1892-93.] Prof. Anglin on Properties of the Parabola. 
35 
On Properties of the Parabola. By Professor Anglin. 
(Read February 6, 1893.) 
In the following pages we propose to obtain expressions for the 
several parts of a parabola (axis, focus, vertex, directrix, tangent at 
vertex, and parameter), whose equation is expressed in the most 
general form in Cartesian co-ordinates. 
Some of the results obtained have already appeared, in which 
cases we shall supply alternative proofs. We shall also endeavour 
to complete the investigation of the whole subject. 
I. When the curve is referred to rectangular axes. 
1. To find the axis, tangent at vertex, svadi parameter. These are 
obtained in Smith’s Conic Sections, § 172, in a very neat manner; 
but we may replace this method with a little variation at the 
beginning. 
(1.) Taking the usual equation 
{ax 4- Pyf -b 2gx 4- 2/y 4- c = 0 , 
we see that ax + I3g = 0 is a diameter since it meets the curve in 
only one point, namely, where it meets the line-2^^r 4- 2/y 4-c = 0 ; 
and this is the tangent at its extremity since it meets the curve in 
the two coincident points given by its equation and {ax 4- /3gY = 0. 
Thus the axis is ax + ^g + \ — 0, where A is a constant ; and since 
the equation to the curve is 
{diameteif 4- tangent at extremity = 0 , 
we obtain A by the condition that these lines are at right angles, 
getting 
^ = (a? + y8/)/(a" + /»")• 
Also, since 
Xa-g ^\p-f _af- Pg 
P -a a2 + ’ 
the tangent at vertex is 
2(a/- !Sg){px - ay) 4- {o? 4- /82)(A2 - c) = 0 ; 
and the parameter is 
