Young — Algebraical Equations of the Third Degree, Sfc. 35 



tion of imaginarity will be satisfied, but not before ; and that the con- 

 dition will continue to be satisfied for every subsequent transformation, 

 throughout the interval \j\, r^], but not after that interval has been 

 passed over. 



It will have been noticed that the foregoing discussion concerns 

 those cubic equations only of which the given coefficients do not 

 themselves supplj^ the required information as to the character of the 

 roots ; that is to say, those equations only in which neither triad of the 

 coefficients satisfies the condition of imaginary roots. 



(10.) In the case in which thej??'s^ triad of the coefficients of the 

 cubic equation P=0 satisfies the condition of imaginarity, the roots of 

 the derived quadratic equation, Q=0, will be imaginary ; and the root 

 of the simple equation, 7^' = 0, will then be the indicator. But if the 

 condition of imaginarity be satisfied by the second triad of the coefl[i- 

 cients, and not by the first also — under which circumstances Q = 

 will have real roots — the roots of the indicating quadratic will be real 

 roots ; and, as already noticed (Art. 6), one of these roots will be posi- 

 tive, and the other negative ; and, as before, the condition of imaginarity 

 will have place throughout the interval between those roots, and will 

 fail to have place for all values outside that interval. But if both 

 triads of the coefficients satisfy the condition of imaginarity, then the 

 first and last terms of the indicating quadratic, the terms 



(.4j'- 3A^A^)x\ and {A^^ - SA^A^), 



will each be preceded by the negative sign ; so that a pair of imagi- 

 nary roots in the equation F -0 would be implied, even should the 

 roots of the indicating quadratic, Q'-3P/" = 0, be themselves imagi- 

 nary; because, then, the first member of this quadratic would be 

 always negative; that is to say, the quadratic expression, 3PP' - Q^ 

 would be positive for every real value of x. As before, the indicator 

 of the pair of imaginary roots would be the root of the simple equation 

 P' = 0, this root being the value of x, for which a derived function 

 vanishes between like signs of the two adjacent derived functions. 



(11.) It is well known that each of the two roots of an equation of 

 the second degree, whenever these roots are unequal, always consists 

 of two distinct parts — the one being a rational number, and the other 

 part being a number, either positive or negative, under the radical 

 sign, with, usually, a real factor prefixed to the radical quantity ; the 

 rational part of each root is always the root of the derived simple 

 equation. 



But when the quadratic is raised to an equation of the third degree, 

 by the introduction of a new simple factor, it is not the case that a 

 root of the equation of the second degree, derived from this cubic 

 equation, will be the rational part of each of the two roots of the 



