I9I2.] OF THE GLOBULAR CLUSTERS. 121 



In spherical coordinates we may take the angle </> for the longitude, 

 6 for the latitude, and r for the radius of the sphere ; and then the 

 required expressions become 



x' — X == r sin 6 cos ^, 



y' — 3' ^ r sin ^ sin </), (4) 



z' — r=:rcos 6. 



The element of mass dm' defined in (i) has the equivalent form 



(r'dx'dy'dz' = u'dr ■ rdO ■ r sin ed<i>. ( 5 ) 



The element of the potential due to this differential element is 



o-';'^ sin ddrddd^ 



r 

 and the general expression for the potential becomes 



(6) 



U=^ r\i4> r smdde r a'rdr. (7) 



4^^ Jo Jo Jo 



If we make use of the equations (i), (4), (5) in equation (2) we 

 may obtain the corresponding expressions for the forces resolved 



along the coordinate axes : 



X= f\os<f)d(j> f sm-ddd f a'dr, 



Jo t/o i/o 



F= r " sin (^d<^ r sin^ Odd f a'dr, (8) 



♦/O t/O I/O 



Z= f^dcf) r cos d sin dde f a'dr. 



t/O t/o t/0 



These expressions will hold rigorously true for any law of density 

 whatever, so long as it is finite and continuous. In the physical 

 universe these conditions always are fulfilled ; and hence if these sev- 

 eral integrals can be evaluated, they will give the potentials and 

 forces exerted on a unit mass by an attracting body such as a cluster 

 of stars, or the spherical shell surrounding the nucleus of a cluster. 

 But before considering the attraction of a cluster in detail, we 



