292] 
A THEOREM IN CONICS. 
483 
and these equations are of course satisfied by p : q : r = A : B : C, since the line 
through (£, 77, £) is a tangent. They are also satisfied by 
l m n 
p : q 1 r ~Aa : Bf3 : Cy* 
as is obvious by substitution, we have therefore 
l m n n 
Ta XJr B/3 y + Cy Z = 0, 
or more simply 
A'P B' v C'£ A 
2 ! a n-- 7 
for the equation of the other tangent through (a, (3, 7), and we have in like manner 
B v A 
7 I + ?J + 7 i = 0 ' 
for the equation of the other tangent through (a, /3', y); the last-mentioned two lines 
intersect in the point X, Y, Z, that is we have 
Xf : Y V : Z£ = 
RG_BC C[A_CA' m A'B AF 
f3y (3'y ‘ 70! ya ’ a/3' a'/3 ’ 
or attending to an above-mentioned equation, we have 
X% : Yrj : = aa! : /3/3' : 77', 
which is the property in question. In the particular case, where the points (a, /3, 7), 
(a', ¡3', y) are the foci, the theorem is an immediate consequence of the well-known 
proposition that the product of the perpendiculars let fall from the two foci upon 
any tangent of the conic is a constant. 
2, Stone Buildings, W.C., 17th March, 1860. 
61—2