1892-93.] Prof. Auglin on Properties of the Parabola. 
41 
Substituting for X its value in the expressions Xp - 2/ and Xa - 2g, 
and reducing, we shall get 
2(a/- 
= + \{aPg - 2ay+ {3aPf- cos m} 
2(a/- P(M-y 
= Ca-f 4- X[a(^f — c^g — ^[Pg + (Sa/Sg — Pf) cos w}. 
3. To find the focus and directrix. 
(1.) By the focus and directrix property. Taking, for the pre- 
sent, the equation to the parabola as 
(ax - biff 4- "Igx -h 2/y 4- c = 0 , 
let the focus be (li^P) and the directrix a; cos a + ?/ cos ^-p = 0, so 
that the equation to the curve is 
{x - Kf -f (^ - lif -1- 2(ir - h){ij - li) cos w = (;r cos a 4- ^ cos -pf^ 
which may be written 
{x sin a - y sin + ^x{p cos a - Z: cos w - h) 
4- 2y(p cos ^-li cos 00 - Z?) -h Id 4- 4- Phk cos w - = 0 . 
Making this identical with the given equation, we have 
sin^a _ sin^/5 cos a - Z; cos co - _p cos - 7? cos w - Z: 
~ff If g 7 
= Qd 4- Id 4- Piik cos o)-p‘^)lc . 
Kow, since the line ax-bg = 0 makes angles a and ^ with the 
axes, we have 
cot a = (6 4- a cos w)/a sin w , cot p = {a + b cos w)/& sin w ‘ 
from which 
sin a = a sin co/y , cos a = (&4- ctcos (jo)/y , 
sin = 5 sin (o/y , cos/5 = (a4- 5 cosw)/y , 
where y^ = 4- 7>^ 4- 2a6 cos w . 
Thus 
h 4- h cos o) =p{ b + a cos to)/y - y sin^w/y^ 
k 4- li cos (0 =p{a 4- b cos co)/y -/ siu^w/y^ ; 
from which 
yVi = byp 4-/ cos (0 - y 
y^k = ayp 4- y cos 0) -/ . 
