58 MOTION IN TERMS OF ACCELERATION. 



The orbit in question 



is an ellipse if JPQ &amp;lt; SP i.e. if V 2 &amp;lt; 



oP 



it is a parabola if \PQ = SP i.e. if F 2 = |^ 



& 



it is a hyperbola if JPQ &amp;gt; P i.e. if F 2 &amp;gt; ^ 



[CHAP. IV. 



56. Motion in a straight line with an acceleration to 



a point in the line varying 

 inversely as the square of 

 the distance. Let a point 

 N move in a straight line OA, 

 starting from A, so that, when 

 ff = X} x = p/x~. 



On OA as diameter de 

 scribe a circle, and let C be 

 its centre, and a its radius; 

 draw NP at right angles to 

 OA, and consider the mo 

 tion of the point P on the 

 circle. 



We shall show that, if P 

 Fig. 37. describes the circle with an 



acceleration towards 0, the 

 point N will have the acceleration named. 



By Example 4 of p. 53 we have , 



acceleration of P = &quot;T &quot; , where li is twice the rate at which OP 

 describes areas about 0. 



To resolve in direction AO multiply by ON /OP and observe 

 that ON: OP = OP: OA. Thus 



acceleration of N = 



Sh*a*ON 8h*a*ON 



(2a.ON) s a ON* 



Hence if we take the point N to start at a distance 2a from 

 and put h 2 = pa, then when ON x,N will have an acceleration 



