432 



PROFESSOR WALLACE ON ANALOGOUS PROPERTIES OF CO-ORDINATES OF 



V - I. 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, their equations, both included in this, of -^ ci/ = a\ may be easily 

 deduced ; and this other property, viz. 1/ semidiaineters he drawn to the extremi- 

 ties of any chord in an ellipse or hyperbola, the sector between them will be bisected 

 by the diameter which bisects that chord. 



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

 problem : 



Problem. 



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

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

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

 curve; and let 



PQ=^, P'Q'=y„ P"Q"=^„ ^"'Q"'^y,, 



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

 It is proposed to express the co-ordinates of each of the four points P, P', V", V' 

 by those of the other three. 



Fig. 2. 



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



2 



the semiconjugate CB=ft; and let c=±^^, the sign + applying in the ellipse, and 



the sign -in the hyperbola. 



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



PD P^D' 

 F^""PD 



