of Curves of the Second Order. 251 



ry = B sin 9 9' = P sin 9 cos 9, and it is evident that any of 

 the preceding equations which were true in the case ot the 

 parabola tf =px mav be transformed so as to apply to the 

 [.^rve y"- = px + qx^- by substituting for the coordinates x 

 and tj of any point in these equations 

 v * 

 I — , and (-!^-~-^ respectively. 



In this way the equation 



y^Vi- ^i (j/-2+i/i) + i^«' = « becomes 

 P'ygyi ?Llf i_/_i!>_+ _i!^'-N + -J^'- = 



and the final equation 



«2 ^3- /^e «3 + «i /32-I33 «i = becomes 



«2 Ai-A2 «3 _ , '»'^"- /^"',, , — r = 



(7+9'«i)(i''+?' «3) (/+?' «■ ) a''+?'«'i) (^'' +2 "3) 0' +9 «i) 

 which gives after reductions 



«2^3-^2«3+«i^2-^3«. = 0, which shows that the theorem 

 is true generally of all the conic sections. 



The preceding method of transformation may be applied to 

 any problem which relates to the intersections only of lines. 

 Thus the equation to the tangents drawn from the point («,^) 

 to the parabola f = p x, the point («, ^) being without the 

 curveis _p (^-af-* (^.r-aj/) (i/-/3) = ; 



therefore the equation to the tangents drawn irom the point 

 (a, /3) to the curve/ = px + qx" is 



or, pM^-a)'-*^^-^-"^) [2'(i/-/^)-!?('3-^-"-^n = <^ 



The property of the inscribed hexagon which has been 

 i)roved leads to many very elegant geometrical constructions ; 

 amongst others it furnishes the simplest method of hnding any 

 number of points in a curve of the second order when hve are 

 .riven. It is probably identical with that of Pascal s mystic 

 r.exagon, upon which he founded a theory of conic sections 

 published in 1640: unfortunately no copy of this work is known 



to exist. -,,/-. if *u 



The theorem in question is deduced by Carnot, troni the 

 following very elegant i)ropcrty of curves of the second order. 

 If tangents A B, A C be drawn to any points B and C in any 

 curve of the second order, if B and C be joined, and any hue 

 P M qr be drawn in any given direction, cutting the chords 

 B C in P the curve in M, anil the tangents produced in q and 

 ,.^ p j^p _- A . M </ . M ?•, A being a constant quantity.— See 

 Carnot's Geometric dc rositioii, p. H6. 



2 K 2 Let 



