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



82. When an hyperbola is described as a central orbit about a focus 

 prove that the rate at which areas are described about the centre is inversely 

 proportional to the distance from the focus. 



83. Prove that the central orbit described with acceleration /x/(distance) 2 

 by a particle projected with velocity Ffrom a point where the distance is R 

 is a rectangular hyperbola if the angle of projection is 



cosec- 1 F/F 2 /2 2 



84. A particle describes an ellipse about a focus, and at any point of the 

 orbit the acceleration begins to be directed to the centre and to vary as the 

 distance, its magnitude being unaltered. Prove that the new orbit is an 

 ellipse having double contact with the old orbit and entirely within it. 



85. A particle describing an ellipse about a focus has its velocity suddenly 

 doubled and turned through a right angle, and proceeds to describe a parabola, 

 the law of the acceleration being unaltered; the axis of the parabola is at 

 right angles to the axis of the ellipse. Prove that the eccentricity of the 

 ellipse is v /2. 



86. A particle describes an ellipse about a focus S starting from one end 

 of the major axis, and arrives at the end of the minor axis in time T. At the 

 end of this time the centre of force is transferred without altering its intensity 

 to the other focus If, and the particle moves for a second interval T under the 

 action of the force to H. Find the position of the particle, and show that if 

 the centre of force were transferred back to S after the second interval T the 

 particle would begin to describe an ellipse of eccentricity (3e &amp;lt;?*)/( 1 +e), where 

 e is the eccentricity of the first ellipse. 



87. A body is revolving in an ellipse of eccentricity J, under the action of 

 a force to the focus S t and when it is at a distance &amp;gt;S P from S equal to the 

 latus rectum, a blow is given to it perpendicular to SP such that its new 

 direction is perpendicular to the major axis. Show that the dimensions of 

 the orbit are unaltered, but the major axis is turned through an angle SPH t 

 where H is the second focus. 



88. A body is moving in a given hyperbola under the action of a force 

 tending to a focus S ; when it arrives at any point P, the force suddenly 

 becomes repulsive: find the position and magnitude of the axes of the new 

 orbit, and show that the difference of the squares of the eccentricities of the 

 new and old orbits is proportional to SP. 



89. Find, when possible, the point in an elliptic orbit about a focus at 

 which if the centre of force were transferred to the empty focus the orbit 

 would be a parabola. Prove that there is no such point unless the eccentricity 

 is greater than ^/S 2. 



90. A particle is describing a circle under a force to a point S on the 

 circumference. At a point P on the circle the force changes to the inverse 

 square, its magnitude being unaltered, and the particle proceeds to describe 

 an ellipse. On PS produced a point Q is taken so that SQ = $SP, QT is 

 drawn perpendicular to the tangent at P, and SQTR is a parallelogram. 

 Show that the middle point of TR is the centre of the ellipse. 



