent photographs. For the sake of comparison we give also the corresponding points for the laws of density and pressure for a sphere of gas following the monatomic law and in convective equilibrium, as developed in the writer's researches on the "Physical Constitution and Rigidity of the Heavenly Bodies” (Astron. Nachr., Nos. 4053, 4104). The nature of these three laws is best understood from the accompanying illustration, Fig. I. 1. The cluster density is greater near the boundary, the curve tending to become asymptotic, as there is no definite boundary to the mass of stars. 2. The cluster density also appears to be relatively greater near the center, so that the curve intersects the monatomic curves in the outer parts of the radius but again unites with them at the center, after falling and pursuing a different course between the surface and the center. 3. As the apparent density of the stars in a cluster is consider 0.8 07 0.6 Q2 01 0.0 1.00 1.00 1.00 O.I 0.87 0.97 0.95 0.2 0.63 0.90 0.80 0.3 0.38 0.74 0.60 /Curve of Density/observed in Star Clusters/Plummer FIG. 1. Illustrating the internal arrangement of a globular cluster. able, and the images spread somewhat on the plate, it is possible that longer photographic exposures or better plates, on which the images do not spread, would give relatively more stars in the region of the middle of the radius of the cluster, and thus bring the law of density for clusters into essential agreement with the monatomic law of density. Further photographic observations, with the best modern instruments, alone would decide this question. A final decision can not be made yet, but in order to have the judgment of the best contemporary astronomical photographer on the subject, I have recently referred the question to Professor W. S. Adams, acting director of the solar observatory at Pasadena, who reports as follows: "I regret that I cannot give answers which would be at all conclusive to your questions regarding the distribution of the fainter stars in star clusters. Up to the present time only a few counts have been made upon our photographs. So far as these go, they do not appear to show any tendency on the part of the fainter stars to predominate around any particular portion of the radius of the cluster, but rather for the distribution to be tolerably uniform. The problem is made difficult by the fact that the central part of our photographs is almost always burned out, so that counting is impossible for some distance along the radius. We have begun, however, to take series of photographs of clusters, giving exposure times with a ratio of 1 to 2.5. These should help greatly in providing an answer to your questions." On the whole the indications are that the capturing process of drawing in stars from without is still going on. This would account for the small density near the outside of the cluster, and also the great central density, the latter being an accumulative effect of the various shells in the course of millions of ages. V. THE POTENTIAL DUE TO A MASS OF GLOBULAR Figure In my "Researches," Vol. II., 1910, I have outlined the process by which the nebulæ form by the aggregation of dust from a distance; and shown that the collecting streams may often take the spiral form, and in this early stage are not of symmetrical figure. The general integrals in Section II. are required to express the attraction of these unsymmetrical masses. But in true sidereal systems as old and fully developed as the globular clusters are known to be, a state of very perfect symmetry has been attained through the oscillations of the entire mass, and the mutual adjustments of the parts of the system, and by the rounding up of the orbits under the secular action of the resisting medium, as implied in Plato's remark that the Deity always geometrizes—ὁ θέος ἀεὶ γεωμέτρει. On this latter process I have dwelt at some length in an address on “The Foundations of Cosmogony," delivered to the St. Louis Academy of Sciences, May 1, 1911, and printed in the Memorie delle Societa degli Spettroscopisti italiani, Rome, Vol. XL., 1911; and in another address entitled "The Evolution of the Starry Heavens," delivered to the California Academy of Sciences, Aug. 7, 1911, and printed in Popular Astronomy for November and December, 1911. Herschel's theory of the spherical figures of clusters (Phil. Trans., 1789, p. 217), conceived as made up of a series of concentric shells of uniform density, but with increasing accumulation towards their centers, is confirmed by modern photographs of various clusters as shown in the accompanying plates from my "Researches," Vol. II. The attraction of a mass of this kind thus becomes similar to that of a sphere made up of concentric homogeneous layers, but with the density increasing towards the center. The integration for the central attraction in these perfectly symmetrical figures thus need not involve or 4, but only the radius r. If σο be the central density of the cluster, and o the density at any point whose distance from the origin of the coördinates at the center is, a shell of density σ and thickness dr will have the mass dm 40x2dx. And the sphere enclosed by this shell will have the mass (30) where 01 is the average density of the enclosed layers included be At the surface of the cluster the gravity of the entire mass will PROC. AMER. PHIL. SOC., LI. 204 F, PRINTED JUNE 5, 1912. where M is the mass of all the stars and the exterior radius of the cluster. If G' be the value of the force of gravity of the cluster at any point below the surface, at a distance r from the center, we shall have The outer shell of the cluster is here neglected as exerting no attraction on a point within, as was long ago established by Newton for homogeneous solid bodies (cf. “Principia,” Lib. I., Prop. XCI., prob. XLV., Cor. 3). To find the ratio of G' to G so as to give the law of central force within the cluster, we have the relation The evaluation of this ratio depends on the integrals between the assigned limits, one corresponding to the entire sphere of radius x', and one to the part of the sphere included within the radius r. Thus the integrals depend on the law of density in the cluster. We have already seen from the researches of Dr. H. von Zeipel, and Mr. H. C. Plummer that the accumulation of density towards the center appears to slightly exceed that of a sphere of monatomic gas in convective equilibrium and fulfilling adiabatic conditions (A. N., 4053, and A. N., 4104). Although the monatomic law may not hold strictly true in clus |