92 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and to O'Q in the form 
Mi'+fMjjBo 
so that the locus of Q is given by 
and is therefore a conic through O 
(L x = 0 ; L 2 = 0), 
and through O' 
(Mj f = 0 ; M 2 = 0). 
(2> 
(3) 
Cor. 2. By assuming the converse theorem (proved later) Maclaurin 
deduces that if P, instead of lying on a straight line, moves on a conic 
through O and O', Q still generates a conic through 0 and O'. 
Dem. 
For a straight line L can then be found, and a point P x moving on 
o l 
it, so that 
Z.P 1 OP = a 
L ? 1 0T = f3' 
are constant angles. 
Hence P 2 OQ and P x O'Q are constant angles; and so, when P x traces 
out l v Q generates a conic through 0 and 0 . (There is, in fact, a 1 — 1 
correspondence between OP and O'P, and between OQ and O Q. • • etc*) 
§ 4. Prop. II 
determines the species and asymptotes of the conic. 
On 00' describe a segment of a circle OKO' containing an angle y so 
that a + /3 + y = a multiple of two right angles. Let it cut l in A and B. 
When P coincides with either A or B the angle at Q in POQO is zero, i.e. 
Q is at infinity on the curve, and OQ (or O'Q) is parallel to an asymptote. 
The angle AOB ( = AO'B) measures the angle between the asymptotes. 
The conic is a hyperbola, a parabola, or an ellipse, according as A and B 
are real and distinct, coincident, or imaginary. 
Cor. 4. The curve cannot be a circle when l is not the line at infinity. 
