I9I2.] OF THE GLOBULAR CLUSTERS. 127 



where a and p are positive numerical coefficients in the form of unde- 

 termined multipHers. 



Now it is significant that to the surface integrals negative terms 

 are attached increasing at rates proportional to the time. The second 

 members of (24) cannot therefore be constant, but must decrease 

 with the time. As U and V depend on the coordinates at the initial 



d dV 



epoch t^, and the derivatives-^ and -^ depend on the same ele- 

 ments, a progressive decrease in the double integrals, to satisfy the 

 right members of (24), implies that the coordinates of the entire 

 system must so change that the surface 5^ decreases. Thus the glob- 

 ular cluster undergoes a secular compression, owing to the accumu- 

 lation of appulses, and the shrinkage of the bounding surface. 



It is well known that under the operation of universal gravitation 

 the bodies of a system, starting from any initial distribution, tend to 

 fall together, so that the potential energy diminishes. When the 

 number of bodies is very large it becomes impossible for the motions 

 to be simply periodic, like that of a planet or comet moving in a 

 Keplerian ellipse ; but although the nature of the non-reentrant orbits 

 cannot be predicted by any known method, it is possible to say that 

 the potential energy of the system tends incessantly to a minimum, 

 while the maximum of the total energy becomes kinetic, and is ex- 

 pended in producing large velocities of the bodies. The left member 

 of (24) therefore incessantly decreases, owing to the exhaustion of 

 the potential energy. This accords with what, on purely mathemat- 

 ical grounds, we found to be the efifect of appulses, on the right 

 member of (24). Hence universal gravitation acts as a clustering 

 pozver, and when the figure of a cluster is rendered globular, the 

 dimensions of the system is further diminished under the secular 

 efifect of appulses and exchanges of velocities going on within the 

 mass of stars. 



In view of the above considerations it is evident that the right 

 member of equations (24) should include independent negative terms 

 to take account of the efifect of general shrinkage, without regard 

 to appulses due to close approach. Thus the final forms of these 



