432 



PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



v' - 1 . The result of my investigation is given in what follows : the only geome- 

 trical properties of the curves supposed known are their definitions by their foci ; 

 from which, then equations, both included in tliis, x^ + c// — d, may he easUy 

 deduced ; and this other property, viz. If semidkuneters be drawn to the extremi- 

 ties of any chord in an ellipse or hyperhola, the sector between tJiem mill be bisected 

 by tlie diameter which bisects that chord. 



3. The formulse sought are to be deduced from the resolution of the following 

 problem : 



Problem. 



Let C be the centre of an ellipse {fig. 1), or hyperbola {fig. 2) ; and let kk' be 

 any diameter in the ellipse, but a transverse diameter in the hyperbola ; 

 and BB' its conjugate ; also let PP'", P'P" be two parallel chords in either 

 curve; and let 



CQ"=a:„ 



CQ" 

 P"'Q" 



= ^3' 



CQ=z, CQ'=«„ 



be the co-ordinates of their extremities P, P', P", P'". 

 It is pro])osed to express the co-ordinates of each of the four points P, P', P", P"' 

 by those of the other three. 



Vie. 2. 



Draw P"'D, P"D' pai-allel to the diameter AC. Put the semitransverse CA=a, 



the semiconjugate CB=ft-, and let c=±|-],the sign + applying in the ellipse, and 



the sign -in the hjrperbola. 



From the similar triangles PDP"', P'D'P", we have 



PD _PD' 

 F^~PD 



