cot 



1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry. 339 



which is equivalent to 



sin' (/ > _ X» 



e'-l~ e'- X'' 

 Again, since the equations of all the four lines PA, PA', PS, PS', are 

 known, the angle between any two of them may be found, viz.y 



*-APA'=^.^^ (155) 



tan SPS-?^^.^_^i— - (156) 



a 1 — e' ( 1 + sm* <p) 



^^^=-^-i^:!*-|-it;^^^^) ^'''^ 



cotS'PA'= |.i±i jcot^--^ sin^l (158) 



l — e I z l-\-e ) 



We have shewn above that 



1 11 1___2 2_ 



p q r s h sin <l> ordinate of P' 

 whence the ordinate of P is a harmonic mean as well between r and 5 

 as between p and q. Again, it is evident that the theorem holds, even if 

 S, S' are not the foci, but any two points on the major axis equidistant 

 from the centre ; for, in that case, instead of putting OS = ae, we have 

 to put OS = ak, where h is a certain constant ; thus, we have the theorem 

 that the ordinate of any point P is a harmonic mean between the in- 

 tercepts made on the minor axis by the two lines joining P to two points 

 on the major axis equidistant from the centre. 

 In order to see whether the formulae 



111.12 



p q r s y 



pq = F, 



hold for any curve other than the conic, let us take the inverse question 

 in a more general form, viz., take as the origin of coordinates, and 

 BOA, OQP any two lines through it. A, B being fixed points ; then, 

 if BQ and AP intersect in E-, required the locus of R, when 



11112 



p q r s y 

 pq = k\ 

 where 0P = j3, 0Q = 5. Let a, /? be the coordinates of R; OA = a, 

 OB = - 6 ; then 



. x-a a-a 



