66 MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



to either OA or OB, these are called the apsidal distances, and 

 the angle between consecutive apses in the order in which the 

 moving point passes through them is always equal to AOB, this 

 is called the apsidal angle. 



The theory just explained is usually stated in the form : 

 There are two apsidal distances and one apsidal angle. 



It is clear that the radius vector is a periodic function of 

 the vectorial angle with period twice the apsidal angle. 



64. Examples. 



1. If the apsidal distances are equal the orbit is a circle described about 

 its centre. 



2. Write down the lengths of the apsidal distances and the apsidal angle 

 for (1) elliptic motion about the centre, (2) elliptic motion about the focus, 

 (3) all the orbits that can be described with a central acceleration varying 

 inversely as the cube of the distance. 



3. Explain the following paradox : Four real normals can be drawn to 

 an ellipse from a point within its evolute, and in Example 6 on p. 53 we 

 found the central acceleration to any point requisite for the description of an 

 ellipse ; there are apparently in this case four apsidal distances and four 

 apsidal angles. 



65. Apsidal angle in nearly circular orbit. Suppose 

 the central acceleration is f(r) at distance r, then a circle of 

 radius c described about its centre is a possible orbit with %h for 

 rate of describing area provided 



; -/&amp;lt;* 



or h* = c 3 /(c). 



Let us suppose the point is at any instant near to the circle, 

 and that it is describing an orbit about the origin with this h. 



The equation of its path is 



ffiu f(r) 



dfr+ &quot;W 



At the instant in question u is nearly equal to - ; if it was 







precisely - , and if the point was moving at right angles to the 

 c 



radius vector, the point would describe the circle of radius c. We 



