18 MOTION OF THE APSIDES. [SECT. HI. 



retarded through as much, the motion at the end of the time 

 will be the same as if no change had taken place. But, as 

 the orbits of the planets are ellipses, this symmetry does not 

 hold : for, as the planet moves unequably in its orbit, it is 

 in some positions more directly, and for a longer time, under 

 the influence of the disturbing force than in others. And, 

 although multitudes of variations do compensate each other 

 in short periods, there are others, depending on peculiar 

 relations among the periodic times of the planets, which do 

 not compensate each other till after one, or even till after 

 many revolutions of both bodies. A periodical inequality of 

 this kind in the motions of Jupiter and Saturn has a period 

 of no less than 918 years. 



The radial force, or that part of the disturbing force which 

 acts in the direction of the line joining the centres of the 

 sun and disturbed planet, has no effect on the areas, but is 

 the cause of periodical changes of small extent in the dis- 

 tance of the planet from the sun. It has already been 

 shown, that the force producing perfectly elliptical motion 

 varies inversely as the square of the distance, and that a 

 force following any other law would cause the body to 

 move in a curve of a very different kind. Now, the radial 

 disturbing force varies directly as the distance ; and, as it 

 sometimes combines with and increases the intensity of the 

 sun's attraction for the disturbed body, and at other times 

 opposes and consequently diminishes it, in both cases it 

 causes the sun's attraction to deviate from the exact law of 

 gravity, and the whole action of this compound central force 

 on the disturbed body is either greater or less than what is 

 requisite for perfectly elliptical motion. When greater, the 

 curvature of the disturbed planet's path, on leaving its peri- 

 helion (N. 64), or point nearest the sun, is greater than it 

 would be in the ellipse, which brings the planet to its aphe- 

 lion (N. 65), or point farthest from the sun, before it has 

 passed through 180, as it would do if undisturbed. So 

 that in this case the apsides, or extremities of the major axis, 



