24I-] PLANE MOTION. ! 3I 



240. To find the equation of the path, or orbit, we write the 

 equations (68) in the form 



and eliminate r 2 by means of (51): 



dt* h dt dP h dt 

 These equations can be integrated separately 



where v ly ^ 2 are the components of the initial velocity. 



Multiplying by 7, x and subtracting, we find, owing to (50), 



(70 



241. The geometrical meaning of this equation is that the 

 radius vector r= V-r 2 +/ 2 drawn from the fixed point O to the 

 moving point P is proportional to the distance of P from 

 the fixed straight line 



(72) 



It represents, therefore, a conic section having O for a focus 

 and the line (72) for the corresponding directrix. 



The character of the conic depends on the absolute value of 

 the ratio of the radius vector to the distance from the directrix ; 

 according as this ratio 



the conic will be an ellipse, a parabola, or a hyperbola. The 

 criterion can be simplified. Multiplying by p/h and squaring, 

 we have 



