120 SEE— DYNAMICAL THEORY [Ap^ii '9, 



clustering power noticed by Herschel to be in progress throughout 

 the sidereal universe. Such an investigation of the central forces 

 governing the motions in clusters is very desirable, because it might 

 be expected to throw light on the mode of evolution of clusters as 

 the highest type of the perfect sidereal system. If it can be shown 

 that a clustering power is really at work, and is of such a nature as 

 to produce these globular masses of stars, it will be less important 

 to consider the details of those systems which have not yet reached 

 a state of symmetry and full maturity ; for the governing principle 

 being established for the most perfect types, it must be held to be 

 the same in all. 



II. General Expressions for the Potential of an Attracting 



Mass. 



If we have a mass M' of any figure whatever, in which the law 



of density is a' = f{x' ,y' ,z'), where (.r',3'',^') are the coordinates 



of the element dm' of the attracting mass, and this element attracts 



a unit mass whose coordinates are {x,y,z) ; then the element of the 



attractmg mass is 



dm' = <j'dx'dy'dz'. (i) 



And the expressions for the forces acting on the unit mass when 

 resolved along the coordinate axes become 



dx 



= X= ^ ^ ^'^^a'dx'dy'drJ, 



dU 



= ^= j j j'-^a'dx'dy'dz'. 



r = V{x' - xf + (y -yf + (,:' - z^- 

 The potential function itself obviously is 



U 



r r r a' dx' dy' ds' , , 



