1892 - 93 .] Prof. Anglin on Properties of the Parabola. 37 
when we have at once 
2(a/- ^g)x = X2/3 - 2X/+ (Be 
and 
2(a/- (Bg)i/ = - 2Xg + ae . 
Substituting for \ its value, and reducing, we shall get 
- Pg){o? + . X 
= /?c(a2 + (B^^f + {ag + (Bf){alBg - (B‘^f- 2a?f) 
^B{af - (Bg){a^ + . tj 
= ac{o? + Pf + {ag + y8/)(a^/- a^g - 2Pg) . 
3. To find the focus and directrix. 
(1.) By the focus and directrix property. Let the focus be {hf) 
and the directrix a? cos ^ + ^sin ^ - ^ = 0, so that the equation to the 
curve is 
{x - hy + (y - hy = {xQ,o&0^-y sin 6 - 
which may be written 
{x &m6-y cos 6^ + 2x{p cos 6 -h) + 2y(p sin 6 -Jc) 
+ + = 0 . 
Making this identical with the given equation, we have 
sin^ 0 cos^ 6 p cos 6 -h psmO -h h^-\- Tc^ —p"^ 
~ p g “ f ~ c ' 
Thus h = pcoB0 - g I {a^ + P) 
and X’ sin ^ - //(a^ + P) , 
... -i/ = {f + g'^)j{o? + Pf - 2p(/sin ^ ^ cos e)j{o? + P ) , 
and 2jK»(/sin ^ ^ cos $){a^ + P) =f -\-g^- c{o? + P ) . 
Now 6j being the inclination of the axis to the initial line, is obtuse ; 
and thus we have 
^p{^f- ^g)sJ^Tp=f B-g'^- c(o? + p) . 
But the directrix is 
(Bx - ay +pPa^ + P = 0, 
and hence it is 
2{a/- (Bg){IBx - ay) +/ + / - c(a" -\-p) = 0. 
Also, 
-{a^-\-P)h = g + jBpPo? + p 
