66 KINETICS OF A PARTICLE. [126. 



(10) A particle of mass m is attracted by two centres O 1} O 2 of 

 equal mass m' and repelled by a third centre (9 3 , whose mass is 

 m" = 2 m'. If the forces are all directly proportional to the respective 

 distances, determine and construct the orbit. 



(n) When a particle moves in an ellipse under a force directed 

 towards the centre, find the time of moving from the end of the major 

 axis to a point whose polar angle is 0. 



126, Force Inversely Proportional to the Square of the Distance : 



f(r)={ji/r* (Newton's law). 



It has been shown in kinematics (Part I., Arts. 229-236) how 

 this law of acceleration can be deduced from Kepler's laws of 

 planetary motion. From Kepler's first law Newton concluded 

 that the acceleration of a planet (regarded as a point of mass 

 m) is constantly directed towards the sun ; from the second he 

 found that this acceleration is inversely proportional to the 

 square of the distance. The motion of a planet can therefore 

 be explained on the hypothesis of an attractive force, 



issuing from the sun. 



The value of /u, which represents the acceleration at unit 

 distance or the so-called intensity of the force, was found to be 

 (Part I., Art. 236; or below, Art. 139) 



and as, according to Kepler's third law, the quantity a B /T 2 has 

 the same value for all the planets, Newton inferred that the 

 intensity of the attracting force is the same for all planets ; in 

 other words, that it is one and the same central force that 

 keeps the different planets in their orbits. 



127. It was further shown by Newton and Halley that the 

 motions of the comets are due to the same attractive force. 

 The orbits of the comets are generally ellipses of great eccen- 



