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V. Analytical Theorem relating to the four Conies inscribed in 

 the same Conic and passing through the same three Points. By 

 A. Cayley, Esq.* 



IMAGINE the four conies determined, and, selecting at plea- 

 sure any three of them, let their chords of contact with the 

 given conic be taken for the axes of coordinates, or lines x=0, 

 y=0, £=0; and then taking for the equation of the given conic 



U = («, b, c,f, g, hjx, y, zf=0 } 



the equations of the selected three conies must be of the form 

 U + /# 2 =0, U + m?/ 2 =0, U + W2 2 = 0, where /, m, n are to be 

 determined in such manner that these conies may have three 

 common points ; the resulting values of /, m, n, and of the coor- 

 dinates of the three common points, that is, the three given 

 points, will of course be functions of the coefficients («, b } c,f,g, h); 

 and the equation of the fourth conic will be of the form 



V + (ix+jy + kz)*=0. 



There is no difficulty in carrying out the investigation. It is 

 found that the coordinates of the given points must be taken to be 



(— /, 9> h ) > if, —ff, h) ; (/, g, —h) 

 respectively, and that, writing as usual 



K = abc - af 2 - % 2 - cK z + %fgh, 

 the equations of the four conies are 



V+(K-abc)~=0, 

 V+(K-abc) y -l = 0, 



V + (K-abcf ¥ =0, 



zY 



V + { K-abc)(j + f + j;)=0. 



It is in fact easy to verify directly that each of these conies 

 passes through the three given points ) but the equations may 

 also be exhibited in the form proper for putting this in evidence. 

 Putting, for shortness, 



9 ti h f f 9 



the equations of the sides of the triangle formed by the given 

 points are X = 0, Y=0, Z=0, and the foregoing equations of 



* Communicated by the Author. 



