140 SEE— DYNAMICAL THEORY [April 19, 



sphere made up of concentric homogeneous shells, but the attraction 

 at every point, including the external surface, is proportional to the 

 radius of the shell in cjuestion. 



Now just as a sphere, either homogeneous or made up of con- 

 centric layers of uniform density, attracts all internal points, in- 

 cluding those at the external surface, with a force proportional to 

 the radius of the shell on which it is situated; so also will a cluster 

 which is condensed towards the center according to any law of 

 density depending wholly on the radius, attract all internal points, 

 including those in the external surface, according to the same law 

 of direct proportionality to the distance from the center. 



When the attractive force varies directly as the distance from the 

 center, the particle so attracted describes an ellipse as was first 

 proved by Newton in the " Principia " (Lib. I., Prop. X., Prob. V.). 

 This case of attraction depending directly on the first power of the 

 distance is also discussed by the analytical method in Vol. II. of my 

 "Researches," 1910, pp. 25-27, where it is shown that the time of 

 revolution is quite independent of the dimensions of the ellipse, but 

 depends wholly on the intensity of the central force. 



For motion in a plane the coordinates of the particle are shown 

 to be defined by the equations : 



V cos -Jr . — — V sin -ylr — 



X — = — sm y lit -\- a cos V ixt, y = ^— sm v [it. (44) 



V ii v> 



As the values of the coordinates are the same at the time t and 



27r _ ^ . . . 27r 



^ -f- — ~- it is evident that the time of revolution is — — , or inversely 

 V/^' VM 



as the square root of /x, where /x is the mass, and exerts the corre- 

 sponding unit of force at unit distance. 



In a cluster with stars arranged according to a law of density 

 depending wholly on the radius, the value of \i. or the force will 

 depend wholly on the radius also, as shown in equation (34). And 

 thus the time of revolution will be independent of the dimensions of 

 the ellipse. Assuming that there is but little relative displacement 

 of the bodies of the clusters, a star situated, therefore, in an outer 



