^^•.;^l'i%W fif^l'ftWipi^n' a minoi* directrix two taHgerits 

 are clrawn,tQ^||)j^ qjMryeiWS^IJng tlie parallel axe in the points.. 



/./.', U)e line /£ii?5Mf(}^tfifeme O Ga constant ratio, or ;^ 



2 b 



=^fjis the minor axis 2 Z. k^^ ^|t]^^^%^pp between:>'thit) 



minor focus and minor directrix. ' = -^ ' 



The very same relation holds when the point is assumed 

 on the common directrix; for if from any point G on the com-*i> 

 mon directrix two tangents are drawn meeting the parallell) 

 axis in the points w', the segment vv' has to the Hne G Fii 

 (drawn to the corresponding focus F) a constant ratio, ortt 



V V ^ CI 



Qp = jT) as the major axis 2 a is to h' the distance between 



the common directrix and corresponding focus. . . ( /v 



XVI. From any point P in a conic section, perpendiculars,, 

 are let fall on the minor directrices, the rectangle under these|'i! 

 perpendiculars is to the square of the semi-diameter pp^ing^ 



through the point P in a constant ratio, or . • • I'-i 



r»'r UT-/ 7-5 ,^, -qiiaiij anio(^ ibiriv/ 



i hn^ \ bIo'iIo & gsqol 



adi ^rasfiJ 109816 PO^ a e flfla arlj aieriw &Jnioq arii 



k being" the disfence between the centre' «fli|i^ ^hAft-i^"^ 

 rectrix. s edi jjniyed ^noiJ 



This theorem is analogous to that from which the ordinar^'-^^ 

 definition of a conic section is usually derived; for let p p' he ^ 

 the focal radii vectores of any point on the curve py, the per*-'' 

 pendiculars from this point on the common directrices, theri*^^ 



pv' 1 a^ F 



iifi ^^ciOii'j'j'6 oittr—f- = ~Y = — 2 — 2" ~ ~2"> '^ ^"^ <i'iiai| io <ion^ B 



'ir, 't jrhlDot'ih ?.f ^ e a a ^^ij^noa Tto sbioo 9di 



Af bemg the distance between the centre and major directrix^m 



a result perfectly analogous to the former. .jjg 



XVII. From any point P in a conic section, let two cords^^o 

 be drawn through the extremities B B' of the minor axe, the^j 

 conic section of which the point O is focus, and which touchesto 

 the cords P B, P B', will also touch the minor directrices of 

 the curve, and the ordinate to the transverse axe through the 

 point P will be the major directrix of this curve. //j vna 



XVIII. Let f be the angle between a pair of tangents -to ndj 

 conic section, pp' the distances of the point of intersection of 

 the tangents to the ordinary foci on the transverse axe, and;rn 

 w the perpendicular from the intersection of the tangents ou.Jj 

 the focal cord passing through one of the points of contact,,lj 

 we shall have the theorem , ,,jj[ 



y;nB rioidw aalj^a p p' sin f = SHu ffyohiooiib adj \(d zyra^ divd 



