SECT, in.] MOTION OF THE APSIDES. 19 



advance in space. When the central force is less than the 

 law of gravity requires, the curvature of the planet's path is 

 less than the curvature of the ellipse. So that the planet, on 

 leaving its perihelion, would pass through more than 180 

 before arriving at its aphelion, which causes the apsides to 

 recede in space (N. 66). Cases both of advance and recess 

 occur during a revolution of the two planets ; but those in 

 which the apsides advance preponderate. This, however, 

 is not the full amount of the motion of the apsides ; part 

 arises also from the tangential force (N. 63), which alter- 

 nately accelerates and retards the velocity of the disturbed 

 planet. An increase in the planet's tangential velocity 

 diminishes the curvature of its orbit, and is equivalent to a 

 decrease of central force. On the contrary, a decrease of the 

 tangential velocity, which increases the curvature of the 

 orbit, is equivalent to an increase of central force. These 

 fluctuations, owing to the tangential force, occasion an alter- 

 nate recess and advance of the apsides, after the manner 

 already explained (N. 66). An uncompensated portion of 

 the direct motion, arising from this cause, conspires with that 

 already impressed by the radial force, and in some cases 

 even nearly doubles the direct motion of these points. The 

 motion of the apsides may be represented, by supposing a 

 planet to move in an ellipse, while the ellipse itself is slowly 

 revolving about the sun in the same plane (N. 67). This 

 motion of the major axis, which is direct in all the orbits 

 except that of the planet Venus, is irregular, and so slow, 

 that it requires more than 109,830 years for the major 

 axis of the earth's orbit to accomplish a sidereal revolution 

 (N. 68), that is, to return to the same stars ; and 20,984 

 years to complete its tropical revolution (N. 69), or to re- 

 turn to the same equinox. The difference between these 

 two periods arises from a retrograde motion in the equinoc- 

 tial point, which meets the advancing axis before it has 

 completed its revolution with regard to the stars. The ma- 

 jor axis of Jupiter's orbit requires no less than 200,610 



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