3 1 6 Review of the Principia of Newton. 



section, or it may never come to an apsis. If, for instance, 

 as has been shown in the last problem, the force be as the 

 cubes of the distances inversely, and the velocity of pro- 

 jection be perpendicular to the radius vector, and such as 

 would be due to an infinite height, the revolving body makes 

 always the same angle with the radius vector, and has no 

 tendency to come to an apsis, after any number of revolu- 

 tions however great, but will revolve in an equi-angular spi- 

 ral, or a circle which is a limit to such spirals. On the oth- 

 er hand, if the force vary in a less inverse ratio than the du- 

 plicate, the revolving body will have a tendency to an apsis, 

 or will come sooner to it than if it were accurately in that 

 ratio, and when the force varies so far from that accuracy as 

 to be in the direct ratio of the distance, the moving body 

 then comes to an apsis by performing half the angular mo- 

 tion it otherwise would in a fixed orbit, and has, in reality, 

 in comparison of this, four apsides. Now if we regard nothing 

 but the mere motion or mutation of the apsides, and the or- 

 bits otherwise to be unchanged ; we may consider that point 

 changed, while the body is moving in an ellipse, and all other 

 circumstances to remain the same. This however cannot 

 take place, unless the orbit itself, or its plane, be changed by 

 a revolution about its center. This compound motion of the 

 orbit and of a body moving in that orbit, would constitute a 

 curve in an immoveable plane; but the curve generated 

 would not be the true curve unless the distances and rela- 

 tive positions of the moveable and immoveable curves were 

 always the same. This condition being supposed, the forces 

 necessary for the retention of a body in an orbit which is 

 itself moveable, may be investigated, if by its compound 

 motion it describe equal areas in equal times about a fixed 

 point. Newton has used this principle for ascertaining the 

 motion of the apsides in elliptical orbits not much differ- 

 ing from circles. Supposing the motion of the apsides to 

 be produced by this compound motion, viz. the motion of a 

 body moving in an ellipse, and a motion of rotation in a cir- 

 cle. This latter motion is additional, or subductive of the 

 former, but the areas generated by it, if the rotatory motion 

 be uniform will always be proportional to those in the fixed 

 ellipsis, and with proper degrees of force, the body may re- 

 volve in this manner about a fixed point as a center. But 

 the forces for its retention in the ellipse and circle, act ac- 

 cording to different laws. If Q be the force in either, fojr 



