134 SEE— DYNAMICAL THEORY [April 19. 



become 



M 



C' = — 2» (33) 



x' 



where ISl is the mass of all the stars and x' 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 attrac- 

 tion on a point within, as was long ago established by Newton for 

 homogeneous solid bodies (cf. " Principia," Lib. I., Prop. XCL, 

 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 



4'r^o J I - \..2 



G' X 



'■fid 



dx 



G 



l'^^/ 



^-5^1(0'- 





(35) 



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 x. 

 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- 



