182 



MR. KOBEET FINLAY ON 



K'y^ + 2 B'xy + C'.r' + 2 D y + 2 E'.r + F' = o, 

 we shall have A' = A% B' = AB, C = 2 B* — AC, 



. • . B'* — A'C = — A^ (B^ — AC) ; 

 hence, if the original curve he an elli^ie the supplementary 

 curve will he an hyperbola, but if the original curve be a 

 parabola the supplementary one loill also be a parabola. 



ig.) If any straight line be drawn through the origin A, 

 meeting the curve (4) in a real point P {x, y), the straight line 

 (3) in a point Q,, and the two straight lines (6) in Rj and R^, 

 we shall have (V, 3) 



PQ : AQ=:— (Ay + Bar + D): D 



PR. .PR,:AR,.AR,= {(B»— AC).r^ + 2(BD — AE)a: 



+ D* — AF} : (D^ — AF) 



and in virtue of these equation (4) becomes 



'PQ\^_.T.. , ^.^ PR. . PR, 

 AR, .AR, 



If the same line meet the curve (5) in a real point P, we 

 shall have 



°'(!-Q)'='^^-"')Il:tl| ®- 



Hence we see that each of the supplementary curves (4) and 

 (5) may be considered as the locus of a point P, such that 



PR, .PR,, a A *' , /P<^\'; .1 1 1, • • 



— — ^ — —~ has a fixed ratio to ( --— ) the algebraic sign of 



AR. .AR, vAQ/ 



this ratio being different for the two curves, but its absolute 



magnitude being the same for both, 



{h.) Let it be required to find the locus of a point P, such 



that if a straight line be drawn from it to a fixed point A, 



meeting the given straight line (3) in Q, and the two parallel 



PR PR 



straight lines (6) in R, and Rj, then _> ' ' 4 n^' shall have a 



AH, . AR„ 



-Q"=(---^)||r:li <^)- 



given ratio to ( r~;r ) 

 VAQ^ 



