;8 KINETICS OF A PARTICLE. [147. 



hence, substituting for v its value from (9), Art. 113, we find, 

 for the differential equation of the orbit, 



As it is often difficult or impossible to perform the integra- 

 tion in finite form, it is of importance to determine the apses 

 and apsidal distances of the orbit. 



147. An apse is a point of the orbit at which the velocity is 

 at right angles to the radius vector drawn from the centre of 

 force ; the length of the radius vector of an apse is called the 

 apsidal distance, and its direction an apsidal line. 



The importance of the apsidal lines lies in the fact that they 

 are lines of symmetry of the orbit, while the apsidal distances 

 are maximum or minimum values of the radius vector. This 

 will be seen from the following considerations. 



By the above formula (34) the velocity is a function of the 

 radius vector alone ; and by (5), Art. in, since sini|r=/W, the 

 angle -fy between radius vector and velocity is also a function of 

 the radius vector alone. It follows that, if the velocity be 

 reversed in direction at any point of the orbit, the same orbit 

 will be described in the opposite sense ; and as at an apse the 

 velocity is perpendicular to the apsidal line, the two portions of 

 the orbit on opposite sides of an apsidal line must be symmet- 

 rical with respect to this line. 



148. The condition for an apse is therefore 



Substituting this value in the above equation (35), the apsidal 

 distances i/u can be found by solving the equation for u. The 

 value of du/dO should also change sign as the particle passes 

 through the apse. 





