464 
ON THE SYMMETRIC FUNCTIONS OF THE ROOTS &C. 
[149 
which is equal to 
a' 3 xfxgxg + &c., 
the symmetric functions x?xgx£, &c. being given by the formulae a 2 = x?x£x£, &c. The 
expression for the Resultant is in each case of the right degree, viz. of the degrees 
6, 3, 2, in the coefficients of the linear, the quadric, and the cubic equations respec 
tively: the two expressions, therefore, can only differ by a numerical factor, which 
might be determined without difficulty. The third expression for the resultant, viz. 
(a, /3, 7$»i, y u #i) • (a, ¡3, y$>s, y», **)•••(«> & 7$®«» 2/e. *«)> 
(where (a?!, ^),... (x 6) y 6 , z 6 ) are the roots of the cubic and quadratic equations) 
compared with the foregoing value, leads to expressions for the fundamental symmetric 
functions of the cubic and quadratic equations, and thence to expressions for the other 
symmetric functions of these two equations; but it would be difficult to obtain the 
actually developed values even of the fundamental symmetric functions. I hope to 
return to the subject, and consider in a general point of view the question of the 
formation of the expressions for the other symmetric functions by means of the ex 
pressions for the fundamental symmetric functions.