122 SEE— DYNAMICAL THEORY [April 19, 



shall first examine the cumulative effect of central forces on the law 

 of density. The problem is intricate and must be treated by methods 

 of great generality, but as it will elucidate the subsequent procedure 

 for determining the attraction of such a mass upon a neighboring 

 point, we shall give the analysis with enough detail to establish clearly 

 the secular effect of close appulses of individual stars upon the figure 

 and internal arrangement of these wonderful masses of stars. 



III. The Cumulative Effect of the Central Forces upon the 

 Figure and Compression of a Globular Cluster of Stars. 



Suppose a globular cluster of stars to be in a moderate state of 

 compression, with density increasing towards the center. Imagine 

 the whole of the mass at the epoch ^^ to be divided into two parts by 

 a spherical surface of radius r, drawn about the center of gravity of 

 the entire system ; and let the external boundary of the cluster be R, 

 so chosen that no star, from the motions existing at the initial epoch, 

 will cross the border r = R. The stars in the outer shell, between 

 the surfaces r and R, with coordinates (-v', 3'',-'), will give rise to a 

 potential U . Those of the nucleus or series of internal shells, be- 

 tween r = o, and r^=r, with coordinates {x,y,z), will give rise to a 

 potential V. Accordingly we have 



, , , a'dx'dy'dz' 



Vix' - xy -^ {y -yf + {z' - zY 



= /// , 



C C C (Tdxdydz 



J J J Vix' — X 



(9) 



V{x' - xf ^ {/ - yf + {z' - zf 

 And the forces resolved along the coordinate axes are 



^^ Y C C C (T'{x'—x)dx'dydz' 



^ ^ " ~ J J J \\x' - xf + [y' - yf +\z' - zf 



^7" J J J [(;r'-;r)^ + (y-jf + (y-.^fji' ^'°) 



•^^ -y CCC <t'{z' -z)dx'dy'dz' 



~^~ J J J \{^r'-x 



r + {y -yf + {^'-^fV' 



