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



7. Interpret the formula v 2 fpp/r so as to show that the velocity at any 

 point P, when a curve is described as a central orbit about a point 0, is that 

 due to falling with constant acceleration, equal to that at P, through one 

 quarter of the chord of curvature in the direction PO. 



53. Elliptic motion about a focus. Let an ellipse of 

 semi-axes a, b be described as a central orbit about a focus S. 

 Let S be the second focus, e the eccentricity, I the semi-latus 

 rectum. 



Let P be any point on the ellipse ; let r and r be the radii 

 vectores drawn from 8 and S to P ; let p and p be the perpen- 



Y 



Fig. 34. 



diculars from S and S on the tangent at P ; let C be the centre 

 and CD the semi-diameter conjugate to CP. 



Then 

 p = CD 3 /ab, rr=CD* t pp =6 2 , r + r = 2a, b* = al. 



Also since Z SPY = Z S PY, we have 



2 = ; , and therefore each of these = , f = 7^ . 

 r r Jrr vD 



Now the acceleration, /, is given by 



tfrab (CD\* _ h^ a = ft_ 

 ~ r) ~^6 2 ~r 2 



\br 



Thus the acceleration varies inversely as the square of the 

 distance r, and, if we write /*/r 2 for it, we have h* = pi. 



