36 Proceedings of the Roynl Irish Academy. 



quadratic equation which enters into the composition of the cubic. 

 For, let ax^ +hx ■{■ c = represent any quadratic equation, then the 

 derived simple equation will be 2aa; + J = ; and if 



then, 



P = {ax"^ + Jx + c){x -\-p), 



Q=ax^ + bx + c + {2ax + b){x + p). 



]N"ow, if the value of x, which makes 2ax + 5-0, namely, the value 

 b_ 



"" ~ " Ya 



of X 



c(x^ + ix ■¥ c - 0, 



-, makes also Q = 0, then we must have for that same value 



which can be the case only when this latter equation has equal roots ; 

 that is to say, only when the first member of it is a complete square ; 

 ■under which circumstance the equation P=0 must have two equal 

 roots. Hence, that root of Q = 0, which is the indicator of a pair of 

 imaginary roots in the cubic equation P-0, — the value oi x, that is, 

 which causes Q to vanish between like signs of P and P' — can never 

 be equal to the real part of the imaginary pair thus indicated. 



[In an equation of degree higher than the third degree, it is possible, 

 when the first member of it has a quadratic factor, the roots of which 

 are imaginary, that a root of the first derived equation may be equal 

 to the real part of the imaginary pair ; but this can happen only under 

 peculiar circumstances. Let Ff- be an equation of degree higher 

 than the third degree, and of which /is a quadratic factor, such that 

 the roots of /= are imaginary. Then, writing i^\ /\ for the first 

 derivees of Ff, the first derivee of Ff will be 



Ff+fF; 



and if this be zero, for the value oi x, which is the root of the simple 

 equation/' =0, then F^ must be zero also ; since /is not zero for that 

 value of X, but is necessarily a positive number. Hence, the value of 

 X, which satisfies the equation /' = 0, cannot possibly also satisfy the 

 derived equation 



F'P-YfF' = Q, 



unless, under the special condition that .r is a root of F'^ = 0, as well as 

 arootof/i=0]. 



(12.) The conclusion in the last Article, namely, that the value of 

 X, which causes Q to vanish between like signs of P and P', can never 

 be equal to the real part of the pair of imaginary roots indicated, may 

 be easily generalized ; and the more comprehensive proposition be stated 

 thu s ; 



