62, 63] 



APSIDAL DISTANCES. 



65 



o 



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, 

 f 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 A T or A T , 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 / 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 AO are bisected by AO. 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 B 

 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 gt 40. 



