32 Proceedings of the Royal Irish Acadcvij/. 



3x^ - 8a; + 3 a:^-4x- + 3x + 2 



Scc^ _ 8a; + 3 3^-4 



1 8^H 64^2 _ 48^ + 9 95,yj + i g^^s _ 1 2a; + 6z - 8, x 3 



or 



82a;« - 48a; + 9 

 = 75a:^-18a;-24 = 3(2o.t-- - 6a; - 8) 

 . • . Q'— 3PP' =7£— 3to+^ = 0, 



the indicating quadratic. 



And since the roots of this are seen to be imaginary, the equation 

 mdicates a pair of real roots in the cubic P = ; and, consequently, 

 that all the roots are real. 



(6.) But the indicating quadratic may always be arrived at in 

 another "way. This other method of proceeding consists in extracting 

 the square root of the product 3PF' ; taking care, however, that the 

 successive terms of the root be the successive terms of the quadratic 

 function Q-.*^ Under this condition, the remainder, at which we shall 

 arrive, when taken with changed signs, and equated to zero, will always 

 be the indicating quadratic. Thus, taking the example just given, 

 where the complete product 3PP' is 



3PP' = 9x^ - 48a;3 + 75^^ _ 1 g^; _ 24, 



and extracting the square root in such way that the terms put one 

 after another, in the root-place, may be 3a;- - 8a; + 3 ; we shall find the 

 remainder, at the end of the operation, to be- 7a;^ + 30a; - 33. This 

 remainder, after changing the signs of its terms, will be 7a;- - 30a; + 33; 

 which, equated to zero, will be the indicating quadratic. 



That the indicating quadratic may always be arrived at in this 

 manner will appear from considering that if we put ^ = to denote this 

 quadratic, that is, if Q'-SPP'^q, it will follow that 3PP' = Qr - q. 

 But in the square root operation described above, oPP' is equal to 

 Or plus the remainder; consequently, q is this remainder taken with 

 changed signs. 



Since all the roots of the foregoing cubic equation are shown by 

 the indicating quadratic to be real roots, we know that the two roots 

 of the derived quadratic equation, Q = 3a;^ - 8.C + 3 = 0, must be sepa- 

 rators of those three roots; the middle one of the three lying between 

 the two real roots of Q = 0. In order to the actual determination of 

 the numerical values of the three roots, the shortest way of proceeding 



* There is, however, no necessity to revert to the expression Q for the first two 

 of the root terms; these same two teiTQS may he arrived at hy following the ordi- 

 narj^ nile for the exti'action of the square root, Q need he referred to only for the 

 t/iird term — the ahsolute niimhcr. 



