28 BARKER : PROJECTILE PARABOLAS. 



is the greatest height reached when a body is thrown vertically 

 upwards with the given speed. These matters can of course 

 be shown analytically with the greatest ease. 



It is to be noted that values of a in the third and fourth 

 quadrants merely repeat curves given by angles in the first 

 and second quadrants respectively ; a fact at once evident on 

 inspection of the equation. 



It will now be easy to remove the restriction of a given 

 vertical plane of projection, and consider the corresponding 

 loci if the paths lie in any vertical plane. Thus, the locus of 

 vertices becomes an ellipsoid of revolution, the locus of foci a 

 sphere, etc., the surfaces being simply those generated by rev- 

 olution of the former loci about the y-axis. 



To guide the eye in following the graph, the latus rectum, 

 and the part of the axis between focus and vertex, of each 

 parabola, have been shown ; also, the axes of the inclined 

 ellipses (loci of latus rectum extremities). In addition, the 

 Roman numerals have been used as in the text. 



