SICT. III. MOTION OF THE APSIDES. 15 



not being directed to the center, occasioas an unequable 

 description of areas, or, what is the same thing, it dis- 

 turbs the motion of the planet in longitude. The tan- 

 gential force sometimes accelerates the planet's motion, 

 sometimes retards it, and occasionally has no effect at all. 

 Were the orbits of both planets circular, a complete 

 compensation would take place at each revolution of the 

 two planets, because the arcs in which the accelerations 

 and retardations take place, would be symmetrical on 

 each side of the disturbing force. For it is clear, that 

 if the motion be accelerated through a certain space, and 

 then 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 un- 

 equably in its orbit, it is in some positions more directly, 

 and for a longer time, under the influence of the dis- 

 turbing 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 compen- 

 sate 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 centers 

 of the sun and disturbed planet, has no effect on the 

 areas, but is the cause of periodical changes of small 

 extent in the distance of the planet from the sun. It 

 has already been shown, that the force producing per- 

 fectly 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 dif- 

 ferent 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 



