1892 - 93 .] Prof. Anglin on Properties of the Parabola. 
45 
The parameter may be found conveniently in two ways. 
(1.) The perpendicular from the focus on the directrix 
= abfldb + ipfi + 6^ - cos co}/c^ = ; 
and thus the parameter = 4a^5%in2(o/c^. 
we have 
(2.) Since 
QV2 = 4SP.PV, 
jp.PV = QV%in 26 >, 
whereat? is the parameter, and 6 = ^^ON(f. 
But 
sin o) = 2 A OQQ' = 20V. QY sin 0 ; 
a%^s>m^oi = ip . OY^. PY ; 
p) = ia%\m^(3ijc^. 
Two methods for finding the tangent at the vertex may also be 
shown. 
(1.) Its equation is 
x{a + b cos w) + y{b + a cos w) - /a = 0 • 
The perpendicular on this from the focus being equal to one-fourth 
of the parameter, we have 
(2.) By expressing the condition that the line shall touch the 
curve. 
The equation to the tangent at {x\ y') being 
{ 2a^6^ + ab(a^ + 6^) cos w - /ac^} jc^ = a^^^in^w/c^ ; 
fxc^jab = {a? + b^) cos w -t- ab(l -f cos^w) 
= (a + b cos o))(b -f a cos w) ; 
and thus the equation is 
xjib -{■ a cos oi) + y ! {a -{• b cos to) = able‘s. 
x/^ax +ylJbif = l, 
will be identical with the above equation, if 
Thus 
1 1 1 
1 
= c^/ab{a -f b cos w)(b -H a cos w) . 
