50-52] CENTRAL ORBIT. 53 



But this resolved part of the acceleration is . 



Hence V - =/ 1 . 



From this equation and the equation vp h we may eliminate 

 v, and obtain the equation 



h 



j f p 



dr 

 Since p = r-j- , we may also write this equation 



When the curve is given we can hence deduce the acceleration 

 to a given point in its plane required for the description of the 

 curve. When /is given in terms of r we can find by integration 

 the (p, r) equation of the curve. 



52. Examples. 



1. Show that, when the orbit is an ellipse described about the centre, the 

 acceleration is proportional to the radius vector. 



2. In the same case show that the velocity at any point is proportional 

 to the length of the diameter conjugate to the diameter through the point. 



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

 with an acceleration to a point C proportional to the distance. Prove that all 

 the elliptic trajectories described have the same director circle. 



Let the tangent at P to one of the trajectories meet the director circle in 

 T, and let Q be the point of contact of the other tangent to this trajectory 

 drawn from T. Prove that the trajectory in question touches at Q an ellipse 

 having C as centre, P as one focus, and CT as the length of the semi-axis 

 major. 



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

 with the given velocity and moving about C with the given central acceleration.] 



4. Show that the central acceleration when a circle is described as a 

 central orbit about a point on the circumference is SA^/r 5 , a being the radius 

 of the circle. 



5. Show that the central acceleration when an equiangular spiral is 

 described as a central orbit about its pole is proportional to r~ 3 . 



6. Show that for an ellipse described as a central orbit about any point 

 in its plane the central acceleration at any point P is proportional to r/q 3 

 where r is the radius vector OP, and q is the perpendicular from P on the 

 polar of 0. 



