52 



MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



Hence xy yx - const., = h say. 



The left-hand member of this equation is the moment of 

 the velocity about the origin, and is therefore equal to the product 

 pv, where v is the velocity of the moving point in any position, 

 and p is the perpendicular from the origin on the tangent to the 

 path at this position. We therefore have 



pv = h. 



Now let s be the arc of the curve measured in the direction in 



which the curve is described from 

 some point B of the curve up to 

 the position P of the moving point 

 at time t, A the area described by 

 the radius vector while the moving 

 point describes the arc s. Then 

 p&s is ultimately twice the area 

 of the infinitesimal triangle de- 

 scribed by the radius vector in the 

 interval A, taken to describe the 

 small arc As, so that ps=2A. 

 Also s = v. 



Hence h, =ps, is twice the rate 

 at which area is described by the 

 radius vector, and, h being con- 

 stant, the radius vector describes 

 areas uniformly, i.e. it describes 

 equal areas in equal times. 



The quantity h is twice the area described in a unit of time. 



The path described by a point P which moves with an 

 acceleration in the line joining P to a point is called a central 

 orbit, and is said to be described about the point 0. 



51. Formula for the central acceleration. Let / be 



the magnitude of the central acceleration at P, supposed directed 

 towards 0, r the radius vector OP, p the radius of curvature of 

 the path at P. 



The resolved part of the acceleration parallel to the normal 



Fig. 33. 



