166 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



29. If a curve is described under a force P tending to the origin and 

 a, normal force N, prove that 



9 d /, T dr\ , d n ^dr\ n 

 p 2 -j- (Nr -=- ) + -=-( Pp* -j- ) = 0. 

 ^ dr \ dp) dr\ ^ dp) 



30. A particle is projected from an apse of Bernouilli s Lemniscate along 

 the tangent with velocity &amp;lt;Jp/a and moves under the action of forces 



(3r/-r 2 ) 3 



to the nearer and further poles respectively, r being the distance from the 

 nearer pole, and / from the further pole. Show that it describes the lemniscate. 



31. A particle P moves under the action of two fixed centres of force 

 &amp;gt;S 15 S 2 producing accelerations /V r i 2 an d /V r 2 2 towards S 1 and S 2 , where r lt r 2 

 are the distances ]P, S 2 P. Prove that if the motion does not take place in a 

 fixed plane there is an integral equation of the form 



( r l 2 0l) ( r 2 2 02) + ^ Cot 01 C t 02 = C (/*! COS 01 + f*2 COS 02) + COnst., 



where 15 6 2 are the angles S^P and ^/S gP, c is the distance S^^ and h is 

 the moment of the velocity about the line of centres. 



32. A thin spherical shell of small radius, moving without rotation, 

 describes a circle of radius R with velocity V about a gravitating centre of 

 force 0, and when its centre is at a point A bursts with an explosion which 

 generates velocity v in each fragment directly outwards from the centre. 

 Prove that the fragments all pass through the line A within a length 



and that if v is small the stream of fragments will form a complete ring after 

 a time approximately equal to ^rrR/v. 



33. Two particles are under the action of forces tending to a fixed point 

 and varying as the distance from that point, the force being the same at the 

 same distance in each case ; the particles also attract each other with a 

 different force varying as the distance between them ; prove that the orbit 

 of either particle relative to the other is an ellipse and the periodic time is 

 27r/V(/* + V) V&amp;gt; and A* denoting the forces on unit mass respectively at unit 

 distance. 



34. A series of particles which attract each other with forces varying 

 directly as the masses and distance are under the attraction of a fixed centre 

 of force which also varies directly as the distance; prove that, if they are 

 projected in parallel directions from points lying on a radius vector passing 

 through the centre of force with velocities inversely proportional to their 

 distances from the centre of force, they will at any subsequent time lie on a 

 hyperbola. 



