16 MOTION OF THE APSIDES. SKCT. 111. 



requisite for perfectly elliptical motion. When greater, 

 the curvature of the disturbed planet's path on leaving 

 its perihelion (N. 64), or point nearest the sun, is 

 greater than it would be in the ellipse, which brings the 

 planet to its aphelion (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 ex- 

 tremities of the major axis, 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 cur- 

 vature 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 re- 

 cede in space (N. 66). Cases both of advance and re- 

 cess 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 alternately 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 alternate re- 

 cess 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 repre- 

 sented, 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 return 

 to the same equinox. The difference between these 



