24 BARKER : 



in. The equation of the locus of the vertices follows 

 upon eliminating a between the expressions for abscissa and 

 ordinate of the vertex. These are, as above 



V^sinacosa V^sin'^a 



X = and y = 



g 2g 



Or, conveniently, since — ^ D 



2g 



X = D 2sinacosa y ^ D sin^a 



which can be put 



D ^ 



X =^ D Sin2a y= (i — COS2tt) 



2 



whence 





^ • 2 ^2 



-— = sin 2a := cOS'"2a 



D' /D\2 



and, adding 





X 



D'' ' / D \2 



2 



Which is the equation of an ellipse, centre at 



(°. !) 



D 

 semi-axes D, 



2 



IV. Now, since in the parabola the vertex is midway 

 between the focus and the directrix, the ordinate of the focus 

 is given by subtracting one-fourth of the latus rectum from 

 the vertex ordinate. That is, F, the ordinate of the focus, is 



