276 MOTION UNDER A VARIABLE FORCE 



If m is the mass of the particle, and if /(r) is the attraction per 

 unit mass when at a distance r from 0, we find that the potential 

 energy of the particle is ~ r 



m I f(r)dr. 



*Jao 



The kinetic energy is |- mv 2 , or 



[/dr\ 2 2 /deVl 



Ht) +r U)l ] 



Expressing that the total energy is constant, we have 



(SMSy+'Jpw*-* <"> 



where E is a constant. 



Equations (109) and (110) lead to the differential equation of 

 the orbit. We have, since r and 6 are both functions of t t 



dr_dr<M 

 di~d0di' 



so that equation (110) may be expressed in the form 



7/1 



and on eliminating from this and equation (109), we have 



Cut 



the differential equation of the orbit. 



LAW OF INVERSE SQUARE 



222. Let us now suppose that the attraction follows the law of 

 the inverse square of the distance, so that 



where JJL is a constant. Then 



()& = -, (li: 



