I9I2.] 



OF THE GLOBULAR CLUSTERS. 



145 



As soon as a part of the shell of thickness dr has been traversed, 

 however, the stars included in the space ^irr-dr will cease to exert at- 

 traction on the moving star; and the further it enters the shell the 

 less central attraction will be exerted from the original focus. As 

 the star quits the shell and enters the hollow space within there 

 will be no central attraction to cause it to describe a Keplerian 

 ellipse. Thus as the radius vector decreases from R to r, the path 

 ceases to be the arc of an ellipse, and becomes a straight line. The 

 body thus moves uniformly across the hollow of the shell, and 

 enters again on the opposite side, with the same velocity it had on 

 quitting the shell. The central attraction of the shell begins to be 

 felt as soon as a layer of thickness dr is trav- 

 ersed, for the space 4.Trr-dr has a mass ^-Kcrr'-dr. 

 and it attracts as if collected at the center of 

 the shell. This force grows till the star emerges 

 from the shell on the outside, when it is equal 

 to that operating at the moment the star first 

 entered the shell. Consequently it will depart 

 from the shell on a Keplerian ellipse exactly 

 similar to that on which it first came in ; and 

 the total external orbit will consist of two ex- 

 actly similar and similarly situated parts of 

 ellipses, joined by straight lines in the hollow 

 of the shell, and within each layer of the shell 

 gradually passing from the arc of an ellipse 

 to a straight line. This path is illustrated by 

 the accompanying figure. 



The orbit here described supposes that 

 no local perturbations have occurred during 

 the complete revolution. Let us now consider 

 the average effect of such perturbations as 

 will occur. These may be best understood by 

 analogy with the average effect of Jupiter on 

 comets crossing his orbit. It is will known that many comets orig- 

 inally traveling in orbits almost parabolic have been thrown within 

 Jupiter's orbit, till quite a large family has been acquired with short 



Fig. 3. Illustrating 

 the capture of an os- 

 cillating star by the 

 action of a spherical 

 shell uniformly filled 

 with stars as in a 

 globular cluster. 



