177-179] PLANETARY THEORY. 159 



Thus the elements of a planet's orbit are : 

 a the semi-axis major, 

 e the eccentricity, 

 i the inclination, 

 ft the longitude of the node, 

 -BT the longitude of the perihelion, 

 e the epoch, 



where e is the longitude of the planet at time t = 0. It is to 

 be noticed that fl is measured in the plane of the Earth's orbit, 

 while e and -sr are measured in the plane of the planet's orbit from 

 a definite line in the plane of the orbit. 



178. Disturbed Elliptic Motion. It would be beyond the 

 scope of this work to explain the methods by which the variations 

 of the elements are determined, but it appears to be not without 

 interest to examine the rates of variation of some of the elements 

 produced by small forces acting in given directions. We shall 

 consider quite generally the rate at which changes are produced in 

 the elements a, e, nr of an elliptic orbit by small tangential and 

 normal forces in the plane of the orbit. 



179. Tangential disturbing force. Suppose a particle P, 

 describing an elliptic orbit about a focus S, receives a small 

 tangential impulse increasing its velocity by Bv. Let R be the 

 distance of the particle from S at the instant, /u,/r 2 the acceleration 

 to S when the distance is r, a + Ba the semi-axis major of the 

 orbit immediately after the impulse. 



We have, by Example 1 of Article 54, 



/2 r 



v z 



Sa 

 giving - = approximately. 



Again, if h is the moment of the velocity about S before the 



