16 MOTION OF THE APSIDES. SECT. III. 



ducing 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 dis- 

 turbed 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 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 extremities 

 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 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 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 curva- 

 ture of the orbit, is equivalent to an increase of central force. 

 These fluctuations, owing to the tangential force, occasion an 

 alternate 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, 



