42 
Proceedings of Boyal Society of Edinlurgli. [sess; 
Hence, observing that 
(/cos CO - + {g cos CO - fy + 2 (/cos (a ~ g){g cos <o -/) cos co 
= (/2 + - 2/^ cos (o) sin^to , 
we have 
(7^2 + 7:^ + 21ik cos co - 
= (/2 + ^2 _ 2fg cos co) sin^co - 2p{af-\- l>g)y sin^co ; 
and thus 
2(a/+ hg)yp =/2 _ 2fg cos co - Cy^. 
But the directrix is 
{b + a cos co)x + (a + & cos oi)y -yp = 0, 
and hence, replacing a and 6 by a and - /?, its equation is 
2(a/- Pg) { (/S - a COS o))x -(a- /3 COS co)y } 
+f^ + 9^~ ^fg cos CO - Cy2 = 0 . 
Also, substituting for yp its value in the equations for the focus, we 
have on reduction 
2{«f-ISg)yVi 
= -p) + + 2a/(/c0S ia-g), 
2{af-fy)f.k 
= -P) - cay® - ^Pg{g cos m -/) . 
(2.) The pair of tangents from {x',y') whose equation is 
^{x, y)cf>{x\ y’) = {{ax + Sy){ox' + Py') + g{x + x') +f{ij + y') + c}^ 
will be at right angles if 
(a2 + — 2aj3 cos co)^(a/, y') 
a{ax' + Sy)+gy + {S(ax' + /3y) +/}2 _ 2{a{ax' + Sf) + g} {S{ax' + /5/) +/} 
and thus the locus of their intersection is 
(a2 + /3^- 2aP cos co)(2ya? + 2/y + c) 
= 2{ax + Pij){ag + Sf- (a/+ /?y) cos co} +/2 + c/2 _ ^fg cos co , 
that is, 
2(a/- Sq){{P - a cos (Ji)x - (a - yScos co)y} 
+/2 + p2 _ 2/^ cos CO- Cy2 = 0 , 
which is the equation to the directrix. 
