KINETICS OF A PARTICLE. 



[137. 



O" 



In the ellipse, the focal radii have a constant sum = 2<z, and lie on 

 the same side of the tangent, making equal angles with it. In the 



hyperbola, they have a con- 

 stant difference = 20, and lie 

 on opposite sides of the tan- 

 gent. 



Hence, determining the 

 point <9" (Fig. 20), which is 

 symmetrical to the centre of 

 force O with respect to the 

 initial velocity, and drawing 



O' 



Fig. 20. 



O 



the line P O", we have only to lay off on this line from P Q a length 

 P O' = (2a r ) ; then O' is the second focus, which for an elliptic 

 orbit must be taken with O on the same side of the tangent PT, and for 

 a hyperbolic orbit on the opposite side. 



137. For a parabola, since e= i, we find, from (19), 



(23) 



The axis of the parabola is readily found by remembering that the 

 perpendicular let fall from the focus on the tangent bisects the tangent 

 (i.e. the segment of the tangent between the 

 point of contact and the axis). Hence, if 

 OT (Fig. 21 ) be the perpendicular let fall 

 from the centre O on the velocity z> , it is 

 only necessary to make TT' = P T, and T' 

 will be a point of the axis. Moreover, the 

 perpendicular let fall from T on OT will 

 meet the axis at the vertex A of the parabola, 

 so that OA = i /. 



138. The relation (21), which must evi- 



Fig. 21 



dently hold at any point of the orbit, can be written in the form 



2a 



(24) 



the upper sign relating to the ellipse, the lower to the hyperbola, while 

 for the parabola, the second term in the parenthesis vanishes (since 

 a = oo). 



