I9I2.] OF THE GLOBULAR CLUSTERS. 129 



nal surface, and thus for the / concentric shells we have 



(l"') = °' 



(27) 



where ir^ =X( (^-i'. V--) is the equation of any surface. 



Moreover, the clusters involve two additional conditions, the first 

 being that in each layer the density o-j shall depend wholly on the 

 radius, and not at all on the angles (</),(9) usually used in polar coor- 

 dinates. If any point in any layer be taken as a pole it suffices to 

 regard simply the new polar angle 9; and the required condition is 



where ctj = i/'(r, ^, ^) is the law of density in any shell, the new angle 

 6 alone being sufficient where there is no fixed pole. The second 

 condition is that the law of density as respects the radius shall be 

 suitable and the same throughout the mass ; so that in every part the 

 form of the density function does not vary as respects the radius : 



"^^^"^ = o. (.,) 



The actual arrangement in any given cluster may not be perfect, but 

 nature always and everywhere works towards the fulfillment of these 

 conditions. 



IV. The Observed Law of Density in Globular Clusters. 



In the Monthly Notices of the Royal Astronomical Society for 

 March, 191 1, Mr. H. C. Plummer, of Oxford, has an important 

 paper " On the Problem of Distribution in Globular Star Clusters." 

 For earlier data on the distribution of stars in clusters he refers to 

 a statistical paper by Mr. W. E. Plummer (Monthly Notices, June, 

 1905, \'ol. LXV., p. 810), and to the much earlier investigations 

 by Professor E. C. Pickering [Harvard Annals, Vol. XXII.) and 

 Professor Solon I. Bailey {Astronomy and Astrophysics, Vol. XII., 

 p. 689). 



