I9I2.] OF THE GLOBULAR CLUSTERS. 133 



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 — 6 6eo^ ael jecofxeTpeL. 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 i, 1911, and printed in the Memorie dellc Societa 

 degli Spettroscopisti italiani, Rome, Vol. XL., 191 1; and in another 

 address entitled " The Evolution of the Starry Heavens," delivered 

 to the California Academy of Sciences, Aug. 7, 191 1, and printed in 

 Popular Astronomy for November and December, 191 1. 



Herschel's theory of the spherical figures of clusters (Phil. 

 Trans., 1789, p. 217), conceived as made up of a series of concen- 

 tric shells of uniform density, but with increasing accumulation 

 towards their centers, is confirmed by modern photographs of vari- 

 ous clusters as shown in the accompanying plates from my " Re- 

 searches," Vol. IL The attraction of a mass of this kind thus be- 

 comes similar to that of a sphere made up of concentric homo- 

 geneous layers, but with the density increasing towards the center. 

 The integration for the central attraction in these perfectly sym- 

 metrical figures thus need not involve 6 or <^, but only the radius r. 



If ctq be the central density of the cluster, and a the density at 

 any point whose distance from the origin of the coordinates at the 

 center is s, a shell of density a and thickness dx will have the mass 



dm = 4ir(j.v-dx. ( 30) 



And the sphere enclosed by this shell will have the mass 



= 47r I (TX^dx = ^ira^x^', (31) 



Jo 



m 



where o-^ is the average density of the enclosed layers included be- 

 tween X = o and .r = .r. Thus we have 



At the surface of the cluster the gravity of the entire mass will 



PROC. AMER. PHIL. SOC. , LI. 2O4 F, PRINTED JUNE 5, I9I2. 



