PROJECTILE PARABOLAS. 23 



Now, if in this equation \' . be constant, while a may 

 assume all real values, we have the equation common to the 

 projectiles in a given plane, for which the speed of projection 

 is the same, the difference between the several curves depend- 

 ing on the angle of projection only. 



11. It can now l)e shown that these parabolas have a 

 common directrix. Recalling that in any parabola the dis- 

 tance between the vertex and directrix is one-fourth of the 

 latus rectum, we proceed to determine the ordinate of the 

 vertex and the \-alue of the latus rectum. The former can 

 be found conveniently bj' determining the abscissa of the 

 vertex as a critical point : then, by substituting this value in 

 the equation of the curve, the corresponding ordinate will be 

 obtained. The procedure is 



2V''C0S"a 



g 2g 2g 



This is the ordinate of the vertex. Xow, from the equa- 

 tion of the parabola ' i > the latus rectum is 

 Therefore D, the directrix ordinate, is 



V'Sina V'COS"a V " 



D = — = — 



2g 2g 2 g 



Thus D is independent of u, and the cur\'es have a 



common directrix whose equation is v =^ — 



2s: 



