170 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



magnitude. Prove that at a point Q on the original ellipse the deviation of 

 the new path, measured along the normal at Q, is 



where H is the second focus, and CB the semi-axis minor. 



54. If, when a particle, describing an elliptic orbit about a focus, is at an 

 end of the axis minor, the centre of force is suddenly shifted a small distance 

 aa towards the particle, the eccentricity e of the orbit will be unaltered, but 

 the axis major will be turned through an angle a*J(e~ 2 - 1). 



55. If the particle (of the last Example) is at an end of the latus rectum, 

 and the centre of force is suddenly shifted a small distance aa towards the 

 centre, show that, to a first approximation, the eccentricity is diminished by a, 

 and the major axis is turned through an angle aa/l, where I is the semi-latus 

 rectum, while the periodic time is unaltered. Also prove that, to a second 

 approximation, the periodic time is increased by 3a% 3 /2 3 of its original 

 value. 



56. If when the particle (of the last Example) is at any point distant r 

 from the centre of force, the centre of force is suddenly shifted a small distance 

 k perpendicular to the plane of the orbit ; prove that the periodic time is 



increased in the ratio 1 +- -j : 1. Also, if the change takes place when the 



particle is at an end of the latus rectum, the angle between the apse line and 

 the radius vector is altered by 



(l-e 2 ) 2 2ea 2 * 



57. A particle is describing an ellipse under a force to a focus $, and, 

 when the particle is at P, the centre of force is suddenly moved a short 

 distance x parallel to the tangent at P. Prove that the axis major is turned 



through the angle ~-^ sin $ sin (9 - <), where G is the foot of the normal, 



6 the angle the normal makes with SG, and $ the angle the tangent makes 

 with SP. 



58. Denning the instantaneous orbit under a central force varying as the 

 distance as that which would be described if the resistance ceased to act, show 

 that, if at any point the resistance produces a retardation /, the rates of 

 variation of the principal semi-axes are given by the equations 



d 



a (a 2 \ 



where v is the velocity and r the radius vector at the instant. 



