so that 



340 A. Mukhopadliyay — Memoir on Plane Analytic Geometry. [No. 3, 



But, since OP =_p, OQ = q, we have 



— a a — a —a a-^h 



whence 



^~a'*=^> (^^^>>(^«0) 



1 12 g (a-h) 

 p q P ah ^ ' 



Hence the theorem that the ordinate is a harmonic mean between the 

 intercepts holds only when a=h, that is, when the line on which the in- 

 tercepts are made is equidistant from the fixed points ; thus, we have the 

 Theorem. — Given two points and a line equidistant from them ; 

 then, taking for axes the given line and the line joining the points, the 

 ordinate of any point is a harmonic mean between the intercepts which 

 the lines joining the point to the given points make on the given line. 



Again, ii pq =7c*, we must have, changing a, jS into x, y in (159) and 

 (160), 



J±_ Jl P 



which may be written 



ab^l?^\a-h)^'=^^ 

 shewing that the theorem holds only when P lies on a conic. In the 

 particular case when the given line is equidistant from the given points, 

 we have a = &, and the conic is 



x' y' , 



— ^ — = 1. 



a'^ k' 

 If the two lines are also at right angles, they are the axes of the conic, 

 and the given constant Jc is the semi-axis-minor. 



II. To determine the position of a point P on an ellipse such that, 

 if the normal at P intersects the minor axis produced in G, the polar of 

 G may subtend a right angle at P. 



Using the same diagram, let the ellipse be 



and P the required point where the eccentric angle is <^, so that the 

 coordinates of P are a cos <p, h sin ^. Then the normal at P is 



jax_ _ Jy_ ^ ^^ 



cos </> sin </> ' 

 so that G is 



(o, - £l^y 



