'912.] OF THE GLOBULAR CLUSTERS. 137 



To illustrate the simple case of a homogeneous sphere^ we re- 

 mark that it attracts a point at its surface with a force 



/= ^-j,2 =^'^<^''=Cr, (39) 



where o- is the density, and r the radius. This equation shows that 

 all points within the sphere are attracted to the center by forces 

 proportional to the radii of the shells on which they are situated, 

 since the external shells exert no attraction on points within. 



Let the solid sphere be set rotating steadily about an axis ; then 

 as the central forces at the various points are proportional to the 

 radii described by the points, there will be no tendency arising from 

 the central attraction for any shell to be displaced with respect 

 to the shells within or without, once the condition of equilib- 

 rium is attained, but the central accelerations will everywhere tend 

 to secure steady motion without relative displacement of the parts 

 of the sphere. The same is true of the centrifugal force, after the 

 adjustment to a suitable figure of equilibrium; for the centrifugal 

 force is equal to v'/r, v being the velocity of the particle and r the 

 radius it describes ; for this gives 



v"^ {T.irrf \TTr 



and as it is common for all particles the force has the same form 

 here as in equation (39). 



What is here proved for the simple case of the homogeneous 

 sphere, will obviously hold also for a sphere made up of concentric 

 spherical shells of uniform density; for the theorem will hold for all 

 the points within. And similarly for ellipsoidal honiaroids, or sphe- 

 roids such as the planets, sun and stars. If any of these masses 

 have attained uniform movement as of rotation, there is no tend- 

 ency to produce a relative displacement of the parts. 



Now the simple equation (39) shows that a similar theorem 

 holds for the internal dynamics of a globular cluster, the component 

 stars of which have attained a state of equilibrium following a 

 definite law of density depending only on the radius. But before 



