EXAMPLES. 165 



21. A particle describes a circle with acceleration pr~ 5 towards a point on 

 the circumference. Prove that it will still describe the circle if acted on by a 

 repulsive force from the centre producing acceleration pa,- 5 provided it starts 

 from rest at a point where r=a (/A/2//)*, a being the radius of the circle. 



22. A particle describes an ellipse under two forces, functions of the 

 distance, one to each focus. If the law of force to one focus is /*r, prove that 

 to the other it must be 



23. An ellipse is described under the action of two forces, one to each 

 focus. Show that the force per unit of mass along the focal radius vector r is 



r) 4 dr ' 

 where 2a is the major axis and v the velocity. 



24. Two centres of force of equal strength, one attractive and the other 

 repulsive, are placed at two points S and H, each force varying inversely as 

 the square of the distance. Show that a particle placed anywhere in the 

 plane bisecting SH at right angles will oscillate in a semi-ellipse of which S 

 and H are foci. 



25. A body is placed at rest in a plane through two fixed centres of force, 

 each varying inversely as the square of the distance, at a point where the 

 forces are equal. Prove that it will oscillate in an arc of an hyperbola if both 

 forces attract, and in the arc of an ellipse if one force attracts and the other 

 repels. 



26. A particle describes a parabola under two forces, one constant and 

 parallel to the axis, and the other passing through the focus ; prove that the 

 latter force varies inversely as the square of the focal distance. Prove also 

 that, if the force through the focus is repulsive, and numerically equal, at the 

 vertex, to the constant force, the particle will come to rest at the vertex ; and 

 find the time occupied in describing any arc of the curve. 



27. A particle describes a circle under the action of forces, tending to the 

 extremities of a fixed chord, which are to each other at any point inversely as 

 the distances r, / from the point to the ends of the chord. Determine the 

 forces, and prove that the product of the component velocities along r and / 

 varies inversely as the length of the perpendicular from the position of the 

 particle to the chord ; also show that the time from one end of the chord to 

 the other is 



a (n a) cos a -f sin a 



V cos 2 a 



where V is the velocity of the particle when moving parallel to the chord, 

 a the radius of the circle, and a the angle between r and /. 



28. A particle moves under the action of a repulsive force /* (u 2 - au 3 ) 

 from a fixed point, and a force n(l/c 2 -u/a) parallel to a fixed line, l/u being 

 the distance from the point. Show that, if it starts from rest at a point where 

 the forces are equal, it describes a parabola of which the point is the focus. 



