164* MOTIONS OF FREE PARTICLES. [CHAP. IX. 



14. A particle is describing a circular orbit of radius a under a force to 

 the centre producing an acceleration f(r) at distance r, and a small increment 

 of velocity Aw is given to it in the direction of motion. Prove that the 

 apsidal distances of the disturbed orbit are 



/(a)+a/(a) 



Prove also that, if the increment of velocity imparted to the particle is 

 directed radially, the apsidal distances are approximately 



ija 

 k * 



15. A particle moves under a central force ^(l + 8cos2^)/r 2 being 

 projected from an apse on the initial line at distance c with velocity VG*/ C ) 5 

 show that the next apsidal distance is c/(l+3). 



16. A particle moves under a central force proportional to u 2 (cu + cos 6)~&quot; 

 towards the centre. Show that the orbit is one of the conies given by the 

 equation (cu + cos #) 2 = a + b cos 2 (6 + a). 



17. A particle moves in a plane under a radial force P and a transverse- 



force T, where 



P=- ^3(3 + 5 cos 20), r=^w 3 sin20; 



prove that a first integral of the differential equation of the path can be 

 expressed in the form 



at) 

 where h Q 2 and C are constants. 



18. A particle moves under the action of a central force P and a transverse 

 disturbing force - /(*). Prove that 



d?u _ P-f(t) 

 ~ 



where F(t) = tf(t)dt. 



19. Prove that in a plane field of force of which the potential referred to 

 polar coordinates is 



a particle, if projected in the proper direction with the velocity from infinity, 

 will describe a curve of the form 



(r - a sin 6)(r b sin 6) = ab, 



i jrovided 



20. A particle of mass m describes a circle (centre C) in period T under 

 the action of a force to a fixed point S. Prove that the force can be resolved 

 into two directed to inverse points 0, in CS and equal respectively to 



co 2 CP* , lew co* 



* 



