PROJECTILES 



211 



En 



of Paths 



173. If we imagine all the parabolas drawn, which can be 

 described by projectiles fired from the point with a given 

 velocity v, we shall obtain a figure similar to fig. 122. The out- 

 side curve obviously separates the points which can be reached 



FIG. 122 



from those which cannot be reached. Thus this is the parabola of 

 which the equation is given in equation (63). A study of fig. 122 

 will now show that this curve is the envelope of the system of 

 parabolas which correspond to the different directions of firing. 



174. The envelope of the system of parabolas can be found more 

 directly by analytical methods. If we write m for tan a in equa- 

 tion (58), we obtain the equation of a parabola of the system in 



the form 



qx* 

 y = m x-j^(l + m 2 ), 



and the whole system is obtained by giving different values to m. 

 The condition for this equation to have equal roots in m is that 



~ 



or, in reduced form, 



= _ _ 



(64) 



If x, y satisfy this relation, two parabolas which only differ 

 infinitesimally pass through x, y, and therefore x, y is a point on 



