Young — Algebraical Equations of the Third Degree, ^r. 33 



will be this. Develope the middle root by continuous approximation : 

 we shall thus arrive at a quadratic equation, the roots of which, when 

 each of them is increased by the root previously developed, will be the 

 other two roots of the proposed cubic. 



(7.) When each triad of the coefficients of a cubic equation fails 

 to satisfy the condition of imaginary roots, and that the roots r^, r,, 

 of the indicating quadratic equation Q^ - 2iPP' = 0, are real and 

 unequal, the roots of the derived quadratic Q=0 will necessarily be 

 real roots. One of these roots, and one only, will be situated in the 

 interval [r,, roj; and if the roots of the cubic be each of them di- 

 minished by that root of Q = 0, the third coefllcient of the trans- 

 formed equation will vanish between like signs. 



For ri, ?'2, are either both positive or both negative, inasmuch as 

 that the first and third terms of the quadratic Qr- 3PP' = are then 

 both positive ;"^^ and, therefore, if ri be that one of these two roots which 

 is the nearer to - oo , the quadi-atic expression Q^ - 3PF' will be 

 positive for every value of x, from the value x = - co up to the value 

 a; = ri ; and also for every value of x, from the value a; = + oo down to 

 the value x = r2. And since the expression changes sign immediately 

 after the passage of a root, and that the changed sign remains perma- 

 nent till the other root passes, it follows that for every value of x, 

 within the limits [^i, r^, the expression Q/ - SPP' must be negative ; 

 whilst for every value of x, outside those limits, it must be positive. 

 This is the same as saying that for every value of x, within the limits 

 C^i) ^2]) the condition of imaginary roots is satisfied, whilst for every 

 value of ;r, outside those limits, the condition fails to be satisfied. At 

 either of the limits, that is, for x = ri, or for x = r.^, the condition of 

 imaginarity is still satisfied ; since, for either of these values of x, 

 3PP' = Q\ But when the roots t\, r^, of the indicating quadratic are 

 equal roots, then we know that these same equal roots belong also to 

 the proposed cubic P= — the imaginarity disappearing with the dis- 

 appearance of inequality between the roots \j\, r^. 



It is impossible, therefore, that Q can vanish between like signs of 

 P and P' outside the limits \j\, n], seeing that if Q could so vanish, 

 Q2 _ ^PP\ for the value of a; which causes Q thus to vanish, would be 

 negative. 



But one of the two roots of Q= must cause Q to vanish between 

 like signs of P and P' ; consequently, this root of Q = must lie he- 

 tween r^ and n. 



It is plain that Q cannot vanish a second time between like signs 

 of P and P' ; since such second evanescence would imply a second pair 

 of imaginary roots in an equation of only the third degree. Hence, the 

 other root of Q = must cause Q, to vanish between imlike signs of P 

 and P' ; so that when this evanescence takes place, the expression 

 Q^-3PP'must be positive. 



* See the expression marked (1) at page 473 of the former Paper. 



R. I. A. PROC SEK n., VOL. II. SCfENrtK. F 



