62, 63] 



APSIDAL DISTANCES. 



65 



Fig. 39. 



63. Apses. An apse is a point of a central orbit at which 

 the tangent is at right angles to the radius vector. 



There is a theory concerning the distribution of the apses 

 when the central acceleration is a single-valued function of the 

 distance, i.e. for the case where the acceleration depends only 

 on the distance and is always the same at the same distance. 



Let A be an apse on a central orbit described about a point 0, 

 / the central acceleration, supposed a 

 single- valued function of distance, TAT' 

 a line through A at right angles to AO. 

 Then a point starting from A at right 

 angles to AO with a certain velocity 

 would describe the orbit, suppose V is 

 this velocity. 



If a point starts from A with velocity 

 V in direction AT or AT', and has the 

 acceleration / towards 0, it describes the 

 orbit ; so that two points starting from A 

 in these two directions with the same velocity V and the same 

 acceleration f describe the same orbit. Since the two points have 

 the same acceleration at the same distance, the curves they de- 

 scribe are clearly equal and similar, and are symmetrically placed 

 with respect to the line AO. Thus the orbit is symmetrical 

 with respect to AO in such a way that chords drawn across 

 it at right angles to A are bisected by A 0. The parts of the 

 orbit on either side of AO are therefore optical images in the 

 line AO. 



Now let the point start from A in direction AT, and let B 

 be the next apse of the orbit that it passes 

 through, also let A' be the next apse after E 

 that it passes through. Then the parts A OB, 

 BOA of the orbit are optical images in the 

 line OB, and the angle AOB is equal to the 

 angle A OB, and the line AO is equal to 

 the line AO. In the same way the next 

 apse the point passes through will be at 

 a distance from equal to OB, and thus 

 all the apses are at distances from equal 

 L. 



Fi g . 40. 



