48-50] UNIFORM DESCRIPTION OF AREAS. 51 



3. In simple harmonic motion given by x=-px starting from # = , 

 prove, by multiplying both sides of the equation by xdt and integrating, that 

 2 =p, (a 2 - # 2 ) for all positions of x. 



4. In the elliptic motion of Article 49 prove that the velocity v at 

 distance r from the centre is given by 



-y 2 +/xr 2 = const., 

 and evaluate the constant. 



5. In the hyperbolic motion of Example 2 prove that the velocity v 

 at distance r from the centre of the hyperbola is given by 



9*s8pfS-{- const 



and evaluate the constant. 



50. Central Acceleration. In the motion just described 

 (Article 48) the resultant acceleration is of magnitude pr and 

 is directed along the radius vector towards the origin, r being 

 the length of this radius vector. An acceleration always directed 

 towards or from a point which occupies a fixed position relative 

 to a frame is called a central acceleration. The point through 

 which the acceleration always passes is called the centre. We 

 shall now prove a general theorem* with reference to central 

 accelerations : 



The path of a point moving with a central acceleration is 

 in a plane through the centre, and the radius vector drawn from 

 the centre to the point describes equal areas in equal times. 



Let the line of the velocity at any instant be drawn, and a 

 plane drawn through the centre and this line. All the circum 

 stances of the motion being symmetrical with regard to this 

 plane, there is no more reason why the point should move out 

 of it on one side than on the other. The point therefore moves in 

 the plane. 



Let the plane of motion be the plane of (x, y\ and let the 

 centre towards or from which the acceleration is directed be 

 the origin. Then, since the acceleration is localised in a line 

 through the origin, its moment about the origin is zero, or we 

 have 



xy - yx = 0. 



The left-hand member of this equation is the differential 

 coefficient of xy yx., so that we have 



5 &amp;lt;4-y*)-a. 



* Due to Newton, Principia, Lib. I. Sect. n. Prop. 1. 



42 



