MOTION ABOUT A CENTER OF FORCE 275 



sweeping out an area in the plane of the orbit. The area described 

 in time dt is the small triangle OPP 1 . We now have 



area described in time dt 

 = area OPP' 



= \OQ vdt 



= l- dt X moment of velocity about 0. 



Thus the area described per unit time is half the moment of the 

 velocity about 0, and this by the theorem of the last section is a 

 constant. Thus we have the theorem : 



Equal areas are described in equal times. 



Differential Equation of Orbit 



221. The theorem just proved, in combination with the theo- 

 rem of the . conservation of energy, enables us to determine the 

 equation of the orbit in which a parti- 

 cle will move. This equation is most 

 conveniently expressed in polar coordi- 

 nates, the center of force being taken 

 as origin. 



If r, are the polar coordinates of the 

 particle, the velocity may be regarded FlG - 137 



7 7/1 



as compounded of a velocity along OP, and a velocity r 

 at right angles to OP. 



The velocity is therefore given by 



The moment of the velocity about is equal to the moment of 

 the second component, for the moment of the first component 



vanishes. Thus the moment of the velocity about is r X r > 

 and since this has a constant value, say h, we have 



r*=h. (109) 



dt 



