40 Mr. T. S. Davies on Pascal's Mystic Hexagram. 



But as x x y x is a point in the conic section denoted by (1.), we 

 have 



ay? + ex? — dy x — e x x +/= — bx x y x , 

 which substituted in the preceding expression gives us 



Again, since Xj Y 2 is the same function of <r 2 y% tnat X 2 Yj 

 is of x x y Xi we shall in the same way obtain 



*i*tr 2/D^ *•!(*•?»* «\W 



It follows from (8.) and (9.) that we have the equation 



X 2 Yj — Xj Y 2 = 



in virtue of the identity of the terms which compose them : 

 and this is the familiar test of the line G H passing through O 

 the origin of coordinates, and furnishing, therefore, a complete 

 proof of the " Pascal." 



Scholium 1. — When the equation is of the above form (1.) 

 with the exception of the last term negative, 



ay 2 + b xy + ex* — dy — ex —f = 0, 



the origin O will be within the curve ; and the Pascal will then 

 become an extension of a theorem of Pappus (prop. 139, 

 book vii.) respecting a quadrilateral figure, to the conic sec- 

 tions generally. 



The proposition in reference to this case may be stated as 

 follows: — 



F 



Let A Z), B C be the diagonals of 

 a quadrilateral inscribed in a conic sec- 

 tion : from A C draw lines A F, C Fto 

 any point F in the arc B D, and from 

 B D to any point E in the arc A C, 

 meeting the former in G H; then the 

 line G H will pass through O the inter- 

 section of A Z), B C. 



Scholium 2. — The theorem of Pappus, above referred to, 

 applies to the case where the conic section, as the locus of E 

 and F, is replaced by the straight lines A C, B D. To deduce 

 this from the preceding investigation, it is only necessary to 

 multiply together the two equations of the lines A C, B D, 

 which gives an equation of the general form (I.), and to which 

 the same process may be applied as that already employed : 

 for the conclusion is deduced from (1.) being the equation of 



