52-54] ELLIPTIC MOTION. 55 



The periodic time in which the ellipse is described is 



54. Examples. 



1. Prove that the velocity v at any point of the ellipse is given by the 

 equation 



/2 1\ 



v*=p, ( --- ). 

 \r aj 



2. Prove that if any conic is described as a central orbit about a focus 

 the acceleration is /u/r 2 towards the focus, and /z = A 2 /7. 



Prove also that when the conic is a parabola v 2 =2/*/r, and when it is a 

 hyperbola v 2 = p (2/r + 1 /a). 



3. Prove that in elliptic motion about a focus S the velocity at any point 

 P is perpendicular and proportional to the radius vector from the other focus 

 to the point TF, where SP produced meets a circle centre S and radius 2a. 



[From the formula in Example 1, this circle is called the "circle of no 

 velocity."] 



4. Prove that the velocity at P can be resolved into two constant 

 components, one at right angles to the radius vector SP, and the other at 

 right angles to the major axis. 



5. Points move from a position P with a velocity V in different directions 

 with an acceleration to a point S varying inversely as the square of the 

 distance. Prove that all the trajectories have equal transverse axes. 



Let the line from P to the second focus of one of the trajectories (supposed 

 elliptic) meet that trajectory in P'. Prove that this trajectory touches at P' 

 an ellipse with $ and P as foci and a definite major axis. 



[This ellipse is the envelope of the trajectories of points starting from P 

 with the given velocity and describing ellipses as central orbits about S.] 



6. To find the time of describing any arc of the ellipse described as a 

 central orbit about a focus. 



Draw the auxiliary circle AQA'. 



Fig. 35. 



