86 Mr. D. Waldie's Experimental Researches 



+ {ys—ys) {yx {y\-y3)+yAy3-y^)+iy4-y3+y^) iy^—yx)) 

 = (yj-yJ iy^-ye) iy^-yx) + (ys-ye) iyx-y^) iyx-y^) 



= , which proves the truth of the theorem in question. 



If yi-ya + 2^2 = ye-yx 

 ys-ye + yi = ya-y^ 



^4 = oTg , ^4 = ^5 J the points 4 and 5 coincide, and by 

 reference to the figure it will easily be seen that it is useless 

 to coincide this case. 



The equation of condition 



yx -y-i + ^3-^4 + y^-ye = o 



has been found upon the supposition, that Xi = 0, x.^ = 0, 

 which simplifies the expressions. But it is easy to show by 

 the transformation of coordinates that if the above equation 

 be true, with such limitations, it is also true in the more ge- 

 neral case, the only limitation required being, that the axis x 

 be parallel to the axis of the parabola ; and then, however 

 the circumscribed hexagon be situated 



yi-y^+ya-yi+ys-ye = o- 



In the Phil. Mag. and Annals, N.S., 1829, vol. vi. p. 249, 1 

 gave a direct proof from the equation to the parabola y^ z^px 

 of Pascal's celebrated property of the inscribed hexagon, 

 and I showed that the proof might be extended to the general 

 equation of the conic sections y"^ z=z p' x -\- q' a?^, by substi- 

 tuting for the coordinates x and y of any point 



X- — , 



-; — ~- and . , respectively. 

 p +q' X p +q X ^ "^ 



By such substitutions all the preceding expressions which 

 are true for the parabola y'^ = px may be extended to the 

 conic sections generally, which are included under the equa- 

 tion y^ — p' x-\-q[ x^. Thus the equation of condition 



yx-y'i + ^3-^4 + ^5-^6 = becomes 



yx y<i ,_ ys , ^^ \ y^ ^_ ■ 



p' + q'xi pf-Vq^x^ p' + ^^s y + ?'^4 P^ + Q.'^^ P'-^9.'^6 



XIV. Experimental Researches on Combustion and Flame. 

 By David Waldie*. 



[Illustrated by Plate I.] 



nPHE subject of combustion has long engaged the attention 



-*- of the most distinguished chemists, and the results of 



their inquiries are incorporated more or less in the various 



* Communicated by the Author. 



