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XXXVI. On some Properties of Curves of the Second Order. 

 By J. W. Lubbock, Esq. F.R.S. <$• L.S.* 

 'ONE of the properties of the conic sections are more ele- 

 gant than those which belong to the inscribed hexagon ; 

 and it is to be regretted that they do not find a place in ele- 

 mentary works on this subject. M. Brianchon has treated this 

 question at some length in the 4th volume of the Joimial de 

 VEcole Polytechnique, and also in the Cori-espo7idance de VEcole 

 PolytecJmiqtie, vol. i. p. 307, and vol. ii. p. 383 ; but he seems 

 to think that it would be next to impossible to give a direct 

 algebraical proof of the fundamental theorem. I have endea- 

 voured to supply this ; and the proof I have given will, I think, 

 be found quite as simple as any which have been obtained by 

 geometrical considerations. 



A geometrical proof is given by Mr. T. S. Davies, in the 

 Philosophical Magazine for Nov. 1826. 



Let (xi 7/,), Gr,3/2) (X33/3) (0:43/4) (.r^j/,) (^gj/g) be any six points 

 in a parabola of which the equation is j/- = j9.r, the cordmate 

 axes being inclined to each other at any given angle ; and let 

 the construction be made, which is indicated by the figure a.^ /3( 

 being the coordinates of the point where the line a:, x^ produced 



cuts the line x^x.,. The points (a, |3,), («2/3^) (a^/S.,) nre in the 

 Bame straight line. 



■ f'""' = £*i:^ because by hypothesis .r„ a, and a^ nre 

 in the same straight line. 



• Cominiinicatoil l)y the Aiitlior. 

 N. S. Vol. G. No. 3i. Oct. 182'.>. 2 K. .r, yi 



