220 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



velocity %*J6 of that requisite for circular motion. Prove that the polar 

 equation of the projection of the path on a plane perpendicular to the axis is 



that the time from one apse to the next is n (2c cosec a)*/^//^ and that the 

 pressure is inversely proportional to the cube of the distance from the vertex. 



127. A particle is projected horizontally along the smooth inner surface 

 of a right circular cone, whose axis is vertical and vertex downwards, the 

 initial velocity being JfighKn^+n}} where h is the initial height above the 

 vertex. Prove that the lowest point of its path is at a height hjn above the 

 vertex. 



128. A right circular cone of vertical angle 2a is placed with one 

 generator vertical and vertex upwards. From a point on the generator of 

 least slope a particle is projected horizontally and at right angles to the 

 generator with velocity v. Prove that it will just skim the surface of the cone 

 without pressure if the distance of the point of projection from the vertex is 



129. A particle is projected horizontally from a fixed point on the interior 

 surface of a smooth paraboloid of revolution whose axis is vertical and vertex 

 downwards. Prove that when it is again moving horizontally its velocity is 

 independent of the velocity of projection. 



130. A particle is projected along a small circle on the surface of a smooth 

 uniform oblate spheroid of semi-axes a and c. Prove that, if it describes the 

 circle with angular velocity , then 



where A and C have their usual meanings in the theory of the attraction of 

 the spheroid. 



131. Prove that, when a body of mass m moves under gravity on a smooth 

 sphere of unit radius, the osculating plane of the path makes an angle 

 tan&quot; 1 (gh/mv 3 } with the normal, h being the moment of momentum about the 

 vertical diameter and v the velocity, the osculating plane always cutting the 

 vertical diameter below the centre. 



132. A particle moves on the inner surface of a smooth bowl in the form 

 of a paraboloid of latus rectum 4a with axis vertical and vertex downwards, 

 being projected along the surface in the horizontal plane through the focus 

 with velocity J(2nag). Prove that the initial radius of curvature of the path 



133. A particle moves inside a smooth paraboloid of revolution whose 

 axis is vertical and vertex downwards, being projected from the level of the focus 

 with velocity due to a height A in a direction making an angle TT with the 

 meridian. Prove that, if I is the latus rectum, the initial radius of curvature 

 of the path is 



v i 



cos tan&quot; 1 r. 

 5 5/4 



