312 A. Mukliopadhj'ay — Memoir on Plane Analytic Geometry. [No. 3, 



-A (a + h) 



Pi^^Pz' 



(ah-h^y 

 A2 



Pl^ Pi^- TZT—i:^ 



(ah-h^y 

 so that (pjS - p^^y = (pi2 +p^2)2 _ 4 pj2 p^2 



A2 j (a-by + U^ 



(ab-h^y 

 The equation (67), therefore, may be written 



xy = ^ (Difference of squares of semi-axes), 

 which is a well-known result. 



If the conic had been originally referred to axes inclined at an angle 

 Q), the equation of the hyperbola referred to the asymptotes would have 

 been 



xy=± — (a — by 4- -i/^^ — 4 cos <o \ h (a-{-b) — ab cos w f I 



4 {ab — h^y L I ) J 



and the right hand side may be proved to be the difference of the squares 

 of the semi-axes given by (53). 



§§. 16 — 20. Laplace's Linear Equation. 



§. 16. Genesis of Laplace's Equation.— The theorem that 



where p is the distance of any point on the curve from a fixed coplanar 

 point, represents a conic is first due, substantially, to Laplace (MScauique 

 Celeste^ Ed. 1878, t. I. p. 177). In integrating the equations for elliptic 

 motion, he gets 



dr = \dx -\r ydy, 

 •which leads to 



r= \-\x-\-yy ; 



Laplace then explicitly adds that " Cette equation, combince avec 

 celles-ci, 



z = ax-{-by, r^ = x^ -{- y'^ -\- z'^ 

 donne une equation du second degre." It is proposed to examine here 

 the geometrical meaning of the arbitrary constants in wliat I have 

 called Laplace's Linear Equation to a conic. 



§. 17. Meaningof the Constants.— That this equation represents 

 a conic may be shewn in various ways, and some additional information 

 regarding the constants may be gained from each standpoint of view. 

 Thus, squaring the equation and putting 



p2 = a.'2 + ?/2, 



