39 
1892-93.] Prof. Anglin on Promrties of the Parabola, 
1. The axis, tangent at vertex, and parameter may be found in a 
manner exactly analogous to that for rectangular axes. 
(1.) The axis being ax-\- Py + X = and writing the equation to 
the curve 
{ax-\-py + Xf^2{g-\a)x-^^f-\p)y + c-\^=^Q, 
the lines 
ax + /Sy + A. = 0, and 2(Aa-^)x+2(X^-/)y + X^-c = 0 
will be at right angles, if 
a{Xa -g) + I3{XI3 -/) - {a(X^ -/) + /3(Xa - p)}cos w = 0, 
or if, 
(a^ + 2a/3 COS 0 ))X = ag + Pf- {af+ pg) COS w . 
Thus, denoting a^ + p^- 2aP cos co by the axis is 
{ax + Py)y^ -\-ag + Pf- (a/+ Pg) cos w = 0 ; 
and, since 
ha-g Xp-f ^ af-Pg ^ 
P — a COS Q) a — P COS (0 \ 
the tangent at vertex is 
2{af- pg){{P - a cos <j))x - {a- P COS 0 ))y} + {X^ - c)y2 = Q . 
To find the parameter, we write equation to curve in form 
{perp. on axisY =p{perp. on tang, at vertex ) , 
where p is the parameter ; that is, 
{ax + py-\- X)^sin2co _ {2(Xa - g)x + 2(X^ -/)?/ + X^ - cjsin w 
aH yS--^ - 2a/3 cos <0 V{ - gf + -ff -S(Xa- g){\j3 -/) cos a, } 
hence 
Py‘" = ^J{{ha-gY + {XP -ff - 2(Xa - g){Xp -/) cos a)}sin w . 
Substituting for Xa-g and Xp~f their values, and observing that 
{p~a cos (J)Y + {a- P cos O))'^ + 2{p - a COS co)(a - P COS co) COS co 
= y^sin^co , 
we get 
p/=2{af- Pg) sin“a./y ; 
and thus 
p = ^{af- Pg) sinVy"- 
(2.) Proceeding similarly to the second method in the rectaii- 
