3*70 Mr. J. J. Sylvester on the solution of 



characteristic of the tangential envelope of the conic, x 9 y 9 z 9 t 9 

 u 9 v as the characteristics of the six points of the circumscribed 

 hexagon> <£ the characteristic of the point in which the line 

 x 9 v meets the line z 9 t ; ay — uu will then be shown to cha- 

 racterize the point in which t, x meets v 9 % ; and thus we see 

 that y 9 u ; t 9 x ; v 9 z 9 the three pairs of opposite sides of the 

 hexagon, will meet in one and the same point, which is Brian- 

 chon's theorem. 



XLVIII. On the solution of a System of Equations in which 

 three Homogeneous Quadratic Functions of three unknown 

 quantities are respectively equaled to numerical Multiples 

 of a fourth Non- Homogeneous Function of the same. By 

 J. J. Sylvester, M.A. 9 F.R.S* 



LET U, V, W be three homogeneous functions of x 9 y 9 z 9 

 and let co be any function of x 9 y 9 z of the ?zth degree, 

 and suppose that there is given for solution the system of 

 equations 



U=A.o> 



V = B.a> 



W=C.a>. 



Theorem. — The above system can be solved by the solution 

 of a cubic equation, and an equation of the rath degree. 

 For let D be the determinant in respect to x 9 y 9 z of 



/V+gY+hW, 



then D is a cubic function of/, g 9 h. Now make D = 



Af+Bg + Ch=0 9 



the ratios of/: g : h which satisfy the last two equations can 

 be determined by the solution of a cubic equation, and there 

 will accordingly be three systems of f 9 g 9 h which satisfy the 

 same, as 



/i gi ll \ 



A gz K 



f 3 £3 ** 

 Now D = implies that/U-f-gV + ^W breaks up into two 

 linear factors ; accordingly we shall find 



(l x x + m } y + x Y z) . (\x + /^ y -f v^) = 

 (l 2 x -f m 2 y -f x 2 z ) (\x + /^ + v 2 z) — 

 {l 3 x + m 3 y + x 3 z) (\ s x + fi 3 y + v 3 z) = 9 

 * Communicated by the Author. 



