PROJECTILE PARABOLAS. 27 



Summarizing the results, we have 



I. Equation of family : 

 y ^ X tana 



2 V'^ cos^a 



II. Equation of directrix : 



y= — = D 



2 or 



III. Equation of locus of vertices: 



k2 





X ' ^ 2 



D^ ^ /D\2 ~" 



IV. Equation of locus of foci : 

 X-' + y^' = D = 



V. Equation of loci of latus rectum extremities: 



(X ± y ^ Dj^ + y^ = D'^ 



VI. Equation of envelope: 



x^ = - 4D (y - D) 



Referring now to the accompanying graph, it will be seen 

 that those members of the family characterized by D = 64 

 (that is, V = 64, if g = 32) for which a has the values o and 

 multiples of 15° have been figured. Various matters of 

 detail might be mentioned, as, for instance, that when a = 45° 

 the horizontal range is maximum ; also, that when a = 90° 

 (vertical projection) the parabola is reduced to a pair of coin- 

 cident straight lines, with focus and vertex at a common 

 point (o, D) on the directrix ; and thus the directrix ordinate 



