of Halley and Fregier. 
157 
Draw the chord OIR across and at right angles to PQ, and let 
f, g be the focal chords parallel to OR, PQ. 
Then 0 1 2 / 01 .IR = PI.IQ/OIJR = gif 
Therefore 01. f— IR.g = OR/(l/f+ 1 /g), 
or 01 ./varies as OR, the chords /and g being at right angles. 
But OR varies as the projection ofi upon the normal at O 1 . 
Therefore, if PQ meets the normal in n, then 01 varies as 
01 I On, and On is constant and n a fixed point. 
2. Another proof is given as a problem in The Ancient and 
Modern Geometry of Conics, page 122 (1881), thus, 
“279. If PQ be a chord of a conic which subtends a right 
angle at a given point 0 on the curve, and MN be the projection 
of the chord upon the tangent at 0, shew that 
PM QN 
~OP 2 ~ OQ 2 = a constant > 
and that PQ passes through a fixed point on the normal at 0.” 
The result follows from the lemma that, if OPO'Q be a rect- 
angle (or parallelogram), and if a line OABC be drawn across 
PQ, OP, O'Q, then 
1_ _1 
OA ~ OB ~ 00 ' 
In the problem, this sum or difference being constant when 
OABC is the normal at 0, it follows that PQ passes through 
a fixed point A on the normal. 
3. Proofs from the Circle. 
a. If 0 be a fixed point on a conic and PQ any chord 
through a fixed point /, then OP, OQ are conjugate lines in an 
involution, and conversely. 
This includes Fregier’s theorem as a special case. 
It may be proved by projecting the conic into a circle with 
I as centre. 
1 Take V on the normal chord ON, draw RVO' to the curve, and let VR, VN cut 
any parallel to 00' in r, n. Lastly let O' coincide with 0. Then 
VR . Vr/f =VN.Vn/h, 
if /, h be the focal chords parallel to VR, VN, and in the limit 
OR/ f cos NOR = ON/h. 
VOL. XI. PT. II. 
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