38 
Proceedings of Royal Society of Edinhurgli. [sess. 
and (a^ + !P)k = -/+ ■; 
hence 
2(a/- yS<^)(a^ + /S=)A = P{f -f) + Pc(a- + /3^) - 2a/jr , 
2(a/- /3^)(a^ + = a{f -f) - ac(a^ + /S^) + 2/3/^- . 
(2.) We may also find the directrix as the locus of the intersec- 
tion of tangents at right angles. 
If (ji(ic,y) = 0 represent the curve, the equation to the pair of 
tangents from {x',f) is 
c}i{x,y)4,{x',g') = {{ax + /3y){ax' -j- fiif) -h g{x -I- x') -]-f{y + y') + c}\ 
which will be at right angles if 
{a^ + P^)g>[x\y') = [a{ax' + ^ 2 j) + g}^ + {/3{ax' + Pf) +f}^, 
and thus the locus of their intersection is 
(a2 -{- p^-){2gx -h 2fg + c) = 2{ax-h py){ag -1- Pf) + 
that is, 
2{af - /Sg)(Rx - ay) +f + g^~ c{a^ + /32) = 0 , 
which is the equation to the directrix. 
(3.) The focus may also be found as the pole of the directrix. 
Making the polar of (x,y), namely, 
{ax -1- /Sg){ax' -I- fy') -f g{x -i- x') +/{y + 'i/) + c = 0 , 
identical with the equation to the directrix, we have 
a(ax +^!/') + cf P{ax + Py’) +/_ (a/- Pg){gx' +fy' + c) 
•2p -2a f + g'^-c{o? + P‘) ’ 
each of which ratios = {(Sg - af)j2{o? + . 
Thus x' and y are given by 
(a^ + ^)(o* + Py) + ag + Pf= 0 I ^ 
2(a^ + p^){gx + fy) +f + rf + c{a? + pr-) = Ol’ 
and hence 
2(a/- pg)(a^ + p^ = p{f -f) + pc{o? + - 2afg 
2(af- pg)(a^ + !P)y = a{f -f) - ac{a^ + [?) + 2pfg . 
II. When the curve is referred to oblique axes. 
The general equation to a parabola is of the same form in the 
oblique as in the rectangular system. 
