EXAMPLES. 81 



106. An ellipse is described as a central orbit about a point on the 

 major axis ; prove that the acceleration at P varies as PL 3 /OP 2 where L is 

 the point of intersection of OP and the diameter conjugate to that passing 

 through P. 



107. When a particle describes an ellipse as a central orbit about any 

 point in its plane the sum of the reciprocals of the velocities at the extremities 

 of any diameter is independent of the position of the point and varies as the 

 periodic time. 



108. Any conic whose centre is C is described as a central orbit about 

 any point R. Prove that the acceleration at P is proportional to CG 3 /EP 2 , 

 CG being drawn parallel to RP to meet the tangent at P in G. 



109. A particle P describes a parabola as a central orbit about a point 

 on the axis ; prove that the acceleration is ^{l/OP+l/Op}~ 3 . OP~\ p being 

 the other point of intersection of OP with the curve ; prove also that the time 

 of passing from one extremity of the ordinate through to the other is 



110. A particle describes a parabola, latus rectum 4a, with an acceleration 

 tending to a point on the axis distant c from the vertex. Prove that the time 

 of moving from the vertex to a point distant y from the axis is proportional 

 to 



111. Prove that any conic can be described by a particle with an 

 acceleration always at right angles to the transverse axis and varying 

 inversely as the cube of the distance from it. 



If a particle is describing an ellipse in this manner, and at one end of one 

 of the equiconjugate diameters the acceleration is suddenly changed in sense 

 without being altered in magnitude, prove that the particle will proceed to 

 describe an hyperbola having the axes of the ellipse as asymptotes. 



112. A particle describes an ellipse with acceleration parallel to a diameter. 

 Show that the acceleration must vary inversely as the cube of the ordinate of 

 the conjugate diameter. 



113. A particle moves with an acceleration fj,y~ 3 towards the axis x, 

 starting from the point (0, k] with velocities 7, V parallel to the axes of #, y. 

 Prove that it will not strike the axis x unless p&amp;gt; V 2 k 2 , and that, in this case, 

 it strikes it at a distance Uk 2 l(^n - Vk] from the origin. 



114. A particle describes a cycloid with an acceleration always perpen 

 dicular to the base, prove that its magnitude is proportional to the inverse 

 fourth power of the radius of curvature at each point of the curve. 



115. Show that a particle can describe an equiangular spiral of angle a 

 and pole S with an acceleration p/SP* 1 whose direction makes a constant 

 angle /3 with the tangent to the spiral provided 



tan a = %(n - 1 ) tan 0. 

 L. 6 



