320 Review of the Principia of Newton. 



of 360°. In the latter case, it will come four times to an 

 apsis, in an entire revolution of 360°. 



3. Again, if the force vary as the cubes of the distances in- 

 versely, the trajectory will every where make such angles 

 with the radius vector, as will never come to an apsis. 



4. If the force be not such as is necessary for the descrip- 

 tion of an accurate ellipsis, it is evident that the body must 

 describe some other curve, whose apsides will be in different 

 places from those of an immoveable ellipse. If that force be 

 in a greater inverse ratio than the inverse duplicate ratio of 

 the distance, the trajectory then verges towards such an one 

 as is produced by forces in the inverse triplicate ratio of the 

 distance, which prevents the body from coming to an apsis 

 as soon as it otherwise would, consequently the motion of the 

 apsides is progressive. 



5. If the force vary in a less inverse ratio of the distance 

 than the duplicate, it then verges to the case where it is directly 

 as the distance, and comes to an apsis in moving only 90°, or 

 over half the angle of revolution necessary for a fixed orbit, 

 the motion of the apsides is then regressive. 



6. By these laws of force variously compounded, bodies 

 may move so as perpetually to form apsides, or so as to have 

 none, or to have a determinate number of them, and then to 

 fly off ad infinitum. Some of these cases it would not be 

 amiss to state. 



1. When the centripetal force is as any power of the dis- 

 tance directly, or less than the first power thereof inversely, 

 the orbit will always have an higher and lower apsis, beyond 

 which the body cannot ascend or descend. 



2. If p, the velocity in the trajectory compared with that of 

 a circle be asp : 1 — and n be the index of the power of the 

 force, according to the distance, then when the centripetal 

 force is as any power of the distance, (whole or broken) betwixt 

 the first and third, the orbit will have two apsides, if p be less 



than \/ , but otherwise only one ; in which last case 



n-\-l 

 the body, after it has passed its apsis, will continue to recede 

 from the centre in infinitum. 



3. When the centripetal force is inversely as any power 

 greater than the third, the orbit can, at most have but one 

 apsis ; but in some cases it will have none at all ; it may go 

 off ad infinitum, or it may revolve in a spiral, approaching the 



