1912.] OF THE GLOBULAR CLUSTERS. 139 



are always exactly equal, and hence if tj^^t.,, we have 



whence 



(42) 



as concluded by Herschel in the Philosophical Transactions, for 

 i8o2, p. 487. 



To establish clearly such actual cases of motion, with the attract- 

 ive force in the direct ratio of the distance from the empty center, 

 where he says the system may revolve together without perturba- 

 tion, and remain permanently convected without a central body, 

 Herschel proceeds to deal first wath two equal double stars re- 

 volving in circles about the common center of gravity of the sys- 

 tem. He next generalizes the procedure by taking two unequal 

 masses, then treats also the cases of motion in elliptic orbits, and 

 finally considers certain types of triple and multiple stars, to which 

 similar reasoning will apply. This paper of Herschel is quite re- 

 markable, and deserving of more attention than it has received. 



VHI. Theorem on the Revolutions of Stars in Clusters. 



It is now obvious that the clusters which have attained a definite 

 law of density depending wholly on the radius will conform to 

 Herschel's Theorem of motion about empty centers, which is also 

 the law for the central motion of particles of a rotating solid. H 

 we imagine a heterogeneous sphere made up of concentric homo- 

 geneous layers, but with the density of the layers increasing towards 

 the center, and take the radii of the layers to be r-^, r^, r^, . . . Vi, and 

 denote by o-^, a^, 0-3, .. . ai the average density of the sphere up to 

 the ith layer inclusive ; then the attraction on points in these several 

 layers will be A^, A^, A^, . . . Ai, as follows: 



4 TTO" T ^ 



A = - -^2- = ^1^ ; ^2 = Q\ ; A = Q\ ; • • • ; ^.■= Q;- (43) 

 3 ' 1 



Thus the constant will vary from layer to layer in a heterogeneous 



